New Universality Class in three dimensions: the Antiferromagnetic $RP^2$ model

TL;DR

This paper reports Monte Carlo simulations of the 3D $RP^2$ model with negative coupling, revealing a second order phase transition and identifying critical exponents, suggesting a new universality class.

Contribution

The study introduces a new universality class in three dimensions through detailed simulation and critical exponent analysis of the $RP^2$ model with negative coupling.

Findings

Second order phase transition observed

Critical exponent $ u=0.784(8)$ measured

Two magnetic-type relevant operators identified

Abstract

We present the results of a Monte Carlo simulation of the $RP^2$ model in three dimensions with negative coupling. We observe a second order phase transition between the disordered phase and an antiferromagnetic, unfrustrated, ordered one. We measure, with a Finite Size Scaling analysis, the thermal exponent, obtaining $\nu=0.784(8)$. We have found two magnetic-type relevant operators whose related $\eta$ exponents are $0.038(2)$ and $1.338(8)$ respectively.

Tables

Table 1. Table 1: Number of Monte Carlo sweeps performed for different lattice sizes. Measures have been taken every 10 sweeps. The integrated autocorrelation times (in sweeps) for both magnetizations and for the energy are also displayed. The statistical errors are below the 5% level. We have discarded in each case about 200 ​ τ χ s 200 subscript 𝜏 subscript 𝜒 𝑠 200\tau_{\chi_{s}} iterations for thermalization.

$L$	MC sweeps($\times {10}^6$)	${\tau }_{{\chi }_s}$	${\tau }_{\chi }$	${\tau }_E$
6	06.71	7.4	5.8	0.60
8	17.07	11.4	7.4	0.73
12	06.51	24.8	12.8	1.02
16	22.14	44.1	21.4	1.30
24	08.77	107.	48.	1.82
32	10.13	179.	87.	2.27
48	03.93	430.	205.	3.10

Table 2. Table 2: Critical exponents obtained from a Finite Size Scaling analysis using data from lattices of sizes L 𝐿 L and 2 ​ L 2 𝐿 2L . In the second row we show the operator used for each column.

	$\nu$		${\eta }_s$		$\eta$
$L$	$d{\xi }_s/d\beta$	$d\xi /d\beta$	${\chi }_s$	$M_s$	$\chi$	$M$
6	0.786(6)	0.790(6)	0.0431(10)	0.0474(9)	1.442(2)	1.447(2)
8	0.785(4)	0.781(4)	0.0375(7)	0.0409(8)	1.413(2)	1.416(2)
12	0.789(8)	0.782(9)	0.0357(17)	0.0382(18)	1.391(3)	1.393(3)
16	0.787(9)	0.781(8)	0.0371(19)	0.0390(19)	1.379(4)	1.381(4)
24	0.77(2)	0.77(2)	0.038(5)	0.038(5)	1.362(8)	1.365(9)

Equations

$$S=\beta\sum_{<ij>}({\mbox{\boldmath$v$}}_{i}\cdot\mbox{\boldmath$v$}_{j})^{2},$$

$${\mathrm{T}}_i^{\alpha \beta }=v_i^{\alpha }{v}_i^{\beta }-\frac{1}{3}{\delta }^{\alpha \beta },$$

$${\mathbf{M}}_s=\frac{1}{V}{\hat{\mathbf{T}}}_{(\pi ,\pi ,\pi )},(\mathbf{M}=\frac{1}{V}{\hat{\mathbf{T}}}_{(0,0,0)}),$$

$$M=⟨\sqrt{\mathrm{tr}{\mathbf{M}}^2}⟩,\chi =V⟨\mathrm{tr}{\mathbf{M}}^2⟩,$$

$$\xi ={\left(\frac{\chi /F-1}{4{\sin }^2(\pi /L)}\right)}^{1/2},$$

$${\beta }_c\in [-2.415,-2,405].$$

$$\begin{array}[]{lll}\xi_{s}&:&\quad\nu=0.788(7)\\ \xi&:&\quad\nu=0.779(6)\ .\end{array}$$

$$⟨O(L,\beta )⟩=L^x{f}_O(\xi (L,\beta )/L)+\dots ,$$

$$Q_O=2^x\frac{f_O\left(\frac{\xi (2L,\beta )}{2L}\right)}{f_O\left(\frac{\xi (L,\beta )}{L}\right)}+\dots .$$

$$\nu =0.784(8).$$

$${\eta }_s=0.038(2).$$

$$\eta =1.338(8),$$

Full text

The theory of Critical Phenomena offers a common framework to study problems in Condensed Matter Physics (CMP) and in High Energy Physics (HEP). In both areas, the concepts of Spontaneous Symmetry Breaking (SSB) and of Universality allow to relate problems in principle very different.

The usual Heisenberg model, associated with the standard ferromagnetic Non Linear $\sigma$ Model (NL$\sigma$M), has a SSB pattern of type $SO(3)/SO(2)$. With the introduction of nontrivial AntiFerromagnetic (AF) interactions the SSB pattern normally changes completely and, usually, frustration is generated. In particular, a SSB pattern shared by several AF models is $SO(3)\times SO(2)/SO(2)$. For instance, some frustrated quantum AF Heisenberg models [1], or the Helimagnets and Canted spin systems [2, 3] are examples of this behavior. Frustrated quantum spin models are specially interesting because of their possible relation with High Temperature Superconductivity [4].

As a general consequence of the Weinberg Theorem the low energy physics of a system is completely determined by its SSB pattern, the effective Lagrangian for the system being the corresponding NL$\sigma$M. In this framework a study has been carried out for $SO(3)\times SO(2)/SO(2)$ in perturbation theory [3], where the main conclusion reached is that the only possible nontrivial critical point in three dimensions is that of $O(4)$. In spite of that, we have found in a nonperturbative lattice formulation of the same model, a critical point with exponents clearly different from those of $O(4)$.

From the HEP point of view, it is of great interest to understand whether AF interactions can generate new Universality Classes. One could even hope that nontrivial antiferromagnetism would be the ingredient needed in order to nonperturbatively formulate interacting theories in four dimensions.

In a previous work [5] we found that, on a three dimensional AF $O(3)$ model, the only new phase transitions generated were first order. We will consider in this letter the $RP^{2}\equiv S^{2}/Z_{2}$ (real projective space) spin model in three dimensions. We place the spins on a cubic lattice with a nearest neighbors interaction:

[TABLE]

where \{\mbox{\boldmathv}_{i}\} are normalized real three-components vectors. The local $Z_{2}$ symmetry \mbox{\boldmathv}_{i}\to-\mbox{\boldmathv}_{i} is preserved even after the SSB (Elitzur’s theorem), and so, the sense of a spin is irrelevant, it is only its direction that matters. It is not hard to see [6], that in the AF case (1) is a lattice discretization of the action $\int{\rm tr}[P{(R^{-1}\partial_{\mu}R)}^{2}]$, where $R\in SO(3)$, and $P$ is the diagonal matrix $\{g,g,-g\}$, with $g$ being the coupling. This is just a particular case of the NL$\sigma$M considered in reference [3].

For $\beta$ positive this model presents a weak first order phase transition which has been used to describe liquid crystals [7]. The ordered phase corresponds to states where all spins are aligned.

For $\beta$ negative there is also an AF ordered phase with a more complex structure [8]. There is a second order phase transition between the disordered phase and an ordered AF one. Let us call a site even (odd) when the sum of its coordinates $x+y+z$ is even (odd). A state where, for instance, all spins on even sites are aligned in a given direction, and those on odd sites lie randomly in the orthogonal plane, has zero energy. So, at $T=0$ the ground state is highly degenerate with a global $O(2)$ symmetry. However, when fluctuations are taken into account, it can be shown in the low temperature limit [6], that both sublattices are aligned in mutually orthogonal directions, as a consequence of Villain’s order from disorder mechanism [9]. We remark that the remaining $O(2)$ symmetry is broken. We will show, with Monte Carlo simulations, that the breaking also holds in the critical region.

In order to discuss the observables measured, let us construct the (traceless) tensorial field $\bf T$ with components

[TABLE]

and its Fourier Transform $\widehat{\bf T}$ in a $L\times L\times L$ lattice with periodic boundary conditions.

The intensive staggered (non staggered) magnetization can be defined in terms of the tensorial field as the sum of the spins on even sites minus (plus) those on odd sites, or equivalently

[TABLE]

where $V$ is the lattice volume. We have observed a phase transition at $\beta\sim-2.41$ for which ${\bf M}_{s}$ is an order parameter (zero value in the disordered phase and a clear nonzero value in the $L\to\infty$ limit in the ordered one). The magnetization ${\bf M}$ is also an order parameter. As these operators correspond to different irreducible representations of the translations group, we will study the scaling properties of each observable independently.

To measure in a Monte Carlo simulation on a finite lattice we have constructed scalars under the $O(3)$ group. For the magnetization and the susceptibility we compute respectively

[TABLE]

and analogously for the staggered observables.

We have also measured the second momentum correlation length, which is expected to have the same scaling behavior at the critical point as the exponential (physical) one, but it is much easier to measure [10]

[TABLE]

where $F$ is the mean value of the trace of $\widehat{\bf T}$ squared at minimal momentum ($2\pi/L$ in direction $x$, $y$ or $z$). To define $\xi_{s}$ we use $\chi_{s}$ and compute $F_{s}$ from $\widehat{\bf T}$ at momentum $(2\pi/L+\pi,\pi,\pi)$ and permutations.

The action (1) is suitable for cluster update methods by using the Wolff’s embedding algorithm [11]. We have checked the performance of both the Swendsen-Wang [12] and the Single Cluster methods [11]. Unfortunately, due to the AF character of the interaction, the critical slowing down is not reduced in any case, as there is always a large cluster that contains most of the lattice sites. In fact, we have measured a dynamic exponent $z\approx 2$ for both methods.

We have also developed a Metropolis algorithm. Near the transition the spin fluctuations are large, and a spin proposal uniformly distributed over the sphere is accepted with nearly 30% probability. So we have used a 3 hits algorithm, reaching a mean 70% acceptance.

Regarding the performance of the three methods mentioned for a given lattice size, the differences are very small in terms of the CPU time. We have selected the Metropolis method which is slightly faster.

In table 1 we display the number of Monte Carlo sweeps performed for the different lattice sizes as well as the integrated autocorrelation times for the observables $\chi$, $\chi_{s}$ and the energy. The total CPU time has been the equivalent of 12 months of DEC Alpha AXP3000 distributed over several workstations.

Every 10 Monte Carlo sweeps we store individual measures of the energy and of the Fourier transform of the tensorial magnetization at suitable momentum values. We have used the spectral density method [13] to extrapolate in a neighborhood of the critical point. The data presented here correspond to simulations at two $\beta$ values ($-2.41$ and $-2.4$). We compute the quantities referred above as well as their $\beta$-derivatives through the connected correlations with the energy.

We have firstly analyzed several quantities that present a peak near the transition point. As we have found that the specific heat does not diverge, we have to limit ourselves to study quantities related with magnetization operators. The advantage of measuring a peak height is that its position also defines an apparent critical point allowing for a very simple and accurate measure. Unfortunately, quantities like the $\beta$-derivatives of the magnetizations or the connected susceptibilities ($\chi_{\rm con}\equiv\chi-VM^{2}$) present their peaks far away from the critical point suffering from large corrections to scaling. For example, $\chi_{\rm con}$ in the $L=16$ lattice peaks at $\beta=-2.29$ where the correlation length $\xi$ is one half of its value at the critical point. This is due to the weakness of the tensorial ordering: the staggered magnetization, for instance, does not reach one half of its maximum until $\beta<-3.5$.

Another possibility is to obtain the infinite volume critical point by other means and, then, to measure the different quantities at this point. By studying the matching of the Binder parameter for the staggered magnetization $V_{M_{s}}=1-\langle({\rm tr}{\bf M}_{s}^{2})^{2}\rangle/(3\langle{\rm tr}{\bf M}_{s}^{2}\rangle^{2})$, as well as that of $\xi_{s}(L,\beta)/L$ and the corresponding non staggered quantities, we conclude that

[TABLE]

To improve the above determination it is necessary a careful consideration of the corrections to scaling. This subject will be discussed in a forthcoming paper [6].

In the case of quantities that change rapidly at the critical point as the magnetizations do, the errors in the determination of the critical point affect very much the results, and this method is not accurate. Nevertheless, we have found important quantities, like the $\beta$-derivatives of the correlation lengths, which are very stable. Both $d\xi_{s}/d\beta$ and $d\xi/d\beta$ should scale as $L^{1+1/\nu}$. Fitting the data from all lattice sizes we obtain an acceptable fit: $\nu=0.793(2)$ with $\chi^{2}/{\rm dof}=2.0/5$ and $\nu=0.787(2)$ with $\chi^{2}/{\rm dof}=5.5/5$; however, if we discard the $L=6$ data the fitted parameters change significantly and change again after discarding the $L=8$ ones. The fits for $L\geq 12$ and $L\geq 16$ agree within errors. We, thus, choose the $L\geq 12$ data for computing $\nu$ but take the statistical error from the fit with $L\geq 16$ (see figure 1):

[TABLE]

To estimate the errors, not considered in (7), associated with the uncertainty in the determination of the critical point we repeat the fits with $\beta$ at the limits of the interval (6). We observe that $\nu$ changes by an amount of a $1\%$.

To avoid the problems reported above, we have also used a method directly based on the Finite Size Scaling ansatz, that allows to write the mean value of any operator $O$ as

[TABLE]

where $\xi(L,\beta)$ is the correlation length measured at coupling $\beta$ in a size $L$ lattice, $f_{O}$ is a smooth operator-dependent function and $x$ depends also on $O$. The dots stand for corrections to scaling.

Measuring $\langle O\rangle$ at the same coupling in lattices $2L$ and $L$ and using (8) we can write for their quotient

[TABLE]

Considering the dependence of $Q_{O}$ on $\rho\equiv\xi(2L,\beta)/\xi(L,\beta)$, we just have to measure at the point where $\rho=2$ to obtain $Q_{O}=2^{x}$ up to corrections to scaling. $x$ can be written in terms of the critical exponents: $x=\gamma/\nu$ for $\chi$, $x=-\beta/\nu$ for $M$, etc.. To compute $\nu$ we can use $O=d\xi/d\beta$ for which $x=1+1/\nu$. As the quantity $\rho$ is an observable and not an external parameter, like $\beta$, this procedure does not require a previous determination of $\beta_{c}$. Another advantage is that the result depends only on measures in just two lattices what allows a better error estimation. In columns 2 and 3 of table 2 we report the results for the thermal exponent $\nu$ obtained from data in lattices $L$ (first column) and $2L$, using as operators the $\beta$-derivatives of the correlation lengths $\xi_{s}$ and $\xi$ respectively. Even in the smaller lattices we do not observe any corrections to scaling. The data from columns 2 and 3 are very correlated statistically and so, by taking the mean value, the errors are only slightly reduced. We select as our best estimation the mean of the results for the lattices 16-32:

[TABLE]

To be compared with the values $\nu(O(3))=0.704(6)$ [14] and $\nu(O(4))=0.748(9)$ [15].

In the case of the magnetic exponents, due to the large slope of $Q$ as a function of $\rho$ (see figure 2), a correct determination of the errors requires to take into account the statistical correlations of the whole data. We obtain for $\gamma_{s}$ and $\beta_{s}$, in the most favorable cases, errors as small as a 0.1%. From these we compute, using the scaling relations $\gamma_{s}/\nu=2-\eta_{s}$ and $2\beta_{s}/\nu=D-2+\eta_{s}$, the values of $\eta_{s}$ with acceptable accuracy (see columns 4 and 5 of table 2). In this case, the corrections to scaling are only significative for the $L=6$ lattice. We quote as our preferred value

[TABLE]

For the non staggered sector, we observe that the usual susceptibility diverges much more slowly than the staggered one ($\gamma_{s}-\gamma\sim 1.02$). The results for $\eta$, using the corresponding scaling relations, are reported in the last two columns of table 2. In this case the corrections to scaling are non negligible for all lattice sizes. As the data fit very well to a linear function of $1/L$ we take as the $L\to\infty$ value

[TABLE]

where the error is half statistical and half due to the possible deviations from linearity. Notice that the large value of $\eta$ means that, at the critical point, the spatial correlation function (G(\mbox{\boldmathr})|_{\beta=\beta_{c}}\sim|\mbox{\boldmathr}|^{-1-\eta}), in the non staggered sector, decreases much faster than in the staggered case.

Our results show that the $O(3)$ symmetry is fully broken in the ordered phase near the critical point. If we discard order $t^{\beta}$ terms ($t$ being the reduced temperature), ${\bf M}$ is zero and the eigenvalues of ${\bf M}_{s}$, are $\{\epsilon,0,-\epsilon\}$ with $\epsilon\propto t^{\beta_{s}}$ ($\beta-\beta_{s}\sim 0.51)$. Consequently, the magnetization tensors of the even and odd sublattices are opposite and the eigenvectors corresponding to the maximum eigenvalues are orthogonal, and similarly for the minimum ones. Considering the order $t^{\beta}$ terms the orthogonality will only hold approximately.

We have studied a spin model in three dimensions with the symmetries of the $O(3)$ group but with very interesting new properties: it presents an ordered vacuum where the $O(3)$ symmetry is fully broken; the transition belongs to a new Universality Class, as the thermal exponent $\nu$ is different from previously known and, finally, the model has two odd (magnetic type) relevant operators with different associated $\eta$ exponents. This may explain why previous perturbative calculations fail to work for this model.

We think that in addition to the interest of the model by itself, the results suggest further studies of related models, like the addition of vector interactions, or four dimensional systems.

We thank Alan Sokal for many enlightening discussions at the beginning of this work, specially regarding the structure of the vacuum in the $T\to 0$ limit and the Finite Size Scaling techniques. We are also indebted to José Luis Alonso, Juan Jesús Ruiz-Lorenzo and Alfonso Tarancón. This work has been partially supported by CICyT AEN93-0604, AEN95-1284-E and AEN93-0776. H. G. Ballesteros and V. Martín-Mayor are MEC fellows.