Renormalization group maps for Ising models in lattice gas variables

Transcription

1 Renormalization group maps for Ising models in lattice gas variables Department of Mathematics, University of Arizona Supported by NSF grant DMS e tgk RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.1/29

2 Outline 1. Introduction - definition of real space RG maps, folklore 2. Past results 3. Real space RG map in lattice gas variables 4. Real space RG map in spin variables 5. Conclusions, future work Preprint in arxiv.org Slides on homepage. e tgk RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.2/29

3 1. Introduction - def of the map Ising type models: spins take on only values ±1. Nearest neighbor interaction More general interaction H(σ) = β <i,j> σ i σ j H(σ) = Y d(y ) σ(y ), σ(y ) = i Y σ i where the sum is over all finite subsets including the empty set. Note that β has been absorbed into the Hamiltonian. RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.3/29

4 Block spins Lattice divided into blocks; each block assigned a block spin variable. Block spins also take on only the values ±1. RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.4/29

5 Definition of map Original spins: σ RG Kernel: T( σ, σ). Satisfies Block spins: σ T( σ, σ) = 1 σ for all original spin configurations σ. Renormalized Hamiltonian H( σ) is formally defined by e H( σ) = σ T( σ, σ) e H(σ) Note: β has been absorbed into the Hamiltonians. Key point: only makes sense in finite volume. RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.5/29

6 Examples of RG map Majority rule with odd number of sites in block. T = 0, 1 only. T( σ, σ) = 1 if the majority of the spins in each block agree with the block spin and T( σ, σ) = 0 otherwise. Majority rule with even number of sites in block. T( σ, σ) is the product over the blocks B of 1 if σ B i B σ i > 0 t( σ B, {σ i } i B ) = 0 if σ B i B σ i < 0 1/2 if i B σ i = 0 where σ B denotes the block spin for block B. RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.6/29

7 Another example- decimation Decimation with b = 2. T( σ, σ) = 0, 1. T = 1 iff original spins not decimated = corresponding block spin. RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.7/29

8 Properties of map T( σ, σ) = 1, e H( σ) = σ T( σ, σ) e H(σ) σ e H( σ) = σ σ e H(σ) So free energy of the original model can be recovered from the renormalized Hamiltonian. Study the critical behavior of the system by studying iterations of the renormalization group map: R(H) = H Remember: R is not even defined from a rigorous point of view. RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.8/29

9 Conjectured behavior of RG map Has a fixed point with stable manifold of co-dim 2 Eigenvalues of linearization > 1 critical exponents next nearest neighbor 0 0 nearest neighbor RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.9/29

10 2. Past rigorous results Existence of map at high temp or large magnetic field Griffiths and Pearce; Israel; Kashapov; Yin Non-existence of map at low temp for various kernels Griffiths, Pearce; Israel; van Enter, Fernández and Sokal Non-existence of map near critical temp for some kernels Existence of map at the critical point for two examples, Haller, K Nearest neighbor model on triangular lattice with specific Kadanoff transformation Decimation with b = 2 for nearest neighbor model on square lattice Goal: Not to determine for each T whether it works or not. Show there is one T that works. RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.10/29

11 Why RG is a good idea < F(σ) > = σ F(σ)e H(σ) σ e H(σ) < F(σ) > σ = σ σ T( σ, σ)e H(σ) T( σ, σ)f(σ)e H(σ) Take H critical in the sense that correlation length of < Introducing block spins moves system off the critical point. Correlation length of < Critical temperature of < > σ should be bounded uniformly in σ. > σ should be lower uniformly in σ. > is infinite. Ould-Lemrabott: MC simulations indicate introduction of block spins lowers critical temp by a factor of 2. K: Proof that < > σ is high temp for MR with 2 by 2 blocks for three block spin configs ferro, checkerboard, strips Strategy RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.11/29

12 Selected past numerical results Lots of Monte Carlo studies using these maps. Swendsen: compute the linearization of the RG map from correlation functions. Avoids computing R itself. In recent years there have been more studies that compute the renormalized Hamiltonian. Brandt,Ron,Swendsen introduced method similar to our approach. Saw significant dependence of H on truncation method. Even though the individual multispin interactions usually have smaller coupling constants than two-spin interactions, the fact that they are very numerous can lead to multispin interactions dominating the effects of two-spin interactions. RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.12/29

13 3. RG map in lattice gas variables RG calculations usually done using the spin variables σ i = ±1. We use lattice gas variables: n i = (1 σ i )/2 which take on the values 0, 1. Note spin value of +1 corresponds to lattice gas value of 0. Notation: σ s for spin variables; value in 1, 1. n s for lattice gas variables; values in 0, 1. Renormalized spins or variables indicated by a bar over them, e.g., σ i, n i. We use σ to denote the entire spin configuration {σ i }. In lattice gas variables H( n) = Y c(y ) n(y ), n(y ) = i Y n i Y summed over all finite subsets of block spins RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.13/29

14 Reference configuration For a finite subset X, n X denotes configuration with all block variables in X equal to 1 and the rest equal to 0. exp( H( n X )) = n T( n X, n) exp( H(n)) Use only such block spin configurations. In infinite volume limit, H( n X )) will diverge, but f(x) = H( n X ) H( n ) should have a finite limit in the infinite volume limit. H( n ) = c( ), ( n (X) = 0 except when X = ). So c( ) V ol. Other c(y ) should have finite limits in the infinite volume limit. f(x) = Y : =Y X c(y ) RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.14/29

15 f(x) = Y : =Y X c(y ) can be solved (standard inversion trick): c(x) = Y : =Y X ( 1) X Y f(y ) Key points Individual c(x) only depends on finite number of f(y ) Computation of f(y ) is easy - high temperature computation. Computation of c(x) does not depend on how many other coefs we compute. RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.15/29

16 Why RG is a good idea - cont Need to compute exp( f(x)) = e H( n X ) e H( n ) = n T( nx, n) e H(n) n T( n, n) e H(n) Let < F(n) > = n F(n)T( n, n)e H(n) n T( n, n)e H(n) exp( f(x)) =< T( nx, n) T( n, n) > So this is computation of a local observable in a probabilty measure with correlation length of order 1. RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.16/29

17 Numerical computation of lattice gas coefs Compute one c(y ) from each translation class. How you compute the f(x) is an interesting story. We have computed about 10, 000 coefficients for first step of MR with 2 by 2 blocks starting with critical nearest neighbor Ising model on square lattice. The model also has an order 8 dihedral group symmetry. Our numerical method breaks this symmetry a priori and we use it as a check on the accuracy. Which sets? Y C For Y = {y 1, y 2,,y n }, Y = n i=1 y i y cm 2, y cm is center of mass. Let Y 1, Y 2 Y N be the sets. (N 10, 000.) We order them so that the coefficients c(y n ) decreases. Approach of this talk applies to general T( σ, σ). Numerical calculations: critical nearest neighbor Ising model on the square lattice, majority rule with 2 by 2 blocks RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.17/29

18 Decay of lattice gas coefs Bottom curve: c(y n ) vs. n. Two lines : c2 n/ Top curve: N i=n c(y n) vs. n e-05 1e ,000 4,000 6,000 8,000 10,000 n RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.18/29

19 4. RG map in spin variables Now consider H in spin variables: H( σ) = Y d(y ) σ(y ), σ(y ) = i Y σ i Use n i = (1 σ i )/2 : H = X c(x) n(x) = X c(x) 2 X i X(1 σ i ) = X = Y c(x) 2 X σ(y )( 1) Y Y :Y X X:Y X ( 1) Y σ(y ) c(x) 2 X d(y ) = ( 1) Y X:Y X c(x) 2 X RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.19/29

20 RG map in spin variables-cont For spin coefficients d(y ) we need the lattice gas coefficients c(x) and hence free energies f(x) for infinitely many X Must introduce some sort of approximation. Y = one set from each translation class of subsets of block lattice H( σ) = d(y ) σ(y + t) Y Y t sum over t is over the translations for the block lattice. Y + t = {i + t : i Y }. Let Y be a finite subcollection of Y. Approximation H( σ) Y Y d(y ) t σ(y + t) Two methods which we will refer to as the partially exact method and the uniformly close method. RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.20/29

21 Partially exact method: H( n) = Y Y c(y ) t n(y + t) Truncate this sum by restricting Y to those in Y: H( n) Y Y c(y ) t n(y + t) Convert this to the spin variables with no approximation. Result: H( σ) Y Y d(y ) t σ(y + t) d(y ) = ( 1) Y c(x) 2 X X:Y X,X+t Y X + t Y means some translation of X is in Y. This approximation agrees exactly with the true H for all configurations σ Y for Y Y. RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.21/29

22 Uniformly close method: Let X be another finite collection of finite subsets which contains at most one set from each translation class. Take X larger than Y. Compute free energies f(x) for X X. error = max X X H( σ X ) Y Y d(y ) t σ X (Y + t) where σ X is the spin configuration which is 1 on X and +1 on all other sites. Choose the coefficients d(y ) to minimize error. RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.22/29

23 RG map in spin variables-cont Think of Y as being fixed. It determines a finite dim space of Hamiltonians Think of X as variable. Larger X more info Our study Y = {Y : Y C H} X = {X : X C f } Study three largest coefs in spin variables : nn, nnn, plaquette Partially exact: plot coef as function of X = Y Uniformly close: plot coef as function of X for several choices of Y RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.23/29

24 Nearest neighbor coefficient - spin variables "p. exact" "unif close: dim=25" "unif close: dim=50" "unif close: dim=100" "unif close: dim=250" "unif close: dim=500" RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.24/29

25 Next nearest neighbor coefficient -spin variables "p. exact" "unif close: dim=25" "unif close: dim=50" "unif close: dim=100" "unif close: dim=250" "unif close: dime=500" RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.25/29

26 Plaquette coefficient - spin variables "p. exact" "unif close: dim=25" "unif close: dim=50" "unif close: dim=100" "unif close: dim=250" "unif close: dim=500" RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.26/29

27 5. Conclusions Lattice gas variables: coefficients only depends on a finite number of values of H. Can be computed very accurately and unambigiously Not discussed: highly accurate method for computing the values of H Lattice gas coefficients decay exponentially over several orders of magnitude with number of terms, but at slow rate. 850 additional terms to see the magnitude reduced by just a factor of 1/2. Spin variables: coefs sensitive to approximation method. Computation of 10, 000 values of H still leaves variation of a percentage for largest coefs Prospects for rigorous results: prove there is fixed point by constructing finite dim approximation in some Banach space... Numerical results suggest that at best such an approach will require a huge number of terms in the finite approximation and at worst the number of terms needed will doom the approach to failure. Wrong representation of H? RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.27/29

28 Future Can you prove anything? Holy grail: rigorous definition of R(H) = H. Existence of fixed point, linearization, etc. First step of map Introducing the block spins makes the correlation length finite. Numerics Further study of decay of coefs of renormalized Hamiltonian Use lattice gas approach to study the map numerically - fixed pt., linearization... Why does decimation fail? Past numerical studies using spin variables produced fairly accurate values for critical exponents. Why is this possible given sensitivity to approximation method? RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.28/29

29 Appendix 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) circles are spins that have yet to be summed over. The red (black) circles are the fixed block spins. B RG in lattice gas varaibles, The Renormalization Group and Statistical Mechanics, July 6-12, UBC, Vancouver, Canada p.29/29