Renormalization group maps for Ising models in lattice gas variables
aa r X i v : . [ m a t h - ph ] M a y Renormalization group maps for Ising models inlattice-gas variables e tgkemail: [email protected] 11, 2018 Abstract
Real-space renormalization group maps, e.g., the majority rule transformation, mapIsing-type models to Ising-type models on a coarser lattice. We show that each coefficientin the renormalized Hamiltonian in the lattice-gas variables depends on only a finite num-ber of values of the renormalized Hamiltonian. We introduce a method which computesthe values of the renormalized Hamiltonian with high accuracy and so computes the co-efficients in the lattice-gas variables with high accuracy. For the critical nearest neighborIsing model on the square lattice with the majority rule transformation, we compute over1,000 different coefficients in the lattice-gas variable representation of the renormalizedHamiltonian and study the decay of these coefficients. We find that they decay exponen-tially in some sense but with a slow decay rate. We also show that the coefficients in thespin variables are sensitive to the truncation method used to compute them. Introduction
Real-space renormalization group transformations were introduced to study critical behaviorin Ising-type models. There has been extensive numerical study of these transformations, andthere is a rich picture of how they are believed to behave. However, there are essentially nomathematical results on these transformations. The usual definition of these transformationsis only formal since it involves an infinite-volume limit which must be proved to exist. Themathematical problem is to show that these renormalization group maps are rigorously definedin a neighborhood of the critical point, and to use them to study the system in a neighborhoodof the critical point. This is a difficult problem and the amount of rigorous progress that hasbeen made is embarrassing. Starting with the critical nearest neighbor Hamiltonian, the firststep of the renormalization group transformation has been proved to be defined for a few specificlattices and transformations [7, 8]. The existence of the transformation well inside the high-temperature phase has been proved by rigorous expansion methods [2, 5, 6]. It is possible toconstruct examples of transformations for which the renormalized Hamiltonian can be provedto be non-Gibbsian, including examples which start from the critical nearest neighbor Isingmodel [16, 17].Even if we start with a finite-range Hamiltonian, after just one step of the renormalizationgroup transformation the renormalized Hamiltonian will be infinite range and have infinitelymany different terms. The conventional wisdom is that they should decay both as the numberof sites involved grows and as the distance between these sites grows, so that the renormalizedHamiltonian may be well approximated by a finite number of terms. In some sense, thisproperty is the raison d’ˆetre of the renormalization group. It should allow one to study criticalphenomena, which are inherently multiscale and so impossible to approximate well by a finitesets of terms, by studying a map of Hamiltonians which can be well approximated by a finitenumber of terms.Swendsen showed that one can compute the linearization of the renormalization group trans-formation about the fixed point from correlation functions that involve the original spins andthe block spins [14]. His method allows one to avoid computing any renormalized Hamiltonians.From the point of view of using the renormalization group to calculate the critical exponents,this was a tremendous advance and was used in a large number of subsequent Monte Carlostudies of the renormalization group. From the point of view of trying to learn more aboutthe renormalized Hamiltonians and the fixed point of the transformation, it had the unfortu-nate side effect that many of these Monte Carlo studies did not compute any renormalizedHamiltonians. In recent years there have been more studies that compute the renormalizedHamiltonian. In particular the the Brandt-Ron representation introduced in [1] and studiedfurther in [9, 10, 11, 12] is similar to the method we use in this paper.The goals of this paper are to give a highly accurate method for computing the renormalizedHamiltonian which works in the lattice-gas representation and to use it to test the conventionalwisdom that the renormalization group transformation is well approximated by a finite numberof terms. Our numerical calculations are done for the critical nearest neighbor Ising model on2he square lattice, and we only consider the renormalized Hamiltonian obtained by a singleapplication of the majority rule transformation using 2 by 2 blocks. However, our approachis quite general and can be applied to other dimensions, lattices and choices of the real-spacerenormalization group transformation.One of the key tenets of the renormalization group is that if we fix a block-spin configurationand study the original system subject to the constraint imposed by the block spins, then thisconstrained system is in a high-temperature phase even if the unconstrained system is at itscritical point. As an extreme case consider the block-spin configuration of all +1’s with themajority rule transformation. The effect of this constraint on the original Ising system is similarto imposing a positive magnetic field, and the constrained system should have a relatively shortcorrelation length. Our computational method for the renormalization group transformationtakes advantage of this property.In the next section we review the definition of real-space renormalization group transforma-tions. In section three we explain our method for computing the renormalized Hamiltonian inthe lattice-gas representation. Some of the details are postponed to section five. We use thismethod to study the decay of the terms in the renormalized Hamiltonian. In section four weconsider how to compute the renormalized Hamiltonian in the more standard spin variables.There are multiple ways to do this, and we will see that the computed value of an individualcoupling coefficient in the renormalized Hamiltonian varies considerable with the method used.We also study the decay of the renormalized Hamiltonian in the spin variables. Section fiveprovides further detail for our method for computing the renormalized Hamiltonian. We con-sider the various sources of error in our computations in section six, and offer some conclusionsin section seven.The significant dependence of coefficients in the renormalized Hamiltonian on the truncationmethod used has been seen before. In particular, Ron and Swendsen observed a change of severalpercent in the nearest neighbor coupling when the number of couplings kept was changed fromsix to twelve [9]. In [10] they wrote “Even though the individual multispin interactions usuallyhave smaller coupling constants than two-spin interactions, the fact that they are very numerouscan lead to multispin interactions dominating the effects of two-spin interactions.” Truncatingthe space of Hamiltonians implies that the linearization of the renormalization group map aboutthe fixed point is also truncated. An early, interesting paper on the effect of this truncation is[13].
In this section we quickly review the definition of real-space renormalization group transforma-tions. We refer the reader to [16] for more detail.Consider an Ising-type model in which the spins take on only the values ±
1. The latticeis divided into blocks and each block is assigned a new spin variable called a block spin. Theexample of the square lattice with 2 by 2 blocks is shown in figure 1. We consider transforma-3ions in which the block spins also take on only the values ±
1. The transformation is specifiedby a kernel T (¯ σ, σ ). Here σ denotes the original spins and ¯ σ the block spins. The kernel isrequired to satisfy X ¯ σ T (¯ σ, σ ) = 1 (1)for all original spin configurations σ . The renormalized Hamiltonian ¯ H (¯ σ ) is formally definedby e − ¯ H (¯ σ ) = X σ T (¯ σ, σ ) e − H ( σ ) (2)(Note that the inverse temperature β has been absorbed into the Hamiltonians in the aboveequation.) This is only a formal definition since we must first restrict to a finite volume inorder to make sense of this equation. Proving that the finite-volume definition of ¯ H has aninfinite-volume limit is essentially an open problem. The condition (1) implies that X ¯ σ e − ¯ H (¯ σ ) = X σ e − H ( σ ) so that the free energy of the original model can be recovered from the renormalized Hamilto-nian. This property allows one to study the critical behavior of the system by studying iterationsof the renormalization group map. In particular, the critical exponents may be related to theeigenvalues of the linearization of the map about its fixed point.One widely studied family of kernels is the family of majority rule transformations. If thereare an odd number of spins in every block, then T (¯ σ, σ ) = 1 if the majority of the spins ineach block agree with the block spin and T (¯ σ, σ ) = 0 otherwise. If there are an even numberof spins in every block, then we let T (¯ σ, σ ) be the product over the blocks B of t (¯ σ B , { σ i } i ∈ B ) = σ B P i ∈ B σ i >
00 if ¯ σ B P i ∈ B σ i < / P i ∈ B σ i = 0 (3)where ¯ σ B denotes the block spin for block B .The general approach presented in this paper applies to all these renormalization groupmaps. The numerical calculations that we will present are for the critical nearest neighborIsing model on the square lattice with the majority rule renormalization group map with twoby two blocks. Real-space renormalization group calculations are usually done using the spin variables σ i = ± n i = (1 − σ i ) / ,
1. Note that we have made the convention that a spin value of +1corresponds to a lattice gas value of 0. Throughout this paper we will use σ ’s for spin variablestaking on the values ±
1, and n ’s for lattice-gas variables taking on the values 0 ,
1. We indicaterenormalized spins or variables with a bar over them, e.g., ¯ σ i , ¯ n i . We use σ to denote the entirespin configuration { σ i } . Likewise, n , ¯ σ and ¯ n denote the corresponding collections of variables.In this section we work entirely in the lattice-gas variables, both for the original Hamiltonianand the renormalized Hamiltonian. We write the renormalized Hamiltonian as¯ H (¯ n ) = X Y c ( Y ) ¯ n ( Y ) (4)where the sum is over all finite subsets including the empty set, and¯ n ( Y ) = Y i ∈ Y ¯ n i X , let ¯ n X denote the block-variable configurationwith all block variables in X equal to 1 and the rest equal to 0. Then eq. (2) saysexp( − ¯ H (¯ n X )) = X n T (¯ n X , n ) e − H ( n ) Note that ¯ n ∅ ( X ) = 0 except when X = ∅ . So ¯ H (¯ n ∅ ) = c ( ∅ ). In particular, c ( ∅ ) will growas the size of the finite volume. The other coefficients c ( Y ) should have finite limits in theinfinite-volume limit. We define f ( X ) by f ( X ) = ¯ H (¯ n X ) − ¯ H (¯ n ∅ )Then f ( X ) should have a finite limit in the infinite-volume limit, and it should be related tothe infinite-volume c ( X ) by f ( X ) = X Y : ∅6 = Y ⊂ X c ( Y ) (5)The system of equations (5) can be explicitly solved for the c ( Y ). We claim that the solutionfor X = ∅ is c ( X ) = X Y : ∅6 = Y ⊂ X ( − | X |−| Y | f ( Y ) (6)This is a standard inversion trick. To verify (6), define c ( X ) by (6). Then for a given X = ∅ , X Y : ∅6 = Y ⊂ X c ( Y ) = X Y : ∅6 = Y ⊂ X X Z : ∅6 = Z ⊂ Y ( − | Y |−| Z | f ( Z )= X Z : ∅6 = Z ⊂ X f ( Z ) X Y : Z ⊂ Y ⊂ X ( − | Y |−| Z | (7)The sum over Y is 1 if X = Z . If Z is a proper subset of X , we claim this sum is 0. To seethis: X Y : Z ⊂ Y ⊂ X ( − | Y |−| Z | = X W : W ⊂ X \ Z ( − | W | = Y i ∈ X \ Z (1 + ( − f ( X ).The important feature of eq. (6) is that the coefficient c ( X ) only depends on a finite numberof free energies f ( Y ), specifically those with Y ⊂ X . As we will see, these free energies can becomputed extremely accurately. So individual coefficients c ( X ) in the lattice-gas variables canbe computed extremely accurately. Moreover, this computation does not depend on how many6erms we decide to keep in the renormalized Hamiltonian. If we increase the number of termswe keep, then the coefficients we have already computed will not change.We have carried out numerical calculations of a large number of the coefficients in thelattice-gas representation for the critical nearest neighbor Ising model on the square latticewith the majority rule renormalization group map with two by two blocks. We need a criterionfor deciding for which Y we will compute c ( Y ). We assume the coefficients will decay as thenumber of sites in Y grows and as the distance between these sites grows. So we need a measureof the size of a set Y . There is no canonical way to define this size. We use the following adhoc quantity. If Y = { y , y , · · · , y n } , then we define S ( Y ) = n X i =1 || y i − c || (8)where c is the center of mass: c = 1 n n X i =1 y i and || || is the usual Euclidean distance in the plane. Note that we do not take a square rootin (8).We claim that if X ′ ⊂ X then S ( X ′ ) ≤ S ( X ). To prove this it suffices to prove the casethat X ′ has one less site than X . Let X be x , x , · · · , x n and X ′ be x , x , · · · , x n − . Let c bethe center of mass of X and c ′ the center of mass of X ′ . So S ( X ′ ) = n − X i =1 || x i − c ′ || The center of mass has the property that it minimizes the above sum. So S ( X ′ ) ≤ n − X i =1 || x i − c || ≤ n X i =1 || x i − c || = S ( X )We fix a cutoff C >
0, and compute c ( Y ) for all Y with S ( Y ) ≤ C . We only need tocompute it for one Y from each translation class, and so there are a finite number of such Y ’s. The bulk of the computation is computing the free energies f ( X ) for X with S ( X ) ≤ C .Using eq. (6) to find the c ( Y ) requires comparatively little computation. The property that X ′ ⊂ X ⇒ S ( X ′ ) ≤ S ( X ), implies that the collection of X for which we must compute f ( X )is just the collection of X with S ( X ) ≤ C .To study how fast the coefficients c ( Y ) decay, we take one coefficient from each translationclass that we have computed and order them so that | c ( Y ) | is decreasing, i.e., | c ( Y n ) | ≥ | c ( Y n +1 ) | .We then plot | c ( Y n ) | as a function of n . This is the bottom curve in figure 2. Note that the7 n Figure 2: The coefficients c ( Y n ) are ordered so | c ( Y n ) | decreases. The bottom curveis | c ( Y n ) | vs. n , and the top curve is the tail P Ni = n | c ( Y n ) | vs. n .vertical axis uses a logarithmic scale. The second quantity plotted (the top curve in the figure)is N X i = n | c ( Y i ) | as a function of n , where N is the total number of Y for which we compute the coefficients.The two lines shown are given by c − n/ for two different values of c . The two curves in thefigure depend on the function S ( Y ) we use to measure the size of sets and the cutoff we use forthis function. However, whatever function and cutoff we use, the resulting curve will be a lowerbound on the curve that would result from computing all the coefficients c ( Y ). In particular,we can make the following observations. The lower curve crosses the horizontal lines at 10 − ,10 − and 10 − at 131,1223 and 4023, respectively. Hence there are at least 131 translationclasses with a coefficient bigger than 10 − , at least 1223 with a coefficient bigger than 10 − ,and at least 4023 with a coefficient bigger than 10 − .In the preceding discussion we used one coefficient from each translation class. In additionto the translation symmetry the model is also symmetric under rotations by 90 degrees andrelections in lattice axes. More precisely, the additional symmetry is the dihedral group of order8. We have repeated the previous study of the decay of the coefficients taking into account thedihedral group symmetry as well as the translational symmetry by taking only one term fromthe above list from each dihedral group symmetry class. The main effect is that the scale onthe horizontal axis is reduced by a factor of 8. This is not surprising since for most subsets,rotations and reflections will generate eight different subsets.From a mathematical perspective, one would like to show that the renormalized Hamiltonianexists in some Banach space. One choice of norm for the Banach space would be X Y :0 ∈ Y | c ( Y ) | One would like to approximate the Hamiltonian by a Hamiltonian with a finite number ofterms. So it is important to see how fast the above sum converges as we include more termsin the Hamiltonian. This is similar to the second plot in figure 2. Note that in this norm eachtranslation class appears | Y | times. So the second plot in figure 2 is in some sense a lowerbound on the decay for the Hamiltonian. Another choice of the norm would be X Y :0 ∈ Y | c ( Y ) | e µ ( Y ) for some measure µ ( Y ) ≥ Y . For this norm the convergence would be evenslower than that seen in the figure.It is worth noting that norms defined using the lattice-gas representation of the Hamiltonianare in general stronger than norms that use the spin variable representation [4]. For example,if we can write the Hamiltonian in the lattice-gas representation (4) with X Y ∋ | c ( Y ) | < ∞ then it is straightforward to show that the Hamiltonian can be represented in the spin variablerepresentation (9) with X Y ∋ | d ( Y ) | < ∞ In the previous section we saw that in the lattice-gas variables there is a natural way to computethe coefficients c ( Y ) in the expansion (4) for ¯ H . In this section we consider the renormalizedHamiltonian in the spin variables: ¯ H (¯ σ ) = X Y d ( Y )¯ σ ( Y ) (9)9ith ¯ σ ( Y ) = Q i ∈ Y ¯ σ i . The sum over Y is over all finite subsets.We can use ¯ n i = (1 − ¯ σ i ) / d ( Y ) in (9) in terms of thelattice-gas coefficients c ( Y ) in (4).¯ H (¯ σ ) = X X c ( X )¯ n ( X ) = X X c ( X ) 2 −| X | Y i ∈ X (1 − ¯ σ i )= X X c ( X ) 2 −| X | X Y : Y ⊂ X ( − | Y | ¯ σ ( Y )= X Y ¯ σ ( Y )( − | Y | X X : Y ⊂ X c ( X ) 2 −| X | = X Y d ( Y ) ¯ σ ( Y )where the spin coefficients d ( Y ) are given by d ( Y ) = ( − | Y | X X : Y ⊂ X c ( X ) 2 −| X | (10)The problem is that to compute the spin coefficient d ( Y ) we need the lattice-gas coefficients c ( X ) for infinitely many X , and so we need the free energies f ( X ) for infinitely many X ’s. Sowe must introduce some sort of approximation.Let Y ∞ be a collection of finite subsets of the renormalized lattice such that one set fromeach translation class is contained in Y ∞ . We can rewrite (9) as¯ H (¯ σ ) = X Y ∈ Y ∞ d ( Y ) X t ¯ σ ( Y + t )where the sum over t is over the translations for the renormalized lattice. Here Y + t denotes { i + t : i ∈ Y } .Now let Y be a finite subcollection of Y ∞ . We want to compute an approximation to theabove of the form ¯ H (¯ σ ) ≈ X Y ∈ Y d ( Y ) X t ¯ σ ( Y + t )We will consider two methods which we will refer to as the “partially exact” method and the“uniformly close” method.For the partially exact method, we begin by noting that we can write ¯ H (¯ n ) as¯ H (¯ n ) = X Y ∈ Y ∞ c ( Y ) X t ¯ n ( Y + t )The approximation is simply to truncate this sum by restricting Y to those in Y :¯ H (¯ n ) ≈ X Y ∈ Y c ( Y ) X t ¯ n ( Y + t )10he c ( Y ) are exact. As we saw in the last section we can compute them from (6) by computingthe free energies f ( X ) for X ∈ Y . We then convert this Hamiltonian to the spin variables withno approximation. The result is that the approximation to ¯ H (¯ σ ) is X Y ∈ Y d ( Y ) X t ¯ σ ( Y + t ) (11)with d ( Y ) = ( − | Y | X X : Y ⊂ X,X + t ∈ Y c ( X ) 2 −| X | (12)where the notation X + t ∈ Y means some translation of X (possibly X itself) is in Y . Thusthis method is equivalent to truncating the exact formula (10) by restricting the sum over X to sets in Y and their translates. In the lattice-gas variables our approximation to ¯ H agreeswith the true ¯ H for all n X such that X ∈ Y . The change from lattice gas to spin variables didnot involve any approximation, so our approximation to ¯ H in the spin variables agrees exactlywith the true ¯ H for all configurations ¯ σ Y for Y ∈ Y . This is the reason for calling this method“partially exact.” It is exact for some of the block-spin configurations.For the uniformly close method let X be another finite collection of finite subsets whichcontains at most one set from each translation class. We compute the free energies f ( X ) for X ∈ X , i.e., we compute ¯ H (¯ σ X ). We define the error of a set of coefficients { d ( Y ) : Y ∈ Y } tobe max X ∈ X | ¯ H (¯ σ X ) − X Y ∈ Y d ( Y ) X t ¯ σ X ( Y + t ) | where ¯ σ X is the spin configuration which is − X and +1 on all other sites. We then choosethe coefficients d ( Y ) to minimize the above error. This is a standard linear programming prob-lem which we solve by the simplex algorithm. We call this the uniformly close approximationsince we have a uniform bound on the difference between our approximation and the exact ¯ H for the block-spin configurations ¯ σ X for X ∈ X . (For other X we cannot say anything abouthow well the approximation does.) If X = Y , then the partially exact approximation makes theabove error zero. We only use the uniformly close approximation for X which are larger than Y . We take the following point of view. We think of Y as being fixed. It determines a finite-dimensional space of Hamiltonians that we use to approximate the renormalized Hamiltonian.We then think of the collection X as being variable. A larger X means we “know” more freeenergies and so have more information to use in computing the approximation. In our studieswe will take the collection Y to be all the subsets Y with s ( Y ) ≤ C ¯ H for some cutoff C ¯ H , and X to be all the X with s ( X ) ≤ C f for some cutoff C f ≥ C ¯ H When we worked in the lattice-gas variables the computation of the coefficients c ( X ) wasunambiguous. The computation of the values of ¯ H (¯ n ) requires some approximations, but as11e will see in section 5 these approximation are well behaved and introduce small errors. Thecomputation of the c ( X ) from the ¯ H (¯ n ) does not require any approximation or truncation. Inthe spin variable representation we now have multiple ways to compute the coefficients d ( X )depending on whether we use the partially exact or uniformly close methods and on the choicesof X and Y . We restrict our study of the spin variable coefficients to studying how thesechoices affect the values of individual coefficients. We focus our attention on three particularcoefficients: the nearest neighbor, the next nearest neighbor and the plaquette. These referto the coefficients of σ i σ j with | i − j | = 1, of σ i σ j with | i − j | = √
2, and of σ i σ j σ k σ l where i, j, k, l are the corners of a unit square. As in the previous section, our numerical calculationsare for the critical nearest neighbor Ising model on the square lattice with the majority rulerenormalization group map with two by two blocks.For the partially exact method we have one parameter - the cutoffs C ¯ H and C f are equaland correspond to the cutoff C of section 3. So we can plot individual coefficients as a functionof C f . For the uniformly close method we have two parameters: the cutoff C ¯ H determines thefinite-dimensional subspace used to approximate the renormalized Hamiltonian and the cutoff C f determines the number of ¯ H (¯ n ) values we use. We plot the coefficients in this case as afunction of C f for several different choices of C ¯ H . The results are shown in figures 3,4, and 5.The variations seen in the three coefficients are roughly comparable in size. Note that whilethe ranges of the vertical axes vary in the three figure, the scales for the vertical axes are allthe same. The variations in the coefficients are on the order of several thousandths. So evenfor these relatively large coefficients, it is difficult to determine the value of the coefficient tobetter than a few percent. For smaller coefficients the variations are somewhat smaller, but asa fraction of the coefficient they are typically larger.These three coefficients (along with many others) have of course been computed before.Two early references are [3, 15]. The point of our study is not the values of these coefficientsbut rather the variation in their values as one varies the method used to compute them.12 Figure 3: The dependence of the nearest neighbor coefficient in the spin variablerepresentation of the renormalized Hamiltonian on the computation method. Thesolid curve is the partially exact method. The dashed curves are the uniformly closemethod with four different choices of C ¯ H Figure 4: The next nearest neighbor coefficient in the spin variable representation ofthe renormalized Hamiltonian 14
Figure 5: The plaquette coefficient in the spin variable15
Computing the free energy
Fix a block-spin configuration ¯ σ . We want to compute the free energy ¯ H (¯ σ ) of the constrainedpartition function exp( − ¯ H (¯ σ )) = X σ T (¯ σ, σ ) e − H ( σ ) Initially we work with the spin variables, but later in this section we will switch to the lattice-gas variables. We only need to do this computation for configurations ¯ σ which are +1 excepton a finite set. Even when the original system is at the critical point, these constrained systemshave relatively short correlation lengths. This is where the real power of the renormalizationgroup becomes apparent. In particular, finite-volume effects in the above computation decayvery quickly as the volume increases.Before we explain our method for the computation, we first indicate how ¯ H (¯ σ ) can becomputed by a Monte Carlo calculation. (We have done such a simulation as a check on themethod we describe later.) Fix a relatively small set of block spins, X . Let V be a finitevolume of block spins containing X which is large enough that the boundary of V is far from X . We include only the factors in the renormalization group kernel corresponding to the blocksin V \ B . For these blocks we take the block spins to be +1. We then run a Monte Carlosimulation of the Ising system with this kernel outside of X . When we sample the simulationwe compute the block-spin configuration on X . This allows us to compute the relative weightsof the possible block-spin configurations on X . From these weights we can then compute the¯ H (¯ σ ) for ¯ σ which are − X .We now turn to our method for computing ¯ H (¯ σ ). It does not involve Monte Carlo meth-ods, and it is much more accurate than the Monte Carlo approach described in the previousparagraph. Everything in the following depends on ¯ σ , but we will not make this dependenceexplicit. Imagine that we have summed over the spins one block at a time in such a way thatwe have reached the state in figure 6. Open circles indicate sites in the original lattice for whichwe have already summed over the spin, and blue (gray) circles represent sites for which we havenot. (Red (solid) circles indicate the block spins which are fixed throughout this computation.)The result of this partial computation of the free energy is a function of the spins in the originallattice with shaded circles. In fact, it only depends on those that are nearest neighbors of aspin with an open circle. We will refer to these spins as boundary spins. The quantity we havecomputed so far is positive, and we write it in the formexp( X X a ( X ) σ ( X ) + ∆ H ) Y B ′ t B ′ (¯ σ, σ )where X is summed over finite subsets of the boundary spins. ∆ H denotes the terms in theHamiltonian that only involve spins with shaded circles. (These terms have not yet entered thecomputation.) The product over B ′ is over the blocks containing shaded circles, and t B ′ (¯ σ, σ )16 Figure 6: We compute ¯ H (¯ σ ) by summing out the original spins one block at a time.The open circles are spins that have been summed over, while the blue (gray) circlesare spins that have yet to be summed over. The red (black) circles are the fixed blockspins.is the factor in the renormalization group kernel for block B ′ . The next step is to sum over thefour spins in the block B and take the logarithm of the result:ln "X σ B exp( X X a ( X ) σ ( X ) + ∆ H ) Y B ′ t B ′ (¯ σ, σ ) The sum over σ B denotes a sum over the spins σ i with i ∈ B . Terms a ( X ) σ ( X ) for which X ∩ B = ∅ pass through this computation trivially. So do the terms in ∆ H which do notinvolve a spin in the block B and the factors t B ′ for B ′ = B . So the computation that we mustactually do is ln X σ B exp( X X : X ∩ B = ∅ a ( X ) σ ( X ) + h ) t B (¯ σ, σ ) h contains the terms in H that only depend on spins with shaded circles and depend onat least one spin in B .To do this computation numerically, we must introduce a truncation. We fix a finite subset D of the boundary sites centered near B . We then restrict the sum over X to X ⊂ D . Weneed to write the result of the truncated computation in the formln X σ B exp( X X : X ∩ B = ∅ ,X ⊂ D a ( X ) σ ( X ) + h ) t B (¯ σ, σ ) = X Y a ′ ( Y ) σ ( Y )The left side only depends on spins in D ′ = D \ B , so the sum on the right may be restrictedto Y ⊂ D ′ . If we define F ( σ ) to be the left side of this equation, then the coefficients are givenby a ′ ( Y ) = 2 −| D ′ | X σ D ′ F ( σ )The amount of computation required grows quite rapidly as D grows for three reasons. First,the number of X with X ⊂ D grows as 2 | D | . Second, the sum over σ D ′ in the above also growsas 2 | D | . Third, the number of Y also grows as 2 | D | . We have found that a ( X ) decays quicklyas the number of sites in X grows. So we can make a further truncation by only keeping terms a ( X ) with | X | less than some specified cutoff. ( | X | denotes the number of sites in X .) Thisgreatly reduces the growth of the computation with D from the first and third effects. But weare still left with the second effect.We can eliminate the second effect by working in the lattice-gas variables. We replace P X a ( X ) σ ( X ) by P X b ( X ) n ( X ). Define F ( n ) = ln X n B exp( X X : X ∩ B = ∅ ,X ⊂ D b ( X ) n ( X ) + h ) t B (¯ n, n ) (13)We need to compute the coefficients in F ( n ) = X X b ′ ( X ) n ( X )As we saw in section 3, they are given by b ′ ( X ) = X Y : ∅6 = Y ⊂ X ( − | X |−| Y | F ( n Y ) (14)where n Y is the configuration that is 1 on Y and 0 off of it. So to compute b ′ ( X ) we only needto compute F ( n Y ) for Y ⊂ X . 18n this approach using the lattice-gas variables we can forget about the set D entirely.Instead we specify a finite collection B of subsets of the boundary spins with the property thatthey intersect B . We then make the approximation X X : X ∩ B = ∅ b ( X ) n ( X ) ≈ X X ∈ B : X ∩ B = ∅ b ( X ) n ( X ) (15)We use (14) to compute b ′ ( X ). It will be nonzero only for X ⊂ D ′ . Before we sum over thenext block of spins, we need to truncate P X b ′ ( X ) n ( X ). We keep only the terms such that X is in B + t where t is the translation that takes the block we just summed over to the block weare summing over next, and B + t denotes the collection of sets of the form X + t for X ∈ B .We take the finite collection B to be all X which intersect B and satisfy S ( X ) ≤ C B where S ( X ) is some size function and C B is some cutoff. We use the size function given by (8) thatwe used for choosing the block-spin sets. In our calculations we take C B = 260 which leads to10 ,
763 sets in the collection B . We discuss the effect of C B on the error in the following section.The above discussion took place in an infinite volume. The region shown in figure 6 is afinite piece of the infinite volume. In practice we can only sum over the spins in a finite numberof blocks. The block-spin configuration ¯ n is of the form ¯ n Y for a finite set Y . We carry out thecomputation in a finite volume which is chosen so that the distance from Y to the boundaryof the finite volume is sufficiently large. We will study how large the finite volume should bein the next section. In this section we study the sources of error in our computations of the f ( X ). There are two:the use of a finite volume to compute infinite-volume quantities and the truncation determinedby the cutoff C B in section 5.We choose the finite volume in which we carry out our calculation as follows. The block-spinconfigurations ¯ n that we consider are of the form ¯ n Y for finite sets Y . We take these sets Y to be centered near the origin and take the finite volume to be a square centered at the origin.The square contains the blocks with centers at (2 i, j ) with − L ≤ i ≤ L , − L ≤ j ≤ L . So theinfinite-volume limit is obtained by taking L → ∞ .To study the finite-volume error in our calculation we do the following. The free energy f ( Y ) depends on L , so we denote it by f L ( Y ). As a measure of the finite-volume error we use1 N X Y | f L ( Y ) − f L − ( Y ) | (16)where the sum is over one element of each translation class with s ( Y ) ≤ N is thenumber of terms in the sum. In this study of the finite-volume error we take C B = 30. Thisis much smaller than the cutoff we used for the main calculations, but the behavior of the19 Figure 7: L is a measure of the size of the finite volume. The quantity plotted is theaverage change in f ( X ) when L is decreased by 1. (See eq. (16).)finite-volume error with L is the same whether we look at this smaller set of coefficients or thelarger set.This average difference as a function of L is shown in figure 7. The vertical scale is log-arithmic, so the approximately linear dependence seen for the smaller values of L indicatesexponential decay of this difference with L . The line shown in the figure is of the form ce − L/ . .Keeping in mind that L corresponds to numbers of blocks and the blocks are 2 by 2, the decaylength of 0 . . L around 10 the difference is dominated by numerical errors. In our simulationswe are very conservative and take L = 15. With this choice the finite-volume error is at thesame level as the numerical error.The translational symmetry of the original model implies that f ( X ) is unchanged if wetranslate X . So we only need to compute f ( X ) for one element of each translation class. Themodel is also invariant under the dihedral group symmetry generated by rotations by mutiplesof π/ f ( X ) breaksthe dihedral symmetry of the lattice, so our values of f ( X ) for X ’s from the same dihedral classare not exactly the same. The dihedral symmetry is only restored when we let L → ∞ and20 Figure 8: We plot the average of f ( X ) − ¯ f ( X ), as defined by eq. (17), and of c ( X ) − ¯ c ( X ), as defined by eq. (18), as a function of the cutoff C B . C B → ∞ . We have already seen that we can take L sufficiently large that the finite-volumeerror is reduced to the order of the numerical error. So we can use the breaking of the dihedralsymmetry to study how the error depends on the cutoff C B .For various choices of C B we compute f ( X ) for the same collection of X as in our maincalculation. Let ¯ f ( X ) be the average of f ( Y ) over one Y from each translation class which isrelated to X by the dihedral symmetry. (The number of terms involved in this average rangesfrom 1 to 8.) The differences f ( X ) − ¯ f ( X ) are a measure of the amount of breaking of thedihedral symmetry and hence of the error in the computation from the cutoff C B . We use theaverage 1 N X Y | f ( Y ) − ¯ f ( Y ) | (17)to quantify the error. As before the sum is over one element of each translation class with s ( Y ) ≤ N is the number of terms in the sum. This quantity is plotted in figure 8 asa function of C B for the free energies f ( Y ). It is the higher set of points. For the coefficients21n the lattice-gas variables we define ¯ c ( X ) analogously, and study the average1 N X Y | c ( Y ) − ¯ c ( Y ) | (18)This quantity is the lower set of points in figure 8. Figure 9: The convergence of four different quantities as C B → ∞ . From top tobottom the four quantities are given by equations (19) to (22).We also study the convergence as C B → ∞ in another way. Let f ∞ ( Y ) denote f ( Y ) for thelargest value of C B which we use, i.e., 260. We then consider1 N X Y | f ( Y ) − f ∞ ( Y ) | (19)This is plotted as a function of C B in figure 9 for the free energy f ( Y ). We also plot1 N X Y | ¯ f ( Y ) − ¯ f ∞ ( Y ) | (20)22s the figure shows, averaging over the dihedral group like this reduces the error somewhat.The figure also includes the analogous plots for the coefficients in the lattice-gas representation,i.e., of the quantities 1 N X Y | c ( Y ) − c ∞ ( Y ) | (21)and 1 N X Y | ¯ c ( Y ) − ¯ c ∞ ( Y ) | (22) We have shown that if we use lattice-gas variables, then the computation of the coefficients inthe renormalized Hamiltonian only depends on a finite number of values of the renormalizedHamiltonian. So this computation does not depend on how we approximate the inherentlyinfinite-dimensional renormalized Hamiltonian by a finite-dimensional approximation. We havealso given a highly accurate method for computing the values of the renormalized Hamiltonianwhich takes advantage of the finite correlation length that results from the introduction of therenormalization group transformation.The renormalized Hamiltonian has infinitely many different terms but the conventionalwisdom is that it may be well approximated by a finite number of terms. In particular, themagnitude of the coefficients should decay as the “size” of the set of lattice sites increases.We studied this for the nearest neighbor critical Ising model on the square lattice under onestep of the majority rule renormalization group transformation. We computed a large numberof coefficients in the lattice-gas variables, ordered them by decreasing magnitude and plottedthem. We found that over several orders of magnitude the coefficients decayed exponentiallywith the number of terms, but the decay rate was slow. It takes about 850 additional terms tosee the magnitude reduced by just a factor of 1 / ,
000 values of the renormalized Hamiltonian, the uncertainty in the spinvariable coefficients due to the different truncation methods is on the order of a percent for thelargest coefficients and even larger as a percentage for some of the smaller coefficients.One might hope to prove theorems about these real-space renormalization group transfor-mations by defining a suitable Banach space of Hamiltonians and then doing a computer aidedproof to show the transformation is defined in some open subset of the Banach space and thereis a fixed point in this subset. Proving there is a fixed point would require constructing anapproximation to the fixed point with a finite number of terms. Our numerical results suggest23hat at best such an approach will require a huge number of terms in the finite approximationand at worst the number of terms needed will doom the approach to failure.Past numerical studies of the two dimensional Ising model using the renormalization grouphave produced fairly accurate values of the critical exponents using a relatively modest numberof terms in the renormalized Hamiltonian. These studies use the spin variables, so their accuracyis surprising given the difficulty we have found in computing the coefficients in the renormalizedHamiltonian accurately. An interesting question is to understand this.
References [1] A. Brandt and D. Ron, Renormalization multigrid (RMG): statistically optimal renor-malization group flow and coarse-to-fine Monte Carlo acceleration,
J. Stat. Phys. ,231-257 (2001).[2] R. B. Griffiths and P. A. Pearce, Mathematical properties of position-spacerenormalization-group transformations,
J. Stat. Phys. , 499-545 (1979).[3] R. Gupta and R.Cordery, Monte Carlo renormalized Hamiltonian Phys. Lett. , 415-417 (1984).[4] R. B. Israel, Convexity in the theory of lattice gases (Princeton University Press, Princeton,1979).[5] R. B. Israel, Banach algebras and Kadanoff transformations, in:
Random Fields (Es-ztergom, 1979), vol. II , J. Fritz, J. L. Lebowitz, D. Sz´asz, (eds.), (North-Holland,1981).[6] I. A. Kashapov, Justification of the renormalization - group method,
Theor. Math. Phys. , 184-186 (1980).[7] T. Kennedy, Some rigorous results on majority rule renormalization group transformationsnear the critical point, J. Stat. Phys. , 15-37 (1993).[8] T. Kennedy and K. Haller, Absence of renormalization group pathologies near the criticaltemperature - two examples, J. Stat. Phys. , 607-637 (1996).[9] D. Ron and R. H. Swendsen, Calculation of effective Hamiltonians for renormalized ornon-Hamiltonian systems, Phys. Rev E , 066128 (2001).[10] D. Ron and R. H. Swendsen, Importance of multispin couplings in renormalized Hamilto-nians Phys. Rev E , 056106 (2002). 2411] D. Ron,R. H. Swendsen and A. Brandt, Inverse Monte Carlo renormalization group trans-formations for critical phenomena, Phys. Rev. Lett. , 275701 (2002).[12] D. Ron,R. H. Swendsen and A. Brandt, Computer simulations at the fixed point using aninverse renormalization group transformation, Physica A , 387-399 (2005).[13] R. Shankar, R. Gupta, and G. Murthy, Dealing with truncation in Monte Carlorenormalization-group calculations,
Phys. Rev. Lett. , 1812-1815 (1985).[14] R. Swendsen, Monte Carlo renormalization group, Phys. Rev. Lett. , 859-861 (1979).[15] R. Swendsen, Monte Carlo calculation of renormalized coupling parameters. I. d = 2 Isingmodel, Phys. Rev. B , 3866-3874 (1984).[16] A. C. D. van Enter, R. Fern´andez and A. D. Sokal, Regularity properties and pathologies ofposition-space renormalization - group transformations: scope and limitations of Gibbsiantheory, J. Stat. Phys. , 879-1167 (1993).[17] A. C. D. van Enter, Ill-Defined Block-Spin Transformations at Arbitrarily High Temper-atures, J. Stat. Phys.83