[ Japanese ]

I put this web page in order to glance at the developments of "Tensor Product State" (TPS)

in the field of statistical physics. It should be noted that from the field theoretical view

point, statistical problems on the lattice is deeply related to quantum physics. Please first

look at the Section 2 of this review. One finds that Baxter's contribution is quite important.

One of the earliest study on MPS/TPS was done by Kramers and Wannier on the study of

the Ising Model: H.A. Kramers and G.H. Wannier: Phys. Rev.

Compared with the Onsager's exact result L. Onsager: Phys. Rev.

approximation gives reasonablly (?) good thermodynamics.

In the following, I pick up some of my contribution, which are listed here, to this subject.[References: Tensor Product State already exists in 1968!!! <- from DMRG Workshop in 2006.]

Baxter: J. Math. Phys. 9, 650 (1968) / Kelland: Can. J. Phys. 54, 1621(1976)

Baxter: J. Stat. Phys. 19. 461(1978) / Tsang: J. Stat. Phys. 20, 95 (1979)

Baxter and I.G. Enting: J. Stat. Phys. 21, 103 (1979)

It is straight forward to apply the K-W approximation to 3D Ising model, if one employs two-

dimensional product of local weights: Prog. Theor. Phys.

This is a kind of tensor product state, which does not contains auxiliary variables. We call it

IRF-type tensor product state. Although the degree of freedom for a local weight is limitted,

optimization of them by way of (free-)energy minimization draws good estimates for phase

transition temperatures. For the application to the ANNNI model, position dependence of each

weight is allowed.

The next step is to introduce auxilially variables, which is often called as ancilla. Since theapplied to 3D Ising Model: Nucl. Phys.

B575(2000) 504-512 (cond-mat/0001083)

q=3, 4, and 5 Potts models: Phys. Rev.E65, 046702 (2002) (cond-mat/0102425)

applied to 3D ANNNI Model: Phys. Rev.B71(2005) 024404 (cond-mat/0210356 ver.3)

applied to 2D Heisenberg Model: (cond-mat/0401115 unpublished) <<<--- Quantum!!!

applied to 3D Ising Model: Acta Phys. Slov. 55 (2005) 141 (cond-mat/0412192)

introduction of ancilla increases the dimension of local weights, the variational minimum

becomes better, while numerical cost increases. In order to decrease the computational effort,

we also considered the pyramidal shape local weight, whch was considered (independently??)

by Klumper et al, Hieida, Sierra and Martin-Delgado. We call this type of variational states as

the vertex-type ones. (With Auxiliary Variables; vertex-Type)

I reached the above formulation through the study of Matrix Product Formulation in DMRG,applied to 3D Ising Model (2-state TPS): Prog. Theor. Phys.

105(2001) No.3, 409-417. (cond-mat/0011103)

applied to 2D transverse field Ising Model: Phys. Rev.E64(2001) 016705 [1-6] (cond-mat/0101360)

applied to 3D Ising Model (3-state TPS): Prog. Theor. Phys.110(2003) No.4, 691-699 (cond-mat/0303376)

which was established by Ostlund and Rommer. Even now my research interest is on MPS, and have

been published following papers

Product Wave Function RG: J.Phys. Soc. Jpn.

64(1995) 4084-4087 (cond-mat/9510004)

MPS and DMRG: Europhys. Lett,43No. 4 (1998) 457-462. (cond-mat/9710310)

Fixed Point of the Finite System DMRG: J. Phys. Soc. Jpn.68(1999) 1537-1540 (cond-mat/9810241)

Introduction from a variational view point: Int. J. Mod. Phys.B13(1999) 1-24. (REVIEW)

Product Wave Function RG and MPS: J. Phys. Soc. Jpn. 75 (2006) 014003 (cond-mat/0509320)

Two-Site shift PWFRG: (arXiv:0807.1798)