Deconfinement transition and dimensional cross-over in the 3D gauge Ising model

TL;DR

This study uses high-precision Monte Carlo simulations to analyze the deconfinement transition in a 3D Z2 gauge theory, confirming the Svetitsky-Yaffe conjecture and exploring dimensional crossover effects.

Contribution

It provides highly accurate critical temperature and exponent data for the 3D Z2 gauge theory, validating theoretical predictions and examining dimensional crossover phenomena.

Findings

Critical temperature matches 3D Ising model results

Critical exponents confirm Svetitsky-Yaffe conjecture

Deviations from Olesen's prediction are around 20%

Abstract

We present a high precision Monte Carlo study of the finite temperature $Z_2$ gauge theory in 2+1 dimensions. The duality with the 3D Ising spin model allows us to use powerful cluster algorithms for the simulations. For temporal extensions up to $N_t=16$ we obtain the inverse critical temperature with a statistical accuracy comparable with the most accurate results for the bulk phase transition of the 3D Ising model. We discuss the predictions of T. W. Capehart and M.E. Fisher for the dimensional crossover from 2 to 3 dimensions. Our precise data for the critical exponents and critical amplitudes confirm the Svetitsky-Yaffe conjecture. We find deviations from Olesen's prediction for the critical temperature of about 20%.

Equations

$$S_W=\sum _n\left\{\tilde{\beta }\left(\sum _i\mathrm{T}{\mathrm{r}}_{\mathrm{f}}(g_{n;0i})+\sum _{i<j}\mathrm{T}{\mathrm{r}}_{\mathrm{f}}(g_{n;ij})\right)\right\},$$

$$g_{n;\mu \nu }=g_{n;\mu }{g}_{n+\mu ;\nu }{g}_{n+\nu ;\mu }^{†}{g}_{n;\nu }^{†}.$$

$$T=\frac{1}{N_{t}a},$$

$$P(\vec{x})\equiv \mathrm{T}{\mathrm{r}}_{\mathrm{f}}V(\vec{x})=\mathrm{T}{\mathrm{r}}_{\mathrm{f}}\prod _{t=1}^{N_t}(g_{\vec{x},t;0}).$$

$$Z_{gauge}(\tilde{\beta })\propto Z_{spin}(\beta )$$

$$\beta =-\frac{1}{2}\log \left[\tanh (\tilde{\beta })\right]$$

$$Z_{gauge}(\tilde{\beta })=\sum _{\{g_{n,\mu }=\pm 1\}}\exp \left(\sum _{n,\mu \nu }\tilde{\beta }{g}_{n;\mu \nu }\right),$$

$$Z_{spin}(\beta )=\sum _{s_i=\pm 1}\exp (-\beta H(s)),$$

$$H(s)=-\sum _{⟨ij⟩}{J}_{⟨ij⟩}{s}_i{s}_j$$

$$ϵ(N_t)=\frac{{\beta }_c-{\beta }_c(N_t)}{{\beta }_c}$$

$$ϵ(N_t)\sim b_1{N}_t^{-\lambda }{N}_t\to \infty$$

$$t=\frac{{\beta }_c-\beta }{{\beta }_c}$$

$$\xi (\beta )\sim b_2{t}^{-\nu }{N}_t\to \infty$$

$$<P(x)P(x+R)>\sim e^{-\sigma (T)R/T}$$

$$\sigma (T)=\sigma \sqrt{1-{\left(\frac{T}{T_c}\right)}^2}$$

$$T_c^2/\sigma =\frac{3}{\pi (d-2)}.$$

$$Z=\sum _{bc}\sum _{s_i=\pm 1}\exp [-\beta H(s,bc)]$$

$$\frac{Z_{ap}}{Z}=\frac{{\sum }_{bc}{\sum }_{s_i=\pm 1}\exp [-\beta H(s,bc)]{\delta }_{bc,ap}}{Z}=⟨{\delta }_{bc,ap}⟩$$

$$\frac{Z_{ap}}{Z_p}=\frac{⟨{\delta }_{bc,ap}⟩}{⟨{\delta }_{bc,p}⟩},$$

$$p_d=\exp (-\beta (1+J_{<ij>}{s}_i{s}_j)).$$

$$\frac{Z_{ap}}{Z_p}=\frac{{\vartheta }_2(0,\tau )+{\vartheta }_3(0,\tau )-{\vartheta }_4(0,\tau )}{{\vartheta }_2(0,\tau )+{\vartheta }_3(0,\tau )+{\vartheta }_4(0,\tau )}$$

$$N_t=c_1{ϵ}^{-\nu }\left(1+c_2{ϵ}^{\Delta }+c_3ϵ...\right)$$

$${ϵ}^{\prime }(N_t)=\frac{T_c(N_t)-T_c}{T_c}=\frac{1/{\beta }_c(N_t)-1/{\beta }_c}{1/{\beta }_c}=ϵ(N_t)(1-ϵ(N_t)...)$$

$$N_t=c_1{\left[ϵ(N_t)\right]}^{-\nu }$$

$$N_t=c_1{\left[{ϵ}^{\prime }(N_t)\right]}^{-\nu }$$

$$a{T}_c=\frac{1}{N_t}{\left({\tilde{\beta }}_c-{\tilde{\beta }}_c(N_t)\right)}^{-\nu }$$

$$\sigma ({\beta }_c)=\sigma ({\beta }_{ref}){\left(\frac{{\beta }_c-\beta }{{\beta }_{ref}-\beta }\right)}^{2\nu }$$

$$\frac{T_c}{\sqrt{\sigma }}=1.2216(24).$$

$$N_x$$

Full text

DFTT 68/95

DAMTP 95-60

Deconfinement Transition and Dimensional Cross-over in the 3D Gauge Ising Model

M. Casellea and M. Hasenbuschb

a Dipartimento di Fisica Teorica dell’Università di Torino

Istituto Nazionale di Fisica Nucleare, Sezione di Torino

via P.Giuria 1, I-10125 Torino, Italy ***e–mail: caselle @to.infn.it

b DAMTP, Silver Street, Cambridge, CB3 9EW, England †††e–mail: M.Hasenbusch@damtp.cam.ac.uk

We present a high precision Monte Carlo study of the finite temperature $Z_{2}$ gauge theory in $2+1$ dimensions. The duality with the 3D Ising spin model allows us to use powerful cluster algorithms for the simulations. For temporal extensions of up to $N_{t}=16$ we obtain the inverse critical temperature with a statistical accuracy comparable with the most accurate results for the bulk phase transition of the 3D Ising model. We discuss the predictions of T. W. Capehart and M.E. Fisher for the dimensional crossover from 2 to 3 dimensions. Our precise data for the critical exponents and critical amplitudes confirm the Svetitsky-Yaffe conjecture. We find deviations from Olesen’s prediction for the critical temperature of about 20 $\%$.

1 Introduction

One of the most interesting predictions of QCD is the existence of a deconfinement transition at some critical temperature $T_{c}$. Finding a precise characterization of this transition is, however, still an open problem. In particular it would be interesting to have some indication on the order of the transition and, in case it is of second order, on its critical indices. The natural framework to pose these questions is that of the finite temperature Lattice Gauge Theories (LGT). In this framework, a seminal contribution was given more than ten years ago by B. Svetitsky and L.Yaffe [1]. They showed that, if the deconfining transition of a given $(d+1)$ dimensional gauge theory is of second order, then its universality class should coincide with that of the $d$-dimensional spin model, with the center of the original gauge group as symmetry group. This result is usually known as the “Svetitsky-Yaffe (SY) conjecture” and has been confirmed in these last ten years by several Montecarlo simulations. Most of these tests were made on LGT with continuous gauge groups (in particular SU(2) and SU(3)) which are obviously the most interesting from a phenomenological point of view, but do not allow in general to reach the high statistics and precision which is needed to test the critical behaviour of a statistical system near a second order phase transition. To make such a test we follow the opposite route, namely to choose the simplest possible nontrivial LGT, and then try to reach in this simplified model the highest possible confidence. Therefore we shall concentrate in the present paper on the three dimensional gauge Ising model. Besides its simplicity there is another important reason to choose exactly the Ising model to test the SY conjecture. The only claim of a failure of the SY conjecture [2] that we are aware of was indeed obtained in a high precision study of the 3d gauge Ising model with temporal extent $N_{t}=8$. Such a result, if confirmed, would obviously imply a deep change in our understanding of the deconfinement transition. In this paper we shall address this question, pushing the analysis up to $N_{t}=16$, again in the 3d Ising gauge model, using a new original algorithm and very high statistics. We shall show that all our results are in perfect and complete agreement with the SY conjecture, we shall also give some argument to justify the apparent disagreement found in [2]. In order to test the SY conjecture we need, first of all, a very precise estimates of the critical deconfinement temperature. With this estimate we can test another interesting conjecture, due to Olesen [3], which is completely independent from the SY one, and is based on the string (or flux tube) description of the infrared regime of QCD.

A seemingly independent, interesting problem, which is actually related to the above one, is that of the so called “dimensional crossover” in layered spin systems. Indeed it is well known that the dimensionality of a system as it approaches the critical point (and hence the universality class to which the system belongs) is only determined by the number of spatial dimensions in which the system has an infinite extension. For instance, the Ising model on a lattice with a finite thickness $N_{t}$ in one direction (and the lattice size in the other two directions, say $N_{s}$, going to infinity) belongs to the same universality class of the ordinary two-dimensional Ising model. As $N_{t}$ increases both the critical indices of the two and of the three-dimensional Ising model can be observed. In particular, it is possible to define so called “effective critical indices” which smoothly interpolate between the $d=2$ to the $d=3$ value of the indices as the adimensional ratio $N_{t}/\xi(\beta)$ (where $\xi(\beta)$ is the bulk correlation length of the system) goes from zero to infinity. The theory of dimensional crossover was developed by T. W. Capehart and M. E. Fisher [4] as a particular case of the more general theory of finite size scaling theory. Their predictions were then compared with high temperature expansions finding in general a good agreement. It is interesting to notice that even if ref. [4] and ref. [1] use different languages and are interested in different systems they follow actually the same type of arguments (scaling and renormalization group) and reach in fact exactly the same conclusions. This is particularly evident in the case of the three-dimensional Ising model, where dimensional cross-over and deconfinement transition are in fact the two faces of the same phenomenon. Duality, which maps the Ising gauge model onto the spin model also relates deconfinement and dimensional cross-over. There is however in [4] a very puzzling result. The critical index $\lambda$ with which the critical temperature on lattices of finite thickness approaches the bulk critical temperature turns out to be different for different boundary conditions. In particular, in the case of periodic boundary conditions in which we are interested here, it does not coincide with the value $\lambda=1/\nu$ (where $\nu\sim 0.63$, is the critical index of the correlation length) which one would expect. This is again a result which, if confirmed, would change our approach to the analysis of the deconfinement transition, since it is a common and general use to extrapolate the finite lattice determinations of the critical temperature to the continuum limit, following the scaling behaviour of the correlation length. In this paper we shall also address this question, showing that the correct answer is indeed $\lambda=1/\nu$ and that the result of [4] is probably an artifact of the strong coupling expansion.

A further reason of interest in the 3d Ising model on layered lattices is that in these last years this model (and in particular its realization on lattices with very few layers) has become a laboratory to test various theoretical methods [5, 6, 7] to find the critical temperature and the critical indices. Due to the high precision of our results we are in the position to test the reliability of these methods and, at least in one case (the MHZ approach [7], see below) to better understand the reason of its failure.

This paper is organized as follows. Sect.2 will be devoted to a brief introduction to finite temperature LGT and to the Svetitsky-Yaffe conjecture. In sect.3 we discuss dimensional crossover following [4], while in sect.4 a brief account of the string model approach to the infrared regime of LGT, and the consequent Olesen conjecture for the deconfinement temperature will be given. In sect.5 we shall describe the model and the algorithm that we use. Finally section 6 will be devoted to a discussion of the results of the simulation and sect. 7 to the comparison with other existing estimates of the critical temperature and of the critical indices.

2 Finite Temperature LGT

2.1 General setting

Let us consider a pure gauge theory with gauge group $G$, defined on a $d+1$ dimensional cubic lattice. In order to describe a finite temperature LGT, we have to impose periodic boundary conditions in one direction (which we shall call from now on “time-like” direction), while the boundary conditions in the other $d$ directions (which we shall call “space-like”) can be chosen freely. We take a lattice of $N_{t}$ ($N_{s}$) spacings in the time (space) direction, and we work with the pure gauge theory, containing only gauge fields described by the link variables $g_{n;\mu}\in G$, where $n\equiv(\vec{x},t)$ denotes the space-time position of the link and $\mu$ its direction. We shall denote the time-like direction with the index $\mu=0$ and the remaining space-like directions with roman indices $\mu=i=1\cdots d$. We choose for simplicity the same bare coupling $\tilde{\beta}$ in the time and space directions. The Wilson action is then

[TABLE]

where ${\rm Tr_{f}}$ denotes the trace in the fundamental representation and $g_{n;0i}$ ($g_{n;ij}$) are the time-like (space-like) plaquette variables, defined as usual by

[TABLE]

In this framework the temperature $T$ is given by:

[TABLE]

where $a$ is the lattice spacing.

In a finite temperature discretization it is possible to define gauge invariant observables which are topologically non-trivial, as a consequence of the periodic boundary conditions in the time direction. The simplest choice is the Polyakov loop, defined in terms of link variables as

[TABLE]

A second important consequence of the periodic boundary conditions in the time direction is that the theory has a new global symmetry (unrelated to the gauge symmetry), with the center $C$ of the original gauge group as symmetry group (hence, if $G$ is abelian, $C$ will coincide with $G$). The Polyakov loop is a natural order parameter for this symmetry.

In $d>1$, finite temperature gauge theories admit a deconfinement transition at $T=T_{c}$, separating the high temperature, deconfined, phase ($T>T_{c}$) from the low temperature, confining domain ($T<T_{c}$). The high temperature regime is characterized by the breaking of the global symmetry with respect to the center of the group. In this phase the Polyakov loop has a non-zero expectation value, and it is an element of the center of the gauge group.

2.2 Svetitsky-Yaffe conjecture

The idea on which the SY conjecture is based is that, if one would be able to integrate out all the gauge degrees of freedom of the original $(d+1)$–dimensional model except those related to the Polyakov loops then the resulting effective theory for the Polyakov loops would be a $d$-dimensional spin system with symmetry group $C$. The deconfinement transition of the original model would become the order–disorder transition of the effective spin system. This effective theory would obviously have very complicate interactions, but Svetitsky and Yaffe were able to argue that all these interactions should be short ranged. As a consequence, if the transition point of the effective spin system is of second order, near this critical point, where the correlation length becomes infinite, the precise form of the short ranged interactions should not be important, and the universality class of the deconfinement transition should coincide with that of the simple spin model with only nearest neighbour interactions and the same global symmetry group. As a consequence all the critical indices describing the two transitions and all the adimensional ratios of scaling quantities should coincide in the limit. This is the case for instance of the ratio of partition functions which will be discussed in sect.5 and of the indices $\nu$ and $\eta$ which will be discussed in sect.6.

This argument holds also for abelian gauge groups. In particular, the deconfinement transition of the (2+1) dimensional gauge Ising model which we shall study in the following should belong to the same universality class as the two-dimensional spin Ising model.

2.3 Duality

A peculiar, interesting, feature of abelian LGT in 2+1 dimensions is that they can be mapped through duality, into (2+1) dimensional spin models, with the same symmetry group (for a review see for instance [8]).

In the case of the Ising model (in which $g_{n,\mu}\in\{\pm 1\}$) this is accomplished by the well known Kramers–Wannier duality transformation which relates the two partition functions:

[TABLE]

where $Z_{gauge}(\tilde{\beta})$ is defined, according to eq.(1) and (2) as

[TABLE]

and

[TABLE]

with

[TABLE]

where $s_{i}\in\{\pm 1\}$, $i$ and $j$ denote nodes of the dual lattice and the sum is extended to the links ${\langle ij\rangle}$ connecting the nearest–neighbour sites. For the moment the couplings $J_{\langle ij\rangle}$ are all chosen equal to $+1$ .

Using the duality transformation it is possible to build up a one–to–one mapping of physical observables of the gauge system onto the corresponding spin quantities. In particular the deconfinement transition in the gauge Ising model becomes the order-disorder transition in the spin Ising model.

It is important to notice that in general the duality transformation does not conserve specific choices of the boundary conditions (bc in the following). In fact, in obtaining eq.(6) (see for instance [8]) one usually assumes to be in the thermodynamic limit and neglects bc. In our discussion it is crucial to impose periodic bc in the time direction in the gauge model, but there is no reason to expect that these hold also for the dual spin model. In fact, it can be shown that the corresponding dual model is then a spin Ising model with fluctuating bc in the time direction, namely a mixture of the two partition functions with periodic and antiperiodic bc respectively. However the antiperiodic bc in the time direction force the formation of a space-like interface in the Ising spin model. As a consequence the antiperiodic free energy will be depressed with respect to the periodic one by a factor proportional to $\exp(-\sigma N_{x}N_{y})$ and in the thermodynamic limit $N_{x},N_{y}\to\infty$ it will become negligible. Hence in this limit the duality transformation actually becomes a mapping between a spin and a gauge Ising model, both with periodic bc in the time direction. This will allow us to make the simulations directly in the spin Ising model with periodic bc (which is a very convenient choice, since for this model (see sect.5) we have a very powerful algorithm to extract the critical couplings) and then to translate the result in the framework of the gauge Ising model, so as to test the SY and Olesen conjectures.

Since this equivalence is of central importance in our work it could be useful to discuss it from a different (but equivalent) point of view. We can immediately see that a mapping between gauge and spin model both with periodic bc is impossible because there is no dual counterpart for the expectation value of an isolated Polyakov loop. Since the Polyakov loop is the order parameter of the deconfinement transition this seems to exclude the possibility to study this phase transition in the dual model. However the dual of the correlator between two Polyakov loops is perfectly well defined. Given any surface in the gauge model bordered by the two Polyakov loops the dual of the correlator is given by $Z_{f}/Z_{spin}$ where $Z_{f}$ is the partition function of the spin model in which the links orthogonal to the surface have been frustrated. If we now take the thermodynamic limit in the spacelike directions (and only in this limit) we can push away as far as we want the two Polyakov loops and obtain, by using the standard factorization theorems, the expectation value of the isolated Polyakov loop.

3 Dimensional cross-over in spin models

In the study of dimensional cross-over in spin models the $N_{t}$ space-like planes are usually called layers. Let us assume that the pure $d$-dimensional model (one layer) has a critical point located at $\beta=\beta_{c}(N_{t}=1)$. If the $d$ spatial dimensions are sent to infinity, keeping the number of layers finite, the system still has a critical point, whose position $\beta=\beta_{c}(N_{t})$ changes continuously with $N_{t}$. The limit $\lim_{N_{t}\to\infty}\beta_{c}(N_{t})$ will coincide with the (bulk) critical point (which we shall simply denote as $\beta_{c}$ in the following) of the $(d+1)$ model. The reduced critical point shift is defined by:

[TABLE]

We can introduce the critical point shift exponent as

[TABLE]

with $b_{1}$ a suitable constant. Let us introduce the reduced temperature (with respect to the bulk transition of the $d+1$ model) :

[TABLE]

Then the bulk correlation length has the following scaling behaviour:

[TABLE]

If we assume that the critical point in the layered system is reached when $\xi(\beta_{c}(N_{t}))\sim N_{t}$ then it immediately follows that $\lambda=1/\nu$ [4]. Hence we expect the value $\lambda^{-1}\sim 0.63$ for the cross-over from the three-dimensional to the two-dimensional Ising model in which we are interested.

In view of this analysis the result suggested in [4] (based on the results of a strong coupling analysis) that $\lambda^{-1}\sim 0.63$ only for $fixed$ bc (in the $N_{t}$ direction), and that $\lambda^{-1}\sim 0.5$ for periodic bc seems indeed rather surprising. It is in fact very unlikely that a change in the bc could affect the above analysis, which is essentially a bulk result. The only known examples of such a change refer to rather degenerate cases like the spherical model or the crossover from the two-dimensional to the one-dimensional Ising model where the choice of the bc can indeed drastically affect the bulk of the model, changing for instance its operator content.

Since our measurements have been made exactly in the case of periodic bc we are in the position to test the result of [4]. We shall give below strong evidence that also for periodic bc one has $\lambda^{-1}\sim 0.63$. Most probably the result reported in [4] was only an artifact of the strong coupling expansion.

4 The Olesen conjecture

It is widely accepted that the infrared regime of a LGT can be described by some sort of effective string theory. This conjecture is a natural consequence of the observation that the colour flux joining a pair of quarks in the confining phase is concentrated inside a tube of finite thickness. According to the conjecture, when the quark and the antiquark are pulled very far apart, the flux tube should behave like a vibrating string [9]. Due to the well known problems related to the Weyl anomaly even the simplest possible string theory, the bosonic string, cannot be consistently quantized in more than one dimension. Depending on the quantization method, one finds either the breaking of rotational invariance or the appearance of interacting longitudinal modes (Liouville fields). However it is possible to show [10] that for large interquark separation these anomalous contributions can be neglected. The resulting effective string theory becomes in this limit an ordinary two dimensional conformal field theory. In particular, in this infrared limit, the bosonic string flows toward the ordinary (two-dimensional) free gaussian model. In the framework of the lattice regularization the interquark potential is given by the expectation value of Wilson loops. Thus, according to the above conjecture, we expect the quantum fluctuations of large Wilson loops to be well described by some suitable effective string model (not necessarily the bosonic one). It can be shown that the contribution to the expectation value of the Wilson loop due to these quantum fluctuations can be written as a power series in the adimensional parameter $(\sigma A)^{-1}$ where $\sigma$ is the ordinary zero temperature string tension and $A$ is the Area of the Wilson loop. By using the effective string model one can then evaluate (at least in principle) these contributions order by order in the expansion parameter. This approach was pioneered in [11] (where it was calculated the first term of the expansion) and its consequences have been successfully tested in various models by several groups during the past years.

This same reasoning can be extended also to the case of the correlator of two Polyakov loops: $<P(x)P(x+R)>$. From the string point of view the only difference with respect to the Wilson loop case is in the bc, which can be easily taken into account. From the gauge theory point of view we have a new important feature: the natural expansion parameter of the theory is now $T/\sqrt{\sigma}$. One can thus hope to obtain an universal relation between $\sigma$ and $T_{c}$ by requiring, for instance, that the temperature dependent string tension $\sigma(T)$ vanishes at the deconfinement temperature $T_{c}$.

This program was initiated by O.Alvarez and R.Pisarski [12], who worked in the $d\to\infty$ limit (where $d$ are the space-time dimensions). Their result was later extended by P.Olesen to any value of $d$ [3]. Assuming the bosonic string model as effective description, they obtained in the $R\to\infty$ limit

[TABLE]

where $\sigma(T)$ is given by:

[TABLE]

and

[TABLE]

We shall denote in the following eq.(16) as the “Olesen conjecture”.

Let us stress again that this result has been obtained assuming the bosonic string as effective model, which could well be a too simple description for the infrared regime of realistic lattice gauge theories. For instance in eq.(16) there is no residual information of the gauge group of the model. The only remaining dependence is on the number of space-time dimensions of the theory. However, recently, exactly in the case of the 3d Ising model, we obtained an impressive confirmation of this bosonic string picture [13]. In order to describe this result let us first make an important observation (see [13] for the details). The bosonic effective string approach described above is completely equivalent to the so called “capillary wave model” (CWM) [14], which describes the behaviour of fluid (or rough) interfaces in statistical systems. This equivalence can be easily understood in the context of the Ising model. The same duality transformation which relates the gauge and the spin models, maps Wilson loops of the gauge model into interfaces in the spin model (modulo the obvious changes in the bc), and maps the effective bosonic string into the CWM. In fact the CWM, in its simplest formulation, describes the interface with an effective Hamiltonian proportional to its area 111More precisely, it is proportional to the variation of the surface area with respect to the classical solution, which in the Ising case is the cristallographic plane., and this is exactly one of the possible ways to define the bosonic string. In [13] we succeeded in evaluating the second term of the expansion in $\sigma A$ in the CWM and found a perfect agreement with the results of a set of high precision simulations. If this agreement would be conserved to higher order of the expansion it would necessarily imply the correctness of the Olesen conjecture. This was one of the initial motivations for the present work. Testing the Olesen conjecture we are in some sense testing the bosonic string picture to all orders (at least in the Ising model).

5 The Numerical Method

5.1 The model

We study the spin Ising model defined by eq.(8) and (9), on a simple cubic lattice with extension $N_{x},N_{y}$ and $N_{t}$ in the three directions $x,y,t$. $N_{t}$ will be always smaller than $N_{x}$ and $N_{y}$, and the bc in the $t$ and $x$ direction will always be periodic. In the $x$ direction we shall implement both periodic and antiperiodic bc. Antiperiodic bc will be obtained by setting the couplings $J_{\langle ij\rangle}=-1$ in the links joining the lowermost and uppermost layers in the $x$ direction, while keeping all the other couplings equal to $+1$. In the following we shall denote the Hamiltonian constructed in this way as $H(s,ap)$, while the Hamiltonian defined as in eq.(9) with all the $J_{\langle ij\rangle}=1$ will be denoted by $H(s,p)$. In the following an important role will be played by the ratio of antiperiodic and periodic partition functions $Z_{ap}/Z_{p}$. The reason of this is twofold: first, as it was shown in [15], it can be used, as the Binder cumulant, to estimate the critical temperature of the system and to extract the critical index $\nu$. Moreover it was shown in [15], that this ratio is more efficient and converges faster than the standard Binder cumulant. Second, in the present case, owing to the exact solution of the two dimensional Ising model, and according to the discussion of the previous sections, we can predict exactly the value of the ratio $Z_{ap}/Z_{p}$ at the critical temperature. This will give us an independent, very precise method to extract the critical temperature.

In order to obtain the ratio $Z_{ap}/Z_{p}$ one must generalize the system and treat also the bc as dynamical variables. The partition function of this generalized system is given by

[TABLE]

where the new index $bc$ can take the two possible values: $bc=\{p,~{}ap\}$. The fraction of configurations with antiperiodic bc is given by:

[TABLE]

and similarly for the periodic ones. Thus we can express the ratio $Z_{ap}/Z_{p}$ as a ratio of observables in the same system:

[TABLE]

which is hence accessible in a single Monte Carlo simulation, provided the bc can be updated efficiently. This is the case for the algorithm which shall be described in the following subsection.

5.2 The algorithm

For the updating of the bc we used a specially adapted cluster algorithm which was introduced in ref. [16]. Let us shortly recall the basic steps of this algorithm.

First the links of the lattice are deleted with the standard probability of the Swendsen-Wang [17] algorithm.

[TABLE]

In the second step one checks whether it is possible to change the type of bc simultaneously with a sign change of certain spins without changing the sign of $J_{<ij>}s_{i}s_{j}$ for any frozen (i.e. not deleted) link.

All clusters (in the sense of the standard Swendsen-Wang algorithm) which have a site in the $x=1$ slice are checked whether they can be uniquely decomposed into a part ”above” and ”below” the $x=1$ to $x=L$ boundary. Technically this is achieved by introducing an additional label when constructing the cluster using the ants in the labyrinth algorithm. The start site of the cluster in the $x=1$ slice gets the label $-1$. The sites joining the cluster get the label of their parent site, except when the $x=1$ to $x=L$ boundary is crossed. Then the new site gets minus the label of the parent site. If at the end of the construction of the clusters all sites connected by a frozen link have consistent labels the type of the bc is altered and the spins are multiplied by their label.

5.3 The Simulations

We alternated the boundary flip with standard single cluster updates [18]. In the case of our largest lattice ($128\times 128\times 16$) we performed 32 single cluster updates per boundary flip. After each boundary flip we performed a measurement of the energy, the magnetisation and the type of the bc. The total number of measurements was 100000 throughout. It is interesting to note that the integrated autocorrelation time $\tau$ for the type of the bc was smaller then $0.5$ due to negative values of the autocorrelation function. A similar observation was already made in [15] for the $L_{x}=L_{y}=L_{t}$ case.

6 Discussion

6.1 Behaviour of $Z_{ap}/Z_{p}$

The first test of the Svetitsky-Yaffe conjecture is given by the behaviour of $Z_{ap}/Z_{p}$. In fact, according to the discussion of sect.2 we expect that at the critical point this ratio coincides with the corresponding one in the two-dimensional Ising model. This can be evaluated by using the exact solution of the model (see for instance [19]) or by using well known conformal field theory techniques (for a review of this techniques see for instance [20]). It depends only on the shape of the two dimensional lattice, namely on the ratio between $N_{y}$ and $N_{x}$. The result is:

[TABLE]

where $\tau=iN_{y}/N_{x}$, and ${\vartheta_{i}(0,\tau)}$ denote the Jacobi theta functions (for the notations see for instance ref. [20]). For completeness we remind that $Z_{p}$ is periodic both in $x$ and $y$ directions, while $Z_{ap}$ is periodic in the $y$ direction and antiperiodic in the $x$ direction. Two interesting limits of this equation are the following: In the $N_{x}\to\infty$ limit we get $\frac{Z_{ap}}{Z_{p}}\to 1$ as expected, since in this limit the bc in the $x$ direction are irrelevant. If $N_{x}=N_{y}$ (square lattice) we have $\frac{Z_{ap}}{Z_{p}}=0.37288488...$ , a result which we shall use in the following.

In order to test this behaviour we fixed $N_{t}=6$, and first simulated the model choosing $N_{x}=N_{y}=12,24,48,96$. We extracted the critical temperature using the usual crossing procedure (see ref. [15]) and found the values quoted in tab.1. For each crossing we also evaluated the ratio $Z_{ap}/Z_{p}$ (third column of tab.1). All the values are in good agreement with the predictions of eq.(21). We decided thus to assume the S-Y conjecture and extract the critical temperature by requiring $Z_{ap}/Z_{p}$ to be equal to $0.37288488...$. The results are reported in tab.2. The gain in precision with respect to the usual crossing method is immediately obvious. As our best estimate of the critical temperature we take the value corresponding to the largest lattice, namely in the present case $\beta_{c}(N_{t}=6)=0.228818(4)$. Fixing this value of $\beta$ we then studied the system for various asymmetric lattices, and extracted the corresponding $Z_{ap}/Z_{p}$ values. The choices of lattice sizes and the results are reported in tab.3 and fig.1 together with the predictions obtained from eq.(21). The agreement is impressive in the whole range of $\tau$ values. Up to our knowledge this is the first example of such a test of the S-Y conjecture and it is probably the most precise and compelling evidence of the validity of the conjecture among those that we shall present in this paper.

6.2 The critical temperature $T_{c}$

We then moved to other values of $N_{t}$ keeping the above described approach. We always studied lattices with the same values of $N_{x}=N_{y}$. We determined the critical temperature $\beta_{c}(N_{t})$ by imposing $\frac{Z_{ap}}{Z_{p}}=0.37288488...$. In tab.4 we report the set of lattices that we studied. For any given $N_{t}$ we chose as our best estimate of the critical temperature that corresponding to the largest value of $N_{x}$. The resulting values of $\beta_{c}$ are listed in the third column of tab.4, while in the fourth column the corresponding values for the dual gauge model are given, obtained by using eq.(6). We shall postpone to sect.7 a detailed comparison of our results with other existing estimates, but we have already reported in tab.4 the two sets of estimates: [2] and [4] which we discussed in the introduction. In particular, in the fifth column we have reported the Montecarlo estimates of [2] (for $N_{t}=2,4,8$) for $\beta_{c}$. They agree with our results, which are however more than one order of magnitude more precise. On the contrary the estimates obtained by using high temperature expansions (last column of tab.4) systematically disagree with our results (except for the $N_{t}=2$ case).

Let us now study the scaling behaviour of these data. Based on general renormalization group arguments we expect corrections to scaling to be of the form

[TABLE]

where $\Delta=\nu\omega$, with $\omega\approx 0.82$ [21] and $c_{1}$ is related to the constant $b_{1}$ which appears in eq.(11) by: $c_{1}=(b_{1})^{\nu}$. The second correction term reflects the ambiguity in the definition of the reduced critical point shift

[TABLE]

First we tried to perform a fit with $c_{1},c_{2},c_{3},\nu$ and $\Delta$ as free parameters, while keeping $\beta_{c}=0.2216546(10)$ [21] fixed. However even when taking all results summarized in table 4 the statistical errors of $c_{2},c_{3}$ and $\Delta$ make the result meaningless. Therefore we chose a much simpler approach: We solved the equations

[TABLE]

and

[TABLE]

for pairs of $N_{t}$ with respect to $c_{1}$ and $\nu$. The results are summarized in table 5. The results obtained from eq.(24) seem to converge faster than those from eq.(25). Therefore we regard $\nu=0.628(3)$ and $1/c_{1}=1.477(18)$ obtained from the pair $N_{t}=12$ and $N_{t}=16$ and using eq.(24) as our best approximation. Comparing the numbers obtained from eq. (24) and eq. (25) we still might expect systematic errors of the order of $0.005$ for $\nu$ and $0.05$ for $1/c_{1}$. Our best estimate agrees perfectly with the existing estimates ($\nu\sim 0.63$) of this critical index for the 3d Ising model, and disagrees with the estimate ($\nu\sim 0.5$) of ref. [4].

In addition this analysis can be performed using the $Z_{2}$ gauge theory inverse temperatures eq.(6) instead of the inverse temperature of the Ising spin model.

[TABLE]

Then we obtain as our best estimates $\nu=0.6251(26)$ and $aT_{c}=2.29(3)$ , to be compared, for instance, with the value $aT_{c}=2.2(1)$ quoted in [22].

6.3 Test of the Olesen Conjecture

In order to obtain an accurate estimate of the ratio $T_{c}/\sqrt{\sigma}$ we need the value of the string tension222As it is well know the string tension of the gauge Ising model and the interface tension of the spin Ising model are related by duality and have the same value. To simplify the discussion, in the following we shall always refer to $\sigma$ as the string tension. at precisely the critical values of $\beta$. A good approximation for these values can be obtained taking as reference values those published in [23] and [13] and interpolating among them with the law:

[TABLE]

where $\beta_{ref}$ is the coupling at which the reference value of $\sigma$ is taken and $\beta$ is as usual the bulk critical point of the 3d Ising model. If $|\beta_{c}-\beta_{ref}|$ is small enough then the uncertainty in the scaling law and in the determination of $\nu$ can be neglected. We can evaluate the error for $\sigma(\beta_{c})$ in a reliable way. The results of this analysis are reported in tab. 6. Once the values of $\sigma(\beta_{c})$ are known one can construct the ratio $T_{c}/\sqrt{\sigma}$ and test if it is stable within the scaling region. These ratios are reported in the last column of tab.6. As it can be easily seen scaling is well established. Unfortunately this procedure cannot be extended up to $N_{t}=14$ and 16, since in that region $\sigma$ has never been estimated. As our final result for the ratio we quote that corresponding to $N_{t}=12$

[TABLE]

To give an idea of the enhancement in precision with respect to the previous estimates we quote the analogous ratio, taken from ref. [22]: ${T_{c}}/{\sqrt{\sigma}}=1.17(10)$.

Our result disagrees from Olesen’s prediction ${T_{c}}/{\sqrt{\sigma}}=\sqrt{3/\pi}=0.977...$ by about 20%. This is not too surprising in view of the remarks made in sect.4 . On the contrary it is rather impressive to see how effective such a simple description can be. It would be very interesting to see if some improved string model could give better estimates for the critical temperature.

6.4 Critical indices

According to the Svetitsky-Yaffe conjecture the critical indices $\nu$ and $\eta$ which characterize the finite temperature transition point at fixed $N_{t}$ must coincide with the corresponding indices of the two dimensional Ising model, whose values are $\nu=1,~{}~{}\eta=0.25$. This is a rather important test since in [2], in a set of high precision simulations of the 3d Ising model with $N_{t}=8$, a rather larger value for $\eta$ was found: $\eta=0.40(9)$. This is indeed the only claim for a failure of the S-Y conjecture of which we are aware. To extract the two indices we used a standard finite size technique, varying the spatial size $N_{x}$ of the lattice. To obtain the index $\nu$ we studied the slope of the $Z_{ap}/Z_{p}$ at the critical point which is expected to scale as $N_{x}^{1/\nu}$. To extract $\eta$ we measured the susceptibility $\chi$ which is expected to scale as $N_{x}^{2-\eta}$. To extract the indices we compared nearby values of $N_{x}$. The results are reported in tab.7. We expect that the measured values approach the correct asymptotic result as $N_{x}$ increases. It is however interesting to notice that in the case of the $\nu$ index this asymptotic value is approached precociously, indeed $all$ the values listed in tab.7 are compatible within the errors with the expected $\nu=1$ value. This indicates that, using the $Z_{ap}/Z_{p}$ slope to extract $\nu$ one has a very small correction to the scaling amplitude, a fact that was already noticed in [15]. On the contrary in the case of the susceptibility we see a large correction to scaling and the expected value is in fact reached only for the largest lattices that we studied. Smaller lattices give systematically higher values of $\eta$. This probably explains the result of ref. [2]. In conclusion we can say that, also for the critical indices, the S-Y conjecture is completely confirmed by our high precision test.

7 Comparison with existing results

As we mentioned in the introduction, in these last years the layered Ising model with low values of $N_{t}$, in particular with $N_{t}=2$, have become a laboratory to test theoretical methods [4, 5, 6, 7] and numerical techniques [2, 24, 25] to estimate the critical indices and the critical temperature of statistical systems. In tab. 8 we have compared our estimates for the critical temperature with those obtained with the mean field transfer matrix approach [5], the Müller-Hartmann-Zittarz method [7], the constrained variational approach [6] and strong coupling expansions [4]. Let us make few observation on these numbers (for a more detailed description of the various approaches we refer to the original literature).

a] In the transfer matrix version of the mean field approximation [5] one actually deals with clusters of infinite extent. As a consequence the determination of the critical temperature is greatly improved with respect to standard mean field results333In the case of the 2d Ising model this approach gives indeed the exact result. This explains the impressive precision of the result for $N_{t}=2$, for which the authors of [5] were also able to estimate the theoretical uncertainty. It is very interesting to observe that our result is in perfect agreement with this theoretical estimate. As $N_{t}$ increases the method becomes less reliable, however in [5] the authors give arguments to support the idea that the $N_{t}=3$ value is still correct within 4 digits, a result which is again in perfect agreement with our estimate.

b] As it can be seen in tab. 8 the Müller-Hartmann-Zittarz method (MHZ) which was first proposed in [7] and studied in this context in [5] gives much worse results for the critical temperature. This is a rather puzzling result since MHZ has some features in common with our approach. MHZ also studies the interface free energy which is related to the logarithm of the ratio $Z_{ap}/Z_{p}$ that we study. They locate the critical point requiring the vanishing of the interface string tension. The main difference between our approach and MHZ is that they approximate the $Z_{ap}/Z_{p}$ ratio with the partition function of a suitable Solid on Solid model (SOS). In this way they neglect overhangs and handles of the interface and exactly this approximation is the reason of the failure as $N_{t}$ increases (while it works perfectly in the case of the pure two dimensional model, where it gives the exact result). This is indeed very interesting, since it gives a further evidence of the relevance of not SOS-like configurations in the physics of two dimensional interfaces. This was already noticed in [26], where the physics of microscopic handles in fluid interfaces was discussed, and also in [13] where, in the framework of the Capillary Wave Model (CWM), the failure of a simple SOS approximation (which in the CW approach becomes a pure free bosonic action) for two dimensional interfaces was shown.

c] The constrained variational approach [6] has been applied up to now only to the $N_{t}=2$ case. The interesting feature of this approach is that it also allows to extract the critical indices $\nu$ and $\gamma$. In particular $\gamma$ is obtained by looking at the variation of the critical temperature as a function of the ratio $\rho$ between the two couplings $\beta_{s}$ and $\beta_{t}$. In the region in which we are interested, which corresponds to $\rho=2$ in the notations of [6] the two indices agree with very high precision with the values: $\nu=1$, $\gamma=7/4$ (which imply $\eta=1/4$) that we have found and that are expected from the SY conjecture.

As $N_{t}$ increases all the theoretical approaches start to give diverging results. For these values of $N_{t}$ the only reliable tool to estimate the critical temperature are the Montecarlo simulations. The existing results are reported in tab.9 and compared with our ones. All the estimates quoted in tab.9 except our ones have been obtained by using the method of Binder cumulants. As it can be seen all the estimates agree perfectly within the errors. Notice however that our results are in general one order of magnitude more precise than the others. This seems to indicate that the method that we have used in this work (which was originally proposed in [15]) is indeed superior to the standard Binder cumulants method.

Acknowledgements

We thank F.Gliozzi, K.Pinn, P.Provero and S.Vinti for many helpful discussions. M. Hasenbusch expresses his gratitude for support by the Leverhulme Trust under grant 16634-AOZ-R8 and by PPARC.

Tables

.

[TABLE]

Tab. 1*   Results of the crossing analysis for $N_{t}=6$. In the first column the two values of $N_{x}$ used to extract $\beta_{c}$, which is reported in the second column. In the last column the value of $Z_{a.p.}/Z_{p.}$ at the crossing point.*

[TABLE]

Tab. 2*   $\beta_{c}$ values for $N_{t}=6$ and various choices of $N_{x}$, obtained by imposing $Z_{a.p.}/{Z_{p.}}=0.37288488...$. In the last column is reported the slope of $Z_{a.p.}/{Z_{p.}}$ at the critical point.*

[TABLE]

Tab. 3*   $Z_{a.p.}/Z_{p.}$ values as a function of the lattice shape. In the third column the values measured in the MC simulation, the first error is the statistical error, the second takes into account the statistical error in the estimate of $\beta_{c}=0.228818(4)$. In the last column the values of eq.(21).*

[TABLE]

Tab. 4*   $\beta_{c}$ values for various $N_{t}$. In the first column the inverse temperature $N_{t}$, in the second column the set of lattices sizes which have been simulated, in the third column our best estimates for $\beta_{c}(N_{t})$ and in the fourth column the corresponding dual values $\tilde{\beta}_{c}(N_{t})$. In the last two column are reported for comparison some existing data for $\beta_{c}(N_{t})$ taken from ref [2] (Montecarlo simulations) and ref [4] (strong coupling expansion).*

[TABLE]

Tab. 5*   The critical index $\nu$ and the constant $1/c_{1}$ as a function of the pair of $N_{t}$ values listed in the first column. In the second and third column the results obtained by using eq.(24). In the last two columns those obtained by using eq.(25).*

[TABLE]

Tab. 6*  The adimensional ratio $T_{c}/\sqrt{\sigma}$ for various values of $N_{t}$. In the second column the critical couplings, in the third column the corresponding values of $\sigma$ interpolated according to eq.(27). The reference values of $\sigma$ for the interpolation are taken from tab.2 of ref. [23] for $N_{t}=6,7,8$ and $10$, while for $N_{t}=12$ we have used the value for $\beta=0.224$ reported in [13].*

[TABLE]

Tab. 7*  The $\nu$ and $\eta$ indices for various values of $N_{t}$. In the second column the pair of $N_{x}$ values used to extract the index (see text). The susceptibility has been evaluated only in a subset of cases.*

[TABLE]

Tab. 8*   Comparison of $\beta_{c}$ values obtained with different theoretical approaches for small values of $N_{t}$. In the first column the inverse temperature $N_{t}$ and in the second column our best estimates for $\beta_{c}(N_{t})$. In the following columns the $\beta_{c}$ values obtained with the mean field transfer matrix approach [5], the Müller-Hartmann-Zittarz method [7], the constrained variational approach [6] and strong coupling expansions [4].*

[TABLE]

Tab. 9*   Comparison of $\beta_{c}$ values obtained with Montecarlo simulations by different groups. In the first column the inverse temperature $N_{t}$ and in the second column our best estimates for $\beta_{c}(N_{t})$. In the following columns the $\beta_{c}$ values obtained in [24], [2] and [25] respectively.*

Figure Caption

.

Fig. 1*   $Z_{ap}/Z_{p}$ values as a function of the ratio $N_{y}/N_{x}$ for $N_{t}=6$, at the critical coupling $\beta_{c}=0.228818$. The crosses denote the numerical estimates reported in Tab. 3, while the solid line is the theoretical expectation according to eq.(21). Errors are not reported since they are smaller than the symbols.*

.