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abstract: 'We introduce and analyze a discontinuous Galerkin method for a time-harmonic eddy current problem formulated in terms of the magnetic field. The scheme is obtained by putting together a DG method for the approximation of the vector field variable representing the magnetic field in the conductor and a DG method for the Laplace equation whose solution is a scalar magnetic potential in the insulator. The transmission conditions linking the two problems are taken into account weakly in the global discontinuous Galerkin scheme. We prove that the numerical method is uniformly stable and obtain quasi-optimal error estimates in the DG-energy norm.'
author:
- |
[Ana Alonso Rodríguez]{}[^1], $\,\,$ [Salim Meddahi]{}[^2]\
and\
[Alberto Valli]{}[^3]
title: 'A discontinuous Galerkin method for the time harmonic eddy current problem [^4]'
---
Introduction
============
In this paper, we present a discontinuous Galerkin (DG) approximation of a time-harmonic eddy current problem. The eddy current approximation of Maxwell equations is obtained by disregarding the displacement current term. It is commonly used in applications related with induction heating, transformers, magnetic levitation and non-destructive testing. These problems often involve composite materials and structures, complex transmission conditions and, eventually, boundary layers due to the skin effect. The ability of | 0 | non_member_992 |
DG methods to handle efficiently unstructured meshes with hanging nodes combined with $hp$-adaptive strategies make them well-suited for the numerical simulation of physical systems related to eddy currents.
The eddy current problem is generally written in terms of either the electric or the magnetic field, cf. [@AVbook]. These two formulations are equivalent at the continuous level but they lead to different numerical schemes. A discontinuous Galerkin method based on a time-harmonic eddy current problem written in terms of the electric field has been analyzed in the pioneering work of Perugia and Schotzau [@PS03]. For the time domain eddy current problem, Ausserhofer et al. introduced in [@ABP09] a formulation based on a magnetic vector potential and propose a numerical method that combines a DG approximation in the conductor with the usual ${\mathrm{H}}^{1}$-conforming Lagrange finite element approximation in the insulator.
Here, we are interested in imposing the magnetic field as primary unknown. The advantage of this approach rests on the reduction of the number of degrees of freedom resulting from the introduction of a scalar magnetic potential in the nonconducting medium. The global formulation of the problem consists in a ${\mathrm{H}}({\mathbf{curl}})$-elliptic problem for vector fields that are curl-free in the insulator $\Omega_I$. | 0 | non_member_992 |
Our DG formulation is obtained by applying for the Laplace equation posed in $\Omega_{I}$ the usual interior penalty finite element method, that can be traced back to [@arnoldIP], see also [@DiPietroErn] and the references cited therein for more details. In the conductor $\Omega_{C}$ we employ, as in [@HPSS05; @PS03], the interior penalty method corresponding to the Nédélec curl-conforming finite element space of the second kind. We point out that the introduction of discrete harmonic fields is necessary when considering domains of general topology. We prove the stability of the resulting combined DG scheme by exploiting the elliptic character of the problem. We also obtain, under adequate regularity assumptions, quasi-optimal asymptotic error estimates. It is worthwhile to notice that the implementation of the DG-method presented here only requires the use of standard shape functions. The curl-conforming finite elements, more precisely, the Nédélec finite elements of the second kind, are only needed for the theoretical convergence results in Section 5.
The outline of this paper is as follows. In Section 2 we derive the model problem used in the finite element approximation. We introduce our DG formulation in Section 3. Finally, Section 4 is devoted to the convergence analysis, and asymptotic error | 0 | non_member_992 |
estimates are provided in Section 5.
The model problem {#s2}
=================
Let $\Omega_C\subset \mathbb{R}^3$ be a bounded polyhedral domain with a Lipschitz boundary $\Gamma$. We denote by $\mathbf{n}_\Gamma$ the unit normal vector on $\Gamma$ that points towards $\Omega_e:= \mathbb{R}^3\setminus \overline \Omega_C$. In order to illustrate the impact of the conductor’s topology in our method, we assume that $\Omega_C$ has a toroidal shape. We notice that the eddy current problem is posed in the whole space with asymptotic conditions on the behaviour of the electric and magnetic fields at infinity. Depending on the nature of the eddy current problem being solved and the geometry involved, a discretization method can be obtained for this problem by either applying a pure finite element approach on a truncated domain or by using a combination of boundary (BEM) and finite elements (FEM), see [@AMV; @Hiptmair; @MS03; @AV09]. The FEM-BEM formulation is posed in the conductor but its implementation is more difficult and it leads to more complex algebraic linear systems of equations. The FEM method needs a large computational domain, but it is simpler and it can provide an alternative in many practical situations. It is the option that we will consider in the following. | 0 | non_member_992 |
To this end, we introduce a bounded domain $D$ containing in its interior $\overline{\Omega}_{C}$ and whose connected boundary $\Sigma= \partial D$ is located at a large enough distance from the conductor $\Omega_{C}$. The bounded domain $\Omega_I:= D\setminus \overline \Omega_C$ represents then the nonconducting region of the computational domain $D$.
Under our assumptions, the first de Rham cohomology group $\mathcal H^1(\Omega_I)$ of $\Omega_{I}$, namely, the space of curl-free vector fields that are not gradients, has dimension one. If we assume that $\Omega_I$ is a polyhedral domain endowed with a tetrahedral mesh, one can use the technique given in [@BRS02] for the explicit construction of a piecewise-linear vector field $\boldsymbol \rho$ spanning $\mathcal H^1(\Omega_I)$ and satisfying $\boldsymbol \rho \times {\bf n}_\Sigma = {\bf 0}$ on $\Sigma$, where ${\bf n}_\Sigma$ denotes the outward unit normal vector to $\Sigma$. For an alternative construction of $\boldsymbol \rho$ see Alonso Rodríguez et al. [@ABGV13].
The eddy current problem formulated in terms of the magnetic field ${\boldsymbol{h}}$ and the scalar magnetic potential $\psi$ reads as follows: $$\label{model0}
\begin{array}{rcll}
\imath \omega \mu {\boldsymbol{h}}+ {\mathbf{curl}}\, {\boldsymbol{e}}&=& \boldsymbol 0 &\text{in $D$}\\[2ex]
{\boldsymbol{e}}&=& \sigma^{-1} ( {\mathbf{curl}}\, {\boldsymbol{h}}- {\boldsymbol{j}}) &\text{in $\Omega_C$}\\[2ex]
{\boldsymbol{h}}&=& \nabla \psi + k \boldsymbol \rho &\text{in $\Omega_I$}\\[2ex]
\psi &=& | 0 | non_member_992 |
0 & \text{on } \Sigma \, ,
\end{array}$$ where ${\boldsymbol{j}}$ is the applied current density, $\mu$ is the magnetic permeability and $\sigma$ is the electric conductivity. In what follows, we assume that $\mu$ and $\sigma$ are positive piecewise constant functions in $\Omega_{C}$ and that $\mu_{|\Omega_I}= \mu_0$ is the permeability constant of vacuum. It follows from the first equation that $$\label{lap}
0={\mathrm{div}}({\boldsymbol{h}}_{|\Omega_I})= {\mathrm{div}}(\nabla \psi + k \boldsymbol \rho)\, \hbox{ in } \Omega_I\, .$$ We point out here that the electric field ${\boldsymbol{e}}$ is not uniquely determined in $\Omega_{I}$. Nevertheless, the tangential components of the magnetic field and the tangential components of any admissible representation of the electric field should be continuous across the interface $\Gamma$, i.e., $$\label{magt}
{\boldsymbol{h}}|_{\Omega_{C}} \times {\boldsymbol{n}}_\Gamma = (\nabla \psi + k \boldsymbol \rho) \times {\boldsymbol{n}}_\Gamma \, .$$ and $$\label{mage}
{\boldsymbol{e}}_{|\Omega_C} \times {\boldsymbol{n}}_\Gamma = {\boldsymbol{e}}_{|\Omega_I} \times {\boldsymbol{n}}_\Gamma.$$ The electric field ${\boldsymbol{e}}$ is considered here as an auxiliary variable, it will be removed from the formulation. Hence, we should deduce from a transmission condition relating ${\boldsymbol{h}}$ and $\varphi$ on $\Gamma$. Applying the surface divergence operator ${\mathrm{div}}_\Gamma$ to both side of and recalling that ${\mathrm{div}}_\Gamma({\boldsymbol{e}}\times {\boldsymbol{n}}_\Gamma)= {\mathbf{curl}}\,{\boldsymbol{e}}\cdot {\boldsymbol{n}}_\Gamma$ we deduce that the field ${\mathbf{curl}}\,{\boldsymbol{e}}$ admits continuous normal components across $\Gamma$. | 0 | non_member_992 |
As a consequence of the first equation of , $\mu{\boldsymbol{h}}$ should also have continuous normal components across $\Gamma$, i.e., $$\label{transe}
\mu {\boldsymbol{h}}\cdot {\boldsymbol{n}}_\Gamma = \mu_{0}(\nabla \psi + k \boldsymbol \rho) \cdot {\boldsymbol{n}}_\Gamma \,.$$ Finally, we deduce from and the property ${\mathbf{curl}}\, \boldsymbol \rho = {\bf 0}$ that $$\int_\Gamma {\boldsymbol{e}}_{|\Omega_C} \times {\boldsymbol{n}}_\Gamma \cdot \boldsymbol \rho =\int_\Gamma {\boldsymbol{e}}_{|\Omega_I} \times {\boldsymbol{n}}_\Gamma \cdot \boldsymbol \rho = \int_{\Omega_I} {\mathbf{curl}}\, {\boldsymbol{e}}\cdot \boldsymbol \rho,$$ thus $$\label{scal}
\int_\Gamma \sigma^{-1} ( {\mathbf{curl}}\, {\boldsymbol{h}}- {\boldsymbol{j}}) \cdot (\boldsymbol \rho \times {\boldsymbol{n}}_\Gamma)= \imath \, \omega \int_{\Omega_I} \mu_0 (\nabla \psi + k \boldsymbol \rho) \cdot \boldsymbol \rho \, .$$
From now on, for the sake of simplicity in notations, ${\boldsymbol{h}}$ will stand for ${\boldsymbol{h}}|_{\Omega_{C}}$. Taking into account , , and , we deduce that the eddy current problem can be formulated in terms of the magnetic field and its scalar potential representation in the insulator in the following form: Find ${\boldsymbol{h}}:\, \Omega_{C}\to \mathbb C^{3}$, $\psi:\, \Omega_{I}\to \mathbb C$ and $k\in \mathbb C$ such that, $$\begin{aligned}
\imath \omega \mu {\boldsymbol{h}}+ {\mathbf{curl}}\, [\sigma^{-1} ( {\mathbf{curl}}\, {\boldsymbol{h}}- {\boldsymbol{j}})] &= \boldsymbol 0 &\text{in $\Omega_C$}\label{ModelProblem1}\\[2ex]
{\boldsymbol{h}}\times {\boldsymbol{n}}_\Gamma & = (\nabla \psi + k \boldsymbol \rho) \times {\boldsymbol{n}}_\Gamma &\text{on $\Gamma$}\label{ModelProblem2}\\[2ex]
\mu\, {\boldsymbol{h}}\cdot {\boldsymbol{n}}_\Gamma & = \mu_0(\nabla \psi + k | 0 | non_member_992 |
\boldsymbol \rho) \cdot {\boldsymbol{n}}_\Gamma &\text{on $\Gamma$}\label{ModelProblem3}\\[2ex]
\int_\Gamma \sigma^{-1} ( {\mathbf{curl}}\, {\boldsymbol{h}}- {\boldsymbol{j}}) \cdot (\boldsymbol \rho \times {\boldsymbol{n}}_\Gamma)&=\imath \, \omega \mu_0 \int_{\Omega_I} (\nabla \psi + k \boldsymbol \rho) \cdot \boldsymbol \rho\label{ModelProblem4}\\[2ex]
{\mathrm{div}}(\nabla \psi + k \boldsymbol \rho) & = 0 &\text{in $\Omega_I$}\label{ModelProblem5}\\[2ex]
\psi &= 0 &\text{on } \Sigma\label{ModelProblem6} \, .
$$
We refer to [@AVbook Section 5] for a proof of the well-posedness of problem -.
The discrete problem {#section3}
====================
Notations
---------
Given a real number $r\geq 0$ and a polyhedron $\mathcal O\subset \mathbb R^d$, $(d=2,3)$, we denote the norms and seminorms of the usual Sobolev space ${\mathrm{H}}^r(\mathcal O)$ by $\|\cdot \|_{r,\mathcal O}$ and $|\cdot|_{r,\mathcal O}$ respectively (cf. [@McLean]). We use the convention ${\mathrm{L}}^2(\mathcal O):= {\mathrm{H}}^0(\mathcal O)$ and ${\bf L}^2(\mathcal O):= [{\mathrm{L}}^2(\mathcal O)]^3$. We recall that, for any $t \in [-1,\: 1 ]$, the spaces ${\mathrm{H}}^{t}(\partial \mathcal O)$ have an intrinsic definition (by localization) on the Lipschitz surface $\partial \mathcal O$ due to their invariance under Lipschitz coordinate transformations. Moreover, for all $0< t\leq 1$, ${\mathrm{H}}^{-t}(\partial\mathcal O)$ is the dual of ${\mathrm{H}}^{t}(\partial\mathcal O)$ with respect to the pivot space ${\mathrm{L}}^2(\partial \mathcal{O})$. Finally we consider $\mathbf{H}({\mathbf{curl}}, \mathcal O):=\{ {\boldsymbol{v}}\in {\mathrm{L}}^2(\mathcal O)^3 \, : \, {\mathbf{curl}}\, {\boldsymbol{v}}\in {\mathrm{L}}^2(\mathcal O)^3\}$ and endow it | 0 | non_member_992 |
with its usual Hilbertian norm ${\lVert{\boldsymbol{v}}\rVert}_{\mathbf{H}({\mathbf{curl}}, \mathcal O)}^2:=
{\lVert{\boldsymbol{v}}\rVert}_{0, \mathcal O}^2 + {\lVert{\mathbf{curl}}\, {\boldsymbol{v}}\rVert}_{0, \mathcal O}^2$.
We consider a sequence $\{{\mathcal{T}}_h\}_h$ of conforming and shape-regular triangulations of $\overline \Omega_C \cup \overline \Omega_I$. We assume that each partition ${\mathcal{T}}_h$ consists of tetrahedra $K$ of diameter $h_K$ and unit outward normal to $\partial K$ denoted ${\boldsymbol{n}}_K$. We also assume that for all $K\in {\mathcal{T}}_h$ we have either $K\subset \overline\Omega_C$ or $K\subset \overline \Omega_I$ and denote $${\mathcal{T}}_h^{\Omega_C}:= {\left\{K\in {\mathcal{T}}_h;\quad K\subset \overline\Omega_C\right\}},\qquad {\mathcal{T}}_h^{\Omega_I}:= {\left\{K\in {\mathcal{T}}_h;\quad K\subset \overline\Omega_I\right\}}.$$ We also assume that the meshes $\{{\mathcal{T}}_h^{\Omega_C}\}_h$ are aligned with the discontinuities of the coefficients $\sigma$ and $\mu$. The parameter $h:= \max_{K\in {\mathcal{T}}_h} \{h_K\}$ represents the mesh size.
We denote by ${\mathcal{F}}_h^0(\Omega_C)$ and ${\mathcal{F}}_h^0(\Omega_I)$ the sets of interior faces of the triangulations ${\mathcal{T}}_h^{\Omega_C}$ and ${\mathcal{T}}_h^{\Omega_I}$ respectively. We also introduce the sets of boundary faces $${\mathcal{F}}_h^\Gamma:= {\left\{F = \overline K\cap\overline{K'};\quad K\in {\mathcal{T}}_h^{\Omega_C},\,\, K'\in {\mathcal{T}}_h^{\Omega_I}\right\}}
\quad
\text{and}
\quad
{\mathcal{F}}_h^\Sigma:= {\left\{F = \partial K \cap \Sigma;\quad K\in {\mathcal{T}}_h^{\Omega_I}\right\}}$$ and consider $${\mathcal{F}}_h^{\Omega_C}:={\mathcal{F}}_h^0(\Omega_C) \cup {\mathcal{F}}_h^\Gamma, \quad {\mathcal{F}}_h^{\Omega_I}:={\mathcal{F}}_h^0(\Omega_I) \cup {\mathcal{F}}_h^\Sigma
\quad \text{and} \quad {\mathcal{F}}_h := {\mathcal{F}}_h^{\Omega_C}\cup {\mathcal{F}}_h^{\Omega_I}.$$ We notice that ${\left\{{\mathcal{F}}_h^\Gamma\right\}}_h$ is a shape regular family of triangulations of $\Gamma$ into triangles $T$ of diameter $h_T$. Finally, we consider the set ${\mathcal{E}}_h$ of | 0 | non_member_992 |
edges $e = \overline T\cap \overline{T'}$ (where $T$ and $T'$ are two adjacent triangles from ${\mathcal{F}}_h^\Gamma$).
Let $\mathcal{O}_h$ be anyone of the previously introduced partitions of $\overline\Omega_C\cup \overline \Omega_I$, $\overline\Omega_C$, $\overline \Omega_I$ or $\Gamma$ and let $E$ be a generic element of the given partition. We introduce for any $s\geq 0$ the broken Sobolev spaces $${\mathrm{H}}^s(\mathcal{O}_h) := \prod_{E\in \mathcal{O}_h} {\mathrm{H}}^s(E)\quad \text{and} \quad \mathbf{H}^s(\mathcal{O}_h) := \prod_{E\in \mathcal{O}_h} \mathbf{H}^s(E)^3\, .$$
For each $w:= {\left\{w_E\right\}}\in {\mathrm{H}}^s(\mathcal{O}_h)$, the components $w_E$ represents the restriction $w|_E$. When no confusion arises, the restrictions will be written without any subscript.
The space ${\mathrm{H}}^s(\mathcal{O}_h)$ is endowed with the Hilbertian norm $${\lVertw\rVert}_{s,\mathcal{O}_h}^2 := \sum_{E\in \mathcal{O}_h} {\lVertw_E\rVert}^2_{s,E}.$$
We consider identical definitions for the norm and the seminorm on the vectorial version $\mathbf{H}^s(\mathcal{O}_h)$. We use the standard conventions ${\mathrm{L}}^2(\mathcal{O}_h):={\mathrm{H}}^0(\mathcal{O}_h)$ and $\mathbf{L}^2(\mathcal{O}_h):=\mathbf{H}^0(\mathcal{O}_h)$ and introduce the bilinear forms $$(w, z)_{\mathcal{O}_h} = \sum_{E\in \mathcal{O}_h} \int_E w_E z_E, \quad \forall w, z \in {\mathrm{L}}^2(\mathcal{O}_h)$$ and $$(\boldsymbol{w}, \boldsymbol{z})_{\mathcal{O}_h} = \sum_{E\in \mathcal{O}_h} \int_E \boldsymbol{w}_E\cdot \boldsymbol{z}_E, \quad \forall \boldsymbol{w}, \boldsymbol{z}\in \mathbf{L}^2(\mathcal{O}_h).$$
Assume that $({\boldsymbol{v}},\varphi,m)\in \mathbf{H}^{1+s}({\mathcal{T}}_h^{\Omega_C})\times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I})\times {\mathbb C}$, with $s>1/2$. Moreover, let us recall that $\boldsymbol \rho$ has been constructed as a piecewise-linear vector field, therefore its restriction to any face $F$ has a meaning. We define ${\mathbf{curl}}_h{\boldsymbol{v}}\in \mathbf{H}^s({\mathcal{T}}_h^{\Omega_C})$ | 0 | non_member_992 |
by $({\mathbf{curl}}_h {\boldsymbol{v}})|_K = {\mathbf{curl}}\, {\boldsymbol{v}}_K$, for all $K\in {\mathcal{T}}_h^{\Omega_C}$; $\nabla_h \varphi \in \mathbf{H}^s({\mathcal{T}}_h^{\Omega_I})$ by $(\nabla_h \varphi)|_K = \nabla \varphi_K$, for all $K\in {\mathcal{T}}_h^{\Omega_I}$. We define also the averages ${\{{\boldsymbol{v}}\}}_{{\mathcal{F}}}\in \mathbf{L}^2({\mathcal{F}}_h^{\Omega_C})$ and ${\{\nabla_h \varphi+m\boldsymbol \rho\}}_{{\mathcal{F}}}\in \mathbf{L}^2({\mathcal{F}}_h^{\Omega_I})$ by $$\label{average1}
\begin{array}{l}
{\{{\boldsymbol{v}}\}}_{{\mathcal{F}}}|_F := {\{{\boldsymbol{v}}\}}_F \hbox{ with}\\ \\
{\{{\boldsymbol{v}}\}}_F:= \left\{
\begin{array}{ll}
({\boldsymbol{v}}_K + {\boldsymbol{v}}_{K'})/2 & \text{if $F=K\cap K'\in {\mathcal{F}}_h^0(\Omega_C)$}\\[.1cm]
{\boldsymbol{v}}_K & \text{if $F\subset \partial K$ and $F \in {\mathcal{F}}_h^\Gamma$},
\end{array} \right.
\end{array}$$ and $$\label{average2}
\begin{array}{l}
{\{\nabla_h\varphi+m \boldsymbol \rho\}}_{{\mathcal{F}}}|_F := {\{\nabla_h \varphi + m \boldsymbol \rho\}}_F \hbox{ with} \\ \\
{\{\nabla_h \varphi + m \boldsymbol \rho\}}_F :=\left\{ \begin{array}{l}
(\nabla \varphi_K + \nabla \varphi_{K'})/2 + m (\boldsymbol \rho_K+\boldsymbol \rho_{K'})/2 \\ \hspace{3cm} \text{if $F=K\cap K'\in {\mathcal{F}}_h^0(\Omega_I)$}\\[.1cm]
\nabla \varphi_K + m \boldsymbol \rho_K \\
\hspace{3cm} \text{if $F\subset \partial K$ and $F \in {\mathcal{F}}_h^\Sigma$}\, ,
\end{array} \right.
\end{array}$$ and the jumps ${\llbracket ({\boldsymbol{v}},\varphi,m) \rrbracket}_{{\mathcal{F}}}\in \mathbf{L}^2({\mathcal{F}}_h^{\Omega_C})$ and ${\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\in \mathbf{L}^2({\mathcal{F}}_h^{\Omega_I})$ by $$\label{jump1}
\begin{array}{l}
{\llbracket ({\boldsymbol{v}},\varphi,m) \rrbracket}_{{\mathcal{F}}}|_F :={\llbracket ({\boldsymbol{v}},\varphi,m) \rrbracket}_F \hbox{ with}
\\ \\
{\llbracket ({\boldsymbol{v}},\varphi,m) \rrbracket}_F:=\left\{ \hspace{-.2cm} \begin{array}{l}
{\llbracket {\boldsymbol{v}}\times {\boldsymbol{n}}\rrbracket}_F:={\boldsymbol{v}}_K \times {\boldsymbol{n}}_K + {\boldsymbol{v}}_{K'}\times {\boldsymbol{n}}_{K'} \\ \hspace{2cm} \text{if $F=K\cap K'\in {\mathcal{F}}_h^0(\Omega_C)$}\\[.1cm]
{\boldsymbol{v}}_K \times {\boldsymbol{n}}+( \nabla \varphi_{K'} + m \boldsymbol \rho_{K'}) \times {\boldsymbol{n}}_{K'}\\ \hspace{2cm} \text{if }F= K\cap K'\in {\mathcal{F}}_h^\Gamma \text{ with } K\in {\mathcal{T}}_h^{\Omega_C}, \,K'\in {\mathcal{T}}_h^{\Omega_I}\, ,
\end{array} \right.
| 0 | non_member_992 |
\end{array}$$ and $$\label{jump2}
\begin{array}{l}
{\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}|_F := {\llbracket \varphi{\boldsymbol{n}}\rrbracket}_F \hbox{ with} \\ \\
{\llbracket \varphi{\boldsymbol{n}}\rrbracket}_F := \left\{
\begin{array}{ll}
\varphi_K {\boldsymbol{n}}_K + \varphi_{K'}{\boldsymbol{n}}_{K'} & \text{if $F=K\cap K'\in {\mathcal{F}}_h^0(\Omega_I)$}\\[.1cm]
\varphi_K {\boldsymbol{n}}_\Sigma & \text{if $F\subset \partial K$ and $F \in {\mathcal{F}}_h^\Sigma$}\, .
\end{array} \right.
\end{array}$$ Similarly, we define the edge averages ${\{{\boldsymbol{v}}\}}_{{\mathcal{E}}}\in \mathbf{L}^2({\mathcal{E}}_h)$ by $${\{{\boldsymbol{v}}\}}_{{\mathcal{E}}}|_e :={\{{\boldsymbol{v}}\}}_e \hbox{ with } {\{{\boldsymbol{v}}\}}_e:=({\boldsymbol{v}}_{K_e} + {\boldsymbol{v}}_{K'_e})/2$$ where $K_e, K'_e\in {\mathcal{T}}_h^{\Omega_C}$ are such that $T=\partial K_e \cap \Gamma\in {\mathcal{F}}_h^\Gamma$, $T'=\partial K'_e \cap \Gamma\in {\mathcal{F}}_h^\Gamma$ and $e = T\cap T'$. We also need to define the edge jumps ${\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\in \mathbf{L}^2({\mathcal{E}}_h)$ by $${\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}|_e := {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_e \hbox{ with }
{\llbracket \varphi{\boldsymbol{t}}\rrbracket}_e :=
\varphi_{K_e} {\boldsymbol{t}}_e + \varphi_{K'_e} {\boldsymbol{t}}'_e\, ,$$ where $K_e, K_e'$ are in this case the elements from ${\mathcal{T}}_h^{\Omega_I}$ such that $T=\partial K_e \cap \Gamma\in {\mathcal{F}}_h^\Gamma$, $T'=\partial K'_e \cap \Gamma\in {\mathcal{F}}_h^\Gamma$ and $e = T\cap T'$. Here, ${\boldsymbol{t}}_e$, ${\boldsymbol{t}}'_e$ are the tangent unit vectors along the edge $e$ given by ${\boldsymbol{t}}_e = ({\boldsymbol{n}}_\Gamma \times \boldsymbol \nu_{T})|_e$ and ${\boldsymbol{t}}_e = ({\boldsymbol{n}}_\Gamma \times \boldsymbol \nu_{T'})|_e$ where $\boldsymbol \nu_{T}$ and $\boldsymbol \nu_{T'}$ are the outward unit normal vector to $\partial T$ and $\partial T'$ respectively that lies on the tangent plane to $\Gamma$.
The DG formulation
------------------
Hereafter, given an integer | 0 | non_member_992 |
$k\geq 0$ and a domain $\mathcal O\subset \mathbb{R}^3$, ${\mathcal{P}}_k(\mathcal O)$ denotes the space of polynomials of degree at most $k$ on $\mathcal O$. For any $m\geq 1$, we introduce the finite element spaces $$\mathbf{X}_h := \prod_{K\in {\mathcal{T}}_h^{\Omega_C}} P_m(K)^3
\quad \text{and} \quad
V_h := \prod_{K\in {\mathcal{T}}_h^{\Omega_I}} \tilde{\mathcal{P}}_{m}(K),$$ where $$\label{tildePm}
\tilde{\mathcal{P}}_{m}(K):=\begin{cases}
{\mathcal{P}}_m(K) & \text{if $\partial K\cap \Gamma \notin {\mathcal{F}}_h^\Gamma$ }\\
{\mathcal{P}}_m(K) + {\mathcal{P}}_{m+1}^T(K) & \text{if $T=\partial K\cap \Gamma \in {\mathcal{F}}_h^\Gamma$}
\end{cases}$$ with ${\mathcal{P}}_{m+1}^T(K)$ representing the subspace of ${\mathcal{P}}_{m+1}(K)$ spanned by the elements of the Lagrange basis corresponding to nodal points located on $T$. It follows that ${\mathcal{P}}_m(K) \subset \tilde{\mathcal{P}}_m(K) \subset
{\mathcal{P}}_{m+1}(K)$ and if $T = \partial K \cap \Gamma \in {\mathcal{F}}_h^\Gamma$ then $\tilde{\mathcal{P}}_m(K)|_T = {\mathcal{P}}_{m+1}(T)$.
Let $h_{\mathcal{F}}\in \prod_{F\in \mathcal{F}_h} {\mathcal{P}}_0(F)$ and $h_{\mathcal{E}}\in \prod_{e\in \mathcal{E}_h} {\mathcal{P}}_0(e)$ be defined by $h_{\mathcal{F}}|_F := h_F$ $, \forall F \in \mathcal{F}_h$ and $h_{\mathcal{E}}|_e := h_e$ $, \forall e \in {\mathcal{E}}_h$ respectively. By virtue of our hypotheses on $\sigma$ and on the triangulation ${\mathcal{T}}_h^{\Omega_C}$, we may consider that $\sigma$ is an element of $\prod_{K\in \mathcal{T}_h^{\Omega_C}} {\mathcal{P}}_0(K)$ and denote $\sigma_K:= \sigma|_K$ for all $K\in \mathcal{T}_h^{\Omega_C}$. We introduce $\mathtt{s}_{\mathcal{F}}\in \prod_{F\in \mathcal{F}_h(\Omega_C)} {\mathcal{P}}_0(F)$ defined by $\mathtt{s}_F := \min(\sigma_K, \sigma_{K'})$, if $F = \partial K \cap \partial K'\in {\mathcal{F}}_h^0(\Omega_C)$ and $\mathtt{s}_F := | 0 | non_member_992 |
\sigma_K$, if $F = \partial K \cap \Gamma\in {\mathcal{F}}_h^\Gamma$. We also need to define $\mathtt{s}_{\mathcal{E}}\in \prod_{e\in \mathcal{E}_h} {\mathcal{P}}_0(e)$ given by $\mathtt{s}_e = \min(\sigma_{K_e}, \sigma_{K'_e})$ where $K_e, K_e\in {\mathcal{T}}_h^{\Omega_C}$ are such that $T=\partial K_e \cap \Gamma\in {\mathcal{F}}_h^\Gamma$, $T'=\partial K'_e \cap \Gamma\in {\mathcal{F}}_h^\Gamma$ and $e = T\cap T'$.
We consider, for $s>1/2$, the Hilbert space $$\mathbf{X}^s({\mathcal{T}}_h^{\Omega_C}) := \left\{{\boldsymbol{v}}\in \mathbf{H}^s({\mathcal{T}}_h^{\Omega_C});\quad {\mathbf{curl}}_h {\boldsymbol{v}}\in \mathbf{H}^{1/2+s}({\mathcal{T}}_h^{\Omega_C})\right\}$$ and define on $\mathbf{X}^s({\mathcal{T}}_h^{\Omega_C}) \times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I}) \times {\mathbb C}$ the sesquilinear forms $$\begin{aligned}
A_h^{\Omega_C}(({\boldsymbol{u}}, \phi,c), &({\boldsymbol{v}}, \varphi,m)) := \imath \omega \left(\mu {\boldsymbol{u}}, {\boldsymbol{v}}\right)_{{\mathcal{T}}_h^{\Omega_C}} + \left(\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{u}}, {\mathbf{curl}}_h {\boldsymbol{v}}\right)_{{\mathcal{T}}_h^{\Omega_C}} \\ &+ \left({\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{u}}\}}_{{\mathcal{F}}}, {\llbracket ({\boldsymbol{v}},\varphi,m) \rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_C}}
+ \left({\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{F}}}, {\llbracket ({\boldsymbol{u}}, \phi, c) \rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_C}}\\ &+ \mathtt{a}^{\Omega_C} \left( \mathtt{s}^{-1}_{{\mathcal{F}}} h_{{\mathcal{F}}}^{-1} {\llbracket ({\boldsymbol{u}}, \phi, c) \rrbracket}_{{\mathcal{F}}}, {\llbracket ({\boldsymbol{v}}, \varphi, m) \rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_C}}\, ,\end{aligned}$$ $$\begin{aligned}
A_h^{\Omega_I}(&({\boldsymbol{u}}, \phi, c), ({\boldsymbol{v}}, \varphi, m)):= \imath \omega \mu_0 (\nabla_h \phi+c \boldsymbol \rho, \nabla_h \varphi+m \boldsymbol \rho)_{{\mathcal{T}}_h^{\Omega_I}} + \dfrac{\mathtt{a}^{\Omega_I}}{\omega \mu_0}\left(h_{{\mathcal{F}}}^{-1} {\llbracket \phi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}, {\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_I}}\\
&- \imath \omega \mu_0\left({\{\nabla_h \phi+c \boldsymbol \rho\}}_{{\mathcal{F}}}, {\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_I}}
- \imath \omega \mu_0\left({\{\nabla_h \varphi+m \boldsymbol \rho\}}_{{\mathcal{F}}}, {\llbracket \phi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_I}}
\\
&- \left({\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{u}}\}}_{{\mathcal{E}}}, {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\right)_{{\mathcal{E}}_h} - \left({\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{E}}}, {\llbracket \phi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\right)_{{\mathcal{E}}_h}
+ \alpha \left( \mathtt{s}^{-1}_{{\mathcal{E}}} h_{{\mathcal{E}}}^{-2} {\llbracket \phi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}, {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}} \right)_{{\mathcal{E}}_h},\end{aligned}$$ and let $$A_h(({\boldsymbol{u}}, p,c), ({\boldsymbol{v}}, \varphi,m)):= A_h^{\Omega_C}(({\boldsymbol{u}}, \phi, c), ({\boldsymbol{v}}, \varphi, m)) | 0 | non_member_992 |
+ A_h^{\Omega_I}(({\boldsymbol{u}}, \phi, c), ({\boldsymbol{v}}, \varphi, m))\, .$$
Let us assume that $\sigma^{-1} {\boldsymbol{j}}\in \mathbf{H}^{1/2+s}({\mathcal{T}}_h^{\Omega_C})$ with $s>1/2$. Then we can define the linear form $L_h(\cdot)$ on $\mathbf{X}^s({\mathcal{T}}_h^{\Omega_C}) \times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I}) \times {\mathbb C}$ by $$L_h(({\boldsymbol{v}}, \varphi)) := (\sigma^{-1} {\boldsymbol{j}}, {\mathbf{curl}}_h {\boldsymbol{v}})_{{\mathcal{T}}_h^{\Omega_C}} + \left( {\{\sigma^{-1} {\boldsymbol{j}}\}}_{{\mathcal{F}}}, {\llbracket ({\boldsymbol{v}}, \varphi, m) \rrbracket}_{{\mathcal{F}}} \right)_{{\mathcal{F}}_h^{\Omega_C}}
- \left({\{\sigma^{-1} {\boldsymbol{j}}\}}_{{\mathcal{E}}}, {\llbracket \varphi {\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\right)_{{\mathcal{E}}_h}.$$
We propose the following DG formulation of problem -: $$\label{DG-FEM}
\begin{array}{l}
\text{Find $({\boldsymbol{h}}_{h}, \psi_h,k_h)\in \mathbf{X}_h\times V_h\times {\mathbb C}$ such that,}\\[2ex]
A_h(({\boldsymbol{h}}_{h}, \psi_h,k_h), ({\boldsymbol{v}}, \varphi, m)) = L_h(({\boldsymbol{v}}, \varphi, m))\quad \forall \, ({\boldsymbol{v}}, \varphi, m)\in \mathbf{X}_h\times V_h\times {\mathbb C}\, .
\end{array}$$
The existence and uniqueness of the solution of this problem is proved in Theorem \[LM\]
We end this section by showing that the DG scheme is consistent.
\[consistency0\] Let $({\boldsymbol{h}}, \psi, k)\in \mathbf{H}({\mathbf{curl}}, {\Omega_C})\times {\mathrm{H}}^1(\Omega_I)\times {\mathbb C}$ be the solution of -. Under the assumption $\sigma^{-1} {\boldsymbol{j}}\in \mathbf{H}^{1/2+s}({\mathcal{T}}_h^{\Omega_C})$ and the regularity conditions $({\boldsymbol{h}},\psi, k) \in \mathbf{X}^s({\mathcal{T}}_h^{\Omega_C})\times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I})\times {\mathbb C}$, with $s>1/2$, we have that $$A_h(({\boldsymbol{h}}, \psi, k), ({\boldsymbol{v}}, \varphi, m)) = L_h(({\boldsymbol{v}}, \varphi, m)) \quad
\forall \, ({\boldsymbol{v}},\varphi, m)\in \mathbf{X}_h\times V_h \times {\mathbb C}.$$
Using again the notation ${\boldsymbol{e}}= \sigma^{-1} ( {\mathbf{curl}}\, {\boldsymbol{h}}- {\boldsymbol{j}})$ and taking into account that ${\llbracket ({\boldsymbol{h}},\psi,k) \rrbracket}_{{\mathcal{F}}} =0$, ${\llbracket | 0 | non_member_992 |
\psi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}=0$, and ${\llbracket \psi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}=0$, it is straightforward to show that $$\begin{gathered}
\label{diff}
A_h(({\boldsymbol{h}}, \psi, k), ({\boldsymbol{v}}, \varphi, m)) - L_h(({\boldsymbol{v}}, \varphi, m)) = \imath \omega \int_{\Omega_{C}} \mu {\boldsymbol{h}}\cdot {\boldsymbol{v}}+ \int_{\Omega_{C}} {\boldsymbol{e}}\cdot {\mathbf{curl}}_h {\boldsymbol{v}}\\
+ \imath \omega \mu_0 \int_{\Omega_I}(\nabla \psi + k\boldsymbol \rho)\cdot (\nabla_h \varphi + m \boldsymbol \rho)
+ \left({\{{\boldsymbol{e}}\}}_{{\mathcal{F}}}, {\llbracket ({\boldsymbol{v}}, \varphi, m) \rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_{C}}}\\ - \imath \omega \mu_0\left({\{\nabla \psi + k \boldsymbol \rho\}}_{{\mathcal{F}}}, {\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_I}}
- \left({\{{\boldsymbol{e}}\}}_{{\mathcal{E}}}, {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\right)_{{\mathcal{E}}_h}.\end{gathered}$$ Integrating by parts in each $K\in {\mathcal{T}}_h^{\Omega_{C}}$ and using yield $$\begin{gathered}
\label{GreenOmegaC}
\int_{\Omega_{C}} {\boldsymbol{e}}\cdot {\mathbf{curl}}_h {\boldsymbol{v}}= \sum_{K\in {\mathcal{T}}_h^{\Omega_{C}}}
\int_K {\mathbf{curl}}\, {\boldsymbol{e}}\cdot {\boldsymbol{v}}-
\sum_{K\in {\mathcal{T}}_h^{\Omega_{C}}} \int_{\partial K} {\boldsymbol{e}}\cdot {\boldsymbol{v}}\times {\boldsymbol{n}}_K \\
= -\imath\omega \int_{\Omega_{C}} \mu{\boldsymbol{h}}\cdot {\boldsymbol{v}}- \sum_{F\in {\mathcal{F}}^0_h(\Omega_{C})}
\int_{F} {\{{\boldsymbol{e}}\}}_F \cdot {\llbracket {\boldsymbol{v}}\times {\boldsymbol{n}}\rrbracket}_F -
\sum_{T\in {\mathcal{F}}_h^\Gamma}
\int_{T} {\boldsymbol{e}}\cdot {\boldsymbol{v}}\times {\boldsymbol{n}}.\end{gathered}$$ Similarly, integrating by parts in each $K\in {\mathcal{T}}_h^{\Omega_I}$ together with and give $$\begin{gathered}
\label{GreenOmegaI}
\imath \omega \mu_{0}\int_{\Omega_I}(\nabla \psi + k\boldsymbol \rho)\cdot (\nabla_h \varphi + m \boldsymbol \rho) = - \imath \omega \mu_{0}\sum_{K\in {\mathcal{T}}_h^{\Omega_I}} \int_K {\mathrm{div}}(\nabla \psi + k \boldsymbol \rho) \varphi \\+\imath \omega \mu_{0}
\sum_{K\in {\mathcal{T}}_h^{\Omega_I}} \int_{\partial K} (\nabla \psi + k\boldsymbol \rho) \cdot {\boldsymbol{n}}_K \varphi
+ m \int_{\Omega_{I}} (\nabla \psi + k\boldsymbol \rho)\cdot \boldsymbol \rho
= \imath \omega \mu_{0} \sum_{F\in {\mathcal{F}}_{h}^0} \int_{F} {\{\nabla \psi + k \boldsymbol \rho\}}_{F}
\cdot | 0 | non_member_992 |
{\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{F}\\
- \imath \omega \mu_{0} \sum_{T\in {\mathcal{F}}^{\Gamma}_{h}}\int_{F} (\nabla \psi + k \boldsymbol \rho)
\cdot \varphi{\boldsymbol{n}}_{\Gamma} + \imath \omega \mu_{0} \sum_{T\in {\mathcal{F}}^{\Sigma}_{h}}\int_{F} (\nabla \psi + k \boldsymbol \rho) \cdot \varphi{\boldsymbol{n}}_{\Sigma} + m \int_{\Gamma} {\boldsymbol{e}}\cdot (\boldsymbol \rho \times {\boldsymbol{n}}_{\Gamma}).\end{gathered}$$ Substituting back and in we obtain $$\begin{gathered}
\label{diff1}
A_h(({\boldsymbol{h}}, \psi, k), ({\boldsymbol{v}}, \varphi, m)) - L_h(({\boldsymbol{v}}, \varphi, m)) = -
\sum_{T\in {\mathcal{F}}_h^\Gamma} \int_T {\boldsymbol{e}}\cdot {\mathbf{curl}}_T \varphi\\
- \imath \omega \mu_0 \sum_{T\in {\mathcal{F}}_h^\Gamma} \int_T \nabla (\psi+ k \boldsymbol \rho) \cdot \varphi {\boldsymbol{n}}_{\Gamma} - \left({\{{\boldsymbol{e}}\}}_{{\mathcal{E}}}, {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\right)_{{\mathcal{E}}_h}.\end{gathered}$$ Finally, using the integration by parts formula $$\sum_{T\in {\mathcal{F}}_h^\Gamma}
\int_{T} {\boldsymbol{e}}\cdot {\mathbf{curl}}_T \varphi =\sum_{T\in {\mathcal{F}}_h^\Gamma}
\int_{T} (\text{curl}_T {\boldsymbol{e}}) \varphi - \sum_{T\in {\mathcal{F}}_h^\Gamma} \int_{\partial T} {\boldsymbol{e}}\cdot \varphi{\boldsymbol{t}}_{\partial T}
=
\int_{\Gamma} (\text{curl}_\Gamma {\boldsymbol{e}}) \varphi -
\left({\{{\boldsymbol{e}}\}}_{{\mathcal{E}}}, {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\right)_{{\mathcal{E}}_h},$$ we deduce from that $$\begin{gathered}
\label{diff1+}
A_h(({\boldsymbol{h}}, \psi, k), ({\boldsymbol{v}}, \varphi, m)) - L_h(({\boldsymbol{v}}, \varphi, m)) = -
\int_{\Gamma} (\text{curl}_\Gamma {\boldsymbol{e}}) \varphi\\
- \imath \omega \mu_0 \sum_{T\in {\mathcal{F}}_h^\Gamma} \int_T \nabla (\psi+ k \boldsymbol \rho) \cdot \varphi {\boldsymbol{n}}_{\Gamma}.\end{gathered}$$ and the result follows from the identity $\text{curl}_\Gamma {\boldsymbol{e}}= {\mathbf{curl}}{\boldsymbol{e}}\cdot {\boldsymbol{n}}$, equation and the transmission condition .
Convergence analysis of the DG-FEM formulation {#section4}
==============================================
The aim of this Section is to prove that the DG-FEM formulation is stable in the DG-norm defined on | 0 | non_member_992 |
$\mathbf{X}^s({\mathcal{T}}_h^{\Omega_C})\times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I})\times {\mathbb C}$ by $$\begin{aligned}
{\lVert({\boldsymbol{v}}, \varphi,m)\rVert}^2 := & {\lVert(\omega\mu)^{1/2}{\boldsymbol{v}}\rVert}^2_{0,\Omega_C} + {\lVert\sigma^{-1/2}{\mathbf{curl}}_h {\boldsymbol{v}}\rVert}^2_{0,\Omega_C} + \omega\mu_0 {\lVert\nabla_h \varphi+ m \boldsymbol \rho\rVert}^2_{0,\Omega_I} \\
+ & {\lVert\mathtt{s}^{-1/2}_{{\mathcal{F}}}h_{{\mathcal{F}}}^{-1/2}{\llbracket ({\boldsymbol{v}},\varphi,m) \rrbracket}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_C}} + \omega\mu_0 {\lVerth_{{\mathcal{F}}}^{-1/2}{\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_I}}\\
+ &{\lVert\mathtt{s}^{-1/2}_{{\mathcal{E}}} h_{{\mathcal{E}}}^{-1}{\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}^2_{0,{\mathcal{E}}_h}.\end{aligned}$$ We also need to introduce $$\begin{gathered}
{\lVert({\boldsymbol{v}}, \varphi,m)\rVert}_{\ast}^2 := {\lVert({\boldsymbol{v}}, \varphi,m)\rVert}^2 +
{\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1}{\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_C}}\\
+ {\lVert\mathtt{s}_{{\mathcal{E}}}^{1/2} h_{{\mathcal{E}}} {\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{E}}}\rVert}^2_{0,{\mathcal{E}}_h}
+{\lVerth_{{\mathcal{F}}}^{1/2} {\{\nabla_h \varphi + m \boldsymbol \rho\}}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_I}}.\end{gathered}$$
The following discrete trace inequality is standard, (see, e.g. [@DiPietroErn Lemma 1.46]).
For all integer $k\geq 0$ there exists a constant $C^*>0$ independent of $h$ such that, $$\label{discreteTrace3D}
h_Q {\lVert v\rVert}^2_{0,\partial Q} \leq C^* {\lVert v\rVert}^2_{0,Q} \quad \forall \, v\in {\mathcal{P}}_k(Q),\quad
\forall Q\in \{{\mathcal{T}}_h,{\mathcal{F}}_h^\Gamma\}.$$
It is used to prove the following auxiliary result.
\[equivalence\] For all $k\geq 0$, there exist constants $C_{\Omega_C}>0$ and $C_{\Omega_I}>0$ independent of the mesh size and the coefficients such that $$\label{discIneq1}
{\lVert\mathtt{s}_{{\mathcal{E}}}^{1/2} h_{{\mathcal{E}}} {\{\sigma^{-1} \mathbf{w}\}}_{{\mathcal{E}}}\rVert} _{0,{\mathcal{E}}_h} +
{\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1}\mathbf{w}\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}} \leq C_{\Omega_C} {\lVert\sigma^{-1/2} \mathbf{w}\rVert}_{0,{\Omega_C}}\, ,$$ for all $\mathbf{w} \in \prod_{K\in {\mathcal{T}}_h^{\Omega_C}}{\mathcal{P}}_k(K)^3$, and $$\label{discIneq2}
{\lVerth_{{\mathcal{F}}}^{1/2} {\{\mathbf{w}\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} \leq C_{\Omega_I} {\lVert\mathbf{w}\rVert}_{0,\Omega_I} \, ,$$ for all $\mathbf{w} \in \prod_{K\in {\mathcal{T}}_h^{\Omega_I}}{\mathcal{P}}_k(K)^3$.
By definition of $\mathtt{s}_{{\mathcal{F}}}$, for any $\mathbf{w} \in \prod_{K\in {\mathcal{T}}_h^{\Omega_C}}{\mathcal{P}}_k(K)^3$, $$\begin{gathered}
\label{transfer0}
{\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1}\mathbf{w}\}}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_C}} =
\sum_{F\in {\mathcal{F}}_h^{\Omega_C}} h_F {\lVert \mathtt{s}_F^{1/2}{\{\sigma^{-1}\mathbf{w}\}}_F \rVert}^2_{0,F}\\
\leq \sum_{K\in {\mathcal{T}}_h^{\Omega_C}} \sum_{F\in | 0 | non_member_992 |
{\mathcal{F}}(K)} h_F {\lVert \mathtt{s}_F^{1/2}\sigma_K^{-1}\mathbf{w}_K \rVert}^2_{0,F}
\leq \sum_{K\in {\mathcal{T}}_h^{\Omega_C}} h_K
{\lVert \sigma_K^{-1/2}\mathbf{w}_K \rVert}^2_{0,\partial K}.\end{gathered}$$
Similarly, $$\begin{gathered}
\label{transfer}
{\lVert \mathtt{s}_{{\mathcal{E}}}^{1/2} h_{{\mathcal{E}}} {\{\sigma^{-1} \mathbf{w}\}}_{{\mathcal{E}}} \rVert}^2 _{0,{\mathcal{E}}_h} =
\sum_{e \in {\mathcal{E}}_h} h_e^2 {\lVert\mathtt{s}_{e}^{1/2} {\{\sigma^{-1} \mathbf{w}\}}_e \rVert}^2_{0,e} \\
\leq \sum_{T\in {\mathcal{F}}_h^\Gamma} \sum_{e\in {\mathcal{E}}(T)} h_e^2 {\lVert\mathtt{s}_{e}^{1/2} \sigma_{K_T}^{-1}
\mathbf{w}_{K_T}\rVert}^2_{0,e}
\leq \sum_{T\in {\mathcal{F}}_h^\Gamma }
h_{T}^2 {\lVert\sigma_{K_T}^{-1/2} \mathbf{w}_{K_T}\rVert}^2_{0,\partial T}\, ,\end{gathered}$$ where $K_T\in {\mathcal{T}}_h^{\Omega_C}$ is such that $T=\partial K_T \cap \Gamma$. It follows from that $$\begin{gathered}
{\lVert\mathtt{s}_{{\mathcal{E}}}^{1/2} h_{{\mathcal{E}}} {\{\sigma^{-1} \mathbf{w}\}}_{{\mathcal{E}}}\rVert}^2 _{0,{\mathcal{E}}_h^{\Omega_I}} \leq C^*
\sum_{T \in {\mathcal{F}}_h^\Gamma} h_{T} {\lVert\sigma_{K_T}^{-1/2} \mathbf{w}_{K_T}\rVert}^2_{0, T} \leq C^*
\sum_{K \in{\mathcal{T}}^{\Omega_C}_h} h_{K} {\lVert\sigma_K^{-1/2} \mathbf{w}_K\rVert}^2_{0, \partial K} \end{gathered}$$ and follows by applying again the discrete trace inequality in the last estimate and in . Finally, for any $\mathbf{w} \in \prod_{K\in {\mathcal{T}}_h^{\Omega_I}}{\mathcal{P}}_k(K)^3$, $$\label{transfer1}
{\lVerth_{{\mathcal{F}}}^{1/2} {\{\mathbf{w}\}}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_I}} = \sum_{F\in {\mathcal{F}}_h^{\Omega_I}} h_F {\lVert{\{\mathbf{w}\}}_F\rVert}_{0,F}^2
\leq \sum_{K\in {\mathcal{T}}_h^{\Omega_I}} h_K {\lVert \mathbf{w}_K \rVert}^2_{0,\partial K}$$ and follows again from .
\[boundedness\] There exists a constant $M>0$ independent of $h$ such that $$| A_h(({\boldsymbol{u}}, \phi, c), ({\boldsymbol{v}}, \varphi, m)) | \leq M {\lVert({\boldsymbol{u}}, \phi, c)\rVert}_* {\lVert({\boldsymbol{v}},\varphi, m)\rVert}$$ for all $({\boldsymbol{u}}, \phi,c)$, $({\boldsymbol{v}},\varphi,m)\in \mathbf{X}^s({\mathcal{T}}_h^{\Omega_C})\times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I})\times {\mathbb C}$, with $s>1/2$.
By the Cauchy-Schwarz inequality, we have that $$\begin{array}{l}
|A_h^{\Omega_C}(({\boldsymbol{u}}, \phi,c), ({\boldsymbol{v}}, \varphi,m))| \\[.1cm]
\qquad \leq \omega {\lVert\mu^{1/2}{\boldsymbol{u}}\rVert}_{0,\Omega_C} {\lVert\mu^{1/2}{\boldsymbol{v}}\rVert}_{0,\Omega_C} +
{\lVert\sigma^{-1/2}{\mathbf{curl}}_h {\boldsymbol{u}}\rVert}_{0,\Omega_C} {\lVert\sigma^{-1/2}{\mathbf{curl}}_h {\boldsymbol{v}}\rVert}_{0,\Omega_C}\\[.1cm]
\qquad + {\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1}{\mathbf{curl}}_h {\boldsymbol{u}}\}}_{{\mathcal{F}}}\rVert}_{0, | 0 | non_member_992 |
{\mathcal{F}}_h^{\Omega_C}}
{\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2} h_{{\mathcal{F}}}^{-1/2} {\llbracket ({\boldsymbol{v}}, \varphi,m) \rrbracket}_{{\mathcal{F}}}\rVert}_{0, {\mathcal{F}}_h^{\Omega_C}}\\[.1cm]
\qquad + {\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1}{\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{F}}}\rVert}_{0, {\mathcal{F}}_h^{\Omega_C}}
{\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2} h_{{\mathcal{F}}}^{-1/2} {\llbracket ({\boldsymbol{u}}, \phi,c) \rrbracket}_{{\mathcal{F}}}\rVert}_{0, {\mathcal{F}}_h^{\Omega_C}}\\[.1cm]
\qquad + \mathtt{a}^{\Omega_C} {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2}h_{{\mathcal{F}}}^{-1/2} {\llbracket ({\boldsymbol{u}}, \phi,c) \rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}} {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2}h_{{\mathcal{F}}}^{-1/2} {\llbracket ({\boldsymbol{v}}, \varphi,m) \rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}}.
\end{array}$$ Applying with $\mathbf{w} = {\mathbf{curl}}_h {\boldsymbol{v}}$ we obtain $$|A_h^{\Omega_C}(({\boldsymbol{u}}, \phi,c), ({\boldsymbol{v}}, \varphi, m))| \leq (1+ C_\Omega + \mathtt{a}^{\Omega_C})\, {\lVert({\boldsymbol{u}}, \phi,c)\rVert}_* {\lVert({\boldsymbol{v}},\varphi,m)\rVert}$$ for all $({\boldsymbol{u}}, \phi,c)$ and $({\boldsymbol{v}},\varphi,m)\in \mathbf{X}^s({\mathcal{T}}_h^\Omega)\times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I}) \times {\mathbb C}$. On the other hand, $$\begin{array}{l}
|A_h^{\Omega_I}(({\boldsymbol{u}}, \phi, c), ({\boldsymbol{v}}, \varphi,m))| \leq \omega \mu_0 {\lVert\nabla_h \phi+c \boldsymbol \rho\rVert}_{0,\Omega_I} {\lVert\nabla_h \varphi + m \boldsymbol \rho\rVert}_{0,\Omega_I} \\[.1cm]
\quad\qquad+ \omega \mu_0 {\lVerth_{{\mathcal{F}}}^{1/2} {\{\nabla_h \varphi+m \boldsymbol \rho\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket \phi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} \\[.1cm]
\quad\qquad+ \omega \mu_0 {\lVerth_{{\mathcal{F}}}^{1/2} {\{\nabla_h \phi+c \boldsymbol \rho\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}}\\[.1cm]
\quad\qquad+ \alpha {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2}h_{{\mathcal{E}}}^{-1} {\llbracket \phi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h} {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2}h_{{\mathcal{E}}}^{-1} {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h}\\[.1cm]
\quad\qquad+ {\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2}h_{{\mathcal{E}}}{\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h} {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2}h_{{\mathcal{E}}}^{-1} {\llbracket \phi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h}\\[.1cm]
\quad\qquad+ {\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2}h_{{\mathcal{E}}}{\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{u}}\}}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h} {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2}h_{{\mathcal{E}}}^{-1} {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h}\\[.1cm]
\quad\qquad+\mathtt{a}^{\Omega_I}{\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket \phi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h}
\end{array}$$ and it follows from (applied with $\mathbf{w}= \nabla_h \varphi+ m \boldsymbol \rho$ ) and (applied with $\mathbf{w}= {\mathbf{curl}}_h {\boldsymbol{v}}$) that $$|A_h^{\Omega_I}(({\boldsymbol{u}}, \phi,c), ({\boldsymbol{v}}, \varphi, m))| \leq (1 + C_{\Omega_I} + C_{\Omega}+ \mathtt{a}^{\Omega_I}+ \alpha) \, {\lVert({\boldsymbol{u}}, \phi,c)\rVert}_* {\lVert({\boldsymbol{v}},\varphi,m)\rVert},$$ which gives the result.
\[discElip\] There exists a constant $\alpha_0>0$ independent of the mesh size and the coefficients such | 0 | non_member_992 |
that if $\min(\mathtt{a}^{\Omega_C}, \mathtt{a}^{\Omega_I}, \alpha)\geq \alpha_0$ then, $$\label{elip}
\text{Re}\left[ (1 - \imath) A_h(({\boldsymbol{v}}, \varphi,m), (\overline {\boldsymbol{v}}, \overline \varphi,\overline m))\right] \geq \dfrac{1}{2} {\lVert({\boldsymbol{v}}, \varphi,m)\rVert}^2 \qquad \forall ({\boldsymbol{v}}, \varphi,m)\in \mathbf{X}_h\times V_h\times {\mathbb C}.$$
By definition of $A_h(\cdot, \cdot)$, $$\label{elip0}
\begin{array}{l}
\text{Re}\left[(1 - \imath) A_h(({\boldsymbol{v}}, \varphi,m), (\overline {\boldsymbol{v}}, \overline \varphi,\overline m))\right] = \omega {\lVert \mu^{1/2} {\boldsymbol{v}}\rVert}_{0,\Omega_C}^2 +
{\lVert\sigma^{-1/2} {\mathbf{curl}}_h {\boldsymbol{v}}\rVert}_{0,\Omega_C}^2\\[.1cm]
\qquad+ 2 \text{Re} \left( {\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{F}}}, {\llbracket (\overline {\boldsymbol{v}}, \overline \varphi,\overline m) \rrbracket}_{{\mathcal{F}}} \right)_{{\mathcal{F}}_h^{\Omega_C}}
+ \mathtt{a}^{\Omega_C} {\lVerth_{{\mathcal{F}}}^{-1/2}{\llbracket ({\boldsymbol{v}}, \varphi,m) \rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}}^2\\[.1cm]
\qquad +\omega \mu_0 {\lVert\nabla_h \varphi+m \boldsymbol \rho\rVert}_{0,\Omega_C}^2 - 2 \omega \mu_0 \text{Re} \left(
{\{\nabla_h \varphi+ m \boldsymbol \rho\}}_{{\mathcal{F}}}, {\llbracket \overline \varphi {\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_I}} \\[.1cm]
\qquad+ \mathtt{a}^{\Omega_I}{\lVerth_{{\mathcal{F}}}^{-1/2}{\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_I}}
- 2 \text{Re} \left( {\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{E}}}, {\llbracket \overline \varphi {\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}} \right)_{{\mathcal{E}}_h}
+ \alpha {\lVerth_{{\mathcal{E}}}^{-1}{\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}^2_{0, {\mathcal{E}}_h}.
\end{array}$$
It follows from the Cauchy-Schwarz inequality and that, $$\label{elip1}
\begin{array}{l}
2 | \text{Re} \left( {\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{F}}}, {\llbracket (\overline {\boldsymbol{v}}, \overline \varphi,\overline m) \rrbracket}_{{\mathcal{F}}} \right)_{{\mathcal{F}}_h^{\Omega_C}}|
\\[.1cm]
\qquad \leq 2 {\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}}
{\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2} h_{{\mathcal{F}}}^{-1/2} {\llbracket ( {\boldsymbol{v}}, \varphi,m) \rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}} \\[.1cm]
\qquad\leq 2C_{\Omega_C} {\lVert\sigma^{-1/2}{\mathbf{curl}}_h {\boldsymbol{v}}\rVert}_{0,\Omega_C} {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2} h_{{\mathcal{F}}}^{-1/2} {\llbracket ( {\boldsymbol{v}}, \varphi, m) \rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}} \\[.1cm]
\qquad\leq \frac{1}{4} {\lVert\sigma^{-1/2}{\mathbf{curl}}_h {\boldsymbol{v}}\rVert}_{0,\Omega_C}^2 + 4 C_{\Omega_C}^2
{\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2} h_{{\mathcal{F}}}^{-1/2} {\llbracket ( {\boldsymbol{v}}, \varphi, m) \rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}}^2.
\end{array}$$ Similarly, by virtue of , $$\label{elip2}
\begin{array}{l}
| 0 | non_member_992 |
2 | \text{Re} \left(
{\{\nabla_h \varphi+m \boldsymbol \rho\}}_{{\mathcal{F}}}, {\llbracket \overline \varphi {\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_I}} | \leq
2{\lVerth_{{\mathcal{F}}}^{1/2} {\{\nabla_h \varphi+ m \boldsymbol \rho\}}_{{\mathcal{F}}} \rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket \overline \varphi {\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}} \rVert}_{0,{\mathcal{F}}_h^{\Omega_I}}\\[.1cm]
\qquad \qquad \leq 2 C_{\Omega_I} {\lVert\nabla_h \varphi+m \boldsymbol \rho\rVert}_{0,\Omega} {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket \varphi {\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}} \rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} \\[.1cm]
\qquad\qquad \leq \frac{1}{2} {\lVert\nabla_h \varphi + m \boldsymbol \rho\rVert}_{0,\Omega}^2 + 4 C_{\Omega_I}^2 {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket \varphi {\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}} \rVert}_{0,{\mathcal{F}}_h^{\Omega_I}}^2.
\end{array}$$ Finally, using we have that $$\label{elip3}
\begin{array}{l}
2 |\text{Re} \left( {\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{E}}}, {\llbracket \overline \varphi {\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}} \right)_{{\mathcal{E}}_h}|\leq
2 {\lVert\mathtt{s}_{{\mathcal{E}}}^{1/2} h_{{\mathcal{E}}}{\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h}
{\lVert\mathtt{s}_{{\mathcal{E}}}^{-1/2} h_{{\mathcal{E}}}^{-1}{\llbracket \varphi {\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h} \\[.1cm]
\qquad \leq
2 C_\Gamma {\lVert\sigma^{-1/2} {\mathbf{curl}}_h {\boldsymbol{v}}\rVert}^2_{0,\Omega_C}
{\lVert\mathtt{s}_{{\mathcal{E}}}^{-1/2} h_{{\mathcal{E}}}^{-1}{\llbracket \varphi {\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h}\\[.1cm]
\qquad\leq \frac{1}{4}
{\lVert\sigma^{-1/2} {\mathbf{curl}}_h {\boldsymbol{v}}\rVert}_{0,\Omega_C}^2 + 4 C_{\Omega_C}^2 {\lVert\mathtt{s}_{{\mathcal{E}}}^{-1/2} h_{{\mathcal{E}}}^{-1}{\llbracket \varphi {\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h}^2.
\end{array}$$ Combining with - and choosing $\alpha_0 = 1/2 + 4 C_\Omega^2 + 4C_{\Omega_I}^2$ we obtain .
We are now in a position to prove the ${\lVert\cdot\rVert}$-stability of the DG scheme .
\[LM\] Assume that $\sigma^{-1} {\boldsymbol{j}}\in \mathbf{H}^{1/2+s}({\mathcal{T}}_h^{\Omega_C})$ and $\min(\mathtt{a}^\Omega, \mathtt{a}^{\Omega_I}, \alpha)\geq \alpha_0$. Then, there exits a unique $({\boldsymbol{h}}_{h},\psi_h,k_h) \in \mathbf{X}_h\times V_h\times {\mathbb C}$ solution of Problem . Moreover if $({\boldsymbol{h}}, \psi,k)\in [\mathbf{H}({\mathbf{curl}},\Omega)\times {\mathrm{H}}^1(\Omega_I)\times {\mathbb C}] \cap [\mathbf{X}^s({\mathcal{T}}_h^{\Omega_C}) \times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I}) \times {\mathbb C}]$ is the solution to - then $$\label{Cea}
{\lVert({\boldsymbol{h}}- {\boldsymbol{h}}_{h}, \psi - \psi_h, k -k_h)\rVert} \leq (1 | 0 | non_member_992 |
+ 2\sqrt{2} M) \inf_{({\boldsymbol{v}}, \varphi)\in \mathbf{X}_h\times V_h} {\lVert({\boldsymbol{h}}- {\boldsymbol{v}}, \psi -\varphi, 0)\rVert}_*.$$
The well posedness of Problem follows immediately from Proposition \[discElip\].
Moreover we deduce from Proposition \[discElip\] and the consistency of the scheme that $$\begin{array}{l}
\frac{1}{2} {\lVert({\boldsymbol{h}}_{h}- {\boldsymbol{v}}, \psi_h - \varphi, k_h-m)\rVert}^2 \\[.1cm]
\qquad \leq \text{Re} \left[ (1 - \imath) A_h(({\boldsymbol{h}}_{h}- {\boldsymbol{v}}, \psi_h - \varphi, k_h-m),({\boldsymbol{h}}_{C,h}- {\boldsymbol{v}}, \psi_h - \varphi, k_h-m)) \right] \\[.1cm]
\qquad = \text{Re} \left[ (1 - \imath) A_h(({\boldsymbol{h}}- {\boldsymbol{v}}, \psi - \varphi, k-m),({\boldsymbol{h}}- {\boldsymbol{v}}, \psi - \varphi, k-m)) \right]
\end{array}$$ for all $({\boldsymbol{v}}, \varphi,m)\in \mathbf{X}_h\times V_h\times {\mathbb C}$. Then from Proposition \[boundedness\] we have $${\lVert({\boldsymbol{h}}_{h}- {\boldsymbol{v}}, \psi_h - \varphi, k_h-m)\rVert} \leq 2 \sqrt{2} M {\lVert({\boldsymbol{h}}- {\boldsymbol{v}}, \psi - \varphi,k-m)\rVert}_\ast.$$ The result follows now from the triangle inequality.
Asymptotic error estimates {#sec5}
==========================
We denote by ${\boldsymbol{\Pi}}_{h,m}^{\text{curl}}$ the $m$-order $\mathbf{H}({\mathbf{curl}}, \Omega_C)$-conforming Nédélec interpolation operator of the second kind, see for example [@NED86] or [@Monk Section 8.2]. It is well known that ${\boldsymbol{\Pi}}_{h,m}^{\text{curl}}$ is bounded on $\mathbf{H}({\mathbf{curl}}, \Omega_C)\cap \mathbf{H}^s({\mathbf{curl}}, {\mathcal{T}}_h^{\Omega_C})$ for $s>1/2$, where $$\mathbf{H}^s({\mathbf{curl}}, {\mathcal{T}}_h^{\Omega_C}) := {\left\{{\boldsymbol{v}}\in \mathbf{H}^s({\mathcal{T}}_h^{\Omega_C});\quad {\mathbf{curl}}_h {\boldsymbol{v}}\in \mathbf{H}^{s}({\mathcal{T}}_h^{\Omega_C})\right\}}.$$ Moreover, there exists a constant $C_1>0$ independent of $h$ such that (cf. [@AVbook]) $$\label{errorInterp1}
{\lVert{\boldsymbol{u}}- {\boldsymbol{\Pi}}_{h,m}^{\text{curl}} {\boldsymbol{u}}\rVert}_{0, \Omega_C} + {\lVert{\mathbf{curl}}({\boldsymbol{u}}- {\boldsymbol{\Pi}}_{h,m}^{\text{curl}}{\boldsymbol{u}})\rVert}_{0, \Omega_C} \leq C_1 h^{\min(s, m)} \big( | 0 | non_member_992 |
{\lVert{\boldsymbol{u}}\rVert}_{s, {\mathcal{T}}_h^{\Omega_C}} + {\lVert{\mathbf{curl}}_h {\boldsymbol{u}}\rVert}_{s, {\mathcal{T}}_h^{\Omega_C}} \big).$$
We introduce $
\mathbf{L}^2_t(\Gamma) = {\left\{\boldsymbol{\varphi}\in \mathbf{L}^2(\Gamma);\,\, \boldsymbol{\varphi}\cdot {\boldsymbol{n}}= 0\right\}}
$ and consider the $m$-order order Brezzi-Douglas-Marini (BDM) finite element approximation of the space $$\mathbf{H}(\text{div}_\Gamma, \Gamma) := {\left\{ \boldsymbol{\varphi}\in \mathbf{L}_t^2(\Gamma); \quad
\text{div}_\Gamma \boldsymbol{\varphi}\in L^2(\Gamma) \right\}}$$ relatively to the mesh ${\mathcal{F}}_h^\Gamma$ (see, e.g. [@Boffi]). It is given by $$\mathcal{\mathbf{BDM}}({\mathcal{F}}_h^\Gamma) = {\left\{\boldsymbol{\varphi}\in \mathbf{H}(\text{div}_\Gamma, \Gamma); \quad
\boldsymbol{\varphi}|_T \in {\mathcal{P}}_m(T)^2,\quad \forall T\in {\mathcal{F}}_h^\Gamma\right\}}.$$ The corresponding interpolation operator $\Pi_{h,m}^{\text{BDM}}$ is bounded on $
\mathbf{H}(\text{div}_\Gamma, \Gamma)\cap \prod_{T\in {\mathcal{F}}_h^\Gamma} {\mathrm{H}}^\delta(T)^2$ for all $\delta>0$ and we recall that it is uniquely characterized on each $T\in {\mathcal{F}}_h^\Gamma$ by the conditions $$\label{BDMfreedom1}
\int_e \Pi_{h,m}^{\text{BDM}} \boldsymbol{\varphi}\cdot {\boldsymbol{n}}_T q = \int_e \boldsymbol{\varphi}\cdot {\boldsymbol{n}}_T q\quad
\forall q \in {\mathcal{P}}_m(e),\quad \forall e\in {\mathcal{E}}(T),$$ $$\label{BDMfreedom2}
\int_T \Pi_{h,m}^{\text{BDM}} \boldsymbol{\varphi}\cdot \mathbf{q} = \int_T \boldsymbol{\varphi}\cdot \mathbf{q}\quad
\forall \mathbf{q} \in {\mathcal{P}}_{m-2}(T)^2 + \mathbf{S}_{m-1}(T),$$ where $\mathbf{S}_{m-1}(T):= {\left\{\mathbf{q}\in \tilde{\mathcal{P}}_{m-1}(T)^2;\quad \mathbf{q}\cdot \begin{pmatrix}x_1\\x_2
\end{pmatrix} = 0\right\}}$ with $\tilde{\mathcal{P}}_{m-1}(T)$ representing the set of homogeneous polynomials of degree $m-1$ and $\begin{pmatrix}x_1\\x_2\end{pmatrix}$ being the local variable on the plane containing $T$.
The commuting diagram property $$\begin{aligned}
\label{commuting2} ({\boldsymbol{\Pi}}_{h,m}^{\text{curl}} {\boldsymbol{u}}) \times {\boldsymbol{n}}_\Gamma &= \Pi_{h,m}^{\text{BDM}} ({\boldsymbol{u}}\times {\boldsymbol{n}}_\Gamma)\end{aligned}$$ holds true for all ${\boldsymbol{u}}\in \mathbf{H}({\mathbf{curl}}, \Omega_C)\cap \mathbf{H}^s({\mathbf{curl}}, {\mathcal{T}}_h^{\Omega_C})$, $s>1/2$, see [@Hiptmair1 section 9] for more details.
For all $K\in {\mathcal{T}}_h^{\Omega_I}$ we define the | 0 | non_member_992 |
local interpolation operator $\tilde\pi_{K,m}:{\mathrm{H}}^{1+s}(K)\to \tilde{\mathcal{P}}_{m}(K)$, $s>1/2$ as follows: recalling the definition of $\tilde{\mathcal{P}}_{m}(K)$ given in
- if $\partial K \cap \Gamma \not \in \mathcal F_h^\Gamma$ then $\tilde{\mathcal{P}}_{m}(K)= {\mathcal{P}}_{m}(K)$ and we take $\tilde\pi_{K,m} = \pi_{K,m}$, where $\pi_{K,m}$ is defined as in [@Monk Section 5.6];
- if $\partial K \cap \Gamma=T \in \mathcal F_h^\Gamma$ then $\tilde{\mathcal{P}}_{m}(K)={\mathcal{P}}_m(K) + {\mathcal{P}}_{m+1}^T(K)$ and $\tilde \pi_{K,m}$ is defined by changing the conditions defining $\pi_{K,m}$ on $T$ and on the edges composing $T$ into $$\label{freedomFb}
\int_T \tilde\pi_{K,m} p q = \int_T p q \qquad \forall q \in {\mathcal{P}}_{m-2}(T)$$ and $$\label{freedomEb}
\int_e \tilde\pi_{K,m} p q = \int_e p q \qquad \forall q \in {\mathcal{P}}_{m-1}(e), \quad \forall e\in {\mathcal{E}}(F)$$ respectively. The remaining degrees of freedom are the same as those defining $\pi_{K,m}$, see [@Monk Section 5.6].
We notice that $
\text{dim}({\mathcal{P}}_m(K) + {\mathcal{P}}_{m+1}^T(K))= \text{dim}({\mathcal{P}}_m(K)) + m+1
$ and the number of degrees of freedom defining $\tilde \pi_{K,m}$ is equal to the number of degrees of freedom of $\pi_{K,m}$ plus $\text{dim}({\mathcal{P}}_{m-2}(T)) - \text{dim}({\mathcal{P}}_{m-3}(T)) = m-1$ additional degrees of freedom on $T$ and one additional degree of freedom on each of the three edges of $T$, which gives a total of $\text{dim}({\mathcal{P}}_m(K)) + m+1$ degrees of freedom. Using this fact, it is | 0 | non_member_992 |
---
author:
- 'Tathagata Banerjee[^1]'
- 'Alex Bernstein[^2]'
- 'Zachary Feinstein[^3]'
bibliography:
- 'bibtex2.bib'
title: Dynamic clearing and contagion in financial networks
---
Introduction {#sec:intro}
============
Financial networks and the contagion of bank failures have been widely studied beginning with the seminal work on financial payment networks by Eisenberg & Noe [@EN01]. The 2007-2009 financial crisis and credit crunch showed the severe impacts that systemic crises can have on the financial sector and the economy as a whole. As the costs of such cascading events is tremendous, the modeling of such events is imperative. Recently there have been significant studies on modeling financial systemic risk and financial contagion. Two major classes of models exist for systemic risk, i.e., those based on network models from [@EN01] and those based on a mean field approach [@fouque2013illustrated; @fouque2015meanfield]. Notably, the network model approach generally is considered in only a static, single time, setting while the mean field approach is considered as a differential system. In this paper we will construct a dynamic extension of the interbank network model of [@EN01] thus closing the gap between these two streams of literature.
Interbank networks were studied first in [@EN01] to model the spread of defaults | 0 | non_member_993 |
in the financial system. In the Eisenberg-Noe framework, financial firms must satisfy their liabilities by transferring assets. One firm being unable to meet its liabilities due to a shortfall of assets can cause other firms to default on some of their liabilities as well, causing a cascading failure in the financial system. The existence and uniqueness of the clearing payments of this baseline model was proven in [@EN01]. That paper additionally provides methods for numerically computing the realized interbank payments. This baseline model has been extended in multiple directions, including bankruptcy costs, cross-holdings, and fire sales. We refer to [@AW_15; @Staum] for reviews of the prior literature. In regards to bankruptcy costs in financial networks, we refer to [@E07; @RV13; @EGJ14; @GY14; @AW_15; @CCY16; @veraart2017distress]. Cross-holdings have been studied in [@E07; @EGJ14; @AW_15; @GHM12]. Fire sales for a single (representative) illiquid asset have been studied in [@CFS05; @NYYA07; @GK10; @AFM13; @CLY14; @AW_15; @AFM16] and for multiple illiquid assets in [@feinstein2015illiquid; @feinstein2016leverage; @feinstein2017currency]. These network models have been implemented by central banks and regulators for stress testing of and studying cascading failures in the banking systems under their jurisdiction, see, e.g., [@Anand:Canadian; @HK:Modeling; @BEST:2004; @ELS13; @U11; @GHK2011].
Mean field models have | 0 | non_member_993 |
also been considered for studying financial contagion and systemic risk. [@fouque2013illustrated] provides a model of agents who revert to the ensemble mean to provide understanding of “systemic risk events” in which many firms fail. Similar mean field diffusion models without controls were studied in, e.g., [@fouque2013stability; @garnier2013a; @garnier2013b]. In contrast, mean field and stochastic games have been proposed for the study of systemic risk in, e.g., [@fouque2015meanfield; @carmona2016delay]. In such models the firms are allowed to borrow from (or lend to) a central bank, the amount of which is optimized to minimize a quadratic cost function. Thus the choice of borrowing and lending provides an optimal control problem beyond the simpler mean field model of [@fouque2013illustrated]. [@NS17] proposes a separate particle system model with mean field interactions.
The current work will focus on adding the time dynamics, which make the mean field models attractive, to the interbank network approach. In fact, the conclusion of [@EN01] provides a discussion of future extensions, one of which is the inclusion of multiple clearing dates. This has been studied directly in [@CC15; @ferrara16]. Additionally, [@KV16] considers a similar approach to model financial networks with multiple maturities. [@feinstein2017currency] further provides another approach to financial networks | 0 | non_member_993 |
with multiple maturities by considering each clearing date as a different asset. All of these works, however, only consider clearing at discrete times. [@sonin2017] presents a continuous-time clearing model that exactly replicates the static Eisenberg-Noe framework. In this work we will present both discrete and continuous-time clearing models. However, our emphasis will be on the derivation and the characterization of the continuous-time model. This in part is motivated by the prospect of unification with the mean-field models as well as traditional financial models which typically employ continuous-time models. Additionally, as we will demonstrate, the continuous-time framework no longer requires monotonicity for existence and uniqueness which is generally assumed for static and discrete-time systems. This is valuable for future works that may model network formation and payments as a non-cooperative game; such games may not satisfy the strong monotonicity assumptions usually considered in static and discrete-time systems, but would likely satisfy the sufficient conditions for the continuous-time framework.
The organization of this paper is as follows. In Section \[sec:setting\] we will provide a review of the static Eisenberg-Noe framework. Of particular interest, in this section, we consider the clearing to be in terms of the equity and losses of the firms, | 0 | non_member_993 |
as considered in, e.g., [@veraart2017distress; @barucca2016valuation] rather than payments as originally studied in [@EN01]. In Section \[sec:discrete\] we propose a discrete-time formulation for the Eisenberg-Noe model. In discrete time we provide results on existence and uniqueness, as well as a numerical algorithm based on the fictitious default algorithm of [@EN01]. We then extend our model to a continuous-time setting in Section \[sec:continuous\]. For continuous time we consider existence and uniqueness of the clearing solutions, and a numerical algorithm for finding sample paths of this clearing solution, under cash flows modeled by Itô processes. We additionally provide conditions for the discrete-time setting to converge to the continuous-time solution as the time step limits to 0. Section \[sec:discussion\] provides discussion on the financial implications of time dynamics in interbank networks. In particular, we find that the static Eisenberg-Noe clearing solution can be recovered in the continuous-time setting by choosing the network parameters precisely. This allows for a notion of determining the true order of defaults as opposed to the fictitious default order discussed in the static literature based on [@EN01]. However, if the continuous-time network parameters are determined to not follow the rules for recreating the static Eisenberg-Noe setting, then the dynamic | 0 | non_member_993 |
and static clearing solutions will generally not coincide. In fact, the set of defaulting and solvent institutions can be altered by rearranging the timing of obligations. As such, using the static Eisenberg-Noe framework for stress testing may result in an incorrect assessment of the health of the financial system. The proofs of the main results are provided in the Appendix.
Static clearing systems {#sec:setting}
=======================
We begin with some simple notation that will be consistent for the entirety of this paper. Let $x,y \in {\mathbb{R}}^n$ for some positive integer $n$, then $$x \wedge y = \left(\min(x_1,y_1),\min(x_2,y_2),\ldots,\min(x_n,y_n)\right)^{\top},$$ $x^- = -(x \wedge 0)$, and $x^+ = (-x)^-$. Further, to ease notation, we will denote $[x,y] := [x_1,y_1] \times [x_2,y_2] \times \ldots \times [x_n,y_n] \subseteq {\mathbb{R}}^n$ to be the $n$-dimensional compact interval for $y - x \in {\mathbb{R}}^n_+$. Similarly, we will consider $x \leq y$ if and only if $y - x \in {\mathbb{R}}^n_+$.
Throughout this paper we will consider a network of $n$ financial institutions. We will denote the set of all banks in the network by ${\mathcal{N}}:= \{1,2,\ldots,n\}$. Often we will consider an additional node $0$, which encompasses the entirety of the financial system outside of the $n$ banks; this node | 0 | non_member_993 |
$0$ will also be referred to as society or the societal node. The full set of institutions, including the societal node, is denoted by ${\mathcal{N}}_0 := {\mathcal{N}}\cup \{0\}$. We refer to [@feinstein2014measures; @GY14] for further discussion of the meaning and concepts behind the societal node.
We will be extending the model from [@EN01] in this paper. In that work, any bank $i \in {\mathcal{N}}$ may have obligations $L_{ij} \geq 0$ to any other firm or society $j \in {\mathcal{N}}_0$. We will assume that no firm has any obligations to itself, i.e., $L_{ii} = 0$ for all firms $i \in {\mathcal{N}}$, and the society node has no liabilities at all, i.e., $L_{0j} = 0$ for all firms $j \in {\mathcal{N}}_0$. Thus the *total liabilities* for bank $i \in {\mathcal{N}}$ is given by $\bar p_i := \sum_{j \in {\mathcal{N}}_0} L_{ij} \geq 0$ and relative liabilities $\pi_{ij} := \frac{L_{ij}}{\bar p_i}$ if $\bar p_i > 0$ and arbitrary otherwise; for simplicity, in the case that $\bar p_i = 0$, we will let $\pi_{ij} = \frac{1}{n}$ for all $j \in {\mathcal{N}}_0 \backslash \{i\}$ and $\pi_{ii} = 0$ to retain the property that $\sum_{j \in {\mathcal{N}}_0} \pi_{ij} = 1$. On the other side of the balance | 0 | non_member_993 |
sheet, all firms are assumed to begin with some amount of external assets $x_i \geq 0$ for all firms $i \in {\mathcal{N}}_0$. The resultant *clearing payments*, under a no priority of payments assumption, satisfy the fixed point problem in payments $p \in [0,\bar p]$ $$\label{eq:EN-p}
p = \bar p \wedge \left(x + \Pi^{\top}p\right).$$ That is, each bank pays the minimum of what it owes ($\bar p_i$) and what it has ($x_i + \sum_{j \in {\mathcal{N}}} \pi_{ji} p_j$). The resultant vector of *wealths* for all firms is given by $$\label{eq:equity}
V = x + \Pi^{\top}p - \bar p.$$ Noting that payments can be written as a simple function of the wealths ($p = \bar p - V^-$), we provide the following proposition. We refer also to [@veraart2017distress; @barucca2016valuation; @banerjee2017insurance] for similar notions of utilizing clearing wealth instead of clearing payments.
\[prop:EN-e\] A vector $p \in [0,\bar p]$ is a clearing payments in the Eisenberg-Noe setting if and only if $p = [\bar p - V^-]^+$ for some $V \in {\mathbb{R}}^{n+1}$ satisfying the following fixed point problem $$\label{eq:EN-e}
V = x + \Pi^{\top}[\bar p - V^-]^+ - \bar p.$$ Vice versa, a vector $V \in {\mathbb{R}}^{n+1}$ is a clearing wealths (i.e., satisfying | 0 | non_member_993 |
) if and only if $V$ is defined as in for some clearing payments $p \in [0,\bar p]$ as defined in the fixed point problem .
We will prove the first equivalence only, the second follows similarly.
Let $p \in [0,\bar p]$ be a clearing payment vector. Define the wealth vector $V$ by , then it is clear that $V^- = \bar p - p$ by definition as well, i.e., $p = \bar p - V^- \geq 0$. Thus from we immediately recover that the wealth vector $V$ must satisfy .
Let $p = [\bar p - V^-]^+$ for some wealth vector $V \in {\mathbb{R}}^{n+1}$ satisfying . By construction we find $$\begin{aligned}
p &= [\bar p - V^-]^+ = \bar p - \left(x + \Pi^{\top}[\bar p - V^-]^+ - \bar p\right)^-
= \bar p - \left(x + \Pi^{\top}p - \bar p\right)^- = \bar p \wedge \left(x + \Pi^{\top}p\right).\end{aligned}$$ We note that $\bar p \geq \left(x + \Pi^{\top}[\bar p - V^-]^+ - \bar p\right)^-$ can be shown trivially.
Due to the equivalence of the clearing payments and clearing wealths provided in Proposition \[prop:EN-e\], we are able to consider the Eisenberg-Noe system as a fixed point of equity and losses rather than | 0 | non_member_993 |
payments. In [@EN01] results for the existence and uniqueness of the clearing payments (and thus for the clearing wealths as well) are provided. In fact, it can be shown that there exists a unique clearing solution in the Eisenberg-Noe framework so long as $L_{i0} > 0$ for all firms $i \in {\mathcal{N}}$. We will take advantage of this result later in this paper. This is a reasonable assumption (as discussed in, e.g., [@GY14]) as obligations to society include, e.g., deposits to the banks.
Discrete-time clearing systems {#sec:discrete}
==============================
Consider now a discrete set of clearing times ${\mathbb{T}}$, e.g., ${\mathbb{T}}= \{0,1,\dots,T\}$ for some (finite) terminal time $T < \infty$ or ${\mathbb{T}}= {\mathbb{N}}$. Such a setting is presented in [@CC15]. For processes we will use the notation from [@cont2013ito] such that the process $Z: {\mathbb{T}}\to {\mathbb{R}}^n$ has value of $Z(t)$ at time $t \in {\mathbb{T}}$ and history $Z_t := (Z(s))_{s = 0}^t$.
(0,9.5) rectangle (6,10) node\[pos=.5\][**Balance Sheet**]{}; (0,9) rectangle (3,9.5) node\[pos=.5\][**Assets**]{}; (3,9) rectangle (6,9.5) node\[pos=.5\][**Liabilities**]{};
(0,7) rectangle (3,9) node\[pos=.5,style=[align=center]{}\][Cash-Flow @ $t = 0$\
$x_i(0)$]{}; (0,6) rectangle (3,7) node\[pos=.5,style=[align=center]{}\][Cash-Flow @ $t = 1$\
$x_i(1)$]{}; (0,2) rectangle (3,6) node\[pos=.5,style=[align=center]{}\][Interbank @ $t = 0$\
$\sum_{j = 1}^n \pi_{ji}(0)p_j(0)$]{}; (0,0) rectangle (3,2) node\[pos=.5,style=[align=center]{}\][Interbank @ $t = | 0 | non_member_993 |
1$\
$\sum_{j = 1}^n \pi_{ji}(1)p_j(1)$]{};
(3,5) rectangle (6,9) node\[pos=.5,style=[align=center]{}\][Cash-Flow @ $t = 0$\
$\sum_{j = 1}^n L_{ij}(0)$]{}; (3,1.5) rectangle (6,5) node\[pos=.5,style=[align=center]{}\][Cash-Flow @ $t = 1$\
$\sum_{j = 1}^n L_{ij}(1)$]{}; (3,0) rectangle (6,1.5) node\[pos=.5,style=[align=center]{}\][Capital\
$V_i(1)$]{};
(0,0) rectangle (3,9); (3,0) rectangle (6,9);
(0,7) – (3,7); (0,6) – (3,6); (0,2) – (3,2); (3,5) – (6,5);
\
(0,6.5) rectangle (6,7) node\[pos=.5\][**Balance Sheet @ $t = 0$**]{}; (0,6) rectangle (3,6.5) node\[pos=.5\][**Assets**]{}; (3,6) rectangle (6,6.5) node\[pos=.5\][**Liabilities**]{};
(0,4) rectangle (3,6) node\[pos=.5,style=[align=center]{}\][Cash-Flow\
$x_i(0)$]{}; (0,0) rectangle (3,4) node\[pos=.5,style=[align=center]{}\][Interbank\
$\sum_{j = 1}^n \pi_{ji}(0)p_j(0)$]{};
(3,2) rectangle (6,6) node\[pos=.5,style=[align=center]{}\][Cash-Flow\
$\sum_{j = 1}^n L_{ij}(0)$]{}; (3,0) rectangle (6,2) node\[pos=.5,style=[align=center]{}\] (t) [Capital\
$V_i(0)$]{};
(7,6.5) rectangle (13,7) node\[pos=.5\][**Balance Sheet @ $t = 1$**]{}; (7,6) rectangle (10,6.5) node\[pos=.5\][**Assets**]{}; (10,6) rectangle (13,6.5) node\[pos=.5\][**Liabilities**]{};
in [0.5]{}
(7,4+) rectangle (10,5+) node\[pos=.5,style=[align=center]{}\][Cash-Flow\
$x_i(1)$]{}; (7,2+) rectangle (10,4+) node\[pos=.5,style=[align=center]{}\] (t+1) [Carry-Forward\
$V_i(0)^+$]{}; (7,0+) rectangle (10,2+) node\[pos=.5,style=[align=center]{}\][Interbank\
$\sum_{j = 1}^n \pi_{ji}(1)p_j(1)$]{};
(10,1.5+) rectangle (13,5+) node\[pos=.5,style=[align=center]{}\][Cash-Flow\
$\sum_{j = 1}^n L_{ij}(1)$]{}; (13.2,1+) rectangle (16,2+) node\[pos=.5,style=[align=center]{}\][Carry-Forward\
$V_i(0)^- = 0$]{}; (13,1.5+) – (13.2,2+) – (13.2,1+) – cycle; (13,1.5+) – (13.2,2+) – (16,2+) – (16,1+) – (13.2,1+) – cycle; (10,0+) rectangle (13,1.5+) node\[pos=.5,style=[align=center]{}\][Capital\
$V_i(1)$]{}; (6,1) – (7,3+); (6,.25) – (14,.25) – (14,1+);
In this setting, we will consider the external (incoming) cash flow $x: {\mathbb{T}}\to {\mathbb{R}}^{n+1}_+$ and nominal liabilities | 0 | non_member_993 |
$L: {\mathbb{T}}\to {\mathbb{R}}^{(n+1) \times (n+1)}_+$ to be functions of the clearing time, i.e., as assets and liabilities with different maturities. The external cash in-flows and nominal liabilities can explicitly depend on the clearing results of the prior times (i.e., $x(t,V_{t-1})$ and $L(t,V_{t-1})$) without affecting the existence and uniqueness results we present, but for simplicity of notation we will focus on the case where the external assets and nominal liabilities are independent of the health and wealth of the firms. Throughout we are considering the discounted cash flows and liabilities so as to simplify notation. In contrast to the static Eisenberg-Noe framework, herein we need to consider the results of the prior times. In particular, if firm $i$ has positive equity at time $t-1$ (i.e., $V_i(t-1) > 0$) then these additional assets are available to firm $i$ at time $t$ in order to satisfy its obligations. Similarly, if firm $i$ has negative wealth at time $t-1$ (i.e., $V_i(t-1) < 0$) then the debts that the firm has not yet paid will roll-forward in time and be due at the next period. For example, consider a network in which obligations come due throughout the day at, e.g., opening, mid-day, and closing, but | 0 | non_member_993 |
that all debts must be cleared by the end of the day. In such a way, the current unpaid liabilities may be paid at a future time, but before the terminal time. That is, a firm can be considered in *distress* at a time if it is unable to satisfy its obligations at that time, but only *defaults* if it has negative wealth at the terminal time. Thus in this paper we primarily focus on the intra-day dynamics rather than the inter-day dynamics. See Figure \[fig:discrete-BS\] for a stylized (snapshot of the) balance sheet example for a firm that has positive wealth at time $0$ that rolls forward to time $1$. The full (actualized) balance sheet for this example with only those two time periods is displayed in Figure \[fig:BS\]; we note that the full balance sheet as depicted considers actualized payments rather than the book value of the obligations.
To incorporate the inter-day dynamics in this framework we can “zero out” a firm before the terminal date if it is deemed to default in much the same as in [@banerjee2017insurance]. A broader framework for dealing with various default mechanisms is discussed in Remark \[rem:loans\]. We can further consider the | 0 | non_member_993 |
Nash game in which firms decide if they will allow debts to be rolled forward in time. In such a setting, if we include a delay for payment due to, e.g., bankruptcy court so that defaulting firms do not pay any obligations until after the terminal time $T$, then the optimal strategy for all firms (up until the terminal time $T$) would be to always allow other firms to roll all debts forward so as to maximize payments.
\[ass:initial\] Before the time of interest, all firms are solvent and liquid. That is, $V_i(-1) \geq 0$ for all firms $i \in {\mathcal{N}}_0$.
We can now construct the total liabilities and relative liabilities at time $t \in {\mathbb{T}}$ as $$\begin{aligned}
\bar p_i(t,V_{t-1}) &:= \sum_{j \in {\mathcal{N}}_0} L_{ij}(t) + V_i(t-1)^-\\
\pi_{ij}(t,V_{t-1}) &:= \begin{cases} \frac{L_{ij}(t) + \pi_{ij}(t-1,V_{t-2})V_i(t-1)^-}{\bar p_i(t,V_{t-1})} &\text{if } \bar p_i(t,V_{t-1}) > 0 \\ \frac{1}{n} &\text{if } \bar p_i(t,V_{t-1}) = 0, \; j \neq i\\ 0 &\text{if } \bar p_i(t,V_{t-1}) = 0, \; j = i \end{cases} \quad \forall i,j \in {\mathcal{N}}_0.\end{aligned}$$ In this way, coupled with the accumulation of positive equity over time, the clearing wealths must satisfy the following fixed point problem in time $t$ wealths: $$\label{eq:EN-discrete}
V(t) = V(t-1)^+ + | 0 | non_member_993 |
x(t) + \Pi(t,V_{t-1})^{\top}\left[\bar p(t,V_{t-1}) - V(t)^-\right]^+ - \bar p(t,V_{t-1}).$$ That is, all firms have a clearing wealth that is the summation of their positive equity at the prior time, the new incoming external cash flow, and the payments made by all other firms minus the total obligations of the firm (including the prior unpaid liabilities). In this way we can construct the wealths of firms forward in time. This can be considered a discrete-time extension of .
We now wish to consider a reformulation of . To accomplish this, we consider a process of cash flows $c$ and functional relative exposures $A$. These we define by $$\begin{aligned}
\nonumber c(t) &:= x(t) + L(t)^{\top}\vec{1} - L(t)\vec{1}\\
\label{eq:discrete-A} a_{ij}(t,V_t) &:= \begin{cases} \pi_{ij}(t,V_{t-1}) &\text{if } \bar p_i(t,V_{t-1}) \geq V_i(t)^-\\ \frac{L_{ij}(t) + \pi_{ij}(t-1,V_{t-2})V_i(t-1)^-}{V_i(t)^-} &\text{if } \bar p_i(t,V_{t-1}) < V_i(t)^-\end{cases} \quad \forall i,j \in {\mathcal{N}}_0.\end{aligned}$$ In the above, $\vec{1} := (1,1,\ldots,1)^{\top}\in {\mathbb{R}}^n$ is the vector of ones. Here we consider $c(t) = x(t) + L(t)^{\top}\vec{1} - L(t)\vec{1} \in {\mathbb{R}}^{n+1}$ to be the vector of book capital levels at time $t$, i.e., the new wealth of each firm assuming all other firms pay in full. We wish to note that the new total liabilities are given | 0 | non_member_993 |
by $L(t)\vec{1}$ and the new incoming interbank obligations are given by $L(t)^{\top}\vec{1}$. We can also consider $c_i(t)$ to be the *net cash flow* for firm $i$ at time $t$. Further, we introduce the functional matrix $A: {\mathbb{T}}\times {\mathbb{R}}^{n+1} \to [0,1]^{(n+1) \times (n+1)}$ to be the relative exposure matrix. That is, $a_{ij}(t,V_t)V_i(t)^-$ provides the (negative) impact that firm $i$’s losses have on firm $j$’s wealth at time $t \in {\mathbb{T}}$. This is in contrast to $\Pi$, the relative liabilities, in that it endogenously imposes the limited exposures concept. In this work the two notions will generally coincide, but for mathematical simplicity we introduce this relative exposure matrix. For the equivalence we seek, we define the relative exposures so that $$L(t)^{\top}\vec{1} + A(t-1,V_{t-1})^{\top}V(t-1)^- - A(t,V_t)^{\top}V(t)^- = \Pi(t,V_{t-1})^{\top}[\bar p(t,V_{t-1}) - V(t)^-]^+$$ for any $V(t) \in {\mathbb{R}}^{n+1}$. This formulation is such that if the positive part were removed from the right hand side, the relative exposures $A$ would be defined exactly as the relative liabilities $\Pi$ by construction. In particular, we will define the relative exposures element-wise and pointwise so as to encompass the limited exposures as in . If $\bar p_i(t,V_{t-1}) > 0$ then we can simplify this further as $a_{ij}(t,V_t) = \frac{L_{ij}(t) | 0 | non_member_993 |
+ a_{ij}(t-1,V_{t-1})V_i(t-1)^-}{\max\{\bar p_i(t,V_{t-1}) , V_i(t)^-\}}$.
Using the notation and terms above we can rewrite with respect to the cash flows $c$ and relative exposures $A$ as $$\begin{aligned}
\nonumber V(t) &= V(t-1)^+ + x(t) + \Pi(t,V_{t-1})^{\top}[\bar p(t,V_{t-1}) - V(t)^-]^+ - \bar p(t,V_{t-1})\\
\nonumber &= V(t-1)^+ + x(t) + L(t)^{\top}\vec{1} + A(t-1,V_{t-1})^{\top}V(t-1)^-\\
\nonumber &\qquad - A(t,V_t)^{\top}V(t)^- - L(t)\vec{1} -V(t-1)^-\\
\nonumber &= V(t-1) + x(t) + L(t)^{\top}\vec{1} + A(t-1,V_{t-1})^{\top}V(t-1)^- - A(t,V_t)^{\top}V(t)^- - L(t)\vec{1}\\
\label{eq:discrete-V} &= V(t-1) + c(t) - A(t,V_t)^{\top}V(t)^- + A(t-1,V_t)^{\top}V(t-1)^-.\end{aligned}$$ For the remainder of this paper we will utilize the cash flow $c$ rather than the external (incoming) cash flow $x$. That is, we will consider financial networks defined by the joint parameters $(c,L)$ as given by the state equations and for wealths and relative exposures.
With this setup we now wish to extend the existence and uniqueness results of [@EN01] to discrete time.
\[thm:discrete\] Let $(c,L): {\mathbb{T}}\to {\mathbb{R}}^{n+1} \times {\mathbb{R}}^{(n+1) \times (n+1)}_+$ define a dynamic financial network such that every bank has cash flow at least at the level dictated by nominal interbank liabilities, i.e., $c_i(t) \geq \sum_{j \in {\mathcal{N}}} L_{ji}(t) - \sum_{j \in {\mathcal{N}}_0} L_{ij}(t)$, and so that every bank owes to the societal node at all times $t | 0 | non_member_993 |
\in {\mathbb{T}}$, i.e., $L_{i0}(t) > 0$ for all banks $i \in {\mathcal{N}}$ and times $t \in {\mathbb{T}}$. Under Assumption \[ass:initial\], there exists a unique solution of clearing wealths $V: {\mathbb{T}}\to {\mathbb{R}}^{n+1}$ to .
\[rem:regularnetwork\] The assumption that all firms have obligations to the societal node $0$ at all times $t \in {\mathbb{T}}$ guarantees that the financial system is a “regular network” (see [@EN01 Definition 5]) at all times.
The analysis of the discrete-time framework can be extended to a probabilistic setting over the filtered probability space $(\Omega,{\mathcal{F}},({\mathcal{F}}(t))_{t \in {\mathbb{T}}},{\mathbb{P}})$. That is, we can consider the clearing wealths in the same manner assuming the cash flow $c: {\mathbb{T}}\times \Omega \to {\mathbb{R}}^{n+1}$ and nominal liabilities $L: {\mathbb{T}}\times \Omega \to {\mathbb{R}}^{(n+1) \times (n+1)}_+$ be adapted processes. Let ${\mathcal{L}}_t^0({\mathbb{R}}^m)$ be the space of ${\mathcal{F}}_t$-measurable random vectors in ${\mathbb{R}}^m$. Let ${\mathcal{L}}_t^p({\mathbb{R}}^m) \subseteq {\mathcal{L}}_t^0({\mathbb{R}}^m)$ for $p \in (0,\infty]$ be the space of equivalence classes of ${\mathcal{F}}_t$-measurable functions $X: \Omega \to {\mathbb{R}}^m$ such that $\|X\|_p := \left(\int_\Omega \sqrt{\sum_{k = 1}^m X_k(\omega)^2} d{\mathbb{P}}\right)^{1/p} < \infty$ for $p < \infty$ and $\|X\|_{\infty} := \operatorname*{ess\,sup}_{\omega \in \Omega} \sqrt{\sum_{k = 1}^m X_k(\omega)^2}$ for $p = \infty$. The following corollary considers the boundedness and measurability properties of the discrete-time clearing wealths. Though | 0 | non_member_993 |
we will not utilize this discrete-time result in this paper, we consider it important to discuss random events to more closely match reality. Further, this result will implicitly appear in the construction and analysis of the continuous-time Eisenberg-Noe formulation of the next section.
\[cor:discrete\] Consider the setting of Theorem \[thm:discrete\] where the random network parameters $(c,L)$ adapted to the filtered probability space $(\Omega,{\mathcal{F}},({\mathcal{F}}(t))_{t \in {\mathbb{T}}},{\mathbb{P}})$. If $c(s) \in {\mathcal{L}}_s^p({\mathbb{R}}^{n+1})$ and $L(s) \in {\mathcal{L}}_s^p({\mathbb{R}}^{(n+1) \times (n+1)}_+)$ for all times $s \leq t$ for some $p \in [0,\infty]$, then the unique clearing solution at time $t$ has finite $p$-norm, i.e., $V(t) \in {\mathcal{L}}_t^p({\mathbb{R}}^{n+1})$.
With the construction of the existence and uniqueness of the solution we now want to emphasize the *fictitious default algorithm* from [@EN01] to construct this clearing wealths vector over time. This algorithm is presented for the deterministic setting; if a stochastic setting is desired then Algorithm \[alg:discrete\] provides a method for computing a single sample path. We note that at each time $t$ this algorithm takes at most $n$ iterations. Thus with a terminal time $T$, this algorithm will construct the full clearing solution over ${\mathbb{T}}$ in $nT$ iterations.
\[alg:discrete\] Under the assumptions of Theorem \[thm:discrete\] in a deterministic | 0 | non_member_993 |
setting the clearing wealths process $V: {\mathbb{T}}\to {\mathbb{R}}^{n+1}$ can be found by the following algorithm. Initialize $t = -1$ and $V(-1) \geq 0$ as a given. Repeat until $t = \max{\mathbb{T}}$:
1. Increment $t = t+1$.
2. \[alg:v\] Initialize $k = 0$, $V^0 = V(t-1) + c(t)$, and $D^0 = \emptyset$. Repeat until convergence:
1. Increment $k = k+1$;
2. Denote the set of insolvent banks by $D^k := \left\{i \in \{1,2,...,n\} \; | \; V_i^{k-1} < 0\right\}$.
3. If $D^k = D^{k-1}$ then terminate and set $V(t) = V^{k-1}$.
4. Define the matrix $\Lambda^k \in \{0,1\}^{n \times n}$ so that $\Lambda_{ij}^k = \begin{cases}1 &\text{if } i = j \in D^k \\ 0 &\text{else}\end{cases}$.
5. \[alg:vk\] Define $V^k = (I - \Pi(t,V_{t-1})^{\top}\Lambda^k)^{-1}\left(V(t-1) + c(t) + A(t-1,V_{t-1})^{\top}V(t-1)^-\right)$.
\[rem:loans\] Note that in the construction of $V^k$ in step of the fictitious default algorithm we utilize the relative liabilities $\Pi(t,V_{t-1})$ in the matrix inverse rather than the relative exposures $A(t,(V_{t-1},V^k))$. This has the added benefit that this definition of $V^k$ is *not* a fixed point problem, which it would be if the relative exposures matrix at time $t$ were considered. This change is possible since, as discussed in the proof of Theorem \[thm:discrete\], | 0 | non_member_993 |
any clearing solution must be in the domain so that the relative liabilities and exposures coincide. This additionally provides the invertibility of this matrix using standard input-output results as discussed in [@EN01; @feinstein2017sensitivity].
We wish to finish up our discussion of the discrete-time Eisenberg-Noe framework by considering some extensions involving loans.
The theoretical framework presented in this paper can be easily extended to incorporate the concepts of loans until some (deterministic) insolvency condition is hit. In particular, we will consider loans made from a central bank or lender of last resort who we will assume are part of the societal node $0$. From this perspective we consider three cases that a firm might be in:
- [**solvent and liquid**]{} in which case the firm has positive equity and pays off its obligations in full;
- [**solvent and distressed**]{} in which case the firm has negative equity, but receives an overnight loan (with interest rate set at the risk-free rate for simplicity) to cover all obligations due on that date; and
- [**insolvent**]{} in which the firm will not receive any loans and is sent to a bankruptcy court.
The determination whether a firm is solvent can be done with an | 0 | non_member_993 |
appropriate exogenous solvency function. We will assume that once a firm is deemed insolvent it can never recover to solvency again. Two possible systems for considering insolvent firms are:
1. [**Receivership:**]{} In such a system, when a firm is deemed insolvent it is placed in receivership so that obligations are payed out on a first-come first-serve basis.
2. [**Auctions:**]{} In such a system, when a firm is deemed insolvent its future assets are auctioned off in order to pay the future liabilities (in a proportional scheme) at the next time point. This will then affect the cash flows $c$ and nominal liabilities $L$, as such we would need to consider $c(t,V_{t-1})$ and $L(t,V_{t-1})$ to truly consider this case. We refer to [@CC15] for a detailed discussion of the auction model for insolvency. The auction system can be interpreted as an internal mechanism for determining bankruptcy costs in contrast to the exogenous parameter in, e.g., [@RV13].
The existence and uniqueness of the clearing solutions in these scenarios require an additional monotonicity property; we can use the notion a speculative system from [@banerjee2017insurance] to get the desired results. This condition encodes the notion that a firm does not benefit from any firm’s | 0 | non_member_993 |
distress.
Continuous-time clearing systems {#sec:continuous}
================================
Consider now a continuous set of clearing times ${\mathbb{T}}$, e.g., ${\mathbb{T}}= [0,T]$ for some (finite) terminal time $T < \infty$ or ${\mathbb{T}}= {\mathbb{R}}_+$. As before, for processes we will use the notation from [@cont2013ito] such that the process $Z: {\mathbb{T}}\to {\mathbb{R}}^n$ has value of $Z(t)$ at time $t \in {\mathbb{T}}$ and history $Z_t := (Z(s))_{s \in [0,t]}$. We will now construct an extension of the continuous-time setting of [@sonin2017] in that we allow for liabilities to change over time and for firms to have stochastic cash flows.
In order to construct a continuous-time model we will begin by considering our network parameters of cash flows and nominal liabilities. Instead of considering $c(t)$ to be the net cash flow at time $t \in {\mathbb{T}}$, we will consider the term $dc(t)$ of marginal change in cash flow at time $t$. Similarly we will consider $dL(t)$ to be the marginal change in nominal liabilities matrix at time $t$; we note that by assumption $dL_{ij}(t) \geq 0$ for all firms $i,j \in {\mathcal{N}}_0$ as, without any payments made, total liabilities should accumulate over time. Our main result in this section (Theorem \[thm:continuous\]) provides existence and uniqueness of the | 0 | non_member_993 |
clearing wealths driven by $(dc,dL)$ when $c(t) = \int_0^t dc(s)$ is an Itô process and $L(t) = \int_0^t dL(s)$ is deterministic and continuous (e.g., $dL$ does not include any Dirac delta functions). This setting, and the results on the continuous-time Eisenberg-Noe model, can be extended to the case in which the cash flows and liabilities are additionally functions of the wealths $V$. For simplicity, in this section we will restrict ourselves so that the parameters are independent of the current wealths. In order to construct a continuous-time differential system, we will consider again the discrete-time setting with explicit time steps ${{\Delta t}}$.
\[ass:society\] The cash flows $c$ are defined by the Itô stochastic differential equation $dc(t) = \mu(t,c(t))dt + \sigma(t,c(t))dW(t)$ for $(n+1)$-vector of Brownian motions $W$ over some filtered probability space $(\Omega,{\mathcal{F}},({\mathcal{F}}_t)_{t \in {\mathbb{T}}},{\mathbb{P}})$. Additionally, the drift and diffusion functions $\mu: {\mathbb{T}}\times {\mathbb{R}}^{n+1} \to {\mathbb{R}}^{n+1}$ and $\sigma: {\mathbb{T}}\times {\mathbb{R}}^{n+1} \to {\mathbb{R}}^{(n+1) \times (n+1)}$ are jointly continuous and satisfy the linear growth and Lipschitz continuous conditions, i.e., there exist constants $C,D > 0$ such that for all times $t \in {\mathbb{T}}$ and cash flows $c,d \in {\mathbb{R}}^{n+1}$ $$\begin{aligned}
\|\mu(t,c)\|_1 + \|\sigma(t,c)\|_1^{op} &\leq C(1 + \|c\|_1)\\
\|\mu(t,c) - \mu(t,d)\|_1 + \|\sigma(t,c) - | 0 | non_member_993 |
\sigma(t,d)\|_1^{op} &\leq D\|c - d\|_1\end{aligned}$$ where $\|\cdot\|_1$ is the 1-norm and $\|\cdot\|_1^{op}$ is the corresponding operator norm. The nominal liabilities $L: {\mathbb{T}}\to {\mathbb{R}}^{(n+1) \times (n+1)}_+$ are deterministic and twice differentiable; for notation we will define $dL(t) = \dot{L}(t) dt$ and $d^2L(t) = \ddot{L}(t) dt^2$. Further, the relative liabilities to society is bounded from below by a level $\delta > 0$, i.e., $\inf_{t \in {\mathbb{T}}} \frac{dL_{i0}(t)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)} = \delta > 0$ for all banks $i \in {\mathcal{N}}$.
We remark that the assumption on the cash flows can be relaxed so long as the stochastic differential equation has a unique strong solution on ${\mathbb{T}}$ and $\mu,\sigma$ satisfy a local linear growth condition and are locally Lipschitz. This relaxation will be applied in Examples \[ex:differentialEN\] and \[ex:dc-dependence\].
In the prior section on a discrete-time model for clearing wealths, we implicitly assumed a constant time-step between each clearing date of ${{\Delta t}}= 1$ throughout. In order to construct a continuous-time clearing model we will begin by making a discrete-time model with an explicit ${{\Delta t}}>0$ term. In fact, this is immediate from the prior construction with a minor alteration to the cash flow term. Herein we construct the net cash flow at | 0 | non_member_993 |
---
abstract: 'A relationship of the random walks on one-dimensional periodic lattice and the correlation functions of the $XX$ Heisenberg spin chain is investigated. The operator averages taken over the ferromagnetic state play a role of generating functions of the number of paths made by the so-called “vicious” random walkers (the vicious walkers annihilate each other provided they arrive at the same lattice site). It is shown that the two-point correlation function of spins, calculated over eigen-states of the $XX$ magnet, can be interpreted as the generating function of paths made by a single walker in a medium characterized by a non-constant number of vicious neighbors. The answers are obtained for a number of paths made by the described walker from some fixed lattice site to another sufficiently remote one. Asymptotical estimates for the number of paths are provided in the limit, when the number of steps is increased.'
author:
- |
$${}$$\
[**N. M. Bogoliubov$^\dagger$, C. Malyshev$^\ddagger$**]{}\
[Steklov Mathematical Institute, St.-Petersburg Department, RAS]{}\
[Fontanka 27, St.-Petersburg, 191023, Russia]{}\
\[0.5cm\] $^\dagger$ e-mail: [*bogoliub@pdmi.ras.ru*]{}\
$^\ddagger$ e-mail: [*malyshev@pdmi.ras.ru*]{}
title: |
$${}$$\
[**The correlation functions**]{}\
[**of the ${\bf XX}$ Heisenberg magnet and**]{}\
[**random walks of vicious walkers** ]{}
---
Introduction {#sec1}
============
The | 0 | non_member_994 |
random walks is a classical problem both for combinatorics and statistical physics. The problem of enumeration of the paths made by the, so-called, *vicious* walkers on the one-dimensional lattice has been formulated and investigated in details by Fisher [@1]. It is supposed that any two vicious walkers, provided both arrive at the same lattice site, annihilate not only one another but all other walkers as well. The problem mentioned still continues to attract considerable attention both of physicists and mathematicians [@2]–[@9]. Closely related problems arise also in the studies of the self-organized criticality [@10], domain walls [@11], and polymers [@12]. In paper [@13] a random walks of the annihilating particles on a ring was considered. In paper [@14] a random turns walks on a semi-axes with a possible creation of the particles at the origin was studied.
It has been shown in [@15], [@16] that the correlation functions, obtained as an averages over the ferromagnetic state of the $XX$ Heisenberg chain, can be used for enumeration of the paths of random walks of vicious walkers. In the present paper the averages of special type are investigated both for the case of ferromagnetic state and for superposition of the eigen-states of | 0 | non_member_994 |
the $XX$ magnet in zero magnetic field. The averages in question play a role of the generating functions of number of paths of the vicious walkers. The calculation of the correlation functions is carried out by means of the functional integration [@17], [@18]. The answers are obtained for the number of paths of a single pedestrian which is travelling from one chosen site to another sufficiently remote lattice site. The asymptotical estimates are obtained for the number of paths in the limit, when the number of steps (and, correspondingly, the number of random turns) is increasing.
The paper is organized as follows. Section \[sec1\] has an introductory character. The Hamiltonian of the model and general calculation of the correlation functions are discussed in Section \[sec2\]. Section \[sec3\] deals with the specific calculations and the corresponding asymptotic estimates. Discussion in Section \[sec4\] concludes the paper.
The model and the correlation functions {#sec2}
=======================================
The $XX$ magnet we are interested in is a particular limit of a more general spin model known as the $XY$ Heisenberg chain, with the Hamiltonian in the transverse magnetic field $h>0$ given by [@19], [@20]: $$\begin{aligned}
&H=H_0+\gamma H_1-hS^z,\\
&H_0\equiv-\sum^{M}_{n,m=1}\Delta^{(+)}_{nm}\sigma^+_{n}\sigma^-_{m},
\\
&H_1\equiv-\frac12\sum_{n,m=1}^M\Delta^{(+)}_{nm}
(\sigma^+_n\sigma^+_{m}+\sigma^-_n\sigma^-_{m}),\qquad
S^z\equiv\frac{1}2\sum_{n=1}^M\sigma^z_n,
\end{aligned}
\label{1}$$ where $S^z$ | 0 | non_member_994 |
is $z$-component of the total spin operator, and $\gamma$ is the anisotropy parameter. The local spin operators $\sigma^\pm_n=(\sigma^x_n\pm i\sigma^y_n)/2$ and $\sigma^z_n$ are given by the Pauli matrices, which depend on the lattice argument $n\in\mathcal M\equiv\{1,2,\dots,M\}$, where $M=0\pmod{2}$. The corresponding commutation relations have the form: $$[\sigma^+_k,\sigma^-_l]\,=\,\delta_{k,l}\sigma^z_l\,,\quad
[\sigma^z_k,\sigma^\pm_l]\,=\,\pm2\delta_{k,l}\sigma^{\pm}_l\,.$$ The introduced *hopping matrix* $\Delta^{(s)}$ is defined by the following entries: $$\Delta^{(s)}_{nm}\equiv\frac12\,
(\delta_{|n-m|,1}+s\delta_{|n-m|,M-1}), \label{2}$$ where ${\delta}_{n,l}$ is the Kronecker symbol, and $s$ can take two values: $s=\pm$. It is assumed that the periodic boundary conditions $\sigma^{\alpha}_{n+M}=\sigma^{\alpha}_n$ are imposed for any $n\in\mathcal M$. The Hamiltonian $H$ is reduced to the Hamiltonian of the $XX$ magnet at zero value of the parameter $\gamma$.
The most general definition of the time $t$ and temperature $T\equiv
1/\beta$ dependent correlation functions of the model under consideration looks as follows: $$G_{j;l}^{ab}(t)\equiv\frac{1}Z
\Tr(\sigma^a_{j}(0)\sigma^b_l(t)e^{-\beta H}),\qquad
Z\equiv\Tr(e^{-\beta H}),
\label{3}$$ where $\sigma^b_l(t)\equiv e^{itH}\sigma^b_le^{-itH}$ and $\Tr$ means the averaging with respect to all eigen-states of the Hamiltonian $H$. In addition, the normalization involves the partition function $Z$. Calculation of the correlators has been carried out in [@21] as averaging over all eigen-functions of the Hamiltonian of the $XX$ magnet. In [@21] the main attention has been paid to a relationship between the correlation functions and | 0 | non_member_994 |
the Fredholm determinants in the thermodynamic limit. In the present paper we shall consider the $XX$ chain only and denote its Hamiltonian by $H$.
To calculate the averages one can use a representation of the canonical Fermi variables $c_j$, $c^{\dagger}_j$, $j\in{\mathcal M}$ through the spin variables [@19], [@20]. The corresponding Jordan–Wigner transformation has the form: $$\sigma^+_n=\biggl(\,\prod_{j=1}^{n-1}\sigma^z_j\biggr)c_n,\qquad
\sigma^-_n=c_n^{\dagger}\biggl(\,\prod_{j=1}^{n-1}\sigma^z_j\biggr),
\qquad n\in{\mathcal M},
\label{4}$$ where $\sigma^z_j=1-2c_j^{\dagger} c_j$. The periodic boundary conditions for the spin variables lead to the following boundary conditions for the Fermi variables: $$c_{M+1}=(-1)^{\mathcal{N}}c_1,\qquad
c^{\dagger}_{M+1}=c^{\dagger}_1(-1)^{\mathcal{N}},
\label{5}$$ where $\mathcal{N}=\sum_{n=1}^Mc_n^{\dagger} c_n$ is the operator of the total number of particles. The Hamiltonian $H$ takes the following form in the fermion representation [@19], [@20] $$H=H^+P^++H^-P^-,
\label{6}$$ where $P^{+}$ ($P^{-}$) are projectors on the states characterized by an even/odd number of fermions: $$P_++P_-=\mathbb I\,,\quad P_+-P_-=(-1)^{\mathcal N}\,.$$ The operators $H^{\pm}$ are formally identical, their superscripts $s=\pm$ point out an appropriate specification of the boundary conditions : $$c_{M+1}\,=\,-s\,c_1\,,\quad c^{\dagger}_{M+1}\,=\,- s\,c^{\dagger}_1\,.$$ To put it differently, the quadratic in the fermion variables operators $H^\pm$ has the following representation: $$H^{\pm}=c^{\dagger}\widehat H^{\pm} c-\frac{Mh}2,\qquad
\widehat H^{\pm}=-\hat\Delta^{(\mp)}+h{\hat I},
\label{7}$$ where the matrices ${\widehat H}^{\pm}$ are expressed through the hopping matrices and $\hat I$ is the unit matrix: $${\widehat H}^\pm\,=\,\left(
\begin{array}{cccccc} h & -1/2 | 0 | non_member_994 |
& & & & \pm1/2 \\
-1/2 & h & -1/2 & & & \\
& -1/2 & h & -1/2 & & \\
& & & \dots & & \\
& & & -1/2 & h & -1/2 \\
\pm 1/2 & & & & -1/2 & h
\end{array}\right)$$ (only non-zero entries are displayed). Besides, the short-hand notations $c^{\dagger}$ and $c$ are used in for the $M$-dimensional row and column with the entries $c_n^{\dagger}$, $c_n$, $n\in\mathcal M$.
In particular, the correlator at $a=b=z$ takes the following form in the representation [@18], [@22], [@23]: $$G_{j;l}^{zz}(t)=1-\frac{2}Z\Tr(c^{\dagger}_{j}c_{j}e^{-\beta H})-
\frac{2}Z\Tr(c^{\dagger}_lc_le^{-\beta H})+
\frac{4}Z\Tr(c^{\dagger}_{j}c_{j}
e^{itH}c^{\dagger}_lc_le^{-(\beta+it)H}).
\label{8}$$ In order to calculate , it is convenient to introduce the generating functional [@18]: $$\mathcal G\equiv\mathcal G(S,T\mid\lambda,\nu)=
\frac{1}Z\Tr(e^Se^{-\lambda H}e^Te^{-\nu H}),
\label{9}$$ where $\lambda$, $\nu$ are the complex parameters, $\lambda+\nu=\beta$. The quadratic operators $S\equiv c^{\dagger}{\widehat S}c$ and $T\equiv c^{\dagger}{\widehat T}c$, used in , are defined by means of the matrices $\widehat S=\diag\{S_1,S_2,\dots,S_M\}$ and $\widehat T=\diag\{T_1,T_2,\dots,T_M\}$. For instance, the last term in right-hand side of is obtained from in the following way: $$\lim_{\substack{S_n,T_n\to0,\\n\in \mathcal M}}\,\lim_{\lambda\to-it}\,
\lim_{\nu\to\beta+it}\frac{\partial}{\partial S_j}\,
\frac{\partial}{\partial T_l}\,\mathcal G(S,T\mid\lambda,\nu).
\label{10}$$ The trace in right-hand side of can be re-written by means of [@18]: $$\Tr(e^Se^{-\lambda H}e^Te^{-\nu H})=
\frac12\,(\mathcal G^+_{\Fa}Z^+_{\Fa}+\mathcal G^-_{\Fa}Z^-_{\Fa}
+\mathcal | 0 | non_member_994 |
G^+_{\Ba}Z^+_{\Ba}-\mathcal G^-_{\Ba}Z^-_{\Ba}),
\label{11}$$ where $$\begin{aligned}
&\mathcal G^{\pm}_{\Fa}Z^{\pm}_{\Fa}\equiv
\Tr(e^Se^{-\lambda H^{\pm}}e^Te^{-\nu H^{\pm}}),
\\
&\mathcal G^{\pm}_{\Ba}Z^{\pm}_{\Ba}\equiv
\Tr(e^Se^{-\lambda H^{\pm}}e^T(-1)^{\mathcal N}e^{-\nu H^{\pm}}),
\end{aligned}
\label{12}$$ and $$Z^{\pm}_{\Fa}\,=\,\Tr(e^{-\beta H^{\pm}})\,,\quad
Z^{\pm}_{\Ba}\,=\,\Tr((-1)^{\mathcal N}e^{-\beta H^{\pm}})\,.$$ Moreover, for the partition function $Z$ we obtain the representation: $$Z=\frac12\,(Z^+_{\Fa}+Z^-_{\Fa}+Z^+_{\Ba}-Z^-_{\Ba})\,.$$ In the thermodynamic limit the terms with the subscript $\Ba$ are mutually compensated, therefore, in order to obtain $\mathcal G$ it is enough to calculate $\mathcal G^{\pm}_{\Fa}$.
The considered fermion representation is characterized by the existence of the Fock state $|0\rangle$ common for both operators $H^{+}$ and $H^{-}$, and satisfying the relations $c_k|0\rangle=0$, $k\in\mathcal M$. However, the corresponding coherent states over $|0\rangle$, $$\begin{array}{rcl}
&&\mid\!z\big\rangle\,\equiv\,\exp \Big(\sum\limits^M_{k=1} c^\dagger_k
z_k\Big) \mid\!0\big\rangle\,\equiv\,\exp (c^\dagger z)\mid\!0\big\rangle\,,\\ [0.5cm]
&&\big\langle z^*\!\mid\,\equiv\,\big\langle 0\!\mid \exp
\Big(\sum\limits^M_{k=1} z^*_k c_k \Big)\,\equiv\,
\big\langle 0\!\mid\exp (z^* c)\,,
\end{array}$$ are different for $H^+$ and $H^-$. Here the short-hand notations $z^*\equiv(z^*_1,\dots,z^*_M)$ and $z\equiv(z_1,\dots,z_M)$ are used for the sets of independent Grassmann parameters $z_k,z^*_k$, $k\in {\mathcal M}$ (it is appropriate to omit the extra index $\pm$ in $z^*$, $z$). Besides, $\sum_{k=1}^{M} c^{\dagger}_kz_k\equiv c^{\dagger} z$, $\prod_{k=1}^M dz_k\equiv dz$. Let us calculate $\mathcal G^{\pm}_{\Fa}Z^{\pm}_{\Fa}$ in using the representation of the trace in the Grassmann integration formalism [@18]: $$\mathcal G^{\pm}_{\Fa}Z^{\pm}_{\Fa}=\int dz\,dz^*\,e^{z^*z}
\langle z^*|e^Se^{-\lambda H^{\pm}}e^Te^{-\nu H^{\pm}}|z\rangle.
\label{13}$$
In order to represent the right-hand side of | 0 | non_member_994 |
this equality as the functional integral, let us introduce $L$ new copies of the coherent states $|x(I)\rangle$, $\langle x^*(I)|$, where $I\in\{1,2,\dots,L\}$. Each of the $2L$ multi-indices $x^*(I)$, $x(I)$ is expressed by $M$ independent Grassmann parameters. Using the decompositions of unity one can represent the right-hand side of as the $(L+1)$-fold multiple integral. In order to express the quasi-periodicity condition it is convenient to introduce the auxiliary variables: $$-{\widehat E}x(0)=x(L+1)\equiv z,\qquad
-x^*(L+1)=x^*(0){\widehat E}^{-1}\equiv z^*,
\label{14}$$ where ${\widehat E}\equiv e^{\widehat S}e^{-\lambda{\widehat H}^{\pm}}e^{\widehat T}$. Tending $L$ to infinity, we obtain the functional integral over the space of the trajectories $x^*(\tau)$, $x(\tau)$, where $\tau\in {\mathbb R}$: $$\mathcal G^{\pm}_{\Fa}Z^{\pm}_{\Fa}=\int e^S\,d\lambda^*\,d\lambda
\prod_{\tau} dx^*(\tau)\,dx(\tau).
\label{15}$$ The action functional $S\equiv\int L(\tau)\,d\tau$ is expressed through the Lagrangian $L(\tau)$: $$L(\tau)\equiv x^*(\tau)\biggl(\frac{d}{d\tau}-
\widehat H^{\pm}\biggr)x(\tau)+
J^*(\tau)x(\tau)+x^*(\tau)J(\tau),$$ where $$J^*(\tau)\equiv\lambda^*(\delta(\tau)\hat I+
\delta(\tau-\nu){\widehat E}^{-1}),\qquad
J(\tau)\equiv(\delta(\tau)\hat I+
\delta(\tau-\nu)\widehat E)\lambda.$$ The integration over the auxiliary Grassmann variables $\lambda^*$, $\lambda$ in guarantees the fulfilment of the constraints . The $\delta$-functions in $J^*(\tau)$, $J(\tau)$ reduce $\tau\in\mathbb R$ to $\tau\in[0,\beta]$. The stationarity conditions $\delta S/\delta x^*=0$, $\delta S/\delta x=0$ result in the following regularized answer [@18]: $$\mathcal G^{\pm}_{\Fa}={\det}\biggl(\hat I+
\frac{e^{(\beta-\nu){\widehat H}^{\pm}}e^{\widehat S}
e^{-\lambda{\widehat H}^{\pm}}e^{\widehat T}-{\hat I}}
{\hat I+e^{\beta{\widehat H}^{\pm}}}\biggr).
\label{16}$$ Furthermore, we substitute into and pass to the momentum representation. | 0 | non_member_994 |
The procedure described can also be applied to other correlators $G_{j;l}^{ab}(t)$ , where $a,b\in\{+,-\}$.
Random walks {#sec3}
============
As it has been shown in [@15], [@16], the flips of spins on a one-dimensional lattice may be associated with a random movements of walkers. Indeed, let us consider a state of the $XX$ Heisenberg chain, which corresponds to the ferromagnetic ordering of $M$ spins: $|\!\!\Uparrow\rangle\equiv\bigotimes_{n=1}^M|\!\!\uparrow\rangle_n$ (i.e., all spins are oriented “up”). Consider the average of the following type: $$F_{j;l}(\lambda)\equiv
\langle\Uparrow\!|\sigma_{j}^{+}e^{-\lambda H_0}
\sigma_{l}^{-}|\!\Uparrow\rangle,
\label{17}$$ where the notation $H_0$ implies that the zero magnetic field $h=0$ is taken in the Hamiltonian , (we shall omit the same subscript for the corresponding matrices ${\widehat H}^{\pm}$ ), and $\lambda\in\mathbb C$ is an “evolution” parameter. “Up” (or “down”) direction of spin corresponds to the empty (or filled) site. Differentiating ${F_{j;l}({\lambda})}$ and expanding the commutator $\lbrack H_0,\sigma_j^{+}\rbrack$ we obtain the difference–differential equation: $$\frac{d}{d\lambda}\,F_{j;l}({\lambda})=
\frac12\,(F_{j+1;l}({\lambda})+F_{j-1;l}({\lambda}))
\label{18}$$ (and similar equation can be also obtained for the fixed index $j$). Solution of the given equation is specified by the boundary conditions imposed on the lattice argument, and by the initial condition at $\lambda=0$.
The average $F_{j;l}(\lambda)$ can be considered as the generating function of the trajectories with random turns that | 0 | non_member_994 |
start at the $l$-th site and end up at the $j$-th site. Indeed, let us introduce the notation $\mathcal D^K_{\lambda}$ for the operator of differentiation of $K$-th order with respect to $\lambda$ at the point $\lambda=0$. The application of $\mathcal D^K_{\lambda}$ to the average leads to the answer: $$\mathcal D^K_{\lambda}\bigl[F_{j;l}({\lambda})\bigr]=
\langle\Uparrow|\sigma_{j}^{+}(-H_0)^K\sigma_{l}^{-}|\Uparrow\rangle=
\sum_{n_1,\dots,n_{K-1}}\Delta^{(+)}_{jn_{K-1}}\dots
\Delta^{(+)}_{n_2n_1}\Delta^{(+)}_{n_1l}.
\label{19}$$ The right hand side of coincides with the entry at the crossing of the $j$-th row and the $l$-th column of the matrix given by the product of $K$ copies of the hopping matrix . Each matrix in this product corresponds to a transition between the two nearest sites of the lattice. After multiplication by $2^{K}$ (this is due to the accepted normalization of the matrix ), the right-hand side of gives the number of the trajectories that consist of $K$ steps and are connecting the $l$-th and $j$-th sites. Let us denote this number by $|P_K(l\rightarrow j)|$.
Let $|P_K (l_1, \dots, l_N\rightarrow j_1, \dots, j_N)|$ be a number of trajectories consisting of $K$ links made by $N$ vicious walkers in the random turns model. Here, the initial and final positions of the walkers on the sites are given respectively by the sequences $l_1 > l_2 | 0 | non_member_994 |
> \dots > l_N$ and $j_1 > j_2 >\dots > j_N$. Let us consider the $N$-point correlation function ($N\leq M$): $$F_{j_1,j_2,\dots,j_N;l_1,l_2,\dots,l_N}({\lambda})=
\langle\Uparrow\!\!|\sigma_{j_1}^{+}\sigma_{j_2}^{+}\dots
\sigma_{j_N}^{+}e^{-\lambda H_0}\sigma_{l_1}^{-}
\sigma_{l_2}^{-}\dots\sigma_{l_N}^{-}|\!\!\Uparrow\rangle.
\label{20}$$ The present correlator is related to enumeration of the admissible trajectories which are traced by $N$ vicious walkers. Indeed, the application of the operator $\mathcal D^K_{\lambda/2}$ to results in the average of the type $$\langle\Uparrow\!\!|\sigma_{j_1}^{+} \sigma_{j_2}^{+} \dots
\sigma_{j_N}^{+}(-2H_0)^K \sigma_{l_1}^{-}\sigma_{l_2}^{-} \dots
\sigma_{l_N}^{-} |\!\!\Uparrow\rangle\,.$$ This average provides the numbers $|P_K (l_1, \dots, l_N\rightarrow j_1, \dots, j_N)|$ that can be established with the help of the commutator $$\lbrack H_0,\sigma_{l_1}^{-}\sigma_{l_2}^{-}\dots\sigma_{l_K}^{-}]=
\sum_{k=1}^K\sigma_{l_1}^{-}\dots\sigma_{l_{k-1}}^{-}
[H_0,\sigma_{l_k}^{-}]\sigma_{l_{k+1}}^{-}\dots\sigma_{l_K}^{-}
\label{21}$$ (in this case, differentiation with respect to $\lambda/2$, instead of $\lambda$, allows to take into account the normalization of the hopping matrix ). The condition of non-intersection of trajectories of the walkers is expressed by the vanishing of the correlation function for any pair of coinciding indices $l_k$ or $j_p$.
Differentiating with respect to $\lambda$ and applying , we obtain the equation: $$\frac{d}{d\lambda}\,F_{j_1,\dots,j_N;l_1,\dots,l_N}(\lambda)=
\frac12\sum_{k=1}^N\bigl(F_{j_1,
\dots,j_N;l_1,l_2,\dots,l_k+1,\dots,l_N}({\lambda})+
F_{j_1,\dots,j_N;l_1,l_2,\dots,l_k-1,\dots,l_N}({\lambda})\bigr).
\label{22}$$ Equation has been considered in [@16] for the case of periodicity with respect to the lattice argument and with the initial condition: $$F_{j_1, \dots, j_N;l_1, \dots, l_N}(0)\,=\,\prod_{m=1}^N\delta_{j_m,l_m}\,.$$ The function $F_{j_1,j_2, \dots, j_N; l_1,l_2, \dots, l_N}(\lambda)$ can be expressed | 0 | non_member_994 |
as the determinant of the matrix consisting of the averages of the type of [@16]: $$F_{j_1,\dots,j_N;l_1,\dots,l_N}({\lambda})=
\det\bigl(F_{j_r;l_s}({\lambda})\bigr)_{1\leq r,s\leq N}.
\label{23}$$
Random walks on the axis {#sec3.1}
------------------------
Let us consider an infinite chain ($M\to\infty$). Then, the modified Bessel function $I_{j-l}(\lambda)$ turns out to be a solution of equation , which respects the condition $F_{j;l}({0})={\delta}_{j,l}$ [@15]: $$F_{j;l}({\lambda})=I_{j-l}({\lambda})=\frac1{2\pi
}\int_{-\pi}^{\pi}d\theta\,e^{{\lambda}\cos\theta}e^{i(j-l)\theta}.
\label{24}$$ There exists the following expansion into the power series for $I_{j-l}(\lambda)$: $$I_{j-l}({\lambda})=\sum_{Q\geq|l-j|}
\frac{1}{\bigl(\frac{Q-j+l}2\bigr)!\,\bigl(\frac{Q+j-l}2\bigr)!}
\biggl(\frac{{\lambda}}2\biggr)^Q,
\label{25}$$ where the summation index $Q$ is subjected to the requirement: $Q+|j-l|=0\pmod{2}$. In the limit of large “time” (${\lambda}\rightarrow\infty$) and for moderate values of $m\equiv|l-j|$, using the known asymptotics for the Bessel function, we obtain for the generating function: $$F_{j;l}({\lambda})\simeq\frac{e^{{\lambda}}}{{\sqrt{2\pi{\lambda}}}}
\biggl(1-\frac{4m^2-1}{8{\lambda}}+\dotsb\biggr),$$ where the decay is governed by the critical exponent $\xi=-1/2$.
Let the number $K$ satisfies the relations $K\geq|l-j|$ and ${K+|j-l|=0\nobreak\pmod{2}}$. Then, differentiation of the series leads to the binomial relation ${|P_K(l\rightarrow j)|}=C_K^L$ for the number of all lattice paths of the “length” $K$ between two sites on the infinite axis: $$|P_K(l\rightarrow j)|\equiv
\mathcal D^K_{\lambda/2}[F_{j;l}({\lambda})]=
\frac{(m+2L)!}{L!\,(m+L)!}\,.
\label{26}$$ Here $L$ denotes the one-half of the total number of turns: $L\equiv(K-m)/2$.
Let us consider now the multi-point correlation function $F_{j_1,j_2, \dots, j_N;l_1,l_2, \dots, l_N}({\lambda})$. As it has been shown above, | 0 | non_member_994 |
$\mathcal D^K_{\lambda/2} [F_{j_1, \dots, j_N;l_1, \dots, l_N}(\lambda)]$ has the sense of the number of trajectories of $N$ vicious walkers each of which has made $K$ steps. A different combinatorial interpretation of this object, however, can be proposed. Really, let us consider a representation of the multi-point correlator in the form of the determinant . Its entries $F_{j_r;l_s}(\lambda)$ in the case of an infinite chain are given by the Bessel function $I_{j_r-l_s}(\lambda)$ . The operator $\mathcal D^K_{\lambda/2}$ acts on the determinant as the differentiation of the product of $N$ functions: $$(f_1(x)f_2(x)\dots f_N(x))^{(K)}=
\sum_{n_1+n_2+\dots+n_N=K}P(n_1,n_2,\dots,n_N)
f^{(n_1)}_1f^{(n_2)}_2\dots f^{(n_N)}_N.
\label{27}$$ The notation $f^{(n)}\equiv d^nf(x)/dx^n$ is used here, and the coefficients $P(n_1,n_2,\dots,n_N)$ are the numbers of permutations with repeats: $$P(n_1,n_2,\dots,n_N)\equiv
\frac{(n_1+n_2+\dots+n_N)!}{n_1!\,n_2!\,\dots n_N!}.
\label{28}$$ Summation in is over all non-negative values of $n_1,n_2,\dots,n_N$, provided their sum is equal to $K$.
Suppose further, that an $N$-dimensional (hyper-)cubic lattice of infinite extension is given, and each site of this lattice is labelled by a set of $N$ numbers. Let $\mathcal T_K(q_1,q_2, \dots
,q_N)$ be the number of the lattice trajectories that can be traced by some walker from the “initial” point $\textbf{\textit{O}}\equiv(0,0, \dots, 0)$ to a point $(q_1,q_2,\dots,q_N)$ in $K$ steps (by a single step the walker can move | 0 | non_member_994 |
to one of the nearest sites). Let all numbers $q_k$ be non-negative, and let the inequality $K\geq q_1+q_2+\dots+q_N$ be fulfilled, which means that the steps that can compensate each other are allowed. Let us denote the number of these steps as $2L$, $$L\equiv\frac{K-q_1-q_2-\dots-q_N}{2}\,.
\label{29}$$ Taking into account , the following formula for the number of paths takes place: $$\mathcal T_K(q_1,q_2,\dots,q_N)=\sum_{L_1+L_2+\dots+L_N=L}
P(q_1+L_1,q_2+L_2,\dots,q_N+L_N,L_1,L_2,\dots,L_N),
\label{30}$$ where summation is taken over all non-negative values of $L_1,L_2,\dots,L_N$, provided that their sum is equal to $L$, and the formula for the number of permutations with repeats is used.
Turning back to the function $F_{j_1,j_2,\dots,j_N;l_1,l_2,\dots,l_N}({\lambda})$ let us define the matrix $(n_{rs})_{1\leq r,s\leq N}$ with the entries $n_{rs}\equiv j_r-l_s$. Then, we arrive to the following
The number of trajectories consisting of $K$ links, which are traced by $N$ vicious walkers on an axis, is expressed through the number of trajectories of the same “length” $K$, which are traced by a single walker travelling over sites of $N$-dimensional lattice of infinite extension: $$\begin{aligned}
|P_K(l_1,\dots,l_N\rightarrow j_1,\dots,j_N)|&\equiv
\mathcal D^K_{\lambda/2}\bigl[F_{j_1,\dots,j_N;l_1,\dots,l_N}(\lambda)\bigr]=
\\
&=
\sum_{S_{a_1,a_2,\dots,a_N}}(-1)^{\mathcal P_S}
\mathcal T_K(n_{a_11},n_{a_22},\dots,n_{a_NN}),
\end{aligned}
\label{31}$$ where summation is taken over all permutations $S_{a_1,a_2,\dots,a_N}\equiv$ $S(\begin{smallmatrix}1,&2,&\dots,&N \\ a_1,&a_2,&\dots,&a_N\end{smallmatrix})$ of the numbers $1,2,\dots,N$, and $\mathcal P_S$ implies a parity of a specific | 0 | non_member_994 |
permutation.
In order to verify one should develop the determinant by a row or by a column and then apply the induction using the relations –.
Let us calculate, for instance, at $N=2$: $$\mathcal T_K(q_1,q_2)=C_{q_1+q_2+2L}^{q_1+L}
\sum_{k=0}^LC^{L-k}_{q_1+L}C_{q_2+L}^{k}=C_K^{q_1+L}C_K^L,
\label{32}$$ where $L=(K-q_{1}-q_{2})/2$ denotes one-half of the total number of turns. Then, using we obtain: $$\mathcal D^K_{\lambda/2}\bigl[F_{j_1,j_2;l_1,l_2}(\lambda)\bigr]=
\mathcal T_K(n_{11},n_{22})-\mathcal T_K(n_{21},n_{12})=
\begin{vmatrix}
C_K^L&C_K^{L+n_{21}}
\\[1mm]
C_K^L&C_K^{L+n_{11}}
\end{vmatrix},
\label{33}$$ where $L=(K-n_{11}-n_{22})/2$ and the equality $n_{11}+n_{22}=n_{12}+n_{21}$ is used.
Representation of the entries $F_{j_1,j_2,\dots,j_N;l_1,l_2,\dots,l_N}(\lambda)$ in the integral form allows to obtain the following expression [@15]: $$\begin{aligned}
F_{j_1,\dots,j_N;l_1,\dots,l_N}({\lambda})={}&
\frac{e^{{\lambda}N}}{N!}\prod_{i=1}^N
\biggl(\,\int_{-\pi}^{\pi}\frac{d\theta_i}{2\pi}\biggr)
e^{-{\lambda}\sum_{k=1}^N(1-\cos\theta_k)}\times{}
\\
&\times
{S}_{\boldsymbol{\pi}}(e^{i\theta_1},e^{i\theta_2},\dots,
e^{i\theta_N})\prod_{1\leq j<k\leq N}
|e^{i\theta_j}-e^{i\theta_k}|^2,
\end{aligned}
\label{34}$$ where $S_{\boldsymbol{\pi}}(e^{i\theta_1},e^{i\theta_2},
\dots,e^{i\theta_N})$ is the Schur function [@24], $$S_{\boldsymbol{\pi}}(x_1,x_2,\dots,x_N)\equiv
\frac{\det(x_j^{\pi_k+N-k})_{1\leq j,k\leq N}}
{\det(x_j^{N-k})_{1\leq j,k\leq N}}\,.
\label{35}$$ The Schur function depends on the partition $\boldsymbol{\pi}=(\pi_1,\pi_2,\dots,\pi_N)$ defined by a sequence of non-negative integers, which are ordered according to non-strict decreasing: $\pi_1\geq\pi_2\geq\dots\geq\pi_N\geq0$. In virtue of translational invariance it is always possible to choose the numbers $l_1$ $>$ $l_2>\dots>l_N\geq-N$ for the initial position of the walkers and to define the elements of the partition by the equalities $\pi_k=l_k+k$. In order to calculate the leading asymptotics of the generating function in the limit ${\lambda}\rightarrow\infty$, let us transform the integral into the following integral [@7], [@25]: | 0 | non_member_994 |
$$\int d^n\theta\prod_{1\leq j<k\leq N}|\theta_j-\theta_k|^2
e^{-{\lambda}/2\sum_{k=1}^N\theta^2_k}=
\frac{(2\pi)^{N/2}}{{\lambda}^{N^2/2}}\biggl(\,\prod_{p=1}^Np!\biggr).$$ It is a special case of the *Mehta integral*, which arises in the theory of the *Gaussian matrix ensembles*. Finally, we obtain the following asymptotics of the generating function for the trajectories traced by $N$ vicious walkers: $$F_{j_1,\dots,j_N;l_1,\dots,l_N}({\lambda})\simeq
\mathcal A\frac{e^{{\lambda}N}}{{\lambda}^{N^2/2}}\,,\qquad
\mathcal A=\frac{\prod_{p=1}^{N-1}p!}{(2\pi)^{N/2}}
\prod_{1\leq j<k\leq N}\frac{l_j-l_k}{k-j}\,,$$ where the well known formula for ${S}_{\boldsymbol{\lambda}}(1,1,\dots,1)$ is taken into account in $\mathcal A$ [@9], [@25]. Therefore, the power-like behavior of $F_{j_1,\dots,j_N;l_1,\dots,l_N}({\lambda})$ is characterized by the exponent $\xi=-N^2/2$.
Random walks over superposition of the eigen-states {#sec3.2}
---------------------------------------------------
The eigen-functions of the $XX$ Hamiltonian, given by the relations , , are constructed as combinations of the states, obtained by “flipping” of $N$ spins in the state $|\!\!\Uparrow\rangle$ [@21]. Indeed, let us consider all admissible strict partitions $\boldsymbol{\mu}=(\mu_1,\mu_2, \dots, \mu_N)$, where $M\geq\mu_1>\mu_2> \dots >\mu_N\geq1$, and establish a correspondence between each partition and an appropriate sequence of zeros and unities: $\bigl\{e_k\equiv e_k(\boldsymbol{\mu})\bigr\}_{k\in\mathcal M}$, where $e_k=\delta_{k,\mu_n}$, $1\leq n\leq N$. The required eigen-function is defined as: $$|\Psi_N(u_1,\dots,u_N)\rangle=
\sum_{\{e_k(\boldsymbol{\mu})\}_{k\in\mathcal M}}
\Upsilon_N(\{u_k\}\!\mid\boldsymbol{\mu})
(\sigma_{M}^{-})^{e_{M}}(\sigma_{M-1}^{-})^{e_{M-1}}\dots
(\sigma_{1}^{-})^{e_1}|\Uparrow\rangle,
\label{36}$$ where summation is taken over all strict partitions $\boldsymbol{\mu}$ of the given type. The number of such partitions is expressed through the number of permutations with repeats : $P(N,M-N)=C_{M}^N$. The wave | 0 | non_member_994 |
functions satisfy the periodic boundary conditions, $$\Upsilon_N(\{u_k\}|\boldsymbol{\mu})\equiv
\det(u_k^{2\mu_l})_{1\leq k,l\leq N}
\label{37}$$ are parametrized by the partitions $\boldsymbol{\mu}$ and by different, up to permutation, sets $\{u_1, \dots, u_N\}$ of solutions of the Bethe equations: $$u_k^{2M}=(-1)^{N-1},\qquad1\leq k\leq N.
\label{38}$$ These solutions have the form: $u_k^2=e^{i2\pi I_k/M}$, where $I_k$ are integers or half-integers (this depends on parity of $N$). Due to the antisymmetry of with the respect to permutations of the parameters $u_k$, it is sufficient to restrict oneself to the strict partitions $M\geq I_1>I_2>\dots>I_N \geq1$ in order to guarantee the single-valuedness of $u_k$. With the help of the corresponding normalized average $$\langle\sigma_{m+1}^{+}e^{-{\lambda}H_0}\sigma_1^{-}\rangle_N\equiv
\frac{\langle\Psi_N|\sigma_{m+1}^{+}e^{-{\lambda}H_0}\sigma_1^{-}
|\Psi_N\rangle}{\langle\Psi_N\mid\Psi_N\rangle}
\label{39}$$ can be represented as a linear combination of $(N+1)$-point generating functions . Therefore, this average is related to the number of random walks of $N+1$ pedestrians. The initial and the final positions of one of them are fixed at $l_1=1$ and $j_1=m+1$, respectively, while for the rest (*virtual*) pedestrians the choice of their initial and the final positions is arbitrary.
Calculation of equation is of interest in the thermodynamic limit, when $M$ and $N$ are growing (their ratio remains finite), which means that the number of virtual pedestrians is increasing. In this limit [@26] $$\widetilde F_{m+1;1}({\lambda})\equiv
\langle\sigma_{m+1}^{+}e^{-{\lambda}H_0}\sigma_1^{-}\rangle_N
| 0 | non_member_994 |
\bigr|_{M,N\gg1}\stackrel{\defa}{=}\Tr^{\prime}(\sigma_{m+1}^{+}
e^{-{\lambda}H_0}\sigma_1^{-}),
\label{40}$$ where the notation $\Tr^{\prime}$ points out that the procedure presented in Section \[sec2\] is used for the calculation of the normalized average. The difference-differential relation, analogous to the equation is valid for ${\widetilde F}_{m+1;1}({\lambda})$ : $$\begin{aligned}
&\frac d{d{\lambda}}\,{\widetilde F}_{m+1;1}({\lambda})=
\frac12\,({\widetilde F}_{m;1}({\lambda})+
{\widetilde F}_{m+2;1}({\lambda}))-
\Tr^{{\prime}}(H_0\sigma_{m+1}^{+}e^{-{\lambda}H_0}\sigma_{1}^{-})-{}\notag
\\
&\qquad{}-
\Tr^{\prime}\biggl(\biggl(\frac{1-\sigma_{m+1}^{z}}{2}\biggr)
\sigma_{m}^{+}e^{-{\lambda}H_0}\sigma_{1}^{-}\biggr)-
\Tr^{\prime}\biggl(\biggl(\frac{1-\sigma_{m+1}^{z}}{2}\biggr)
\sigma_{m+2}^{+}e^{-{\lambda}H_0}\sigma_{1}^{-}\biggr).
\label{41}\end{aligned}$$ The form of the present equation makes it possible to suppose that the average ${\widetilde F}_{m+1;1}({\lambda})$ can also be of interest as a generating function of the random walks.
Let us turn to the calculation of ${\widetilde F}_{m+1;1}({\lambda})$ in the fermionic representation . It is convenient to reduce the problem to calculation of the generating function of the form: $$\widetilde{\mathcal G}\equiv\Tr^{\prime}
(e^Sc_{m+1}e^{-\lambda H_{0}}c^{\dagger}_1e^{-\nu H_{0}}),
\label{42}$$ where the operator $S$ is defined just like in (i.e., by means of the matrix $\widehat S = \break \diag\{S_1,S_2,\dots,S_M\}$). Indeed, the functional ${\widetilde F}_{m+1;1}({\lambda})$ corresponds to the choice of $\nu=0$ and $S_k=-i\pi\theta(m-k)$, where $\theta(m-k)$ is the Heavyside function, $\theta(0)=1$. The second term in the right-hand side of corresponds to the differentiation by $\nu$ at the point $\nu=0$, in the third term we put $\nu=0$ and differentiate with respect to $S_{m+1}$. In both cases we put $S_k=-i\pi\theta(m-k)$. Taking into account the | 0 | non_member_994 |
fact, that the contribution of the terms labelled by the index $\Ba$ in is negligible at sufficiently large $M$ and $N$, we approximately obtain: $$\begin{aligned}
&\widetilde{\mathcal G}\approx
\biggl[\tr(e^{-{\lambda}{\widehat H}^{0}}{\hat e}_{1,m+1})-
\frac d{d\alpha}\biggr]\det({\hat I}+\widehat{\mathcal M}_1+
\alpha\widehat{\mathcal M}_2)\bigr|_{{\alpha}=0},
\\
&\widehat{\mathcal M}_1+\alpha\widehat{\mathcal M}_2\equiv
e^{-\nu\widehat H^{0}}e^{\widehat S}e^{-\lambda{\widehat H}^{0}}
({\hat I}+\alpha\hat e_{1,m+1}e^{-\lambda{\widehat H}^{0}}),
\end{aligned}
\label{43}$$ where $\hat{ e}_{1,m+1}\equiv
({\delta}_{1,n}{\delta}_{m+1,l})_{1\leq n,l\leq M}$. The matrix $\widehat H^{0}$ is used instead of $\widehat H^{\pm}$ since $s$ can be replaced by zero for the sufficiently large $M$.
The relation is written in the coordinate representation. In order to pass to the momentum representation it is convenient to use certain formulas provided in [@22]. Keeping the matrix notations as in , we obtain the answer for ${\widetilde F}_{m+1;1}({\lambda})$ (in the limit $M\to\infty$, the corresponding operations should be understood in the sense of the operations over the corresponding integral operators [@26]): $${\widetilde F}_{m+1;1}({\lambda})=
\det({\hat I}+\widehat{\mathcal U}_m)
\biggl[\tr(e^{-{\lambda}\hat{\varepsilon}_{0}}\breve e_{1,m+1})-
\tr\biggl(\frac{\widehat{\mathcal V}_{m}}{{\hat I}+
\widehat{\mathcal U}_m}\biggr)\biggr]
\label{44}$$ (the notation $\tr$, for instance, corresponds to the trace of $M\times M$ matrices). The matrices $\widehat{\mathcal U}_m$, $\widehat{\mathcal V}_{m}$, ${\breve e}_{1,m+1}$ are given by the corresponding momentum representations of the matrices $\widehat{\mathcal M}_1$, $\widehat{\mathcal M}_2$, ${\hat
e}_{1,m+1}$ . However, we shall need explicit expressions only for the | 0 | non_member_994 |
traces $\tr\widehat{\mathcal U}_m$ and $\tr\widehat{\mathcal V}_{m}$ (see below). In the momentum representation, $\hat{\varepsilon}_{0}$ is a diagonal matrix of the eigen-energies of the $XX$ model at $h=0$ [@21]. Formally expanding ${\widetilde F}_{m+1;1}({\lambda})$ in the powers of $\widehat{\mathcal U}_m$ we shall obtain the answer in two lowest orders: $$\begin{aligned}
&{\widetilde F}_{m+1;1}({\lambda})\approx
F_{m+1;1}({\lambda})+F_{m+1;1}({\lambda})
\tr\widehat{\mathcal U}_m-\tr\widehat{\mathcal V}_{m},
\nonumber
\\
&\tr\widehat{\mathcal U}_m=(M-2m)F_{1;1}({\lambda}),
\label{45}
\\
&\tr\widehat{\mathcal V}_{m}=F_{m+1;1}(2{\lambda})-
2\sum_{l=1}^{m}F_{m+1;l}({\lambda})F_{l;1}({\lambda}),
\nonumber\end{aligned}$$ where the notation $F_{j;l}({\lambda})$ implies the relations . Although $M$ and $m$ are chosen to be large enough, the ratio $m/M$ is assumed to be finite. Equation is fulfilled in each order separately by the terms presented in ${\widetilde F}_{m+1;1}$ .
By an analogy with the ferromagnetic case, let us act on ${\widetilde F}_{m+1,1}({\lambda})$ by the operator $\mathcal D^K_{\lambda/2}$. Then, in the first order we shall obtain the relation . In the second order, the answer is of the following form: $$MC_K^LC_K^L-C_K^L\sum_{l=0}^KC_K^l+2\sum_{l=1}^m
\begin{vmatrix}
C_K^{L+l-1}&C_K^L
\\
C_K^L&C_K^L
\end{vmatrix}.
\label{46}$$ By virtue of , the result of the application of $\mathcal D^K_{\lambda/2}$ to the second order function $F_{j_1,j_2;l_1,l_2}(\lambda)$ is connected, as a particular case of , with the number of the two-dimensional paths $\mathcal T_K$, and is expressed through the corresponding determinant. It means that it will be appropriate | 0 | non_member_994 |
to express in the following equivalent form: $$\begin{aligned}
&(M-K)|P_{K}(l\rightarrow l+m)|^2+{}
\notag \\
&\qquad{}+
\mathcal D^K_{\lambda/2}\begin{bmatrix}
2\displaystyle\sum_{l=1}^m
\begin{vmatrix}
F_{m+1;l}({\lambda})&F_{m+1;1}({\lambda})
\\
F_{l;l}({\lambda})&F_{l;1}({\lambda})
\end{vmatrix}
-\displaystyle\sum_{l=0}^K
\begin{vmatrix}
F_{m+L;l}({\lambda})&F_{m+1;1}({\lambda})
\\
F_{l;l}({\lambda})&F_{l;L}({\lambda})
\end{vmatrix}
\end{bmatrix}.
\label{47}\end{aligned}$$ In other words, the result of application of $\mathcal D^K_{\lambda/2}$ to in the second order can be reformulated in terms of the random walks of the two pedestrians (see and ) and the squared number of walks of a single pedestrian. The summation by the index $l$ in $\tr\widehat{\mathcal V}_{m}$ can be interpreted as the summation over positions of a virtual walker in .
Using the equation , one can represent in terms of the number of trajectories on a two-dimensional lattice: $$(M-2(m+1))\mathcal T_K(m,0)+
\sum_{l=0}^m\mathcal T_K(m-l,l)-\sum_{l=1}^L\mathcal T_K(m+l,l)-
\sum_{l=1}^L\mathcal T_K(l,m+l).
\label{48}$$ In this relation various lattice trajectories of $K$ links are enumerated. All these trajectories start at the same point $\textbf{\textit{O}}=(0,0)$ while they terminate on the segments of the dashed broken line which connects the points $(L,L+m)$, $(0,m)$, $(m,0)$, and $(L+m,L)$ (see figure). Formally, the sign of the sum is not definite though its asymptotics is positive, in general. An analogous description is expected in the higher orders as well.
Typical configuration.
Let us estimate the behavior of the number of | 0 | non_member_994 |
paths, which is given by the representation , in the limit, when the number of links $K=m+2 L$ increases. We shall assume, that the restriction $1\ll m\ll L$ is valid which means that $m$ increases moderately in the comparison with the increase of the number of turns $L$: for instance, let $L$ increase as $m^2$. Using the known asymptotical expansion of the logarithm of the gamma-function (see Appendix) [@27], one can estimate the binomial coefficient $C_K^L$. Restricting ourselves by the first order of smallness, we obtain: $$C_K^L\approx\frac{2^K}{{\sqrt{\pi L}}}e^{-m^2/(4L)}
\biggl(1-\frac{m}{2L}\biggl(1-\frac{m^2}{4L}\biggr)\biggr)\approx
\frac{2^K}{{\sqrt{\pi L}}}\biggl(1-\frac{m^2}{4L}\biggr)
\sim\frac{2^{2 L}}{{\sqrt{\pi L}\,}}.
\label{49}$$ The second approximate equality in takes place if $L$ is increasing faster than $m^2$. The estimate characterizes an increase of the number of the trajectories for a single pedestrian.
The third term in can be written as $2A(m,L)C_K^LC_K^L$, where $$A(m,L)\equiv-m+\sum_{l=1}^m\frac{(L+m+2-l)_{l-1}}{(L+1)_{l-1}}\,.
\label{50}$$ Standard notation $(\alpha)_{n}$ for the Pochhammer’s symbol is used in [@27]. Applying again an expansion of the logarithm of the gamma-function (A1), we can estimate $A(m,L)$ : $$A(m,L)\simeq mZ_1(m,L)-Z_0(m,L)+\mathcal{O}(m^{-1}),
\label{51}$$ where $$\begin{aligned}
&Z_0(m,L)\equiv e^{m^2/4L}\biggl(1+\frac{m}{L^2}\sum_{l=0}^{m/2}
e^{-l^2/L}\biggl(\frac{m^2}{4}-l^2\biggr)\biggr),
\\
&Z_1(m,L)\equiv-1+e^{m^2/4L}\,\frac{2}{m}\sum_{l=0}^{m/2}e^{-l^2/L}.
\end{aligned}
\label{52}$$ Let the values $m$ and $L$ increase with the ratio $L/m^2$ being finite and of order of unity. It can be shown (by means of | 0 | non_member_994 |
numerical check as well), that the coefficient functions $Z_0(m,L)$ and $Z_1(m,L)$ remain finite in this case, and the contribution of $Z_0(m,L)$ is negligible in comparison with $m Z_1(m,L)$ in . One can use Eqs. and in order to estimate in the leading approximation: $$\frac{2^{4L}}{\pi L}\,e^{-m^2/2L}(M+2mZ_1(m,L)-
(\pi L)^{1/2}e^{m^2/4L}).
\label{53}$$ Because of the behavior of the coefficient $Z_1(m,L)$, the corresponding contribution in may turn out to be comparable with $M$. The relation demonstrates that the description of the random walks considered in the representation of the superposition of the eigen-states is more complicated than the one in the ferromagnetic case. This description can be regarded as a simultaneous walks of the initial (i.e., principal) and virtual pedestrians. The ending points of the trajectories belonging to all the three segments of the dashed broken line on the figure (see the representation of two-dimensional random walks ) correspond to comparable contributions into the estimate . In certain cases, characterized by the limiting behavior of the ratio $m^2/L$, the contribution of the segment between the points $(m,0)$ and $(0,m)$ can become dominating.
Conclusion {#sec4}
==========
It is shown that the correlation functions of the $XX$ Heisenberg magnet, calculated over the superposition of the eigen-states, as | 0 | non_member_994 |
well as over the ferromagnetic state, are connected with enumeration of the trajectories made by the walkers moving on the lattice. A relationship is established between the number of trajectories made by a several vicious walkers and the number of paths made by a single random turns walker on a lattice of a dimension equal to the number of the vicious walkers. Differentiation of the generating function, calculated over the superposition of the eigen-states, demonstrates a more complicated combinatorial picture than that of the ferromagnetic case. In particular, the set of the paths made by a single pedestrian is replaced by the set of trajectories made simultaneously by the principal and virtual (both vicious) pedestrians. An estimate is obtained for the number of trajectories made both by the principal and the virtual pedestrians.
Acknowledgement {#acknowledgement .unnumbered}
===============
This paper was partially supported by the Russian Foundation for Basic Research, No. 07-01-00358, and by the Russian Academy of Sciences program ,,Mathematical Methods in Non-Linear Dynamics”.
Appendix {#appendix .unnumbered}
========
Asymptotic expansion for the logarithm of the gamma-function at large $|z|$ and $|\arg z |< \pi$ has the form [@27]: $$\begin{array}{rcl}
\log\Gamma
(z\,+\,\alpha)&=&\displaystyle{\Bigl(z\,+\,\alpha\,-\,\frac12\Bigr) \log
z\,-\,z\,+\,\frac12\,\log(2\pi)}\nonumber\\[0.4cm]
&+&\displaystyle{\sum\limits_{p=1}^n
(-1)^{p+1}\,\frac{B_{p+1}(\alpha)}{p(p+1)}\,z^{-p}\,+\,
\mathcal{O}\Bigl(\frac1{z^{n+1}}\Bigr)}\,,
\end{array}\eqno(A1)$$ where $n= 1, | 0 | non_member_994 |
2, 3, \dots$. The Bernoulli polynomials $B_{n}(\alpha)$ ($A1$) are defined as follows: $$B_n(\alpha)\,=\,\sum\limits_{l=0}^n C_n^l\,B_l\,\alpha^{n-l}\,,$$ where $C_n^l$ are the binomial coefficients, and $B_l$ are the Bernoulli numbers. The first Bernoulli polynomials $B_{p}(\alpha)$ look as follows: $$\begin{aligned}
B_0(\alpha)\,=\,1\,,\qquad B_1(\alpha)\,=\,\alpha\,-\,\frac12\,,\qquad
B_2(\alpha)\,=\,\alpha^2\,-\,\alpha\,+\,\frac16\,,\nonumber\\[0.4cm]
B_3(\alpha)\,=\,\alpha^3\,-\,\frac32\,\alpha^2\,+\,\frac12\,\alpha\,,\qquad
B_4(\alpha)\,=\,\alpha^4\,-\,2 \alpha^3\,+\,\alpha^2\,-\,\frac1{30}\,,
\nonumber\end{aligned}$$ where the Bernoulli numbers $B_l$ are used: $$\begin{aligned}
B_0\,=\,1\,,\qquad B_1\,=\,-\,\frac12\,,\qquad
B_2\,=\,\frac16\,,\qquad B_4\,=\,-\,\frac1{30}\,.
\nonumber\end{aligned}$$ 0.5cm
[99]{}
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A. J. Guttmann, A. L. Owczarek, X. G. Viennot, *J. Phys. A: Math. Gen.*, **31**, 8123–8135 (1998).
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| 0 | non_member_994 |
INTRODUCTION
============
Because of the macroscopic degeneracy of single-particle states in a Landau level, neither disorder nor electron-electron interactions in a two-dimensional electron system (2DES) can be treated perturbatively in the quantum Hall regime. This is the [*raison d’être*]{} for the many interesting and surprising phenomena[@general] which have arisen in quantum Hall physics. Theories of quantum Hall physics usually include either only interactions or only disorder, although both are always present. In particular, it is common to include only disorder in studies of the integer quantum Hall effect (IQHE), which generally focus on the sudden jump in the Hall conductivity between values separated by $e^2/h$, and it is common to include only interactions in studies of the fractional quantum Hall effect (FQHE), which generally focus on the ability of interactions to create charge gaps at partial Landau level fillings. The competition between interactions and disorder has often, but not always,[@Green; @Nederveen] been neglected, in part because of the lack of easily manageable analytical and numerical tools that can deal with both simultaneously. In this paper we address an instance in which this competition is particularly direct and can be successfully addressed with elementary techniques.
At Landau level filling factor $\nu=1$, | 0 | non_member_995 |
the ground state of a disorder-free 2DES is a strong ferromagnet,[@Sondhi] [*i.e.*]{} it is completely spin-polarized by a Zeeman field of infinitesimal strength. In practice, of course, the field experienced by a 2DES in the quantum Hall regime is [*not*]{} infinitesimal; however the field’s Zeeman coupling to the electron spin is typically very weak compared to other energy scales. (In referring to these systems as ferromagnets we are emphasizing that they remain spin polarized in the limit of zero Zeeman splitting. In experimental systems,[@Barrett] typical values of the interaction and Zeeman energy scales are $\sim 160 $ K and $\sim 3$ K respectively. The spin-splitting produced by this bare Zeeman coupling is usually negligible in paramagnetic states.) The quantum Hall ferromagnet has a large gap for charge excitations, and hence has a robust quantum Hall effect. For typical Zeeman coupling strengths, its elementary charged excitations are topologically charged spin textures (Skyrmions) containing several flipped spins.[@general; @Sondhi; @Fertig] Large Skyrmions have a Hartree energy cost smaller that those of conventional quasiparticles and, because the spins align locally, only slightly higher exchange energy. [@Brey] Because Skyrmions are the lowest-energy charged excitations, the global electron spin polarization is expected to decrease rapidly as | 0 | non_member_995 |
$|1-\nu|$ increases. This has been observed in nuclear magnetic resonance (NMR) experiments. [@general; @Barrett; @Fertig] On the other hand, the $\nu=1$ state of non-interacting disordered electrons differs qualitatively. The ground state is a compressible paramagnet with no Knight shift and no gap for charged excitations. For zero Zeeman coupling, quasiparticle states at the Fermi energy are quasi-extended and cause the Hall conductivity to suddenly jump by $2 e^2/h$ as this filling factor is crossed; $\nu=1$ is in the middle of a Hall ‘riser’, instead of being at the middle of a Hall plateau. The competition between disorder and interactions at $\nu=1$ can be addressed[@olderhfdis] using the Hartree-Fock approximation which has the virtue of being exact[@general] in both the non-interacting and the non-disordered limits. Experimental information on this competition comes primarily from transport and NMR studies. Early NMR studies[@Barrett] of weak-disorder quantum Hall ferromagnets, yielded relatively featureless lineshapes and Knight shifts in good agreement with Hartree-Fock (HF) theory estimates[@Fertig; @Brey; @Cote] of Zeeman-coupling and filling-factor dependent Skyrmion sizes. (Effective field theory estimates[@Sondhi; @Rajaraman] are not accurate in the case of typical Zeeman coupling strengths.) More recent experiments[@Barrett_private] paint a more complex picture, in part because the measurements were performed at lower | 0 | non_member_995 |
temperatures where the signal is not motionally averaged.[@Jairo; @Yalegrouppapers] It is now clear that disorder plays a role in the interpretation of these experiments, even when it is weak. At stronger disorder, as the non-interacting limit is approached, the spin-polarization must eventually vanish. In our model calculations we find that as the interaction strength is increased relative to disorder at $\nu=1$, the 2DES ground state suffers a continuous phase transition a from paramagnetic to a ferromagnetic state. Depending on the details of the disorder model, a second continuous phase transition to a fully spin-polarized incompressible strong ferromagnet with a gap for charged excitations may occur at still stronger interactions. For the disorder models we use, the fully polarized state is reached when the Coulomb energy scale is approximately twice the Landau-level-broadening disorder energy scale. Away from $\nu=1$, screening by mobile charges reduces the importance of disorder and the system reaches maximal spin-polarization at smaller interaction strengths. The maximally polarized ground state at moderate interaction strengths is best described as a glass of localized conventional quasiparticles formed in the $\nu=1$ fully polarized vacuum. Only for stronger interactions do we find a phase transition to a state with non-collinear magnetization in which | 0 | non_member_995 |
the localized particles have Skyrmionic character.
We organize this paper as follows. In Sec. II we summarize our implementation of finite-size HF theory in the lowest Landau level (LLL). In Sec. III and IV we present and discuss our numerical results for calculations at $\nu=1$ and $\nu\ne 1$. The possibility, discussed in recent work,[@Nederveen] that at $\nu=1$ disorder will induce reduced-size Skyrmion-anti-Skyrmion pairs in the ground state is specifically addressed in Sec. III. Finally, in Sec. V we present our conclusions.
Hartree-Fock Theory in the LLL
==============================
The HF approximation allows the interplay between disorder and interactions to be addressed while retaining a simple independent-particle picture of the many-body ground state. For the current study, the use of this approximation is underpinned by the fact that it reproduces the exact ground state at $\nu=1$ in both weak interaction and strong interaction[@general] limits. HF theory is a self-consistent mean field theory, and as such it has strengths and shortcomings, which we discuss later. In this section, we outline the basic formalism of HF approximation calculations in the LLL limit.
In a strong magnetic field, the Landau level splitting is very large and, since excitations to higher Landau levels are effectively forbidden | 0 | non_member_995 |
at the low experimental temperatures, we follow the common practice of considering only LLL states. Neglecting the frozen kinetic-energy degree of freedom, the Hamiltonian in second quantization is written as $${\cal H}={\cal H}_I+{\cal H}_{dis}+{\cal H}_Z\,,$$ where ${\cal H}_I$ is the interaction part of the Hamiltonian $$\begin{aligned}
{\cal H}_I&=&\frac{1}{2}\int d{{\bf r}}\int d{{\bf r}}' \sum_{\sigma\, \sigma'}
v_I({{\bf r}}-{{\bf r}}') \hat\psi^\dagger_\sigma({{\bf r}})
\hat\psi^\dagger_{\sigma'}({{\bf r}}') \hat\psi_{\sigma'}({{\bf r}}') \hat\psi_\sigma({{\bf r}})\,\,,\end{aligned}$$ ${\cal H}_{dis}$ is the external disorder part of the Hamiltonian $$\begin{aligned}
{\cal H}_{dis}&=&\int d{{\bf r}}\sum_{\sigma}v_{E}({{\bf r}})
\hat\psi^\dagger_\sigma({{\bf r}})\hat\psi_\sigma({{\bf r}})\,,\end{aligned}$$ and ${\cal H}_Z$ is the Zeeman term $${\cal H}_Z=-\frac{1}{2}g\mu_B \int d{{\bf r}}\sum_{\sigma \sigma'}
\hat\psi^\dagger_{\sigma'}({{\bf r}}') \hat\psi_{\sigma}({{\bf r}}')
\vec{\tau}_{\sigma' \sigma}\cdot\vec{B}(\vec r)\,,$$ with $\sigma=\uparrow,\downarrow$, $v_I$ and $v_E$ being the Coulomb interaction and disorder potentials respectively, and $\tau_i$ being the Pauli matrices. Here we also define the Zeeman coupling strength as $\tilde g=g\mu_B B/(e^2/\epsilon l)$ for later reference. We chose the the Landau gauge elliptic theta functions as our basis $$\phi_m(x,y)=\frac{1}{\sqrt{L_y l \sqrt{\pi}}}
\sum_{s=-\infty}^\infty e^{i\frac{1}{l^2}x_{m,s}y}
e^{-\frac{1}{2 l^2}(x-x_{m,s})^2}\,\,,$$ where $x_{m,s}=\frac{2\pi m l^2}{L_y}+s L_x$, $m,m'=1,\dots,N_\phi$, $N_\phi=A/(2\pi l^2)$, and $l$ is the magnetic length which we set equal to $1$ for simplicity. These wavefunctions satisfy the semi-periodic boundary conditions $\phi_m(x,y)=\phi_m(x,y+L_y)$ and $\phi_m(x+L_x)=\exp(+iL_x y/l^2)\phi_m(x,y)$.
We consider the HF single particle states to be | 0 | non_member_995 |
a linear combination of the up and down spin states of these basis functions $$|\alpha\rangle=\sum_{m,\sigma}\langle m \sigma|\alpha\rangle \,\,
|m \sigma\rangle\,\,.$$ Before writing down the Hamiltonian matrix in the HF approximation, we introduce several notation simplifying definitions, closely following previous HF studies. [@olderhfdis] The expectation value of the particle density (in momentum space) is given by $$\begin{aligned}
\langle \rho({{\bf q}})\rangle&=&\sum_\alpha n_F(\epsilon_F-\epsilon_\alpha)\langle \alpha
|e^{-i{{\bf q}}\cdot{{\bf r}}}|\alpha\rangle\\
&\equiv&N_\phi e^{-\frac{1}{4}q^2}
\sum_{\sigma,\sigma'} \delta_{\sigma,\sigma'} \Delta_{\sigma'\,\sigma}({{\bf q}})
e^{-i\frac{q_x q_y}{2}}\,\,,\end{aligned}$$ where we define $$\Delta_{\sigma'\,\sigma}({{\bf q}})\equiv\frac{1}{N_\phi}
\sum_{m,m'}\delta_{(x_{m'},x_m+q_y)}e^{-iq_x x_m}
\rho_{\sigma'\,\sigma}(x_{m'}|x_m),$$ and $$\begin{aligned}
\rho_{\sigma'\,\sigma}(x_{m'}|x_m)
&=&\sum_\alpha n_F(\epsilon_F-\epsilon_\alpha) \langle
m'\sigma'|\alpha\rangle\langle\alpha|m\sigma\rangle\,.\end{aligned}$$ Here $\delta_{(x_{m'},x_m+q_y)}$ is a periodic Kronig delta function, [*i.e.*]{} it is nonzero for $x_{m'}=x_m+q_y+s\,L_x$ for any integer s. With these definitions we can write the Hamiltonian matrix in the HF approximation in a compact form which is simple to diagonalize numerically $$\begin{aligned}
&&\langle m \sigma|{\cal H}|m' \sigma'\rangle_{HF} =
\sum_{{{\bf q}}\epsilon {\rm BZ}}
\left\{(\gamma\Delta_0({{\bf q}}){\rm U_H}({{\bf q}})+{\rm U_{\rm dis}}({{\bf q}})) {\rm I}+\right.
\nonumber\\
&&\left.\frac{\gamma}{2}(\Delta_0({{\bf q}}){\rm I}+\vec{\Delta}({{\bf q}})\cdot\vec{\tau})
{\rm U_F}({{\bf q}})\right\}
\delta_{(x_m,x_{m'}+q_y)}e^{+iq_x x_{m'}}\nonumber\\
&&-\frac{1}{2}\tilde g \hat B\cdot \vec{\tau}\,,\end{aligned}$$ where $\Delta_\alpha({{\bf q}})={\rm Tr}\{\Delta_{\sigma \sigma'}({{\bf q}})
\tau^{\alpha}\}$, and $\hat B$ specifies the orientation of the external magnetic field. The various effective potentials which appear here are defined as $${\rm U_{dis}}({{\bf q}})=\frac{1}{A}\sum_{{\bf G}}e^{-\frac{1}{4}|{{\bf q}}+{{\bf G}}|^2}v_E({{\bf q}}+{{\bf G}}) e^{
| 0 | non_member_995 |
\frac{i}{2}(q_x+G_x)(q_y+G_y)}\,\,,
\label{Udis}$$ $${\rm U_H}({{\bf q}})=\frac{1}{2\pi}\sum_{{{\bf G}}} e^{-\frac{1}{2}|{{\bf q}}+{{\bf G}}|^2}
\frac{2\pi e^2}{|{{\bf q}}+{{\bf G}}|}(1-\delta_{{{\bf q}}+{{\bf G}},0})\,\,,$$ and $${\rm U_F}({{\bf q}})=-\frac{1}{A }\sum_{{{\bf q}}'}
e^{-\frac{1}{2}|{{\bf q}}'|^2}e^{i{q'}_x q_y-iq_x q'_y}
\frac{2\pi e^2}{|{{\bf q}}'|}(1-\delta_{{{\bf q}}',0})\,,
\label{U_F}$$ with ${{\bf G}}=(\frac{2\pi N_\phi}{L_x} n_x,\frac{2\pi N_\phi}{L_y} n_y)$.
For $v_E({{\bf r}})$, we choose a white noise potential without spatial correlation, $\langle\langle v_E({{\bf r}})v_E({{\bf r}}')\rangle\rangle=\sigma^2 \delta({{\bf r}}-{{\bf r}}')$. The density of states in the non-interacting limit has been calculated exactly by Wegner for this distribution of the disorder potential,[@Wegner] yielding a full width at half maximum of approximately $1.06 \sigma$. In our calculations the parameter $\gamma=e^2/\epsilon\sigma$ specifies the relative strength of interactions and disorder broadening. This type of disorder potential distribution is characterized by a single parameter $\sigma$, which we use as our unit of energy. As we discuss later, our results are insensitive to correlations in the disorder potential on length scales smaller than $l$, but would change in some respects for disorder potentials which are smooth on the magnetic length scale. The HF equations are solved by an iterative approach which can create difficulties which must be addressed. The HF equations generally have many solutions that correspond to different, usually metastable, extrema of the HF energy functional. The challenge is | 0 | non_member_995 |
to locate the true global minimum. In particular, the iteration process will not break any symmetries of the Hamiltonian which are not broken by the starting charge and spin-densities, even though the global minimum of the HF energy functional frequently does break at least some of these symmetries. To counter such problems, it is usually a good idea to introduce small artificial terms in the Hamiltonian which break the continuous symmetries and help the iterative process to reach the lowest energy state. In this problem, the iterative process is also hampered by severe convergence problems at zero temperature connected with the localization of HF quasiparticle wavefunctions and the long range nature of the Coulomb interactions. A small change in the energy of a particular orbital may involve a substantial rearrangement of the charges. These problems can be mitigated by always working at a temperature which is comparable to the finite-size quasiparticle energy level spacing and which scales to zero as the system size increases. The Zeeman term in the Hamiltonian, for which we choose typical experimental values, reduces the SU(2) spin symmetry to a U(1) symmetry. In order to break the continuous U(1) symmetry we introduce an artificial (but very | 0 | non_member_995 |
small) local magnetic field at the center of our simulation cells which points in the x-direction. It is this space-dependent field, required on purely technical grounds, which has motivated developing the formalism in a manner which permits non-constant Zeeman coupling strengths. To ensure that this artificially field and the finite temperature do not affect the final solution, we lower the magnitude of these terms until no change is seen in local charge and spin densities, or in HF quasiparticle energies.
The phase diagrams discussed in Secs. III and IV, were obtained by starting from the non-interacting case and incrementing the interaction strength $\gamma$, taking as the starting densities the self-consistent densities from the previous $\gamma$ value. There is, of course, some hysteresis involved in this process so we do a backwards sweep on $\gamma$ once we have reached the maximum interaction strength for a given run. If they differ, we use the smaller of the values obtained in upward and downward sweeps for the energy per particle. The energy per particle is obtained using the expression $$\begin{aligned}
&&\frac{E}{N}=\frac{1}{\nu}\sum_{{{\bf q}}\in {\rm BZ}}{\rm U}_{\rm dis}({{\bf q}}) {\Delta_0}^*({{\bf q}})
+\frac{\gamma}{2\nu}\sum_{{{\bf q}}\in {\rm BZ}}{\rm U}_{\rm H}({{\bf q}}) |\Delta _0({{\bf q}})|^2\\
&&+{\rm U}_{\rm F}({{\bf q}})
| 0 | non_member_995 |
\left(|\Delta _{\uparrow\uparrow}({{\bf q}})|^2
+|\Delta _{\downarrow\downarrow}({{\bf q}})|^2+|\Delta_{\uparrow
\downarrow}({{\bf q}})|^2+| \Delta_{\downarrow\uparrow}({{\bf q}})|^2\right)\\
&&-\frac{\gamma}{2} \tilde g \hat B\cdot \vec{P}_{tot}\,,\end{aligned}$$ where $ \vec{P}_{tot}$ is the total global spin polarization. The local spin magnetization density, which we calculate as well, is given by $$\langle S_i({{\bf r}})\rangle=
\frac{\hbar}{2}
\sum_{m',m,\sigma,\sigma'} \rho_{\sigma \sigma'}(x_{m'}|x_m)\tau^i_{\sigma \sigma'}
\phi_m^*({{\bf r}})\phi_{m'}({{\bf r}})\,\,.$$ We define the local spin polarization as $\langle P_i({{\bf r}})\rangle=2\langle S_i({{\bf r}})\rangle/(\hbar
\langle \rho({{\bf r}})\rangle)$. Note that in this case $\langle |\vec P({{\bf r}})| \rangle$ does not have to be equal to $1$ since the system is compressible except in the limit of very large $\gamma$ where $\langle |\vec{P}({{\bf r}})|\rangle\rightarrow 1$ for all ${{\bf r}}$.
Our criteria for convergence is that $$\delta \Delta\equiv\frac{1}{N_\phi^2} \sum_{{{\bf q}}\in {\bf BZ}}
\sum_{\sigma \sigma'}
|\Delta_{\sigma \sigma'}^{i}({{\bf q}})-\Delta_{\sigma \sigma'}^{i-1}({{\bf q}})|^2< 1\times
10^{-6}\,,$$ where $i$ stands for the $i$th iteration. We have performed calculations for several disorder realizations at different values of $\nu$ for system sizes of $N_\phi=16-32$. The finite size effects come mainly from the effective exchange potential $U_F({{\bf q}})$ and have been studied in detail previously.[@Allan_finite] The main effect is on the interaction part of the energy per particle and it is well understood and easily corrected. We believe that the physics of the phase transitions | 0 | non_member_995 |
observed in these calculations is not affected qualitatively by finite-size effects. Our qualitative conclusions are based on persistent features which are obtained for several different disorder realizations. The values of $\gamma$ at which the various transitions and cross overs we discuss below take place, do not change by more than $5\%$ for different realizations. The results we present here are for one particular disorder realization.
Results at $\nu=1$
==================
At $\nu=1$ the disorder free ($\gamma \to \infty$) 2DES has[@general] a $S =N/2$ ground state. The $S_z=S=N/2$ member of this multiplet is a single Slater determinant and can therefore be obtained by solving Hartree-Fock equations self-consistently. It is only in recent years that samples which are sufficiently clean to reach, or at least nearly reach, complete spin polarization have been grown.[@Barrett] The collective behavior producing such a ground state was not exhibited in earlier samples which had more disorder in the form of unintended impurities, interface dislocations, and, in modulation doped samples, the potential from remote ionized donors. Fig. \[phase\_diag2\] summarizes the HF theory results we have obtained for the dependence of the spin polarization on interaction strength. The calculations were performed for a realistic value of the Zeeman coupling strength, | 0 | non_member_995 |
$\tilde g=0.015$, and at a very small value, $\tilde g=0.0018$. Extrapolating from these two values to $\tilde g =0$, allows us to identify parameter values for which spontaneous spin polarization occurs, [*i.e.,*]{} values for which the ground state is ferromagnetic. We find that ferromagnetism occurs for $\gamma \gtrsim 0.5$ in the Hartree-Fock approximation; at smaller values of $\gamma$ the single-particle disorder term dominates and yields a spin-singlet ground state. Notice that the spin susceptibility, which may be estimated from the difference between the spin-polarizations at the two $\tilde g$ values, is small in the singlet state, and becomes large as the phase transition to the ferromagnetic state is approached. For the specific finite-size disorder realization we have studied, complete spin polarization is reached at a finite value of $\gamma \sim 1.5$. At larger values of $\gamma$, the system has a finite gap for charge excitations. We must be aware, however, that the HF approximation overestimates the tendency of the system to order so the interaction strength at both transition points should be taken as lower limits. In addition, any physically realistic disorder potential is likely to have rare strong disorder regions which prevent the fully polarized state from being reached.
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In our calculations, there is a wide region of interaction strengths $\gamma$ for which partially spin-polarized states occur. In this regime our HF ground states nearly always have non-collinear magnetic order. We show local spin polarization and charge density profiles of typical partially polarized states in Figs. \[local\_pol2\] and \[local\_dens2\]. The origin of the reduced spatially integrated spin polarization is partly due to variation of spin-orientation, but principally due to a reduction in the average value of the of the [*magnitude*]{} of the local spin polarization. This point is illustrated in Fig. 1 (open circles) and may be inferred from Fig. \[local\_pol2\] (a). The reduction in spin-polarization is due to the occurrence of doubly-occupied orbitals, [*i.e.*]{} to disorder induced charge fluctuations which cannot be accurately described in models which include only the spin degree of freedom. The charged excitations of the ground state in this regime are ungapped and involve population of localized quasiparticle states. We also remark that local density profiles at these relatively small $\gamma$ values, illustrated in in Fig. \[local\_dens2\] (a), follow the effective disorder potential smoothed by the form factor for lowest Landau level electrons. Rapid spatial variation components in the white noise model disorder potential | 0 | non_member_995 |
have little effect on the electronic state. The relationship between electron number density and the Pontryagan index density of the local spin orientation, valid for slow spin-orientation variation and nearly constant charge density,[@general] is [*not*]{} valid in this regime. Still, the collective nature of the 2DES manifests itself in the nonzero spin polarization density perpendicular to the Zeeman field. As interactions strengthen further, the local charge density smoothes out favoring the minimization of Coulomb energy at a cost in disorder energy. (See Fig. 2 (b) and 3 (b) for $\gamma\approx 1.5$.)
Experimentally, the effects of disorder can be seen most directly in the NMR spectral line shape obtained at the lowest possible temperatures where the spin profile is frozen on the experimental time scale. [@Jairo; @Yalegrouppapers] The NMR intensity spectrum in this regime is given by $$I(f,\gamma)\propto \int d{\bf r} \rho_N(z)e^{
-\frac{1}{2\sigma^2}(2\pi f -2\pi K_s \rho_e(z) \langle \vec{S}({{\bf r}};\gamma)\rangle)}
\,,
\label{NMRspectra}$$ with $\sigma=9.34 {\rm ms}^{-1}$ and $K_s\sim 25{\rm KHz}$. Here $\rho_N(z)$ is the nuclear polarization density and $\rho_e(z)$ is the electron density envelope function in the quantum well. The evaluation of such spectra has been outlined elsewhere;[@Jairo; @Yalegrouppapers] here we simply show results for several interaction strengths in Fig. \[NMR2\]. | 0 | non_member_995 |
The parameters used in Eq. \[NMRspectra\] are the same as the ones used in Ref. .
Note that the quantity usually identified experimentally as the Knight shift, the location of the peak in the NMR spectrum in Fig. \[NMR2\], does not match the global polarization. This Knight shift measurement always overestimates the global polarization. In order to obtain the global polarization of the system from the measured spectrum one has to extract the first moment of a normalized spectrum.[@Jairo] One sees from the NMR spectrum at $\gamma=1.25$ that disorder induced spin density variation leads to a broadening of the maximum peak and can even lead to secondary peaks at lower Knight shift frequencies. Note, however, that features corresponding to negative Knight shifts, corresponding to regions of reversed electronic spins, are unlikely because the typical size of such regions is small and because they are also obscured by the finite width of the quantum wells which trap the 2DES. Our calculations demonstrate that care must be taken in interpreting low temperature NMR data in the quantum Hall regime.
The partially polarized regime can also be studied experimentally by measuring the transport activation gap. Provided that weak Zeeman coupling can be ignored, | 0 | non_member_995 |
the extended quasiparticle states are expected to be precisely at the Fermi level when the 2DES is in a paramagnetic state. The Hall conductivity should jump from $0$ to $2 e^2/h$ at $\nu=1$. In the ferromagnetic state, the majority-spin extended quasiparticle state will be below the Fermi level, the majority-spin extended state will be above the Fermi level and the Hall conductivity at $\nu=1$ should be quantized at $\sigma_{xy} = e^2/h$. The spontaneous splitting of the two extended state energies is experimentally accessible and should exhibit interesting power law critical behavior as the ferromagnetic state is entered. This transport gap should vary monotonically with the global spin polarization, although the precise relationship between these quantities is not trivial.
It is interesting to note that the Skyrmion-anti-Skyrmion pairs predicted recently by Nederveen and Narazov [@Nederveen] do not appear in our calculations. We do not conclude that these objects cannot appear at $\nu=1$; we would expect them, for example, if we choose a disorder model with relatively large potential variations, but only on a length scale much larger than the Skyrmion size. In this case the NL$\sigma$ model considerations in Ref. should be applicable. Our calculations demonstrate rather clearly however, that charge | 0 | non_member_995 |
density variation at $\nu=1$ does not necessarily, or even usually, require the existence of well defined Skyrmion quasiparticles.
Results at $\nu\ne 1$
=====================
In clean (large $\gamma$) samples where full polarization is observed at $\nu=1$, the global polarization decays rapidly with $|1-\nu|$.[@Barrett] It is generally accepted that this property is a unique signature which experimentally establishes the thermodynamic stability of Skyrmion collective quasiparticles. In the strong disorder limit, on the other hand, spontaneous spin polarization does not occur at any filling factor near $\nu = 1$.
The global polarization results for $\nu \ne 1$ in Fig. \[phase\_diag3\], illustrate how the system interpolates between these two extrema. As the interaction strength $\gamma$ is increased from 0 to 2, the behavior is similar to the $\nu=1$ case. For strong disorder charge variation is dominant, and small spin polarizations occur primarily because many single particle orbitals are occupied by both up and down spin electrons. Charge variation is the dominant response to disorder, and it continues to play an important role at all interaction strengths. At sufficiently large $\gamma$, our finite size systems reach a state with the maximum spin polarization allowed by the Pauli exclusion principle. This maximally polarized state is reached | 0 | non_member_995 |
earlier than in the case at $\nu=1$ ($\gamma\sim1.4-1.6$) because, we believe, a larger number of charged quasiparticles are available to screen the random potential. At this point the system forms what we refer to as a conventional quasiparticle glass (CQG). The conventional Laughlin quasiparticles are initially localized in the deepest minima (or maxima for $\nu<1$) of the effective disorder potential and as the interaction strength increases, or equivalently the depth of the disorder potential wells becomes smaller, the charged quasiparticles rearrange themselves locally into a quasi-triangular Wigner crystal pinned by the strongest of the disorder potential extrema. At larger $\gamma$ we observe a transition from a CQG to a Skyrmion glass. The location of this transition is marked by a reduction of the global polarization from its maximally polarized value. For a specific disorder realization the point of cross over from the CQG to the Skyrmion glass, as illustrated in Fig. \[phase\_diag3\], depends on filling factor and $\tilde g$. The dependence of the transition point on $\tilde{g}$ in this regime can be approximated by considering a simple model for a single Skyrmion trapped at a disorder potential extrema. We approximate its energy by $$E(K)=U(K-K_0)^2+g^*\mu_B B K+\sigma AK\,,
\label{model}$$ where $K$ | 0 | non_member_995 |
is the number of spin flips per Skyrmion, $\sigma$ is the strength of the disorder potential and $A$ is a phenomenological parameter. The first two terms determine the optimal Skyrmion size in the absence of disorder.[@UKmodel_Allan] The form for the third term reflects the property that Skyrmions with smaller $K$ are smaller and will be able to concentrate more strongly close to the potential extrema. This simple model gives an estimate of the interaction strength at which $K > 0$ Skyrmions first become stable $$\gamma^*=\frac{A}{2K_0U/(e^2/\epsilon \ell)-\tilde{g}}\,.
\label{gammastar}$$ The parameters $U$ and $K_0$ can be estimated[@UKmodel_Allan] for filling factor $\nu=1.25$ as $U/(e^2/\epsilon \ell)\sim0.014$ and $K_0\sim 1$. Using our numerical result for where the transition occurs at $\tilde{g}=0.0018$, we estimate that $A\sim 0.1$. From this, one obtains an estimate of $\gamma^*\sim 7$ for the cross over point from conventional quasiparticles to Skyrmions at $\tilde{g}=0.015$. This is in reasonable agreement with the actual cross over point $\gamma^*\sim 10$ (see Fig. \[phase\_diag3\], the transition is out of scale in Fig. \[phasediag\]) given the simplicity of the model. These estimates of the maximum disorder strength at which Skyrmion physics is realized could be checked by performing NMR experiments in samples where electron density, and hence | 0 | non_member_995 |
the interaction strength, is adjusted by the application of gate voltages.
For a particular realization of the disorder potential, particle-hole symmetry is broken in a finite system and is recovered only in the limit of very large $\gamma$. The particle-hole symmetry relation for the global spin polarization is i.e. $(1-\epsilon) P_z(\nu=1-\epsilon)=(1+\epsilon) P_z(\nu=1+\epsilon)$ where $\epsilon
<1$. At large $\gamma$ this relation is approximately satisfied. Also in this limit, the Pontryagan relation between the local density profile and the local spin density [@general] becomes accurate. In the clean limit, the Skyrmion system crystallizes in a square lattice for the filling factors considered here. (The Skyrmion crystal is triangular[@UKmodel_Allan] for $\nu$ very close to 1.) The disordered Skyrmion glass state has very smooth fluctuations of the local spin density, compared to the CQG, although both lattices are pinned by the disorder potential. We remark that quantum fluctuations in Skyrmion positions are not accounted for in HF theory, and it is quite possible that even in this limit the ground state is a liquid rather than a crystal. [@Juanjo] As noted in Ref. it is possible that the broken U(1) symmetry of the Skyrmions orientation order predicted by Hartree-Fock does not survive quantum fluctuations. | 0 | non_member_995 |
We show an example of the CQG in Fig. \[local\_pol3\] (a) and \[local\_dens3\] (a), and of the quasi Skyrmion lattice state in Fig. \[local\_pol3\] (b) and \[local\_dens3\] (b). Note that we find, in agreement with Nederveen and Narazov, [@Nederveen] a shrinking of the Skyrmion size as disorder broadening increases. This effect may help explain the appearance of a “tilted plateau” centered around $\nu=1$ in the Knight shift vs. filling factor data. [@Barrett_private] Rare highly disorder regions in the sample may localize and reduce the effective size of the few Skyrmions present at these filling factors. This would allow the bulk of the sample to be fully polarized at $\nu\ne 1$ and give rise to a Knight shift equivalent to the one at $\nu=1$. The plateau is tilted because of the change in fully polarized density as pointed out in Ref. .
Discussion
==========
We have used the Hartree-Fock approximation to study the competition between interactions and disorder near Landau level filling factor $\nu=1$. At a qualitative level our results can be summarized by the schematic zero-temperature phase diagram shown in Fig. \[phasediag\], which is drawn for the case of small but non-zero Zeeman coupling. Distinct ground states can be distinguished | 0 | non_member_995 |
by different values for the quantized Hall conductivity, $\sigma_{xy}$, by the presence or absence of spontaneous spin-polarization perpendicular to the direction of the Zeeman field, and by the presence or absence of a gap for spin-flip excitations. At small $\gamma$ (strong disorder), the electronic state is paramagnetic (denoted as PC in Fig. \[phasediag\]), there is no spin-polarization in the absence of Zeeman coupling, and the Hall conductivity is expected to jump from $0$ to $2 e^2/h$ as the filling factor $\nu$ crosses the $\nu=1$ line. For a small Zeeman coupling, there will be a small splitting between the majority-spin and minority-spin extended state energies and the zero-temperature Hall conductance should have a narrow intermediate $e^2/h$ plateau centered on $\nu=1$. However, we do not expect that this plateau will be observable at accessible temperatures, and have indicated this in Fig. \[phasediag\] by using a thick line to mark the $0$ to $2 e^2/h$ phase boundary. At somewhat larger $\gamma$ there is a phase transition at zero Zeeman energy between paramagnetic and ferromagnetic states (denoted as FC in Fig. \[phasediag\]). In our calculations this transition occurs at a larger value of $\gamma$ at $\nu=1$ than away from $\nu=1$. As $\gamma$ increases in | 0 | non_member_995 |
the ferromagnetic state, we expect that the separation between majority-spin and minority-spin extended state levels will increase rapidly so that the $\nu=1$ integer quantum Hall plateau will broaden and become observable. At still larger $\gamma$, we find a transition to a state with the maximum spin polarization allowed by the Pauli exclusion principle. At $\nu \le 1$, this is full spin-polarization. In these states, marked ‘SG’ for spin-gap in Fig. \[phasediag\], the differential spin-susceptibility vanishes. For realistic disorder models, it seems likely that in the thermodynamic limit there will always be rare high-disorder regions in the sample which prevent maximal spin-polarization from being achieved. For this reason, the phase transition we find in our finite systems likely indicates a crossover from large to small differential spin-susceptibility in macroscopic systems; we have therefore marked this transition by a dashed line. Finally at the largest values of $\gamma$ (weakest disorder) the physics for $\nu$ near $1$ is dominated by Skyrmion quasiparticles which emerge from the $\nu=1$ ferromagnetic vacuum. In this regime, the system develops spontaneous spin-polarization in the plane perpendicular to the direction of the Zeeman field. In Fig. \[phasediag\] we have labelled this regime NCF for non-collinear ferromagnet.
This phase diagram | 0 | non_member_995 |
is intended to represent the filling factor interval $0.85 \le \nu \le 1.15$, over which fractional quantum Hall effects are not normally observed and it appears likely that Hartree-Fock approximation calculations are able to represent interaction effects. Some of our findings may help explain the striking tilted plateau feature observed in the NMR spectra [@Barrett_private] near $\nu=1$. Nevertheless, we have found rich structure in the crossover between non-interacting and disorder-free limits of the $\nu=1$ quantum Hall effect which helps explain the difficulty experienced in attempting to construct a simple interpretation of low-temperature NMR spectra. Our calculations motivate experimental studies of $\nu=1$ transport activation energy studies near the paramagnetic to ferromagnetic phase transition.
Helpful conversations with S.E. Barrett, Luis Brey, and Tatsuya Nakajima are greatly acknowledged. This work was supported by the National Science Foundation under grants DMR-9714055 and DMR-9820816.
For an introduction to Skyrmions and related topics see S.M. Girvin, [*The Quantum Hall Effect: Novel Excitations and Broken Symmetries*]{} in Les Houches Summer School 1998 (to be published by Springer Verleg and Les Editions de Physique, 1999), and references therein. S.M. Girvin and A.H. MacDonald, in [*Perspectives in Quantum Hall Effects: Novel Quantum Liquids in Low-Dimensional Semiconductor Structures*]{}, edited by | 0 | non_member_995 |