11410PHYS401200

https://github.com/pcchen/11410PHYS401200/

pcchen@phys.nthu.edu.tw

Links

• https://exercism.org/
• https://sci-tai.vm.nthu.edu.tw/signin
• https://github.com/smorita/TRG_Ising_2D
• https://smorita.github.io/TN_animation/
• https://itensor.org/docs.cgi?page=book/trg
• https://github.com/Cytnx-dev/Cytnx
• https://www.scipost.org/SciPostPhysCodeb.53

2D Ising Model

Z(2x2)(T=1, h=0) = 5973.916645008712

Assignment-1

For 1D Ising model

• Find/Derive analytical expression for transfer matrix at any temperture T and external field h.
• Find/Derive analytical expression for free energy per site for L-sites system.
• Code: Use exact summation to evaluate partition function, and free energy per site for L-sites system.
• Code: Use transfer matrix to evaluate partition function, and free energy per site for L-sites system.
• Code: Plot free energy per site v.s. T, with different L.
• Code: Plot specific heat per site v.s. T, with different L.

For 2D Ising model

• Code: Use exact summation to evaluate partition function, and free energy per site for Lx * Ly system.
• Code: Use transfer matrix to evaluate partition function, and free energy per site for Lx * Ly system.
• Code: Plot free energy per site v.s. T, with (Lx,Ly)=(2,2), (3,2), (2,3,), (3,3), (3,4), (4,3), (4,4), etc
• Code: Plot specific heat per site v.s. T, with different (Lx,Ly)=(2,2), (3,2), (2,3,), (3,3), (3,4), (4,3), (4,4), etc

Assignment-2

For 2D Ising model

• Construct rank-2 tensor (=matrix) M
• Construct rank-4 tensor T from M
• When temp=1=J, element of T includes
  [[4.76220e+00 0.00000e+00 0.00000e+00 3.62686e+00 ]
  [0.00000e+00 3.62686e+00 3.62686e+00 0.00000e+00 ]
  [0.00000e+00 3.62686e+00 3.62686e+00 0.00000e+00 ]
  [3.62686e+00 0.00000e+00 0.00000e+00 2.76220e+00 ]]
  
• Contract four T tensors to get Z_2x2
• Use numpy.linalg.svd and numpy.linalg.eigh to find M-matrix from W-matrix.

Assignment-3

• Use list and np.array to create vector=rank-1 tensor and matrix=rank-2 tensor.
• Perform vector dot and matrix multiplication, study/plot time v.s. dimension.
• Try numpy.tensordot and numpy.einsum.

Assignment-4

• Perform SVD on nxn matrix and truncate it, see how error increases as you truncate more singular values.