“Boundary Operator” in the Matrix Product States

Isao MaruyamaOsaka University, Japan

DMRG @ Kyoto

“Boundary Operator” in the Matrix Product States

Collaborators and three works

with H. Katsura (Gakusyuin University, Japan)

with H. Ueda (Osaka Univ.), K. Okunishi (Niigata Univ.)

with M. Orii(Osaka Univ.), H. Ueda

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• H.Katsura, I.Maruyama, J. Phys. A. 43. 175003(2010)• I.Maruyama, H.Katsura, J. Phys. Soc. Jpn. 79, 073002 (2010)

• Poster Session: '' New Approaches to Get the Property of Quantum Spin Systems in the Thermodynamic Limit ''

• Poster Session: '‘ Entanglement Entropy and Energy Accuracy for the Small System Size: MPS, TTN, and MERA''

Collaborators and three works(2)




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Exact solution, Bethe ansatz

Numerical calculation

Collaborators and three works




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Exact solution, Bethe ansatz

Numerical calculation

Uniform Matrix Product State(MPS)

Site dependent MPS

Uniform MPS

Uniform MPS with the boundary matrix Ω

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For the S=1/2 spin chain with L sites,

Ostlund, Rommer PRL.75.3537 (1995)


MPS

Uniform MPS


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MPS

Uniform MPS


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As mentioned by Ostlund, Rommer PRL.75.3537 (1995),

Translational Operator T

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T shifts one-site left

translational invariance

Then, when is the boundary matrix Ω important??

As mentioned by Ostlund, Rommer PRL.75.3537 (1995),

Translational Operator T

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translational invariance

Then, when is the boundary matrix Ω important??

the boundary matrix Ω

T shifts one-site left

Exact solution tells us,…

Outline

Back Ground Matrix Product States (MPS)

Bethe Ansatz

What we have done in the two papers

Key words Domain wall boundary condition

Quantum Transfer matrix

5 vertex model

fixed particle number

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F. C. Alcaraz and M. Lazo, J. Phys. A 37 (‘04).

Coordinate BA

algebraic BA

Factorizing

Matrix-product BA

Bethe Ansatz(BA)

J. M. Maillet and J. S. de Santos: q-alg/9612012.

H. Bethe: Z. Phys. 71 (1931) 205.

Back GroundDMRG @ Kyoto

Verstraete, Cirac, Phys. Rev. Lett. 104, 190405 (2010)

Matrix Product State (MPS)

DMRG PWFRG TEBD


•Tensor Product State•TTN, PEPS, MERA•Continuous MPS

Exact ground state• Direst Product of Spin singlet• VBS state

in Majumdar-Ghosh model or Shastry-Sutherland model

In AKLT model

0


Coordinate BA

algebraic BA

Factorizing

Matrix-product BA

Bethe Ansatz(BA)


H. Bethe: Z. Phys. 71 (1931) 205.




DMRG PWFRG TEBD





In AKLT model

0


Coordinate BA

algebraic BA

Factorizing

Matrix-product BA

Bethe Ansatz(BA)


H. Bethe: Z. Phys. 71 (1931) 205.




DMRG PWFRG TEBD



Exact ground state• Direct Product of Spin singlet• VBS state


In AKLT model

0


Coordinate BA

algebraic BA

Factorizing

Matrix-product BA

Bethe Ansatz(BA)


H. Bethe: Z. Phys. 71 (1931) 205.




DMRG PWFRG TEBD



Exact ground state• Direct Product of Spin singlet• VBS state


In AKLT model

0


Coordinate BA

algebraic BA

Factorizing F Matrices

Matrix-product BA

Bethe Ansatz(BA)


H. Bethe: Z. Phys. 71 (1931) 205.

Our workDMRG @ Kyoto



DMRG PWFRG TEBD





In AKLT model

0


Coordinate BA

algebraic BA

Factorizing F Matrices

Matrix-product BA

Bethe Ansatz(BA)


H. Bethe: Z. Phys. 71 (1931) 205.

Our workDMRG @ Kyoto



DMRG PWFRG TEBD





In AKLT model

In Heisenberg chainJ. Phys.A.43.175003

0

Matrix Product BAAlcaraz and Lazo’s work.

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F. C. Alcaraz and M. Lazo, J. Phys. A 37 (‘04). Algebraic relations among matrices is given.

Explicit form of A is given. 2n dimension.F. C. Alcaraz and M. Lazo, J. Phys. A 39 (‘06))

Questions and Motivations:1. Why the dimension of matrices is 2n? Not 2L?2. What is an explicit form of Ω?

Spin ½ Heisenberg Hamiltonian ( L sites, n down spins )

Why the boundary matrix is required

a Bethe state with momentum P

Alcaraz and Lazo require

as one of the algebraic relations

If P=0, Ω=1 satisfies the relation above.

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Then, we can neglect the boundary matrix Ω!?→ the answer is NO.

Answers:2n due to the six vertex model / Ω is the DWBC

XXZ model = 6-vertex model (2D statistical model)

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( Others =0 )

Questions and Motivations:1. Why the dimension of matrices is 2n? Not 2L?2. What is an explicit form of Ω?



Bethe state( L sites, n down spins )

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( Others =0 )




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( Others =0 )




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( Others =0 )




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( Others =0 )




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( Others =0 )




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( Others =0 )

MPS is




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( Others =0 )

MPS is

The dimension of matrix is 2n

Answer!




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( Others =0 )

MPS is




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( Others =0 )

MPS is

Domain Wall Boundary Condition(DWBC)

Answer!




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( Others =0 )

MPS is

E.g.:Suzuki Trotter decomposition M.Suzuki PTP.56.1454, PRB.31.2957

Quantum transfer matrix

5 vertex model

Obtained MPS does not agree with the Matrix Product BA.

We need a simplification via a“gauge” transformation.

We found that F is given by

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5 vertex model

After the simplification

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6 vertex model

We found this expression agrees with the Matrix Product BA

Two solvable models

Heisenberg spin chain

Lieb-Liniger model

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We have studied two solvable models

: 1D Bose gas with point interaction

Experimental realization in trapped one-dimensional gases

NATURAE.429.277, SCIENCE.305.1125

I.Maruyama, H.Katsura, J. Phys. Soc. Jpn. 79, 073002 (2010)

H.Katsura, I.Maruyama, J. Phys. A. 43. 175003(2010)

Method

Artificial discretization L=Na

Bethe state on a lattice with finite n particles (n down arrows, N-n up arrows= )

MPS on a lattice

Continuous limit N→∞, a→0:

Problem: ∞ number of in MPS !

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Same as spin ½ except for boson’s ∞ d.o.f.

Method

Artificial discretization L=Na

Bethe state on a lattice with finite n particles (n down arrows, N-n up arrows= )

MPS on a lattice

Continuous limit N→∞, a→0:

Problem: ∞ number of in MPS.

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Same as spin ½ except for boson’s ∞ d.o.f.

Due to , it is diagonal matrix

Another notation:

Exact continuous MPS (cMPS)

We obtain…

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• path ordered operator : P• reflection(creation) operator

• momentum operator

• boundary operator to fix the particle number.

“world line” graph for continuous MPS

Continuous time Loop algorithm

“world line” graph for continuous MPS

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Beard, Wiese PRL.77.5130

Artificial discretization MPS ST-

deconposition

Continuous limit N→∞ Trotter number

→∞

Continuous MPS

Continuous-(imaginary) time

Simplification due to S-matrix comes from crossing “world line”

http://takayama.issp.u-tokyo.ac.jp/Topics/C-1-0804.html

Summary

Key words Domain wall boundary condition

Quantum Transfer matrix

5 vertex model

fixed particle number

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Ω

Ω

Uniform MPS with the Boundary matrix for the Bethe ansatz




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State with the total momentum P

Domain wall boundary condition in the Bethe ansatz

fixing the particle number in the Bethe ansatz.

Periodicity of Spontaneous translational symmetry broken state

Choice of the principal eigenvalues of the transfer matrix for the uniform MPS in the thermodynamic limit

Importance of the Boundary Matrix Ω

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Periodicity and the boundary matrix.

For AF Ising model, Neel states there are doubly degenerated ground state with

spontaneous translational symmetry breaking.

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Uniform MPS

With the boundary matrix

The periodicity is controlled by the boundary operator

END

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“Boundary Operator” in the Matrix Product States