# Correlations and order parameter at a Coulomb-crystal phase transition in a three-dimensional dimer model

Grégoire Misguich, Vincent Pasquier, Fabien Alet

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Grégoire Misguich, Vincent Pasquier, Fabien Alet. Correlations and order parameter at a Coulomb-crystal phase transition in a three-dimensional dimer model. Physical Review B, 2008, 78, pp.100402(R). 10.1103/PhysRevB.78.100402. hal-00323012

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Correlations and order parameter at a Coulomb-crystal phase transition
in a three-dimensional dimer model

Grégoire Misguich,¹,* Vincent Pasquier,¹ and Fabien Alet²

¹Institut de Physique Théorique, CEA, IPhT, CNRS, URA 2306 F-91191 Gif-sur-Yvette, France

²Laboratoire de Physique Théorique, IRSAMC, Université Paul Sabatier, CNRS, 31062 Toulouse, France

(Received 11 July 2008; published 3 September 2008)

The three-dimensional classical dimer model with interactions shows an unexpected continuous phase transition between an ordered dimer crystal and a Coulomb liquid. A detailed analysis of the critical dimer and monomer correlation functions points to a subtle interplay between the fluctuations of the crystal order parameter and the “magnetic” degrees of freedom present in the Coulomb phase. The distribution probability of the crystal order parameter suggests an emerging continuous $O(3)$ symmetry at the critical point.

DOI: 10.1103/PhysRevB.78.100402

PACS number(s): 75.10.−b, 64.60.−i, 05.50.+q

I. INTRODUCTION

The Berezinskii-Kosterlitz-Thouless transition¹ is an ex-
ample of critical phenomenon which cannot be understood
from the conventional Landau-Ginzburg-Wilson (LGW)
point of view, based on spontaneously broken symmetries
and critical fluctuations of the associated order parameter.²
Many quantum systems in one-dimension and classical sys-
tems in two-dimension (2D) exhibit such type of continuous
phase transition, which is understood as a proliferation of
topological defects. The recent suggestion³ that some phase
transitions in $D>2$ could also be outside the LGW
framework triggered an important activity, both on the
field-theoretical⁴ and numerical⁵⁻⁷ sides.

The classical dimer model on the cubic lattice⁸,⁹ is one of
the simplest three-dimensional (3D) models with a transition
which does not seem to fit in the LGW picture,⁹ although the
field theory of the critical point has not yet been elucidated.
Configurations of the model are hard-core dimer coverings
of a cubic lattice.⁸ Each dimer occupies two sites and each
site belongs exclusively to one dimer. In addition, each con-
figuration has an energy equal to −1 times the number of
square plaquettes with two parallel dimers.⁹ At low tempera-
ture T, this interaction favors the sixfold degenerate regular
(crystalline) arrangements of dimers. At high T, the lattice
symmetries are restored (dimer liquid), but dimer-dimer cor-
relations decay algebraically with distance (with the same r⁻³
dependence as the interaction energy of two magnetic
dipoles).⁸ This so-called *Coulomb phase* has also been found
in other 3D models¹⁰ and is relevant to some pyrochlore
compounds (see for instance Ref. 11).

The most intriguing properties of this dimer model con-
cern the phase transition between the low-*T* crystal and the
Coulomb phase. Previous simulations⁹ indicated that the
transition is *continuous* (second order). This is already in
contradiction with the LGW approach, which generically
predicts a *first-order* transition in the present case. The main
question, still open at present, is to understand what are the
fields needed for a description of the long-distance properties
at the critical point, and what is the associated field theory. In
this Rapid Communication, we report on numerical investi-
gations of correlation functions and order parameters in the
vicinity of the critical temperature $T_c$. We show that the di-

polar character of the dimer correlations is gradually lost
when approaching the transition, and find an unexpected
probability distribution of the order parameter, with a pos-
sible emerging *continuous* symmetry. Related to rotations of
a three-component order parameter, this *O*(3) symmetry is
not present at the microscopic level. Terms which make the
symmetry group discrete at the lattice scale (a dimer can only
take six orientations) may be irrelevant in the renormaliza-
tion group (RG) sense at the critical point, but clearly rel-
evant in the ordered phase. This finding is a step toward a
description of the transition in the continuum limit, and es-
tablishes a tight connection with previous models (quantum⁶
and classical⁵) discussed in the context of non-LGW
transitions.

II. DIMER CORRELATIONS: FROM DIPOLAR
TO SPIN-SPIN FORM

In Coulomb phases, it is useful to introduce a magnetic
field⁸ defined on each bond by $B_{ij}=-B_{ji}=1$ if a dimer sits
between sites $i \in A$ and $j \in B$ ($A$ and $B$ are the two sublat-
tices) and $B_{ij}=0$ otherwise. When considered as a vector
field **B**, its lattice divergence satisfies $\nabla \cdot \mathbf{B} = (-)\mathbf{r} = \pm 1$, as
each site is occupied by exactly one dimer. Coarse graining
over a large number of sites, the contributions of both sub-
lattices cancel out to give $\nabla \cdot \mathbf{B} = 0$. In the continuum limit, **B**
is thus written using a vector potential: $\mathbf{B} = \nabla \times \mathbf{A}$. At high $T$
and long distances, this zero-divergence constraint captures
the essential aspects of the dimer hardcoreness.⁸ The Cou-
lomb phase is then described by the simplest action compat-
ible with the gauge invariance $\mathbf{A} \to \mathbf{A} + \nabla \Lambda$ ($\Lambda$ is an arbitrary
function), that is $S = \frac{1}{2} \int d^2\mathbf{r} (\nabla \times \mathbf{A})^2(\mathbf{r})$. Here the rigidity $K^{-1}$
depends on $T$. This effective action leads to dipolar long-
distance correlations:⁸

$$
\langle B^\alpha(0) B^\beta(\mathbf{r}) \rangle^c \sim \frac{1}{K(T)} \frac{3 r^\alpha r^\beta - \delta^{\alpha\beta} r^2}{\mathbf{r}^5}. \quad (1)
$$

$K^{-1}$ can be measured numerically through $LK^{-1} = \langle (\phi^z)^2 \rangle$; the
fluctuations of the total flux $\phi^z = \int dx dy B^z(\mathbf{r}=[x, y, z_0])$ going
through the sample (independent of $z_0$). Reference 9 showed
that $K^{-1}$ vanishes continuously when approaching the transi-
tion at $T_c \approx 1.675$, as magnetic-field fluctuations are costly
Correlations and order parameter at a Coulomb-crystal phase transition
in a three-dimensional dimer model

Grégoire Misguich,¹,* Vincent Pasquier,¹ and Fabien Alet²

¹Institut de Physique Théorique, CEA, IPhT, CNRS, URA 2306 F-91191 Gif-sur-Yvette, France

²Laboratoire de Physique Théorique, IRSAMC, Université Paul Sabatier, CNRS, 31062 Toulouse, France

(Received 11 July 2008; published 3 September 2008)

The three-dimensional classical dimer model with interactions shows an unexpected continuous phase transition between an ordered dimer crystal and a Coulomb liquid. A detailed analysis of the critical dimer and monomer correlation functions points to a subtle interplay between the fluctuations of the crystal order parameter and the “magnetic” degrees of freedom present in the Coulomb phase. The distribution probability of the crystal order parameter suggests an emerging continuous $O(3)$ symmetry at the critical point.

DOI: 10.1103/PhysRevB.78.100402

PACS number(s): 75.10.–b, 64.60.–i, 05.50.+q

I. INTRODUCTION

The Berezinskii-Kosterlitz-Thouless transition¹ is an ex-
ample of critical phenomenon which cannot be understood
from the conventional Landau-Ginzburg-Wilson (LGW)
point of view, based on spontaneously broken symmetries
and critical fluctuations of the associated order parameter.²
Many quantum systems in one-dimension and classical sys-
tems in two-dimension (2D) exhibit such type of continuous
phase transition, which is understood as a proliferation of
topological defects. The recent suggestion³ that some phase
transitions in $D>2$ could also be outside the LGW
framework triggered an important activity, both on the
field-theoretical⁴ and numerical⁵⁻⁷ sides.

The classical dimer model on the cubic lattice<sup>8,9</sup> is one of
the simplest three-dimensional (3D) models with a transition
which does not seem to fit in the LGW picture,<sup>9</sup> although the
field theory of the critical point has not yet been elucidated.
Configurations of the model are hard-core dimer coverings
of a cubic lattice.<sup>8</sup> Each dimer occupies two sites and each
site belongs exclusively to one dimer. In addition, each con-
figuration has an energy equal to −1 times the number of
square plaquettes with two parallel dimers.<sup>9</sup> At low tempera-
ture T, this interaction favors the sixfold degenerate regular
(crystalline) arrangements of dimers. At high T, the lattice
symmetries are restored (dimer liquid), but dimer-dimer cor-
relations decay algebraically with distance (with the same r<sup>-3</sup>
dependence as the interaction energy of two magnetic
dipoles).<sup>8</sup> This so-called *Coulomb phase* has also been found
in other 3D models<sup>10</sup> and is relevant to some pyrochlore
compounds (see for instance Ref. 11).

The most intriguing properties of this dimer model con-
cern the phase transition between the low-*T* crystal and the
Coulomb phase. Previous simulations⁹ indicated that the
transition is *continuous* (second order). This is already in
contradiction with the LGW approach, which generically
predicts a *first-order* transition in the present case. The main
question, still open at present, is to understand what are the
fields needed for a description of the long-distance properties
at the critical point, and what is the associated field theory. In
this Rapid Communication, we report on numerical investi-
gations of correlation functions and order parameters in the
vicinity of the critical temperature *T*<sub>c</sub>. We show that the di-

polar character of the dimer correlations is gradually lost
when approaching the transition, and find an unexpected
probability distribution of the order parameter, with a pos-
sible emerging *continuous* symmetry. Related to rotations of
a three-component order parameter, this *O*(3) symmetry is
not present at the microscopic level. Terms which make the
symmetry group discrete at the lattice scale (a dimer can only
take six orientations) may be irrelevant in the renormaliza-
tion group (RG) sense at the critical point, but clearly rel-
evant in the ordered phase. This finding is a step toward a
description of the transition in the continuum limit, and es-
tablishes a tight connection with previous models (quantum<sup>6</sup>
and classical<sup>5</sup>) discussed in the context of non-LGW
transitions.

II. DIMER CORRELATIONS: FROM DIPOLAR
TO SPIN-SPIN FORM

In Coulomb phases, it is useful to introduce a magnetic
field⁸ defined on each bond by $B_{ij}=-B_{ji}=1$ if a dimer sits
between sites $i \in A$ and $j \in B$ ($A$ and $B$ are the two sublat-
tices) and $B_{ij}=0$ otherwise. When considered as a vector
field **B**, its lattice divergence satisfies $\nabla \cdot \mathbf{B} = (-)\mathbf{r} = \pm 1$, as
each site is occupied by exactly one dimer. Coarse graining
over a large number of sites, the contributions of both sub-
lattices cancel out to give $\nabla \cdot \mathbf{B} = 0$. In the continuum limit, **B**
is thus written using a vector potential: $\mathbf{B} = \nabla \times \mathbf{A}$. At high $T$
and long distances, this zero-divergence constraint captures
the essential aspects of the dimer hardcoreness.⁸ The Cou-
lomb phase is then described by the simplest action compat-
ible with the gauge invariance $\mathbf{A} \to \mathbf{A} + \nabla \Lambda$ ($\Lambda$ is an arbitrary
function), that is $S = \frac{1}{2} \int d^2\mathbf{r} (\nabla \times \mathbf{A})^2(\mathbf{r})$. Here the rigidity $K^{-1}$
depends on $T$. This effective action leads to dipolar long-
distance correlations:⁸

$$
\langle B^\alpha(0) B^\beta(\mathbf{r}) \rangle^c \sim \frac{1}{K(T)} \frac{3r^\alpha r^\beta - \delta^{\alpha\beta} r^2}{\mathbf{r}^5}. \quad (1)
$$

$K^{-1}$ can be measured numerically through $LK^{-1} = \langle (\phi^z)^2 \rangle$; the fluctuations of the total flux $\phi^z = \int dx dy B^z(\mathbf{r}=[x, y, z_0])$ going through the sample (independent of $z_0$). Reference 9 showed that $K^{-1}$ vanishes continuously when approaching the transition at $T_c \approx 1.675$, as magnetic-field fluctuations are costly
MISGUICH, PASQUIER, AND ALET

PHYSICAL REVIEW B 78, 100402(R) (2008)

<figure><img src="image_1.png" /><figcaption>FIG. 1. (Color online) (a) Dimer correlations $\langle d_0^x d_r^y \rangle$ as a function of $|\mathbf{r}|$ and multiplied by $|\mathbf{r}|^3$ for a few selected $T$ above $T_c$ (system size $L=64$). For this orientation, the spin contribution vanishes. (b) Current correlations $\langle J_0^x J_r^y \rangle$ of the ICLM as a function of $|\mathbf{r}|$ and multiplied by $|\mathbf{r}|^3$ at $T=0.5$ (in the Coulomb phase) and $T_c=0.33305$ (system size $L=64$). Correlators at $T_c$ have been multiplied by 50. Simulations of the dimer model (ICLM) were performed with the algorithm of Ref. 12 (Ref. 14).</figcaption></figure>

when approaching the crystal. Although the dipolar form dominates at high $T$, we find that this contribution to the dimer-dimer correlations (Eq. (1)) progressively disappears when approaching $T_c$. In Fig. 1(a), the correlations $\langle d_0^x d_r^y \rangle^c$ for $\mathbf{r}=[i,i,0]$ are multiplied by $|\mathbf{r}|^3$ to highlight the dipolar contribution and its strong reduction when going from $T=4$ to $T=1.68$ (just above $T_c$). This is expected since $K^{-1}$ vanishes when approaching $T_c$. However, the magnetic action does not account for an important qualitative feature. Setting $\alpha=\beta=x$ in Eq. (1), the prefactor $a$ of the dipolar contribution $a/r^3$ depends on the lattice direction (for instance $a=0$ for $\mathbf{r}=[i,i,i]$ and $a=1/2$ for $\mathbf{r}=[i,i,0]$), whereas the actual correlations close to $T_c$ appear to be almost isotropic [see Fig. 2(a)].

Equation (1) misses an important physical ingredient: the crystal-like dimer correlations associated to the order parameter of the symmetry-breaking phase. At each site, we define a three-component vector $\vec{n}$ by $n^\alpha(\mathbf{r}) = \pm(-1)^{r^\alpha}$ if the dimer sitting in $\mathbf{r}$ points in the direction $\pm\alpha$, and 0 otherwise. This ensures that $\vec{n}$ is “ferromagnetically” correlated in the crystal phase ($\langle\vec{n}\rangle = \pm\vec{e}_x, \pm\vec{e}_x, \pm\vec{e}_y$ in the six ground states). Making

<figure><img src="image_2.png" /><figcaption>FIG. 2. (Color online) Dimer correlations $\langle d_0^x d_r^y \rangle^c$. (a) $T \approx T_c$ and just above, for different orientations of $\mathbf{r}$ ($L=128$). (b) A few selected $T$ above $T_c$ ($L=64$). The lattice direction $\mathbf{r}=[i,i,i]$ is such that the dipolar contribution vanishes.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>FIG. 3. (Color online) Monomer correlation as a function of $r/\xi$ in log-log scale for a few $T$ just above $T_c$, assuming $\xi=(T-T_c)^{-1/2}$ and $T_c \approx 1.6743$ ($L=128$).</figcaption></figure>

the assumption that long-distance fluctuations of $\vec{n}$ are described by a three-component “spin” model [i.e., absence of coupling between real space $r^\alpha$ and internal spin $n^i$ indices, as in $O(n)$ models], the angular dependence of the correlations become $\langle n^\alpha(0)n^\beta(\mathbf{r}) \rangle^c \sim \delta^{\alpha\beta}$. It turns out that in a wide temperature range around $T_c$, the connected dimer-dimer correlations $\langle d^\alpha(0)d^\beta(\mathbf{r}) \rangle^c$ are well approximated by a dipolar contribution, plus a “spinlike” contribution:

$$ \frac{(-1)^r 3r^\alpha r^\beta - \delta^{\alpha\beta} \mathbf{r}^2}{K} + (-1)^{r^\alpha} g(|\mathbf{r}|) \delta^{\alpha\beta}. \quad (2) $$

Taking $\alpha=x, \beta=y$, and $\mathbf{r}=[i,i,0]$ [Fig. 1(a)], the “spin” term vanishes and only the dipolar part remains, whereas for $\alpha=\beta=x$ and $\mathbf{r}=[i,i,i]$ the dipolar part vanishes and one is left with $g(r)$. The latter is displayed in Fig. 2(b), where strong and slowly decaying correlations are found close to $T_c$. At $T_c$, $g(r)$ is very likely algebraic and compatible with $\sim 1/r$, although it is difficult to give an error bar on the exponent.

A more quantitative analysis can be done on the monomer correlator $Z(r,T)$, defined as the ratio of the partition function with two fixed test monomers at distance $r$, by the bare partition function. In the Coulomb phase, $Z(r,T)$ converges to a finite value when $r \to \infty$.⁹ As shown in Fig. 3, $Z(r,T)/Z(\infty,T)$ is well fitted by a function $F(x)$ of the single parameter $x=r\sqrt{T-T_c}$, suggesting the existence of a length $\xi(T) = (T-T_c)^{-1/2}$, diverging at $T_c$ with an exponent $\nu$ compatible with 0.5. By construction $F(x) \to 1$ as $x \to 1$, indicating that for distances $r \lesssim \xi(T)$, $Z(r,T)$ decays in a way consistent with $\sim 1/r$. At distances large compared to the lattice spacing, but small compared to some (diverging) correlation length $\xi(T)$, the correlations are thus qualitatively not of Coulomb type, although $T > T_c$. Below $\xi$, the correlations (of both dimers and monomers) are compatible with an exponent $\eta$ close to zero. Noteworthy, these values are those obtained from independent thermodynamic measurements.⁹

It is instructive to compare these results with the correlations of a simpler model: the integer current loop model (ICLM).¹³,¹⁴ There, the allowed configurations are integer-valued currents $J_{ij} = -J_{ji} \in \mathbb{Z}$ defined on each bond $(ij)$ of the cubic lattice (analogous to the dimer magnetic field $\mathbf{B}$), with the constraint that the net current arriving at each site is zero: $\text{div} \vec{J} = 0$. The energy of each configuration is $E = \frac{1}{2} \sum_{(i,j)} (J_{ij})^2$. This model also has a Coulomb phase at high $T$, with large
MISGUICH, PASQUIER, AND ALET

PHYSICAL REVIEW B 78, 100402(R) (2008)

<figure><img src="image_2.png" /><figcaption>FIG. 1. (Color online) (a) Dimer correlations $\langle d_0^x d_r^y \rangle$ as a function of $|\mathbf{r}|$ and multiplied by $|\mathbf{r}|^3$ for a few selected $T$ above $T_c$ (system size $L=64$). For this orientation, the spin contribution vanishes. (b) Current correlations $\langle J_0^x J_r^y \rangle$ of the ICLM as a function of $|\mathbf{r}|$ and multiplied by $|\mathbf{r}|^3$ at $T=0.5$ (in the Coulomb phase) and $T_c=0.33305$ (system size $L=64$). Correlators at $T_c$ have been multiplied by 50. Simulations of the dimer model (ICLM) were performed with the algorithm of Ref. 12 (Ref. 14).</figcaption></figure>

when approaching the crystal. Although the dipolar form dominates at high $T$, we find that this contribution to the dimer-dimer correlations (Eq. (1)) progressively disappears when approaching $T_c$. In Fig. 1(a), the correlations $\langle d_0^x d_r^y \rangle^c$ for $\mathbf{r}=[i,i,0]$ are multiplied by $|\mathbf{r}|^3$ to highlight the dipolar contribution and its strong reduction when going from $T=4$ to $T=1.68$ (just above $T_c$). This is expected since $K^{-1}$ vanishes when approaching $T_c$. However, the magnetic action does not account for an important qualitative feature. Setting $\alpha=\beta=x$ in Eq. (1), the prefactor $a$ of the dipolar contribution $a/r^3$ depends on the lattice direction (for instance $a=0$ for $\mathbf{r}=[i,i,i]$ and $a=1/2$ for $\mathbf{r}=[i,i,0]$), whereas the actual correlations close to $T_c$ appear to be almost isotropic [see Fig. 2(a)].

Equation (1) misses an important physical ingredient: the crystal-like dimer correlations associated to the order parameter of the symmetry-breaking phase. At each site, we define a three-component vector $\vec{n}$ by $n^\alpha(\mathbf{r}) = \pm(-1)^{r^\alpha}$ if the dimer sitting in $\mathbf{r}$ points in the direction $\pm\alpha$, and 0 otherwise. This ensures that $\vec{n}$ is “ferromagnetically” correlated in the crystal phase ($\langle\vec{n}\rangle = \pm\vec{e}_x, \pm\vec{e}_x, \pm\vec{e}_y$ in the six ground states). Making

<figure><img src="image_3.png" /><figcaption>FIG. 2. (Color online) Dimer correlations $\langle d_0^x d_r^y \rangle^c$. (a) $T \approx T_c$ and just above, for different orientations of $\mathbf{r}$ ($L=128$). (b) A few selected $T$ above $T_c$ ($L=64$). The lattice direction $\mathbf{r}=[i,i,i]$ is such that the dipolar contribution vanishes.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIG. 3. (Color online) Monomer correlation as a function of $r/\xi$ in log-log scale for a few $T$ just above $T_c$, assuming $\xi=(T-T_c)^{-1/2}$ and $T_c \approx 1.6743$ ($L=128$).</figcaption></figure>

the assumption that long-distance fluctuations of $\vec{n}$ are described by a three-component “spin” model [i.e., absence of coupling between real space $r^\alpha$ and internal spin $n^i$ indices, as in $O(n)$ models], the angular dependence of the correlations become $\langle n^\alpha(0)n^\beta(\mathbf{r}) \rangle^c \sim \delta^{\alpha\beta}$. It turns out that in a wide temperature range around $T_c$, the connected dimer-dimer correlations $\langle d^\alpha(0)d^\beta(\mathbf{r}) \rangle^c$ are well approximated by a dipolar contribution, plus a “spinlike” contribution:

$$ \frac{(-1)^r 3r^\alpha r^\beta - \delta^{\alpha\beta} \mathbf{r}^2}{K} + (-1)^{r^\alpha} g(|\mathbf{r}|) \delta^{\alpha\beta}. \quad (2) $$

Taking $\alpha=x, \beta=y$, and $\mathbf{r}=[i,i,0]$ [Fig. 1(a)], the “spin” term vanishes and only the dipolar part remains, whereas for $\alpha=\beta=x$ and $\mathbf{r}=[i,i,i]$ the dipolar part vanishes and one is left with $g(r)$. The latter is displayed in Fig. 2(b), where strong and slowly decaying correlations are found close to $T_c$. At $T_c$, $g(r)$ is very likely algebraic and compatible with $\sim 1/r$, although it is difficult to give an error bar on the exponent.

A more quantitative analysis can be done on the monomer correlator $Z(r,T)$, defined as the ratio of the partition function with two fixed test monomers at distance $r$, by the bare partition function. In the Coulomb phase, $Z(r,T)$ converges to a finite value when $r \to \infty$.⁹ As shown in Fig. 3, $Z(r,T)/Z(\infty,T)$ is well fitted by a function $F(x)$ of the single parameter $x=r\sqrt{T-T_c}$, suggesting the existence of a length $\xi(T) = (T-T_c)^{-1/2}$, diverging at $T_c$ with an exponent $\nu$ compatible with 0.5. By construction $F(x) \to 1$ as $x \to 1$, indicating that for distances $r \lesssim \xi(T)$, $Z(r,T)$ decays in a way consistent with $\sim 1/r$. At distances large compared to the lattice spacing, but small compared to some (diverging) correlation length $\xi(T)$, the correlations are thus qualitatively not of Coulomb type, although $T > T_c$. Below $\xi$, the correlations (of both dimers and monomers) are compatible with an exponent $\eta$ close to zero. Noteworthy, these values are those obtained from independent thermodynamic measurements.⁹

It is instructive to compare these results with the correlations of a simpler model: the integer current loop model (ICLM).¹³,¹⁴ There, the allowed configurations are integer-valued currents $J_{ij} = -J_{ji} \in \mathbb{Z}$ defined on each bond $(ij)$ of the cubic lattice (analogous to the dimer magnetic field $\mathbf{B}$), with the constraint that the net current arriving at each site is zero: $\text{div} \vec{J} = 0$. The energy of each configuration is $E = \frac{1}{2} \sum_{(i,j)} (J_{ij})^2$. This model also has a Coulomb phase at high $T$, with large
current fluctuations and dipolar $\langle J^a J^b \rangle$ correlations. At low $T$, small current loops dominate and eventually $J_{ij}=0$ everywhere at $T=0$ (a state which does not have a dimer analog). Through a standard duality transformation,¹⁵ the ICLM maps exactly onto an $O(2)$ spin model with Villain interaction.¹⁶ In contrast to the dimer model, the phase transition of the ICLM $[T_c^{ICLM}=0.333\,05$ (Ref. 14)] is known to belong to the 3D-XY universality class. The Coulomb phase of the ICLM corresponds to the ordered phase in the angular variables $\theta_r$, and the dipolar correlations are a direct consequence of the Goldstone mode for these dual spins. At high $T$, this dictionary works also well for the dimer model: $K^{-1}$ is the stiffness of a dual $O(2)$ ferromagnet. The monomer correlation $Z(r)$ corresponds to the spin-spin correlation $\langle \exp(i\theta_0 - i\theta_r) \rangle$, and $Z(\infty)$ (finite in the Coulomb phase and vanishing at $T_c$) to the square of the order parameter (magnetization) $|\langle e^{i\theta_0} \rangle|^2$. The crucial difference with the dimer model is the absence of spontaneously broken symmetry in the low-$T$ phase of the ICLM, and thus the absence of an order parameter similar to $\vec{n}$. This does not only make the critical exponents of the dimer model different from that of 3D XY,⁹ but marks a qualitative difference concerning the correlation in the Coulomb phase close to $T_c$ [for $r \le \xi(T)$]. The current correlations of the ICLM remain dipolar down to $T_c$. Both in the Coulomb phase and at $T_c$, they decay as $\sim 1/r^3$ with a geometric prefactor which does depend on the direction of $\mathbf{r}$ [see the large-$|\mathbf{r}|$ behavior in Fig. 1(b)]. The ICLM obeys Eq. (1) without the need for an additional spinlike contribution. Only the overall amplitude is reduced close to $T_c$, with $1/K(T_c) \sim 1/L$ [see renormalization factor in Fig. 1(b)]. The fact that dimer correlations are dominated by nondipolar contributions close to $T_c$ gives additional support for this transition not being of 3D XY nature, as naively expected from the high-$T$ side.

### III. SYMMETRY IN THE CRYSTAL ORDER PARAMETER AT $T_c$

The results above point to the importance of fluctuations of the crystal order parameter $\vec{n}$ at the transition. We therefore looked at the distribution probability of $\vec{N}=L^{-3}\sum_{\mathbf{r}}\vec{n}(\mathbf{r})$ close to $T_c$. For large systems, the probability distribution of $(N^x, N^y, N^z=0)$ is almost circular [see the distribution for $L=128$ and $T=1.673$ in Fig. 4(b)]. Some small squarelike anisotropy is still detectable but we argue below that it could be a finite-size effect and the asymptotic form of the distribution spherically symmetric in the thermodynamic limit. To quantify this, we define

$$C_4 = \frac{1}{2} \langle \cos(4 \arg[N^x + iN^y]) + \cos(4 \arg[N^y + iN^z]) \\ + \cos(4 \arg[N^z + iN^x]) \rangle, \quad (3)$$

which measures the $O(3)$ character of the angular distribution of $\vec{N}$. If $\vec{N}$ is distributed isotropically, $C_4$ vanishes in the thermodynamic limit. If $\vec{N}$ preferentially points along the axis directions, one expects $C_4>0$. In the crystal phase $C_4 \to 1$ when $L \to \infty$, signaling that the direction of the order

<figure><img src="image_4.png" /><figcaption>FIG. 4. (Color online) (a) Spherical character of the order parameter $\vec{N}$ distribution ($C_4$) as a function of $1/L$ for different $T$. The data for $T=1.6$ have been divided by 2. (b) Probability distribution $P(N^x, N^y, |N^z| \le 0.03)$ as a function of $N^x$ and $N^y$ for several temperature and system sizes. The range is $N^{x,y} \in [-0.3, 0.3]$ for $L<128$ and $[-0.2, 0.2]$ for $L=128$.</figcaption></figure>

parameter $\vec{N}$ is locked along one crystal axis [see the $C_4$ ($T=1.66$) and the associated distributions in Figs. 4(a) and 4(b)]. In the Coulomb phase, $\vec{N}$ is typically very small ($\langle \vec{N}^2 \rangle \sim L^{-3}$) and distributed in a Gaussian way, thus with spherical symmetry. The nontrivial result is the behavior of $C_4$ at $T_c$. Figure 4(a) suggests that $C_4(T=T_c, L) \to 0$ when $L \to \infty$, and thus indicate an $O(3)$ symmetric distribution of the order parameter.¹⁷

For $T$ slightly below $T_c$, Fig. 4(a) reveals that $C_4(T, L)$ has a minimum for a system size $L_0(T)$. This minimum shifts to larger size when approaching $T_c$ from below. $L_0(T<T_c)$ thus defines a length scale, diverging at $T_c$, up to which the discrete symmetry breaking becomes weaker at long distances.

This is similar to the 3D XY model with $Z_{q \ge 4}$ anisotropy,¹⁵,¹⁸ where the $q$-fold anisotropy is dangerously irrelevant and makes the (two-component) order parameter apparently $U(1)$ symmetric in the ordered phase, up to some length scale diverging at $T_c$. A crucial difference here is that the order parameter $\vec{n}$ has three components, and a RG analysis of the $O(3)$ model predicts that cubic anisotropies, such as $(n^x)^4+(n^y)^4+(n^z)^4$, should be relevant.¹⁹ The latter anisotropic $O(3)$ model is however qualitatively different from the model studied here, as its high-$T$ phase is a featureless paramagnet. From this point of view, the classical spin model of Ref. 5 (where a constraint equivalent to $\text{div } \mathbf{B}=0$ is enforced on spin configurations) may be closer to our study since it also possesses a high-$T$ Coulomb phase. Discrete anisotropies seeming irrelevant for the dimer critical point, both models would be predicted from the symmetry point of view to share the same universality class. However, based on the values of the critical exponents, this is not the case.

### IV. DISCUSSION

The numerical observations of this Rapid Communication emphasize the unconventional nature of the Coulomb-crystal
current fluctuations and dipolar $\langle J^a J^b \rangle$ correlations. At low $T$, small current loops dominate and eventually $J_{ij}=0$ everywhere at $T=0$ (a state which does not have a dimer analog). Through a standard duality transformation,¹⁵ the ICLM maps exactly onto an $O(2)$ spin model with Villain interaction.¹⁶ In contrast to the dimer model, the phase transition of the ICLM $[T_c^{ICLM}=0.333\,05$ (Ref. 14)] is known to belong to the 3D-XY universality class. The Coulomb phase of the ICLM corresponds to the ordered phase in the angular variables $\theta_r$, and the dipolar correlations are a direct consequence of the Goldstone mode for these dual spins. At high $T$, this dictionary works also well for the dimer model: $K^{-1}$ is the stiffness of a dual $O(2)$ ferromagnet. The monomer correlation $Z(r)$ corresponds to the spin-spin correlation $\langle \exp(i\theta_0 - i\theta_r) \rangle$, and $Z(\infty)$ (finite in the Coulomb phase and vanishing at $T_c$) to the square of the order parameter (magnetization) $|\langle e^{i\theta_0} \rangle|^2$. The crucial difference with the dimer model is the absence of spontaneously broken symmetry in the low-$T$ phase of the ICLM, and thus the absence of an order parameter similar to $\vec{n}$. This does not only make the critical exponents of the dimer model different from that of 3D XY,⁹ but marks a qualitative difference concerning the correlation in the Coulomb phase close to $T_c$ [for $r \le \xi(T)$]. The current correlations of the ICLM remain dipolar down to $T_c$. Both in the Coulomb phase and at $T_c$, they decay as $\sim 1/r^3$ with a geometric prefactor which does depend on the direction of $\mathbf{r}$ [see the large-$|\mathbf{r}|$ behavior in Fig. 1(b)]. The ICLM obeys Eq. (1) without the need for an additional spinlike contribution. Only the overall amplitude is reduced close to $T_c$, with $1/K(T_c) \sim 1/L$ [see renormalization factor in Fig. 1(b)]. The fact that dimer correlations are dominated by nondipolar contributions close to $T_c$ gives additional support for this transition not being of 3D XY nature, as naively expected from the high-$T$ side.

### III. SYMMETRY IN THE CRYSTAL ORDER PARAMETER AT $T_c$

The results above point to the importance of fluctuations of the crystal order parameter $\vec{n}$ at the transition. We therefore looked at the distribution probability of $\vec{N}=L^{-3}\sum_{\mathbf{r}}\vec{n}(\mathbf{r})$ close to $T_c$. For large systems, the probability distribution of $(N^x, N^y, N^z=0)$ is almost circular [see the distribution for $L=128$ and $T=1.673$ in Fig. 4(b)]. Some small squarelike anisotropy is still detectable but we argue below that it could be a finite-size effect and the asymptotic form of the distribution spherically symmetric in the thermodynamic limit. To quantify this, we define

$$C_4 = \frac{1}{2} \langle \cos(4 \arg[N^x + iN^y]) + \cos(4 \arg[N^y + iN^z]) \\ + \cos(4 \arg[N^z + iN^x]) \rangle, \quad (3)$$

which measures the $O(3)$ character of the angular distribution of $\vec{N}$. If $\vec{N}$ is distributed isotropically, $C_4$ vanishes in the thermodynamic limit. If $\vec{N}$ preferentially points along the axis directions, one expects $C_4>0$. In the crystal phase $C_4 \to 1$ when $L \to \infty$, signaling that the direction of the order

<figure><img src="image_5.png" /><figcaption>FIG. 4. (Color online) (a) Spherical character of the order parameter $\vec{N}$ distribution ($C_4$) as a function of $1/L$ for different $T$. The data for $T=1.6$ have been divided by 2. (b) Probability distribution $P(N^x, N^y, |N^z| \le 0.03)$ as a function of $N^x$ and $N^y$ for several temperature and system sizes. The range is $N^{x,y} \in [-0.3, 0.3]$ for $L<128$ and $[-0.2, 0.2]$ for $L=128$.</figcaption></figure>

parameter $\vec{N}$ is locked along one crystal axis [see the $C_4$ ($T=1.66$) and the associated distributions in Figs. 4(a) and 4(b)]. In the Coulomb phase, $\vec{N}$ is typically very small ($\langle \vec{N}^2 \rangle \sim L^{-3}$) and distributed in a Gaussian way, thus with spherical symmetry. The nontrivial result is the behavior of $C_4$ at $T_c$. Figure 4(a) suggests that $C_4(T=T_c, L) \to 0$ when $L \to \infty$, and thus indicate an $O(3)$ symmetric distribution of the order parameter.¹⁷

For $T$ slightly below $T_c$, Fig. 4(a) reveals that $C_4(T, L)$ has a minimum for a system size $L_0(T)$. This minimum shifts to larger size when approaching $T_c$ from below. $L_0(T<T_c)$ thus defines a length scale, diverging at $T_c$, up to which the discrete symmetry breaking becomes weaker at long distances.

This is similar to the 3D XY model with $Z_{q \ge 4}$ anisotropy,¹⁵,¹⁸ where the $q$-fold anisotropy is dangerously irrelevant and makes the (two-component) order parameter apparently $U(1)$ symmetric in the ordered phase, up to some length scale diverging at $T_c$. A crucial difference here is that the order parameter $\vec{n}$ has three components, and a RG analysis of the $O(3)$ model predicts that cubic anisotropies, such as $(n^x)^4+(n^y)^4+(n^z)^4$, should be relevant.¹⁹ The latter anisotropic $O(3)$ model is however qualitatively different from the model studied here, as its high-$T$ phase is a featureless paramagnet. From this point of view, the classical spin model of Ref. 5 (where a constraint equivalent to $\text{div } \mathbf{B}=0$ is enforced on spin configurations) may be closer to our study since it also possesses a high-$T$ Coulomb phase. Discrete anisotropies seeming irrelevant for the dimer critical point, both models would be predicted from the symmetry point of view to share the same universality class. However, based on the values of the critical exponents, this is not the case.

### IV. DISCUSSION

The numerical observations of this Rapid Communication emphasize the unconventional nature of the Coulomb-crystal
phase transition. On one side, the presence of a diverging length scale below which correlations are not Coulomb-like cannot be interpreted by the sole high-T magnetic point of view. From the ordered side, we exhibited a length scale below which the crystal order parameter seems to fluctuate in an unexpected O(3) symmetric way. Besides the difficulty to build a LGW theory from the corresponding order parameters,⁹ those are clear qualitative indications that individual (magnetic or crystalline) descriptions cannot capture the physics involved at the transition. This points to the existence of a critical regime where the interaction between the magnetic degrees of freedom at play in the Coulomb phase and the spin degrees of freedom which order in the crystal dominates the long-distance physics. This phenomenology is similar to that of a non-compact $CP^1$ (NCCP1) model,³,⁵ which precisely couples spin and gauge degrees of freedom. The NCCP1 theory is characterized by two continuous symmetries: $SU(2)$ for spin rotations and $U(1)$ associated to the magnetic-flux conservation. In our case, the flux conservation is implemented at the lattice level and we argued that

continuous spin rotations emerge at the critical point. This is similar to the 2D quantum spin system studied in Ref. 6, but there the spin rotation symmetry is explicit and a gauge flux conservation is argued to emerge dynamically at the critical point. Despite all these common ingredients between NCCP1 theories and our dimer model, there is a clear mismatch between our critical exponents and those attributed so far to NCCP1 critical points.⁵,⁶ So, the actual field theory describing the dimer model transition remains to be uncovered. Besides its theoretical importance, the issue of the universality class of this transition will also become experimentally relevant if a Coulomb-crystal phase transition was observed in nature, as in a frustrated (icelike) magnet for instance.

## ACKNOWLEDGMENTS

G.M. acknowledges K. S. Kim for useful discussions. Calculations were performed at CCRT/CEA (project 575) using the ALPS libraries.²⁰

*gregoire.misguich@cea.fr

¹V. L. Berezinskii, Sov. Phys. JETP 32, 493 (1971); J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973); J. M. Kosterlitz, ibid. 7, 1046 (1974).

²See, e.g., L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd ed. (Pergamon, New York, 1980).

³T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004); T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, Phys. Rev. B 70, 144407 (2004).

⁴D. L. Bergman, G. A. Fiete, and L. Balents, Phys. Rev. B 73, 134402 (2006); O. I. Motrunich and T. Senthil, ibid. 71, 125102 (2005); P. Ghaemi and T. Senthil, ibid. 73, 054415 (2006); K. Gregor and O. I. Motrunich, ibid. 76, 174404 (2007); T. Grover and T. Senthil, Phys. Rev. Lett. 98, 247202 (2007); O. I. Motrunich and A. Vishwanath, arXiv:0805.1494 (unpublished).

⁵O. I. Motrunich and A. Vishwanath, Phys. Rev. B 70, 075104 (2004).

⁶A. W. Sandvik, Phys. Rev. Lett. 98, 227202 (2007); R. G. Melko and R. K. Kaul, ibid. 100, 017203 (2008).

⁷A. Kuklov, N. Prokof'ev, and B. Svistunov, Phys. Rev. Lett. 93, 230402 (2004).

⁸D. A. Huse, W. Krauth, R. Moessner, and S. L. Sondhi, Phys. Rev. Lett. 91, 167004 (2003).

⁹F. Alet, G. Misguich, V. Pasquier, R. Moessner, and J. L. Jacobsen, Phys. Rev. Lett. 97, 030403 (2006).

¹⁰J. Villain, Solid State Commun. 10, 967 (1972); S. V. Isakov, K. Gregor, R. Moessner, and S. L. Sondhi, Phys. Rev. Lett. 93, 167204 (2004); A. Banerjee, S. V. Isakov, K. Damle, and Y. B. Kim, ibid. 100, 047208 (2008); T. S. Pickles, T. E. Saunders, and J. T. Chalker, arXiv:0708.3791 (unpublished).

¹¹S. T. Bramwell, M. J. Harris, B. C. den Hertog, M. J. P. Gingras, J. S. Gardner, D. F. McMorrow, A. R. Wildes, A. Cornelius, J. D. M. Champion, R. G. Melko, and T. Fennell, Phys. Rev. Lett. 87, 047205 (2001); C. L. Henley, Phys. Rev. B 71, 014424 (2005); S. V. Isakov, R. Moessner, and S. L. Sondhi, Phys. Rev. Lett. 95, 217201 (2005).

¹²A. W. Sandvik and R. Moessner, Phys. Rev. B 73, 144504 (2006).

¹³M. Wallin, E. S. Sørensen, S. M. Girvin, and A. P. Young, Phys. Rev. B 49, 12115 (1994).

¹⁴F. Alet and E. S. Sørensen, Phys. Rev. E 67, 015701(R) (2003).

¹⁵J. V. José, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson, Phys. Rev. B 16, 1217 (1977).

¹⁶J. Villain, J. Phys. (Paris) 36, 581 (1975).

¹⁷Since $T=1.675$ may be slightly below the actual $T_c$, $C_4(T=1.675)$ may start to increase for $L \ge 128$.

¹⁸J. Lou, A. W. Sandvik, and L. Balents, Phys. Rev. Lett. 99, 207203 (2007).

¹⁹J. M. Carmona, A. Pelissetto, and E. Vicari, Phys. Rev. B 61, 15136 (2000).

²⁰A. F. Albuquerque et al., J. Magn. Magn. Mater. 310, 1187 (2007); M. Troyer, B. Ammon, and E. Heeb, Lect. Notes Comput. Sci. 1505, 191 (1998); See http://alps.comp-phys.org
MISGUICH, PASQUIER, AND ALET

PHYSICAL REVIEW B 78, 100402(R) (2008)

phase transition. On one side, the presence of a diverging length scale below which correlations are not Coulomb-like cannot be interpreted by the sole high-T magnetic point of view. From the ordered side, we exhibited a length scale below which the crystal order parameter seems to fluctuate in an unexpected O(3) symmetric way. Besides the difficulty to build a LGW theory from the corresponding order parameters,⁹ those are clear qualitative indications that individual (magnetic or crystalline) descriptions cannot capture the physics involved at the transition. This points to the existence of a critical regime where the interaction between the magnetic degrees of freedom at play in the Coulomb phase and the spin degrees of freedom which order in the crystal dominates the long-distance physics. This phenomenology is similar to that of a non-compact $CP^1$ (NCCP1) model,³,⁵ which precisely couples spin and gauge degrees of freedom. The NCCP1 theory is characterized by two continuous symmetries: $SU(2)$ for spin rotations and $U(1)$ associated to the magnetic-flux conservation. In our case, the flux conservation is implemented at the lattice level and we argued that

continuous spin rotations emerge at the critical point. This is similar to the 2D quantum spin system studied in Ref. 6, but there the spin rotation symmetry is explicit and a gauge flux conservation is argued to emerge dynamically at the critical point. Despite all these common ingredients between NCCP1 theories and our dimer model, there is a clear mismatch between our critical exponents and those attributed so far to NCCP1 critical points.⁵,⁶ So, the actual field theory describing the dimer model transition remains to be uncovered. Besides its theoretical importance, the issue of the universality class of this transition will also become experimentally relevant if a Coulomb-crystal phase transition was observed in nature, as in a frustrated (icelike) magnet for instance.

## ACKNOWLEDGMENTS

G.M. acknowledges K. S. Kim for useful discussions. Calculations were performed at CCRT/CEA (project 575) using the ALPS libraries.²⁰

*gregoire.misguich@cea.fr

¹V. L. Berezinskii, Sov. Phys. JETP 32, 493 (1971); J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973); J. M. Kosterlitz, ibid. 7, 1046 (1974).

²See, e.g., L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd ed. (Pergamon, New York, 1980).

³T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004); T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, Phys. Rev. B 70, 144407 (2004).

⁴D. L. Bergman, G. A. Fiete, and L. Balents, Phys. Rev. B 73, 134402 (2006); O. I. Motrunich and T. Senthil, ibid. 71, 125102 (2005); P. Ghaemi and T. Senthil, ibid. 73, 054415 (2006); K. Gregor and O. I. Motrunich, ibid. 76, 174404 (2007); T. Grover and T. Senthil, Phys. Rev. Lett. 98, 247202 (2007); O. I. Motrunich and A. Vishwanath, arXiv:0805.1494 (unpublished).

⁵O. I. Motrunich and A. Vishwanath, Phys. Rev. B 70, 075104 (2004).

⁶A. W. Sandvik, Phys. Rev. Lett. 98, 227202 (2007); R. G. Melko and R. K. Kaul, ibid. 100, 017203 (2008).

⁷A. Kuklov, N. Prokof'ev, and B. Svistunov, Phys. Rev. Lett. 93, 230402 (2004).

⁸D. A. Huse, W. Krauth, R. Moessner, and S. L. Sondhi, Phys. Rev. Lett. 91, 167004 (2003).

⁹F. Alet, G. Misguich, V. Pasquier, R. Moessner, and J. L. Jacobsen, Phys. Rev. Lett. 97, 030403 (2006).

¹⁰J. Villain, Solid State Commun. 10, 967 (1972); S. V. Isakov, K. Gregor, R. Moessner, and S. L. Sondhi, Phys. Rev. Lett. 93, 167204 (2004); A. Banerjee, S. V. Isakov, K. Damle, and Y. B. Kim, ibid. 100, 047208 (2008); T. S. Pickles, T. E. Saunders, and J. T. Chalker, arXiv:0708.3791 (unpublished).

¹¹S. T. Bramwell, M. J. Harris, B. C. den Hertog, M. J. P. Gingras, J. S. Gardner, D. F. McMorrow, A. R. Wildes, A. Cornelius, J. D. M. Champion, R. G. Melko, and T. Fennell, Phys. Rev. Lett. 87, 047205 (2001); C. L. Henley, Phys. Rev. B 71, 014424 (2005); S. V. Isakov, R. Moessner, and S. L. Sondhi, Phys. Rev. Lett. 95, 217201 (2005).

¹²A. W. Sandvik and R. Moessner, Phys. Rev. B 73, 144504 (2006).

¹³M. Wallin, E. S. Sørensen, S. M. Girvin, and A. P. Young, Phys. Rev. B 49, 12115 (1994).

¹⁴F. Alet and E. S. Sørensen, Phys. Rev. E 67, 015701(R) (2003).

¹⁵J. V. José, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson, Phys. Rev. B 16, 1217 (1977).

¹⁶J. Villain, J. Phys. (Paris) 36, 581 (1975).

¹⁷Since $T=1.675$ may be slightly below the actual $T_c$, $C_4(T=1.675)$ may start to increase for $L \ge 128$.

¹⁸J. Lou, A. W. Sandvik, and L. Balents, Phys. Rev. Lett. 99, 207203 (2007).

¹⁹J. M. Carmona, A. Pelissetto, and E. Vicari, Phys. Rev. B 61, 15136 (2000).

²⁰A. F. Albuquerque et al., J. Magn. Magn. Mater. 310, 1187 (2007); M. Troyer, B. Ammon, and E. Heeb, Lect. Notes Comput. Sci. 1505, 191 (1998); See http://alps.comp-phys.org