# Anyons in generalized Luttinger liquid models

K.-V. Pham, M Gabay, P Lederer

► To cite this version:

K.-V. Pham, M Gabay, P Lederer. Anyons in generalized Luttinger liquid models. EPL - Europhysics Letters, 2000, 51 (2), pp.161-167. 10.1209/epl/i2000-00526-5. http://dx.doi.org/10.1209/epl/000526-5

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Anyons in generalized Luttinger liquid models.

K.-V. Pham, M. Gabay and P. Lederer

Laboratoire de Physique des Solides, associé au CNRS

Université Paris-Sud

91405 Orsay, France

Schulz and Shastry recently introduced a new class of exactly solvable one dimensional models with non-Fermi liquid physics. First, we show that this behavior is caused by the existence of anyonic excitations with charge $\frac{\delta}{2\pi}$ where $\delta$ is the phase shift of the potential. Since $\delta$ is an arbitrary real number, these anyons may therefore carry an irrational charge. Next, in view of this result we revisit the general classification of non Fermi-liquid two-species gapless 1D models: while it is usually thought that $c = 1 \times c = 1$ theories form a four parameter space, we establish that the most general non relativistic conformal field theory has ten parameters. This defines a single larger universality class with sub-manifolds associated with the standard Luttinger liquid (with spin-charge separation), the Schulz-Shastry class of models, and gapless models without spin-charge separation such as the Hubbard model in a magnetic field.

The experimental observations of many anomalous properties in the normal state of high $T_c$ superconductors, have triggered a renewed interest in non-Fermi liquid (NFL) ground states of many electron systems. In one dimensional (1D) metals, the concept of Landau quasiparticle breaks down and electronic states are described instead in terms of collective modes: a striking consequence of this effect is the so-called spin-charge separation [?]. A construction of the associated holon and spinon excitations by bosonization was given in Ref. ? . In addition, correlation functions exhibit non-trivial power law decays; the fixed point associated with this critical behavior corresponds to the Luttinger liquid (LL)[? ] universality class. In 2D, the concept of anyons (particles with fractional statistics [? ] ) has proved particularly fruitful in the construction of highly correlated collective electronic states. In this paper, starting from a 1D model introduced by Schulz and Shastry (SS), we show that there exists close connections between the LL and 1D anyonic models[? ]. Recently, Schulz and Shastry (SS) introduced a new class of exactly solvable one-dimensional models[? ] where the standard Luttinger liquid (LL) is recovered in a limiting case. This larger universality class was found using a series of exactly solvable lattice and continuum models involving two species of fermions labelled by a spin index $\sigma = \pm$, coupled by a gauge potential acting on particles of opposite spin. The simplest model they consider is[? ]:

$$H = \sum_{i\sigma} (p_i + \sigma A_{\sigma}(x_{i\sigma}))^2 \qquad (1)$$

where $A_{\sigma}(x) = \sum_j V(x - x_{j,-\sigma})$. The model is trivially solvable because one can make a (singular) gauge transformation leading to the non-interacting hamiltonian: $H' = e^{iS}He^{-iS} = \sum_{i\sigma} (p_i)^2$. However the physics remains non-trivial because boundary conditions are not preserved by the diagonalization: if $\delta = \int_0^L V(x)dx$ is the phase shift of the SS potential, momenta must be quantized as

$$k_{i\sigma} = \frac{2\pi}{L} n_{i\sigma} + \sigma \frac{N_{-\sigma}\delta}{L} \qquad (2)$$

$(n_{i\sigma}$ is an integer) and SS find that critical exponents are in general anomalous. If $\delta/2\pi$ is an integer, the model describes non interacting fermions. But otherwise, the hamiltonian (??) yields properties expected for a non-Fermi liquid : in the long distance limit, correlation functions decay as power laws with exponents depending on the value of $\delta$. In the solution of the SS models, one might infer from the pseudomomentum parametrization that elementary excitations are trivial and simply consist of electrons dressed by the gauge potential, which attaches flux tubes to each electron. Inasmuch as flux tubes are seen only by particles of the opposite spin, the commutation relations for fermions of a given spin species are unchanged by the gauge potential. Yet, as we stated above, hamiltonian (??) predicts non Fermi-liquid behavior; our paper first establishes that in the low-energy limit, this property stems from the existence of stranger elementary excitations: anyons carrying a fractional charge $\delta/2\pi$, where $\delta$ is the phase shift of the gauge potential. Conversely we show that anyon models in 1D lead to NFL behavior: a marginal operator connected to the anyonic statistics gives rise to a new universality class, differing from that of the standard LL; whereas the usual LL is described in terms of a four parameter gaussian model ($u_c, u_s$ the charge and spin velocities, $K_c, K_s$ the charge and spin LL parameters), the most general LL is a ten parameter theory, defining a larger universality class. Three distinct sub-manifolds can be identified and we exhibit that connected to anyon models.
In the following, we consider a bosonized version of the SS model corresponding to hamiltonian (??) obtained by mapping the low-energy spectrum to a gaussian model[? ]:

$$
H^{ss} = \frac{v_F}{2} \int dx \sum_{\sigma} (\partial_x \Phi_{\sigma})^2 + \left( \Pi_{\sigma} - \sigma \frac{\delta}{\pi} \partial_x \Phi_{-\sigma} \right)^2 \quad (3)
$$

$$
\rho_{\sigma} = -\frac{1}{\sqrt{\pi}} \partial_{x} \Phi_{\sigma}, \quad j_{\sigma} = \frac{1}{\sqrt{\pi}} \Pi_{\sigma} \qquad (4)
$$

$$
[\Phi_{\sigma}(x), \Pi_{\sigma'}(y)] = i\delta_{\sigma\sigma'}\delta(x-y) \quad (5)
$$

$\rho_{\sigma}$ and $j_{\sigma}$ are the charge and current density parametrized by the standard phase fields $\Phi_{\sigma}$ and $\Pi_{\sigma} = \partial_{x}\Theta_{\sigma}$ of LL theory. $\Theta_{\sigma}$ is the superfluid phase; the bosonization formulas are as usual:

$$
\Psi_{\sigma, \varepsilon}(x) = \exp i\sqrt{\pi}(\Theta_{\sigma} - \varepsilon\Phi_{\sigma}) \exp i\varepsilon k_{F}x, \quad (6)
$$

where $k_F$ is the Fermi momentum and $\varepsilon = \pm$ indexes the right and left components of the fermion fields.
We now introduce the shifted fields:

$$
\tilde{\Theta}_{\sigma} = \Theta_{\sigma} - \sigma \frac{\delta}{\pi} \Phi_{-\sigma},
$$

$$
\tilde{\Pi}_{\sigma} = \partial_{x} \tilde{\Theta}_{\sigma}
$$

which are canonical conjugates of the field $\Phi_{\sigma} : [\Phi_{\sigma}(x), \tilde{\Pi}_{\sigma'}(y)] = i\delta_{\sigma\sigma'}\delta(x-y)$; the hamiltonian $H^{ss}$ becomes simply
the standard boson hamiltonian with $K_c = 1 = K_s$:

$$
H^{ss} = \frac{v_F}{2} \int dx \sum_{\sigma} (\partial_x \Phi_{\sigma})^2 + (\tilde{\Pi}_{\sigma})^2
$$

In the non-interacting case, the two Fermi points can be decoupled: there is a chiral separation. More precisely:

$$
H = H_{+} + H_{-} = \frac{v_F}{4} \sum_{\epsilon=\pm} \int dx \sum_{\sigma} \partial_x (\Theta_{\sigma\epsilon})^2
$$

where $\Theta_{\sigma\epsilon} = \Theta_{\sigma} - \epsilon\Phi_{\sigma}$ is a chiral field propagating in the direction $\epsilon$ ($\Theta_{\sigma\epsilon}(x,t) = \Theta_{\sigma\epsilon}(x - \epsilon v_F t)$); $[H_+, H_-] = 0$ means the fields $\Theta_{\epsilon\sigma}$ are dynamically independent. If one considers the Fourier transforms of the fermion fields $\Psi_{\sigma\pm}(x)$ given by eq.(??), then it is clear that

$$
[H_{\epsilon}, \Psi_{\sigma\epsilon'}^{+}(q)] = \left( v_F |q| + \frac{\pi v_F}{L} \right) \Psi_{\sigma\epsilon'}^{+}(q) \delta_{\epsilon\epsilon'}
$$

(in other words free electrons are the eigenstates of the non-interacting hamiltonian). We now turn to the SS
hamiltonian where the previous statement will prove less trivial; consider the following operators and hamiltonians:

$$
\Psi(Q_{\sigma\epsilon}) = \exp\left(-i\sqrt{\pi}Q_{\sigma\epsilon}(\tilde{\Theta}_{\sigma} - \epsilon\Phi_{\sigma})\right) \quad (7)
$$

$$
H_{\epsilon}^{ss} = \frac{v_F}{4} \int dx \sum_{\sigma} \partial_x (\tilde{\Theta}_{\sigma\epsilon})^2 \qquad (8)
$$

Again, $\tilde{\Theta}_{\sigma\epsilon} = \tilde{\Theta}_{\sigma} - \epsilon\Phi_{\sigma}$ are dynamically independent fields and $[H_{+}^{ss}, H_{-}^{ss}] = 0$, while

$$
[H_{\epsilon}^{ss}, \Psi^{+}(Q_{\sigma\epsilon})(q)] =
$$

$$
\left( v_F |q| + \frac{\pi v_F}{2L} \left( Q_{\sigma\epsilon}^2 + \left( Q_{\sigma\epsilon} + \sigma \frac{\delta}{\pi} Q_{\sigma\epsilon} \right)^2 \right) \right) \Psi^+(Q_{\sigma\epsilon})(q) \quad (9)
$$

$Q_{\sigma\epsilon}$ is the charge carried by the anyon state $\Psi(Q_{\sigma\epsilon})$[? ]. Only some of these states are physical as we show below. The structure of the Fock space, which has been ignored so far, is what determines the physical states. In the zero charge sector, the excitations of $H^{ss}$ are phonons, whose expressions are obtained from the non-interacting case by replacing $\Theta_{\sigma}$ by $\tilde{\Theta}_{\sigma}$. The charged sectors of the Fock space are direct products of Hilbert spaces for the spin up electrons and spin down electrons; according to the bosonization formulas (??), a complete basis of physical states is given by the operators:

$$
O_{\{N_{\sigma, \epsilon}\}} = \prod_{\sigma \epsilon} (\Psi_{\sigma, \epsilon}^{+})^{N_{\sigma \epsilon}}
$$
where {$N_{\sigma, \epsilon}$} are four arbitrary integers corresponding to the number $N_{\sigma, \epsilon}$ of electrons with spin $\sigma$ created at the Fermi momentum $\epsilon k_F$. These states can be rewritten in terms of the shifted fields $\tilde{\Theta}_{\sigma\epsilon}$ as:

$$O_{\{N_{\sigma, \epsilon}\}} = \prod_{\sigma} \exp \left( -i\sqrt{\pi} \left( N_{\sigma,+} + \frac{\sigma\delta}{2\pi} \sum_{\epsilon} N_{-\sigma,\epsilon} \right) \tilde{\Theta}_{\sigma,+} \right) \\ \times \exp \left( -i\sqrt{\pi} \left( N_{\sigma,-} - \frac{\sigma\delta}{2\pi} \sum_{\epsilon} N_{-\sigma,\epsilon} \right) \tilde{\Theta}_{\sigma,-} \right)$$

Since the fields $\tilde{\Theta}_{\sigma\epsilon}$ are dynamically independent this means that the state created by $O_{\{N_{\sigma, \epsilon}\}}$ is unstable and will decay. The new eigenbasis consists of four generators {$\Psi^+(Q_{\sigma\epsilon}), \sigma = \pm, \epsilon = \pm$} carrying the charges

$$Q_{\sigma\epsilon} = \left( N_{\sigma,\epsilon} + \epsilon \frac{\sigma\delta}{2\pi} (N_{-\sigma,\epsilon} + N_{-\sigma,-\epsilon}) \right). \qquad (10)$$

Since $N_{\sigma, \epsilon}$ is an integer while $\delta$ is an arbitrary number, this means $Q_{\sigma\epsilon}$ will not in general be an integer and may even assume irrational values. The above construction proves that fractional states are involved in any transition to any given charge sector and completely determines the values of the charges $Q_{\sigma\epsilon}$: no other values are allowed.

To illustrate the previous result, we consider the effect of adding a spin up electron at the right Fermi point $k_F$: ($N_{\uparrow+} = 1$; $N_{\uparrow-} = 0 = N_{\downarrow+} = N_{\downarrow-}$) (for the sake of clarity we use here $\sigma = \uparrow, \downarrow$ instead of $\pm$). This state decays into: 1) $\Psi^+(Q_{\uparrow+} = 1) = \exp(-i\sqrt{\pi}\tilde{\Theta}_{\uparrow,+}$; this operator is the product of an electron operator by the vortex operator $\exp(i\sqrt{\pi}\frac{\delta}{2\pi}\Phi_{\downarrow}$ which implements flux attachment. 2) $Q_{\uparrow-} = 0$ and therefore adding the up spin does not produce a left-moving counterpart to the previous state 1) in our particular example. 3) The state $\Psi^+(Q_{\downarrow+} = -\frac{\delta}{2\pi})$ is also generated: it is a charge $-\frac{\delta}{2\pi}$ anyon moving to the right. 4) Finally, a similar state with charge $\frac{\delta}{2\pi}$ is created and represents an anyon moving to the left. We note that operator 1) creates a "dressed" electron which is in fact an anyon propagating to the right: type 1) operators for a given spin species (say, up) anticommute but have anyonic commutation relations with the anyon of the other spin species (say, down).

Furthermore, equation (??) also shows that the entire spectrum of fractional states can be built from the "dressed" charge one anyon, and the second charge $\frac{\delta}{2\pi}$ anyon, which together generate all the other excitations.

Let us explain why we might have expected the existence of some exotic anyon with a charge proportional to $\delta$ in the SS hamiltonian (??). The addition of one electron with spin up has two effects on the pseudoparticle distribution (??). Firstly, this adds a pseudomomentum of spin up and physically this means we will have a charge one excitation akin to composite fermions: the only difference is that instead of having an electron to which an even number of flux tubes are attached, we will have an electron and a flux of the opposite spin, proportional to $\delta$ attached to it[?]. Since the flux is not an integer in units of the flux quantum, this excitation will in general be an anyon (with charge one). But secondly, the addition of a spin up electron leads to a reshuffling of the Fermi sea of spin down particles since the total phase shift created by the SS gauge potential is proportional to the number of particles with up spin (eq. (??)): the pseudomomentum distribution for spin down evidently changes when there is addition of a spin up electron. For spin down particles, this corresponds to a pure current excitation without addition of spin down electrons; let us now ask ourselves what is the nature of the object carrying that current excitation. The current is $j \propto \delta$ and can be written in terms of a left-moving carrier and a right-moving one as: $j = u_+Q_+ - u_-Q_-$ where $u_\pm$ are the velocities of the carriers. Besides, $Q_+ + Q_- = 0$ since no spin down electron has been added: in other word, this pure current excitation consists of a quasiparticle-quasihole pair. The velocities $u_\pm$ cannot depend on $\delta$ as should be clear from eq. (??) which shows that the linearization of the spectrum does not change the Fermi velocity: $u_\pm = v_F$. This then implies that $Q_+ \propto \delta$. The spectrum of the SS hamiltonian must therefore accommodate (at least in the low-energy limit) anyon excitations carrying a fractional charge proportional to the phase shift $\delta$.

The pseudo-unitary transformation that Schulz and Shastry used to build their class of models is a generalized Jordan-Wigner transformation. From this standpoint the SS models can be analyzed in the general context of singular gauge transformations, which have provided the physics of strongly correlated fermions with many new concepts in two dimensions: anyons, Chern-Simons transmutations, composite bosons[?], Jain's composite fermions[?], the anomalous metal at $\nu = 1/2$. The SS models apply similar concepts to 1D physics: however the SS models are not the simplest applications of singular gauge transformations to 1D because they mix flux tube attachment with spin. We now turn to a spinless fermion version of the SS models which captures the essential physics of the SS universality class without the complication of spin:

$$H[u, K, \delta] = \frac{u}{2} \int dx \frac{(\partial_x \Phi)^2}{K} + K \left( \Pi - \frac{\delta}{\pi} \partial_x \Phi \right)^2$$

Exactly like for the SS hamiltonian, the shifted momentum $\Pi$ means that particles acquire an Aharonov-Bohm phase proportional to $\delta$ each time they see other particles. This gauge potential is not trivial since it produces a
statistical transmutation of all the particles into anyons. In particular if $\delta = \pi$ we are just making a Jordan-Wigner transformation and are therefore turning fermions into (hard core) bosons: as is well known, this is far from being innocuous because it changes the spectrum of the gaussian LL hamiltonian (the zero modes obey different selection rules[?]). For arbitrary $\delta$, the spectrum is also affected, which leads for instance to a change in the way right-moving fermion Green functions decay with distance $G_R(x) \sim \exp(i(k_F - \delta/\pi)x/x^\alpha$, $\alpha = \frac{1}{2K} + \frac{K}{2}(1 - \frac{\delta}{\pi})^2$: in this expression it is not possible to eliminate $\delta$ via a redefinition of $K$ reflecting the fact that the model does not reduce to a standard effective LL.

For $K=1$, the previous hamiltonian can be derived by bosonization of the model:

$$H_{anyon} = \sum_{i} (P_{i} + \delta\rho(x_{i}))^{2}$$

This hamiltonian is the exact counterpart of the 2D free anyon model [?]. In contrast with the 2D many-anyon problem which is still unsolved, our 1D model is exactly solvable; its low-energy properties can also be conveniently recovered through bosonization: this is a major difference with the 2D model. An analysis similar to the one presented above shows that that the elementary excitations of the 'free' anyon problem ($K=1$ and $\delta \neq 0$), differ from the initial anyon (the electron dressed by flux tubes): one finds other anyon states, with non-integral charge. This shows that the 'free anyon model' defined through the singular gauge potential is already strongly correlated, even in the absence of two-body interactions. Details will be reported elsewhere [?]. Let us try to understand this new model from a broader perspective. In a relativistic framework, the gaussian model has only one marginal operator: $\partial_\mu \Phi \partial^\mu \Phi$ and accordingly the critical surface is described by a single parameter, the compactification radius; in condensed matter physics, relativistic covariance is not necessary: this means that in that case we can construct three marginal operators, respectively $(\partial_x \Phi)^2$, $\Pi^2 = (\partial_0 \Phi)^2$ and $\Pi \partial_x \Phi$. Thus, the most general $c=1$ CFT (critical single species 1D theory) will be a three-parameter theory: this is precisely what we found for the generalized LL hamiltonian $H[u, K, \delta]$ which depends on three independent parameters associated with the three marginal operators mentioned above. The third marginal operator $\Pi \partial_x \Phi$ is usually not taken into account, probably because it can be gauged away by a suitable transformation. One may be deceived into thinking that this operator has no physical effect. This is however incorrect: the gauge transformation is actually singular, and leads to multivalued wavefunctions.

We now apply these insights to the standard LL, the hamiltonian of which is a quadratic form of the charge $(\partial_x \Phi_c, \Pi_c = \partial_x \Theta_c)$ and spin $(\partial_x \Phi_s, \Pi_s = \partial_x \Theta_s)$ operators. We clarify below the status of the SS universality class with respect to that of the standard LL. In the absence of relativistic covariance there are ten allowed marginal operators and these fall into three categories: 1) the four standard marginal operators: $(\partial_x \Phi_c)^2, (\partial_x \Phi_s)^2, \Pi_c^2, \Pi_s^2$; they define the usual LL with spin-charge separation; 2) four mixed operators corresponding to flux-tube attachment: (a) $(\partial_x \Phi_c) \Pi_c, (\partial_x \Phi_s) \Pi_s$ which are the counterparts of the operators considered in the spinless LL, (b) $(\partial_x \Phi_c) \Pi_s, (\partial_x \Phi_s) \Pi_c$: the antisymmetric combination of these operators to which one adds the four standard operators corresponds to the SS universality class; (3) two operators mixing spin and charge: $\partial_x \Phi_c \partial_x \Phi_s$ and $\Pi_c \Pi_s$.

The generalized Luttinger liquids obtained by considering the first two classes of marginal operators are all exactly solvable because the four mixed operators can be absorbed by a singular gauge transformation. The third class leads to a solvable Tomonaga-Luttinger model if and only if the matrices associated with the quadratic forms $a(\partial_x \Phi_c)^2 + b(\partial_x \Phi_s)^2 + c\partial_x \Phi_c \partial_x \Phi_s$ and $\alpha\Pi_c^2 + \beta\Pi_s^2 + \gamma\Pi_c\Pi_s$ commute $(a, b, c, \alpha, \beta, \gamma$ are the marginal coupling constants). A physical realization is provided by the Hubbard model in a magnetic field: a striking property of this model is that spin-charge separation is not obtained anymore, no matter how small the magnetic field is; an orthogonality catastrophe with the standard LL occurs and indeed the finite-size spectrum of that model, which can be determined by Bethe Ansatz, does not correspond to that of the standard Luttinger liquid (class 1)[?]; the spectrum can however be mapped onto that of an hamiltonian which incorporates the third class of marginal operators [?]: $a(\alpha\partial_x \Phi_c + \beta\partial_x \Phi_s)^2 + b(\beta\partial_x \Phi_c - \alpha\partial_x \Phi_s)^2 + c(\alpha\Pi_c + \beta\Pi_s)^2 + d(\beta\Pi_c - \alpha\Pi_s)^2$. This completes the characterization of $c=1 \times c=1$ CFT in a non-relativistic context: one has a 10 parameter critical surface instead of the 4 parameter theory found for instance in string theories.

In this paper, we first established that the SS hamiltonian admits (in the low-energy limit) anyonic excitations carrying a charge proportional to the phase shift of the SS gauge potential. This illustrates the non-trivial effect of singular gauge transformations. Then we showed that reciprocally, anyonic 1D models lead to NFL behavior. They constitute a new universality class, distinct from that of the standard LL. This property stems from the fact that the anyonic operator is a marginal operator. The Schulz-Shastry model is a member of that new class and the standard Luttinger liquid is recovered in a limiting case. This new universality class falls into the wider context of $c=1 \times c=1$ critical theories, which is a ten parameter theory. The authors dedicate this paper to Heinz Schulz who introduced
them to the SS model. K-V P. thanks B. Pioline for information on string theories.

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[2] K.-V. Pham, M. Gabay, P. Lederer, Eur. Phys. Journ. B **9**, 573 (1999); ibid, cond-mat/9909295

[3] F. Wilczek, *Fractional Statistics and Anyon Superconductivity*, Ed. F. Wilczek, 1990, World Scientific, Singapore.

[4] In this paper, anyons are defined as particles which acquire a non-trivial phase under the exchange of positions: in terms of operators, commutation relations are $O(x)O(y) + \exp i\theta O(y)O(x) = 0$; 2D exchange statistics uses the same definition. This is equivalent to flux-tube attachment;

[5] H. J Schulz, B. S. Shastry, Phys. Rev. Lett. **80**, 1924 (1998).

[6] We will call that model, the SS hamiltonian.

[7] The gauge transformation $U = \exp iS$ is singular because it does not preserve boundary conditions. It is easy to show that $S\{x_{1\sigma}, \dots, x_{i\sigma} + L, \dots\} - S\{x_{1\sigma}, \dots, x_{i\sigma}, \dots\} = \sigma N_{-\sigma} \delta$ where $N_\sigma$ is the number of fermions with spin $\sigma$ and $\delta$ is the phase shift. This leads to the modified expression for the momentum $k_{i\sigma}$.

[8] Pictorially, an electron and a current of particles of the opposite spin orbiting around it.

[9] The commutation relations of this operator show it is an anyon; there is also a mutual statistics between particle of opposite spins, i.e. a statistical phase when particles of opposite spin are exchanged.

[10] K-V Pham, M. Gabay, P. Lederer, unpublished.

[11] S. C. Zhang, H. Hansson, S. Kivelson, Phys. Rev. Lett. **62**, 82 (1989); S. C. Zhang, Int. J. Mod. Phys. B **6**, 25 (1992)

[12] J. K. Jain, Phys. Rev. Lett. **63**, 199 (1989)

[13] F. D. M. Haldane, Phys. Rev. Lett. **47**, 1840 (1981)

[14] H. Frahm, V. Korepin, Phys. Rev. B **43**, 5643 (1991).