a mathematical approach to universality in two dimensions

Physica A 279 (2000) 250–259www.elsevier.com/locate/physa

A mathematical approach to universality intwo dimensionsThomas Spencer∗

School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA

Keywords: Ising; Universality; Statistical mechanics; Renormalization

Dedicated to Joel Lebowitz with great admiration

1. Introduction

The nearest-neighbor Ising model in two dimensions was solved by Lars Onsager in1944 [1]. It was the �rst example of a model with a phase transition which could bestudied in detail and which yielded critical exponents di�erent from those of mean �eldtheory. Since Onsager’s seminal work, the model has been analyzed and reexpressedin many di�erent ways. Kaufmann [2] simpli�ed Onsager’s work using spinor repre-sentations and Schultz et al. [3] showed that the transfer matrix could be expressed interms of noninteracting fermions. Other derivations relied on the relation of the Isingmodel to complete dimer covers of an auxiliary lattice [4,5] or combinatorial methods[6]. See [7–9] for a review.All the above solutions rely heavily on the nearest-neighbor interaction together

with the absence of a magnetic �eld. Next to nearest-neighbor perturbations or 4 spinplaquette perturbations destroy the integrability of the model. Nevertheless, it is widelybelieved that such perturbations do not change the critical exponents. Thus, the longdistance behavior of correlations near Tc should be proportional to those calculatedby Onsager. This is a special case of the principle of universality which asserts thatcritical exponents and scaling limits depend only on a few general features of the modelsuch as symmetry and dimension. However, one must be careful about such assertionssince it is known that two independent Ising models in 2D coupled with a four spin

∗ Fax: +1-609 951 4459.E-mail address: [email protected] (T. Spencer)

0378-4371/00/$ - see front matter c© 2000 Published by Elsevier Science B.V. All rights reserved.PII: S 0378 -4371(00)00011 -X

T. Spencer / Physica A 279 (2000) 250–259 251

interactions equivalent to Baxter’s eight vertex model [10] which has non-universalbehavior even for small coupling.In this review, I shall describe recent, joint work with Haru Pinson [11] which

proves a form of universality for perturbations of the 2D nearest-neighbor model. Theseperturbations are required to be even and local. Under these conditions, we show thatthe speci�c heat and energy–energy correlations near Tc behave as they do in thenearest-neighbor case. We cannot yet treat spin–spin correlations.To state our results more precisely, let �⊂Z2 be a large box with periodic boundary

conditions. For each lattice site j∈� let �j =±1. The energy of a spin con�gurationfor a perturbation of the nearest-neighbor model is given by

H��(�; K) ≡ K

∑|j−j′|=1j; j′ ∈�

�j�j′ + �HI : (1.1)

The perturbation HI is required to be a local functional of an even number of spins.For example, a pair interaction plus a local 4 spin interaction may be written as

HI = a∑

i; j∈�Ji; j�i�j + b

∑p

∏j∈p

�j; (1.2)

where p is a plaquette and j∈p are the 4 vertices of the plaquette. The letters a and bdenote constants. The couplings Ji; j are assumed to be translation invariant and decayexponential fast in |i − j|.The partition function at inverse temperature K is de�ned by

Z��(K) =

∑�j=±1j∈�

eH��(�; K): (1.3)

For a periodic box � de�ne correlations

〈�A〉��(K) =∑

�=±1 eH��(�; K)�A

Z��(K)

: (1.4)

where �A =∏

j∈ A �j and A is a �nite subset of Z2. The subscript � is dropped in thelimit the box goes to in�nity.

Theorem 1. For small |�|; there exists Kc(�) depending smoothly on � such that

|〈�x�x′�y�y′〉(Kc)− 〈�x�x′〉(Kc)〈�y�y′〉(Kc)| ∼= C1(|x − y|+ 1)2

for some positive constant C1. Here; x and x′ denote nearest-neighbor sites.

Under very general conditions the free energy

f�(K) = lim�↑Z2

log∑

� eH��(�; K)

|�| (1.5)

exists and is independent of boundary conditions.

252 T. Spencer / Physica A 279 (2000) 250–259

For the nearest-neighbor case f0(K) has a singularity in K at exactly one temperatureKc(0). Moreover, the speci�c heat de�ned by

@2

@K2f0(K) ≈ Clog(|� − �0c |−1) (1.6)

has a logarithmic divergence.

Theorem 2. For |�| small and |K − Kc(�)|6|�|4,|〈�x�x′�y�y′〉(K)− 〈�x�x′〉(K)〈�y�y′〉(K)|6A1e−A2|K−Kc(�)‖x−y|; (1.7)

where the points x; x′ and y; y′ are nearest-neighbor pairs.

B1log(|K − Kc(�)|−1)6| @2

@K2f�(K)|6B2log(|K − Kc(�)|−1) (1.8)

for some positive constants A1; A2; B1; B2.

Remark. By using transfer matrix methods we can also obtain a lower bound on theenergy-energy correlation when x and y are in lattice directions. This shows that thecorrelation length exponent �= 1 in the even sector.

2. Outline of Proof

The nearest-neighbor Ising model in 2 dimensions can be expressed in terms of“Gaussian” Grassman integrals. This means that the action is quadratic in the Grass-mann �elds. Grassmann �elds are anti-commuting variables which provide a very con-venient algebraic formalism for dealing with determinants and Pfa�ans. Perturbationsof the nearest-neighbor case can also be expressed in terms of Grassmann integralsbut the perturbation introduces terms of degree 4 and higher in the action. Thus, themodel is no longer solvable. Nevertheless, we prove [11] that such perturbations areirrelevant provided that there are no more than 2 independent Grassmann �elds perlattice site. In the case of Baxter’s eight vertex model, 4 independent Grassman �eldsper site are required.Our proof uses a rigorous renormalization group analysis and follows the formalism

developed by Feldman et al. [12] which seems well suited to our problem. Thesetechniques have their roots in the earlier work of [13,14] on the Gross–Neveu quantum�eld theory. See also [15]. Since we are working with Grassmann variables, there areno “large �eld” problems and the Hadamard inequality is used to estimate remaindersand large products of �elds.

2.1. Grassmann Integrals

Grassmann variables j indexed by j∈{1; 2; : : : ; N} are anticommuting variablessatisfying

j1 j2 + j2 j1 = 0; 2j = 0: (2.1)


The j commute with scalars and even polynomials in . All functions of { j} arepolynomials. For an ‘-tuple J = ( j1; : : : ; j‘) we de�ne the monomial

J = j1 j2 : : : j‘ ; and set |J |= ‘: (2.2)

If |J |=0; J ≡ 1. The Grassmann integral ∫N − ≡ ∫ − d N · · · d 1 is a linear functionalde�ned on polynomials in by∫

N

I = 0 06|I | ¡ N

= 1 I = (1; 2; : : : ; N ): (2.3)

Let A be an N × N anti-symmetric matrix and set

( ; A ) =N∑j; k

Ajk j k : (2.4)

If N = 2n we have∫Ne1=2( ; A ) =

∫N

∏j1¡j2

(1 + Aj1j2 j1 j2 )

=12n

1n!

∑�

(−1)�A�(1)�(2)A�(3)�(4) : : : A�(2n−1)�(2n)

= Pf (A): (2.5)

The sum is over all permutations � and Pf (A) is the Pfa�an of A. Recall that Pf (A)2=det A. If N is odd (2.5) vanishes.We de�ne the analogue of the Gaussian expectation for Grassmann variables with

covariance C =−A−1,

〈P〉C =∫

P d�C( ) = Pf (A)−1∫NP e1=2( ; A ); (2.6)

where P is an even polynomial in . We refer to (2.6) as the “Gaussian” integral orexpectation of “covariance” C since

〈1〉C = 1; 〈 j j2〉C = C( j1; j2): (2.7)

To de�ne a generating function, let �j denote an independent family of anti-commutingvariables. Then as in the Gaussian case we have

〈e∑

�j j〉= exp12 (�; C�): (2.8)

By di�erentiating (2.8) in � and then setting �=0 we can calculate all correlations in . Derivatives act linearly on polynomials in j. If F is a monomial with a k factor,write F = kF ′. Then @F=@ k = F ′. If F has no k factor then @F=@ k = 0.

2.2. The Partition Function

To obtain a Grassmann representation for the partition function of the nearest-neighborIsing model, we associate to each lattice site 4 independent Grassmann variables,


(Hx; �Hx; Vx; �Vx) ≡ �x and de�ne an ultralocal “Gaussian measure” d�(�x) by

d�(�x) = exp[ �HxHx + �VxVx + �Vx �Hx + Hx �Vx + Vx �Hx + VxHx] dHx d �Hx dVx d �Vx:

(2.9)

The bar superscript just labels an independent Grassmann variable and∫�i

x�jx d�(�x) =±1 for i 6= j: (2.10)

Consider the nearest-neighbor model with variable coupling Kb, where b= ( j; j′) is anearest-neighbor bond. Let tb = tanhKb. Then the partition function is

Z�({Kb}) =∑

�j=±1

∏b⊂�

eKb�j�j′

=∏b⊂�

cosh Kb

∑�j=±1

∏b

(1 + tb�j�j′)

= 2|�|∏b⊂�

cosh Kb

∫ ∏x

etx; x+1 �HxHx+1+tx; x+i �VxVx+i d�(�x): (2.11)

Here x + 1; (x + i) is the site one unit to the right (above) x. The integrability of themodel is re ected by the quadratic nature of our expression in Grassmann �elds.

Remark. To establish (2.11), we expand both sides in powers of tb = tjj′ . The mainproblem is to show that each term contributing to the right side of (2.11) is positive.This can be done in two dimensions using Kasteleyn’s lemma. See [4,5,7,11,16]. Thethree-dimensionsal Ising model also has a Grassmann integral representation [16] butthe action is not quadratic but quartic.

For K constant, we may write the last factor of (2.11) in the form∫exp(�; DK�)

∏x

d�x: (2.12)

The matrix DK is a �nite di�erence Dirac operator whose inverse is the “covariance” ofthe “Gaussian”integrand (2.11). For constant K the inverse can be explicitly computedin Fourier space and is bounded except when K = Kc. When K 6= Kc; D−1

K has aGreen’s function decaying at an exponential rate proportional to m(K) ∼= |K − Kc|,

|D−1K (x; y)| ∼= exp(−m(K)|x − y|): (2.13)

At Kc we have

|∇dxD

−1Kc(x; y)| ∼= Const

(|x − y|+ 1)d+1 (2.14)

so each derivative of D−1Kcproduces an extra power of (|x−y|+1)−1. If K is random,

then the corresponding �nite di�erence Dirac operator has random coe�cients. Theanalysis of such operators has been carried out when the randomness is constant alonglattice lines [7].


For �= 0, the local energy �x�x+1 corresponds to

�x�x+1 → cosh−2(K) �HxHx+1 + tanhK (2.15)

in the Grassmann variables. This can easily be seen by di�erentiating the left-handside of (2.11) in Kx; x+1. Spin-spin correlations are non-local and hence more di�cultto analyze. Using the above correspondence and (2.14) we see that at Kc

〈�x�x+1�y�y+1〉 − 〈�x�x+1〉〈�y�y+1〉=[coshKc]−4[〈 �HxHx+1Hy �Hy+1〉C − 〈Hx �Hx+1〉C〈Hy �Hy+1〉C]∼= Const(|x − y|+ 1)2 ; (2.16)

where C =D−1Kc.

Perturbations of the nearest-neighbor models can also be expressed in Grassmannvariables using (2.11). For example, the plaquette perturbation described in the intro-duction can be written as exp �

∑p Fp(�) where Fp is a polynomial of degree 4 in �x

where x belongs to the plaquette p.Before proceeding with the analysis of the perturbation F , we �rst make a linear

change of variables

Hx; �Hx; Vx; �Vx → x; � x; �x; ��x: (2.17)

For � = 0, the �, �� variables are massive for all K and they remain so even with asmall perturbation �F . Thus, we can integrate the � �elds leaving us with a ”Gaussian”measure d�0( � ; ) and an interaction W 0( � ; ). This reduction to two independent�elds is an important step which cannot be performed for Baxter’s eight vertex model.Let C0(p) be the Fourier transform of the “covariance” of d�0( ; � ). It has at most

a single pole near p= 0 where it takes the form

C0(p) =D(�)p2

(ip1 + p2 im0(K; �) + O(p2)

−im0(K; �) + O(p2) ip1 − p2

): (2.18)

We may assume that W 0 has no quadratic terms in ; � since they may be incorporatedinto the measure of d�0. To leading order the critical temperature K is the root ofm0(K; �) = 0. The decay properties of C0 are also given by (2.13) and (2.14).In order to understand why W 0( ; � ) is irrelevant at the perturbative level, consider

a quartic monomial in � ; localized near x:

Wx = x x′ � x� x′ : (2.19)

Since we have only two independent Grassmann �elds and 2x = � 2x = 0

Wx = x(∇ )x � x(∇ � )x (2.20)

and therefore the Wick ordered Wx correlations have rapid decay

|〈: Wx : : Wy :〉C |6qConst

|x − y|p ; (2.21)

where p= 8. Each �nite di�erence (∇ ) produces an extra factor of |x − y|−1. If Wwere a product of independent �elds, no derivatives could be introduced and p= 4


in (2.21) indicating that our perturbation is marginal. The mass contribution m(k) � is relevant but its coe�cient is de�ned to vanish at Kc. Terms of the form ∇ aremarginal and can be absorbed into the Gaussian measure. Their only e�ect is to makea slight adjustment of the prefactor D in (2.18).

2.3. A renormalization group step

We now sketch the structure of a renormalization group step. The covariances andrenormalized actions Wn are de�ned inductively. Given a covariance Cn we decomposeit into the fast and slow modes at scale Ln:

Cn = Cfn + Cs

n; (2.22)

where the Fourier transform C(p)

Cfn∼= (1− e−L2np2 )Cn(p) (2.23)

is a good approximation to Cfn for small p. The length L¿5 and � must be chosen so

that �L2 is small. Let a denote a Grassmann �eld. If d�fn (a) is the ”Gaussian measure”

of covariance Cfn , then∣∣∣∣

∫axay d�f

n (a)∣∣∣∣ = |Cf

n (x; y)|

6CLn e

−C′=Ln|x−y| (2.24)

with C and C′ positive constants. Moreover, we have∣∣∣∣∫

aJ d�fn (a)

∣∣∣∣6( CLn )|J |=2L−nd( J ) ≡ Sn( J ); (2.25)

where recall (2.2), |J | is the number of �elds in aJ and d(J) is the number of derivativesappearing in aJ . (These derivatives do not explicily appear in (2.1).)We de�ne W

n+1to be

eWn+1(c) =

∫eW

n(c+a) d�fn (a)∫

eWn(a) d�fn (a)

; (2.26)

where c and a denote independent Grassmann �elds. To leading order Wn+1

is

0Wn+1(c) =

∫(Wn(c + a)−Wn(a)) d�f

n (a) (2.27)

and the leading mass term MWn+1 and DWn+1 are de�ned by

MW = im∑

� x x = im( � ; ) (2.28)

and

DW = ( � ; A · ∇ ) + ( ; B · ∇ ) + ( � ; �B · ∇ ); (2.29)


where (; ) is the l2 inner product and A and B are vectors in R2. The mn+1(K) andthe coe�cients An+1; Bn+1; Cn+1; �Dn+1 and their complex conjugates are chosen so thatif W2 denotes the terms of W of degree 2 then

0Wn+12 ( � ; )− DWn+1 −MWn+1

may be expressed in terms of second-order derivatives of the �eld x. Now wede�ne

Wn+1 = Wn+1 − DWn+1; (2.30)

d�n+1 =eDWn+1

d�Csn∫

eDWn+1d�Csn

(2.31)

and Cn+1 is the covariance of d�n+1. The norm ‖ · ‖n at scale n is de�ned by

‖Wnl; d‖n =

∑|J |=l

d( J )=d

|Wn( J )|Sn( J )eC′=Ln−1diam( J ): (2.32)

Here∑′ is the sum over the multi-index J with one point of J , say j1, �xed and

diam( J ) is the diameter of {ji} in J . Recall S( J ) is de�ned by (2.25).Our main estimate, proved by induction on n, is that for � small

L2(n−1)‖Wnl ‖n6

�DLn�l (2.33)

for all terms except the leading mass term MWn. This estimate is used to prove that atlong distances the Wn become irrelevant so that the long distance behavior is governedby the “Gaussian” Grassmann integral. See Ref. [11] for details.

3. Continuous spin models

The preceeding analysis may be adapted to certain continuous spin models on Z2.Let sj denote a real valued spin and de�ne the Hamiltonian

H� =12

∑

j∈�(∇s)2j +

∑j∈�

V (sj)

(3.1)

with V a symmetric double well potential. For example the potentials

V1(sj) = �(s2j − b2)2; (3.2)

e−V2(sj) = e−�(sj+b)2 + e−�(sj−b)2 (3.3)

both give rise to local Gibbs distributions that are sharply concentrated near sj = ±bwhen � is large. In the case of V2, the partition function

Z� =∫e−H�(s)�j∈� dsj (3.4)


can be evaluated as an Ising sum proportional to

∑�j=±1

exp

[∑�

Ji; j�i�j

]; (3.5)

where

Jij =b2�2

−�+ �(i; j) (3.6)

and � denotes the lattice Laplacian. Each �j =±1 corresponds to the �rst or secondterm of the right-hand side of (3.3). Note that Jij is virtually nearest-neighbor when �is large since if |i − j|〉1, Jij is proportional to �−(|i−j|−1). Hence, Theorems 1 and 2can be applied. We expect that V1 can be analyzed in a similar way except that therewill also be multi-spin interactions generated.If we take the continuum limit in one direction, the partition function (3.4) then

becomes the Feynman–Kac representation of a chain of quantum anharmonic oscillatorsdenoted Hq:

HNq =

12

N∑i

[− d2

d x2i+ (xi − xi+1)2 + V (xi)

](3.7)

with V = V1, in the limit N → ∞. Let EN0 (b)〈EN

1 (b)〈EN2 (b) · · · denote the eigenvalues

of HNq which depend on b through (3.2). Then for large � we expect to prove that in

the limit of large N , EN0 (b)=N has a logarithmic singularity in its second derivative at

some bc and

EN2 (b)− EN

0 (b) = Const|b− bc|: (3.8)

To establish such a result, we must map the lattice �eld system onto an Ising model,which is then expressed in Grassmann variables. The continuum limit in one directionproduces anisotropy which should pose no di�culties for our estimates.

Remark. In dimensions four and above, the eigenvalue separation (3.8) is known tobe proportional to |b− bc|1=2 for small �.

References

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1232.[3] T. Schultz, D. Mattis, E. Lieb, Two-dimensional Ising model as a soluble problem of many Fermions,

Rev. Mod. Phys. 36 (1964) 856.[4] C. Hurst, H. Green, J. Chem. Phys. 33 (1959) 1059.[5] P.W. Kasteleyn, Dimer Statistics and phase transitions, J. Math. Phys. 4 (1963) 287.[6] M. Kac, J. Ward, A combinatorial solution of the two-dimensional Ising model, Phys. Rev. 88 (1952)

1332.[7] B. McCoy, T. Wu, The two-dimensional Ising Model, Harvard University Press, Cambridge, MA, 1973.[8] J. Palmer, C. Tracy, Two-dimensional Ising correlations: the SMJ analysis, Adv. Appl. Math. 4 (1)

(1983) 46.


[9] C. Itzykson, J. Drou�e, Statistical Field Theory: 1, Cambridge University Press, Cambridge 1989.[10] R. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, New York, 1982.[11] H. Pinson, T. Spencer, Universality and the two dimensional Ising model, preprint.[12] J. Feldman, H. Kn�orrer, E. Trubowitz, A Representation for Fermionic Correlation Functions, Comm.

Math. Phys. 195 (1998) 465.[13] K. Gawedzki, A. Kupiainen, Gross- Neveu model through convergent perturbation expansions, Comm.

Math. Phys. 102 (1) (1985) 1.[14] J. Feldman, J. Magnen, V. Rivasseau, R. S�en�eor, Massive Gross-Neveu model: a rigorous perturbative

construction, Phys. Rev. Lett. 54 (14) (1985) 1479.[15] G. Benfatto, G. Gallavotti, Renormalization Group, Princeton University Press, Princeton, 1995.[16] S. Samuel, The use of anticommuting variable integrals in statistical mechanics, J. Math. Phys. 21

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a mathematical approach to universality in two dimensions

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