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Physica A 321 (2003) 108–113www.elsevier.com/locate/physa

Duality and multicritical point oftwo-dimensional spin glassesHidetoshi Nishimoria ;∗, Koji Nemotob

aDepartment of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku,Tokyo 152-8551, Japan

bDivision of Physics, Hokkaido University, Sapporo 060-0810, Japan

Abstract

We present a conjecture on the exact location of the multicritical point of models of spinglasses on the two-dimensional square lattice using duality transformation. The results not onlyagree very well with numerical estimates but also conform excellently to the existing knowledgeon annealed systems and to the arguments using gauge symmetry.c© 2002 Elsevier Science B.V. All rights reserved.

PACS: 05.50.+q; 75.10.Nr

Keywords: Spin glass; Ising model; Potts model; Multicritical point; Conjecture; Exact location

1. Introduction

Determination of the precise location of the multicritical point is a target of ac-tive current research in the theory of spin glasses [1–11]. There exist, however, onlyvery limited number of exact results known on this and related problems for ;nite-dimensional spin glasses [12]. In the present contribution, we develop a duality argu-ment to predict the location of the multicritical point in models of spin glasses on thesquare lattice [13]. We shall present several reasons that our conjecture can well bethe exact solution to this problem.

2. Duality for the random Zq model

The ;rst system we treat is a random Zq model with gauge symmetry whichincludes the ±J Ising model and the Potts gauge glass. Following the notation of

∗ Corresponding author.E-mail address: nishi@stat.phys.titech.ac.jp (H. Nishimori).

0378-4371/03/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.doi:10.1016/S0378-4371(02)01791-0

H. Nishimori, K. Nemoto / Physica A 321 (2003) 108–113 109

Wu and Wang [14], the partition function is

Z =∑{�i}

exp

∑〈ij〉V (�i − �j + Jij)

: (1)

Here, �i(=0; 1; : : : ; q−1) is the q-state spin variable, Jij(=0; 1; : : : ; q−1) is the quenchedrandomness, and V (·) is a periodic function with period q. The sum in the exponentruns over neighbouring sites on the square lattice. The ±J Ising model correspondsto q = 2, and the Potts model with random chirality [15] is recovered by settingV (x) = �x;0 (mod q).

It is straightforward to generalize the duality transformation in the Wu–Wang for-malism [14] to the random model, and the result is

Z =∑{�}

′exp

∑〈ij〉V (�ij + Jij)

= qN∑{�}

′exp

∑〈ij〉V (�ij) +

2�iq�ijJij

; (2)

where �ij denotes �i− �j and �ij is the bond variable on the dual lattice correspondingto �ij, and eV is the Fourier transform of eV . The prime in the sum indicates thatthe bond variables �ij and �ij are constrained such that their sums over a closed loopvanish [14]. Thus the duality transformation consists in replacing the exponential inthe middle expression of (2) with that of the ;nal one:

eV (�+J ) ↔ eV (�)+2�i�J=q : (3)

Complex interactions in the dual coupling for spin glasses can already be seen inthe usual duality relation for the coupling constant of the Ising model:

e−2K = tanhK : (4)

Replacement of K with −K on the right-hand side implies an addition of �i in theexponent on the left-hand side. Eq. (3) is a Zq-generalization of this relation.

3. Invariant subspace

Average over quenched randomness is dealt with by the replica method. A gener-alization of the existing argument [16] suggests that it is convenient to consider thefollowing quantity

�=

∑q−1l=0 pl

(∑q−1�=0 eV (�+l)

)n∑q−1

l=0 pl enV (l)

; (5)

110 H. Nishimori, K. Nemoto / Physica A 321 (2003) 108–113

where pl is the probability that Jij takes the value l. It is not diGcult to check that theduality transformation (3) changes � into �=qn=�. The subspace de;ned by �=qn=2 inthe parameter space (p1; p2; : : : ; T; : : :) is therefore left invariant by the duality transfor-mation: A point in this subspace is transformed into another point in the same subspacealthough the latter point does not necessarily have real values of Boltzmann weights.In general it is possible to consider complex values of Boltzmann weights (or param-eters such as the temperature); the subspace �= qn=2 embedded in the complex-valuedparameter space maps onto itself by the duality transformation (3). We shall considerthe cross section of this subspace with the phase diagram with real parameters, whichwe shall call the invariant subspace in the following.

One should be careful in identifying the invariant subspace with a critical surface[17]. We discuss this problem later; let us see here what the invariant subspace lookslike in the quenched system, which is obtained by taking the limit n→ 0 of the relation�= qn=2:

q−1∑l=0

pl log

q−1∑

�=0

eV (�+l)−V (l)

=

12

log q: (6)

4. Consequences

We now apply (6) to a few examples and discuss the consequences. The ;rst caseis the ±J Ising model (q= 2) with V (0) = �J , V (1) =−�J and p0 = p, p1 = 1− p.The invariant subspace (6) then reads

p log(1 + e−2�J ) + (1 − p) log(1 + e2�J ) = 12 log 2 : (7)

This curve is plotted in Fig. 1 in the p–T phase diagram. It is observed (and canbe con;rmed analytically) that the curve reaches the point with minimum p on theNishimori line [18] e−2�J = (1− p)=p, from which and (7), the minimum point pc isfound to satisfy

− pc logpc − (1 − pc) log(1 − pc) =log 2

2(8)

or pc = 0:889972. This coincides with numerical estimates of p at the multicriticalpoint with high precision: 0.8905(5) [1], 0.886(3) [2], 0.8872(8) [3], 0.8906(2) [4]and 0.8907(2) [5].

The next example is the q-state Potts gauge glass with V (0) = �J , V (1) = V (2) =· · ·=V (q−1)=0 and p0 =1− (q−1)p, p1 = · · ·=pq−1 =p, where we have followedthe notation of Jacobsen and Picco [6]. The invariant subspace (6) is, in this case

{1 − (q− 1)p} log{1 + (q− 1)e−�J} + (q− 1)p log(q− 1 + e�J ) =log q

2:

(9)

H. Nishimori, K. Nemoto / Physica A 321 (2003) 108–113 111

Fig. 1. Invariant subspace of the ±J Ising model is the curve drawn by the solid line. Shown dashed is theNishimori line.

This formula reduces to the Ising counterpart (7) when q=2 under appropriate changesof energy scale and notation (exchange of p and 1 − p). The curve speci;ed by (9)reaches the extremum point in the p–T phase diagram again on the Nishimori linep= 1=(e�J + q− 1) [15]. In particular, for q= 3, this extremum point pc satis;es

− (1 − 2pc) log(1 − 2pc) − 2pc logpc =log 3

2; (10)

whose numerical value pc = 0:0797308 coincides very well with the location of themulticritical point, 0.079–0.080 [6].

It is possible to apply the same argument to the random Ising model with a generaldistribution function P(Jij) of exchange interactions [16]. The result for the invariantsubspace is∫ ∞

−∞dJij P(Jij) log(1 + e−2�Jij) =

log 22

: (11)

This formula applied to the Gaussian model with P(Jij)˙ e−(Jij−J0)2=2J 2represents a

curve in the phase diagram similar to that of the ±J model in Fig. 1 and again hasthe smallest J0=J on the Nishimori line �J 2 = J0 [18]. This point is at J0=J = 1:02177which is very close to the multicritical point evaluated numerically (Horie, Nemoto,Hukushima and Ozeki, private communication).

5. Discussions

What do all these results mean? One possibility is that the exact locations of multi-critical points have been derived for the above models. The present argument using thesubspace invariant under duality is, in general, not guaranteed to give the exact phaseboundary or multicritical point. Nevertheless we have several reasons to conjecture thatour result may be exact, in particular for the multicritical point.

112 H. Nishimori, K. Nemoto / Physica A 321 (2003) 108–113

The ;rst reason is that the invariant subspace � = qn=2 coincides with the exactphase boundary for the ±J Ising model in the case of n = 1 and 2 [19]. It shouldbe noticed that this coincidence exists only above the multicritical point in the phasediagram for the n = 2 case because there are two mutually dual (but not self-dual)critical curves below the multicritical point. This fact may be related to the strangeshape of the curve below the multicritical point in Fig. 1. The second reason is theimpressive agreement with numerical estimates explained above, which seems to usbeyond an accidental coincidence. The third evidence is the fact that the Nishimoriline appears naturally from the present duality formalism. It is well established that themulticritical point is located on the Nishimori line for models with gauge symmetry[12]. The natural emergence of the Nishimori line from arguments without explicit useof gauge transformation strongly suggests that something deep may be hidden behindthe scene.

There are of course several problems that deserve special caution. As pointed out byAharony and Stephen [17], duality does not yield ;xed points of the transformation forrandom models whereas ;xed points of duality are often identical to critical points innon-random systems. Aharony and Stephen thus argued that duality in random systems,unable to identify ;xed points, is not a useful tool of analysis. We are suggesting herethat an invariant subspace, if not ;xed points, may be of some use in random systems.

Another problem is whether or not the whole invariant subspace (6) or (11) coincideswith the phase boundary. It is dangerous to accept this identi;cation at least below themulticritical point in the phase diagram because of the ±J Ising model with n = 2as mentioned above. The origin of disagreement of our result (7) near p = 1 withthe perturbative calculation of Domany [20] also needs explanation. It is possible thatonly the location of the multicritical point has been predicted correctly by the presentargument. Comparison of our formula (7) with numerical evaluations of the criticalcurve above the multicritical point [1,21] suggests that such may be the case. Sincethe multicritical point is a location where the enhanced symmetry on the Nishimoriline in the replicated system [19] or in the supersymmetric formulation [8,22] coexistswith the present symmetry under duality transformation, something special can wellhappen there.

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