Vortex condensation in two-dimensional non-abelian spin models

Volume 112B, number 4,5 PHYSICS LETTERS 20 May 1982

VORTEX CONDENSATION IN TWO-DIMENSIONAL NON-ABELIAN SPIN MODELS

Sorin SOLOMON and Yoel STAVANS Department o f Nuclear Physics, Weizmann Institute o f Science, Rehovot, Israel

and

Eytan DOMANY Department o f Electronics, ICeizmann Institute o f Science, Rehovot, Israel

Received 12 February 1982

We study by Monte Carlo simulation a family of two-dimensional models of coupled "matter" S and gauge a = -+ 1 fields. As the gauge coupling hvaries, the model interpolates between the Helsenberg (h = ~) and P2 (h = 0) models. We present evidence for vortex condensation at f'mite temperatures Tc(h), below which the thermodynamic functions have the same behavior as those of the Heisenberg model. For a range of h values a first-order transition is found at Tc(k). Since similar results were found for the same model with X Y spins known to have a Kosterlitz-Thouless transition, we cannot rule out the existence of an infinite-order transition at Tc(h) for the model with three component spins.

A few years ago it was suggested [1-4] that the global topology of the gauge group has a strong in, fluence on the phase diagram of four-dimensional gauge models. The scenario was that the dynamics of models with nontrivial topological groups (especially zrl) was governed by their (singular) topological ob- jects (vortices). This suggestion triggered increasing interest in studies of phase diagrams of four-dimensional gauge theories and two<limensional spin models believed to be their analogs [5-9] . For the abelian models it was indeed shown that vortex Conden- sation takes place at a finite value of the coupling constant, and that a phase transition is induced [10 -12 ] . However, the abelian models happened to ex- hibit asymptotic slavery (algebraic decay of correla. tions at low temperatures). This fact may have been as relevant to the occurrence of the phase transition as the vortices themselves [7].

In (continuous) non-abelian models there is evidence for asymptotic freedom (disorder at arbitrarily low temperatures) [13-16]. There are many argu- ments to the effect that this may induce the condensation of vortices already at T c = 0, and prevent in this way the occurrence of a phase transition [7,17]. However, no rigorous proof was ever presented.

Thus commonly accepted facts and beliefs could

be summarized as follows: (1) Strong support for vortex condensation at

finite temperatures in abelian theories. (2) Doubts as to whether the mechanism survives

in non-abelian theories, substantiated by considerable evidence for many non-abelian theories having only one disordered phase. However, no well-established results existed for non-abelian theories with vortices.

In a previous letter [17] it was suggested that the way to decisively resolve the dynamical role of vortices is to compare models that have identical local topology and dynamics, but differ in global topology. Among the simple examples of such models were SU(N)/Z N versus SU(N) and in particular S0(3) versus SU(2). The P2 model [called there O' (3)] was then compared to the 0(3) (Heisenberg) model through a hamiltonian strong-coupling expansion [5]. In the P2 model, two neighboring spins have the same energy when one of the pair is reversed, i.e.:

E(Si, ~ ) = E(Si, -~ ).

The Pad~ approximants showed a zero of the/3 function for the P2 model, in contradistinction to the lack thereof in the 0(3)model. Thus, it was concluded that these two models have indeed very different phase structures.

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Subsequently, even stronger indications of sharp differences between the SO(3) versus SU(2) gauge models in four dimensions were found using Monte Carlo methods [18-21] . On the other hand, extensive numerical work on the P2 model in two dimensions did not reveal energy hysteresis, thus excluding the possibility of a first-order transition [22]. We present here MC results fo ra family o fP 2 spin models in two dimensions, which clearly indicate a dramatic increase in the vortex density over a very narrow temperature range. We interpret this as a vortex condensation transition. We do not attempt to decisively resolve here the question whether this sharp condensation is accompanied by a "thermodynamic" phase transition or not, but rather, present our numerical results. Furthermore, we argue that on the basis of Monte Carlo simulations this question cannot be decisively resolved, for reasons discussed below.

We study the thermodynamic functions (behavior) • of a family of models interpolating between the P2 model and the 0(3) (Heisenberg) model. The Heisenberg model is non-abelian, does not have vortices and has no transition. The P2 model is non-abelian, has the same local topology as the Heisenberg model but it does have vortices. Since the parameter X of our family of models controls the chemical potential of the vortices, its variation allows a smooth interpolation between the P2 and Heisenberg models. We also present results for the X Y analog of the above family. The X Y model is abelian, has topological excitations, and is known to have a transition. However, our family of models allows us to study the effect of varying the chemical potential of the half-charged vortices (while keeping that of the integer vortices constant).

The models considered and their respective thermodynamic functions are defined as follows:

The basic variables of our model are an n-component unit vector field S(r), and an Ising-like gauge field o(r, r +/2); the former are associated with the sites r, and the latter with the finks (r, r + p) of a square lattice.

Defining for each plaquette the product

7r(r) = o(r, r + p) o(r + p, r + p + ~)

X o(r +/~ + 0, r + ~) o(r + P, r ) , (1)

we can write the hamiltonian of our model as

~C=- ~_1 S(r )o(r , r+fOS(r+f~) -X~lr ( r ) . (2)

When n(r) = - 1 , the plaquette is "frustrated, and one expects a vortex centered on the plaquette.

To see this note that in order to have a single frustrated plaquette at r, one has to introduce a string of a(r') = - 1 values of length L extending from r to the boundary of the system. The lowest energy state of the S variables with this assignment of o variables is a vortex configuration where S rotates by ~r as one tra- verses a path around the frustrated plaquette. The energy of this P2 vortex is "~ln L, while for comparison, the ferromagnetic state has energy "~L.

Consider some special cases for n = 3. For X = 0, by summing over o one finds that the model is equiv- alent to the P2 with interaction In cosh (S(r).S(r + It)). For X = 0% no vortices are allowed and the system can be gauged to a configuration with all the links positive, that is to the 0(3) model.

The thermodynamic functions we compute are: (1) The specific heat per spin

cN _~2 C-NXB N [<9~2)-<9~ N)21, (3)

where.l~ = 1/kBT and the brackets ( ) indicate ther- mal averages:

(0) = ~ 0 [{S}, {a)] exp{-/3~}/Z. {S(r),,,(r,r')}

(2) The measure of fluctuations of the tensor:

Q~¢(r) = S(CO(r)S(¢)(r) 5- ,

as given by

(4)

i

and in particular the susceptibility

(5)

X = ~ X at~t3- a,/3

We compute also the vorticity:

1 V = ~ ~ [1 - ~r(r)]/2.

(6)

(7)

We studied the model (2) also with n = 2, i.e. X Y

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variables, in this case, for ~ = 0 the effective interaction is In cosh(S(r)" S(r')). This allows for configurations with half integer topological charge. Again, such configurations are associated with frustrated plaquettes. For X = oo the model has effective interaction S(r),S(r') and admits only integer charged topological configurations.

Our results are presented by means of figs. 1 - 5 shown below. The main features we found can be summarized as follows:

Ground state. For ~ < X1 = - ( 1 - l /x/2) the ground state is one with all plaquettes frustrated; the spins form a pattern of alternating right- and left-handed vortices, with (SFS/.)2 = 1/2 for all near- est neighbor pairs. When ~ > k l the ground state is one in which all plaquettes are non-frustrated, and all spins are aligned.

First-order transition. For a range of values, ~1 < ~ <~ ~2 ~ - 0 . 2 6 , we observe a £trst-order transition as a l ine/ 'cOt ) is crossed between a phase with vanishingly small vorticity (V = 0) and a phase with practically all plaquettes carrying a vortex (V = 1). This first-order transition shows a clearly defined temperature hysteresis loop; two phases, with V ~ 0 or +1 "coexisting" over a wide range of temperatures. Simulations starting at high temperatures and slowly cooling, lead to the V = 1 state, which is meta- stable; on the other hand, starting in the ground state and heating the system slowly, one follows different V(T) and E(T) curves (fig. 1). It is interesting to note that such a first-order transition was observed for the X Y version of the model as well; in that case it is a first-order transition from a disordered to a "massless" phase. We understand this first-order transition in the fOllowing manner. It appears for h < 0; therefore the gauge coupling term in ~g prefers frustra- tion: however, the energetic cost (caused by the S ' S term; for a frustrated plaquette, not all bonds can be satisfied) is too high, and the ground state is unfrustrated. As the temperature is raised, spins tend to disorder, thereby decreasing this cost or "penalty" for frustrating a plaquette. However, once some plaquettes are frustrated, they prefer to be grouped in large clusters. Thus, opon heating, we observe a sud- den jump of the vorticity; V jumps from practically zero to practically one (fig. la). The temperature at which this occurs is interpreted as Tc('h ).

Vortex condensation. For k > k 2 no hysteresis

V

I.O

0,5

0

Q5 r

O.4

[ I

X =- 0.27 • cooling

(a) o.oo-o~

I ; I 0.1 0.2 Q3

T

X =- 0,27

• cooling o heating

0.3

<E>

0.2

0.1

,//" (b) I I

0.1 0.2 Q3

T

Fig. 1. First-order t ransi t ion in the mode l o f eq. (2); T -1 is the coefficient of the SoS o term. Upon heating and cooling the system follows different vorticity (a) and energy (b) curves.

is observed. However, we fmd that for temperatures T < To(k), vorticity is practically zero, and starts to depart from this value over a narrow temperature range. We interpret this as a vortex condensation transition; the line Te(k ) is shown in fig. 2. The state- ments made above are correct "for both n = 3 and n

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0.6

0.4

0.3

0.2

O.I

I I I

o ~, : 0.0 (n:2)

• ,~=-0.15 (n:3)

• X= 0.0 (n=5)

+ X=0.5 (n=3)

• *. )',= 1.0 (n=3)

I I I I I

QI 0.2 0.3 Q4 Q5 0.6 0.7 0.8 T

Fig. 2. Vorticity as a function of T for various values of X, for the XY (n = 2) and the Heisenberg (n = 3) cases.

= 2. However, in the XY case the increase of vort ici ty as a function of temperature is less dramatic. Note also that , in the X Y case, we measure only vorticity associated with +½ topological charge. On the basis o f our data we cannot say whether there is a sharp transit ion at Te(X ) from a phase with an exponen- tially small number o f vortices that are t ightly bound in pairs, to a phase where some pairs have dissociated. This is known to be the case for the XYmode l . Never- theless, we dear ly see that below Te(X ) vort ici ty i s negligible, and therefore for T < Tc(X ) the model (2) wi th f'mite X should behave precisely as the model with X = ~ . This is indeed the case, as seen from our calculation of various thermodynamic func-

tions. For the X Y case Koster l i tz -Thouless transitions

occur on bo th the X = 0 and X = oo lines, associated with condensation of vortices wi th topological charge of +_.{, or +1, respectively. Thus the phase diagram in- dicated on fig. 3a is expected; as long as To(k) < TKT CA = oo), the transit ion follows Tc(~). But when Te(h), i.e. the temperature at which vortices wi th +{ charge become abundant , is above TKT(h = ~ ) , w e expect

/.0

0.5

I I I I I

X Y ( X ) Models wl

• 411 iii

I / _ / _ J _ $ 3_ 0.2 0.4 0.6 0.8 f~.O

T

1.0

0.5

I I i 1 I i

Pz IX) M o d e l s

b)

0.2 Q4 0.6 0.8 1.0 1.2 T

Fig. 3. (a) Phase diagram of the XY(h) model: the massless phase is separated from the condensed phase by a line of transitions. For low temperature there is a short segment of a first-order line that turns into a line of Kosterlitz-Thouless transitions. For T < TKT, this line is expected to lie close to the solid line in the figure, at which vortices of ±{ charge condense. As TKT is approached, the phase boundary is ex-

• peered to lie close to the dashed line. (b) Phase diagram of the model (2) with Heisenberg spins. At low temperatures a first-order transition is observed. The solid line corresponds to numerically observed vortex condensation; however, it is not clear whether a real transition (i.e. singularities of some thermodynamic functions) is associated with this line.

the transit ion line to be near TCA) = TKTCA = oo). This is so since for T < TcCA ) we expect the model to behave like its ~ = oo version, which is, in the X Y case, known to have a transition at TKT ~ = o~).

Thermodynamic functions. We measured the spe.

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cific heat and susceptibility of our system. It should be noted that we calculated × using eqs. (5),(6), and assuming (Q~a) = 0. For temperatures below TccA) we indeed found X to coincide (within numerical accura- cy) with that of the appropriate ~ = oo (Heisenberg or X Y ) model. It is important to note that in the vicinity of TccA) (for most ~ values studied) the correlation length ~H of the Heisenberg model is very large. For example, for the P2 model C A = 0), To(0) -~ 0.2, and for this value [16], ~H "~ 0.01 exp(2zr/T)/(1 + 27r/T)

1010. Thus it is no wonder that in this region, numerical simulations of systems with linear size L < ~H show very strong scaling of X with size. However, this scaling does not mean at all that the P2 model has a massless low-temperature phase; it only means that for T < TccA), with negligible vorticity, the model is identical to the Heisenberg model which happens to have a very large ~JH (small mass). Our data for X are presented in fig. 4; note that for each X, X indeed

joins smoothly that of ~ = oo in the vicinity of TccA) as determined from fig. 2. We found some interesting structure of the specific heat. For k = o% C,~ (T) has a single peak. We noticed that vortex condensation is accompanied by another peak that follows TccA) as

is varied. Thus, for example for k = 5 we find two peaks of C s (T); the one at low temperature is that of the Heisenberg model, and in addition, another appears near To(5 ). Neither peak scales with size, and therefore Cx(T ) does not diverge in the thermodynamic limit. Some of our specific-heat data are presented in fig. 5.

Details o f Monte Carlo procedure. A standard Metropolis procedure was used, with periodic boundary conditions. The spins S(r) ar e updated by visit- ing sequentially each site r 0. Given the old value Sold(r) of the spin to be updated, the candidate for the new value is chosen with equal probability among the unit vectors S that fulfill the condition IS'Sol d (r)l

5 0 0 ~ - - ~ '

• ,%. = ~0 2 0 x 2 0 v X=0.6 2 0 x 2 0 x k=O 2 0 x 2 0

;k=cO IOx I0 X ~ ~ ),.=0.5 IOx I0 .oo , \

5(?

×

'i

o.2 o:4 o16 0.8 Lb "1"

Fig. 4. Susceptibility versus temperature for the model of eq. (2) with Heisenberg spins, for various values of X and two lattice s/zes. For temperatures below Tc(X), i.e. when the vorticity drops to zero, the susceptibility becomes practically identical to that of the model with ~ -- ~; precisely in this region scaling with size (due to the large correlation length of the Heisenberg model) is observed.

t I /2" I I I

• X=0

o X=I

. X=5

\+ "-+.~

l o.z o.4 o6 o.8 "~ 2 ~ 6 8

T

Fig. 5. Specific heat per spin for the model of eq. (2) with Heisenberg spins. For X ; 0 and 1 the peak occurs near Tc(~.), the temperature where Harp increase of vorticity was observed. For increasing 2, values this peak moves to higher temperatures (Tpeak _~ 3 for ?~ = 5) and the low-temperature peak, characteristic of the X = oo (Heisenberg) model appears as well. None of the observed peaks scale with size.

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> p. In some runs the condition was S.Sold(r) > p. No significant differences were observed between these two versions.

During part of the runs the value o f p was fixed at the beginning of the run. In the rest of the runs, the value of p was floating as to insure an acceptance ratio between 0.15 and 0.4. No difference was observed between the two cases. However, in the me- tastable regions, different values o f p seem to introduce a bias by giving preference to one or another of the competing phases.

Af te r updating the spin S on a given site r, the o's associated with adjacent links are updated. In some of the runs, all four links (oi, i+# , {li.i+v ) are submitted to updating attempts. In the rest o t the runs, only the two links oi, i+t~and oi,i+vwere updated. No significant differences were obtained between these two pro- cedures. After updating each degree of freedom, the new values o f the extensive physical quantities asso- Ciated with the new lattice configuration were computed.

The mean values o f these quantities and their fluctuations were computed over different sequences of the total MC generated Markov chain.

The reliability of the results was established by comparing values of physical quantities provided by independent MC sequences.

For high precision computations of X, C,E and V we used runs that reached 50000 steps/spin. For the study of hysteresis we used cooling/heating rates of AkT/(steps/spins) <<. 0.005/104.

Summary of results. We found a first-order transition for a range of negative values, both for the X Y (n = 2) and Heisenberg/P 2 (n = 3) case. In the former case this is a first-order transition between a disordered and a massless phase. F o r n = 3 this transition line separates two phases with exponential decay of correlations, but one with a very low, and the other with a very high vortex concentration. This first- order transition disappears for ~ > X2; for such values we fired a line Tc(X) such that for T < Tc(~) the vortex concentration is practically zero, and therefore all thermodynamic functions are those of the appropriate ~ = oo model. For the X Y case this implies that the massless and disordered phases are separated by a line TKTCA ) given, to a good approximation by TKT(X) = Min(Tefh), TKT(~k = oo)). For the n = 3

case, although we cannot completely rule out the existence of a transition on the Tc(X ) line, our data suggest that for X > X 2 (i.e, the end of the first-order line) no sharp phase transition exists.

While this work was completed, we received pre- prints from Fukugita et al., and from Kogut and Sinclair, who studied closely related models by Monte Carlo. Our results are in agreement with these pre- prints.

We thank T. Banks, H, Meirowitz, H. Rubinstein and A. Schwimmer for useful discussions. This re- search was supported in part by the US-Israel Bi- national Science Foundation, Jerusalem, Israel.

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Vortex condensation in two-dimensional non-abelian spin models