Transfer matrix method for next-nearest-neighbor Ising models

Background

The transfer matrix (TM) method calculate the partition function and correlation length of a semi-infinite stripe of $L \times \infty$, obtained from the first a few eigenvalues. It also gives the marginal probability distribution of every possible state on a layer $L$. Combining these one can also generate the equilibrium configuration of strip width L and arbitrary length N.

Summary

Language: C++ (std=c++11)

Publications:

• Yi Hu and Patrick Charbonneau. "Resolving the two-dimensional axial next-nearest-neighbor Ising model using transfer matrices." Physical Review B, 103, 094441 (2021). http://doi.org/10.1103/PhysRevB.103.094441

• Yi Hu and Patrick Charbonneau. "Numerical transfer matrix study of frustrated next-nearest-neighbor Ising models on square lattices." Physical Review B, 104, 144429 (2021). http://doi.org/10.1103/PhysRevB.104.144429

File descriptions

• TMat/ : transfer matrix computation core
• DumpConfig/ : dump equilibrium configurations
• tmat.make : makefile script for transfer matrix calculation
• dumpconfig.make : makefile script for configuration planting

Compilation

The code requires Eigen3

The code also requires Spectralib v0.9. Note that v1.x the API has changed.

Install these libs in system, or set the header path in $INCLUDE environment variable. Run make -f tmat.make to compile for tmat and make -f dumpconfig.make for dumpconfig. Modify makefile scripts when necessary.

Running

Transfer matrix calculation

The binary "tmat" takes one parameter which is the input file. It computes for the eigenvalue, right and left eigenvectors output as "X_val.dat", "X_vec.dat", "X_vecl.dat".

An example input file is illustrated below:

modeltype = B # model type. A: ANNNI; B: BNNNI and 3NN; D: DNNI
dim = 2       # dimensionality. 2 or 3
axialdir = 6  # indices of the type of boundary condition (see source)
neigs = 1     # number of eigenvalues been calculated
savevec = 1 1 # options (1 or 0) to save vectors or computing layer magnetization (or not)
outfile = 2DYBM  # prefix for output files
size = 12        # system width L
J = 0.5          # inverse temperature beta * J 
kappa = 0.8 0    # kappa1 and kappa2 values
h = 0            # external field

For 2D (dim = 2):

• modeltype support A, B, D for the ANNNI, BNNNI+3NN, and DNNI model, respectively.

• For the ANNNI model, axialdir supports

  • 0: y-double-layer
  • 1: $\perp$-TM, full $2^L \times 2^L$ transfer matrix
  • 2: $\parallel$-TM, full $4^L \times 4^L$ transfer matrix
  • 5: $\perp$-TM, reduced size
  • 6: $\parallel$-TM, reduced size
  • 9: $\perp$-TM, open boundary
  • 13: $\perp$-TM, reduced size, open boundary
  • 17, 18: $\perp$-TM, a pair of additional up-up or up-down spins in each side
  • 19, 20: $\perp$-TM, two pair of additional up-up or up-down spins in each side

• For the BNNNI/3NN model, axialdir supports

  • 3: $/$-TM, full $2^L \times 2^L$ transfer matrix
  • 6: $\parallel$-TM, reduced size
  • 7: $/$-TM, open boundary
  • 13: $\perp$-TM, reduced size

• For the DNNI model, axialdir supports

  • 1: $\perp$-TM, full $2^L \times 2^L$ transfer matrix
  • 2: $\parallel$-TM, full $4^L \times 4^L$ transfer matrix
  • 5: $\perp$-TM, reduced size
  • 6: $\parallel$-TM, reduced size
  • 17, 18: $\perp$-TM, a pair of additional up-up or up-down spins in each side

Configuration planting

The binary "dumpconfig" takes three parameters, for example, "./dumpconfig input_ymannni.dat 1 100". The first parameter is the input file, the second is the number of configurations to generate, the third is the row of configurations.

It generates an equilibrium configuration, e.g.,

000000000000
000000000000
111111111111
111111111111
011100000000
000000000000
100011111111
111111111111
111111111111
000000000000

where 0s are up spins and 1s are down spins.