Post on 07-May-2022
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
DMRG @ Kyoto
• 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)
with H. Katsura (Gakusyuin University, Japan)
with H. Ueda (Osaka Univ.), K. Okunishi (Niigata Univ.)
with M. Orii(Osaka Univ.), H. Ueda
DMRG @ Kyoto
• 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''
Exact solution, Bethe ansatz
Numerical calculation
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
DMRG @ Kyoto
• 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''
Exact solution, Bethe ansatz
Numerical calculation
Uniform Matrix Product State(MPS)
Site dependent MPS
Uniform MPS
Uniform MPS with the boundary matrix Ω
DMRG @ Kyoto
For the S=1/2 spin chain with L sites,
Ostlund, Rommer PRL.75.3537 (1995)
Uniform Matrix Product State(MPS)
MPS
Uniform MPS
Uniform MPS with the boundary matrix Ω
DMRG @ Kyoto
For the S=1/2 spin chain with L sites,
Ostlund, Rommer PRL.75.3537 (1995)
Uniform Matrix Product State(MPS)
MPS
Uniform MPS
Uniform MPS with the boundary matrix Ω
DMRG @ Kyoto
For the S=1/2 spin chain with L sites,
Ostlund, Rommer PRL.75.3537 (1995)
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
DMRG @ Kyoto
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
DMRG @ Kyoto
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
Ostlund, Rommer PRL.75.3537 (1995)
•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
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
Ostlund, Rommer PRL.75.3537 (1995)
•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
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
Ostlund, Rommer PRL.75.3537 (1995)
•Tensor Product State•TTN, PEPS, MERA•Continuous MPS
Exact ground state• Direct Product of Spin singlet• VBS state
in Majumdar-Ghosh model or Shastry-Sutherland model
In AKLT model
0
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
Ostlund, Rommer PRL.75.3537 (1995)
•Tensor Product State•TTN, PEPS, MERA•Continuous MPS
Exact ground state• Direct Product of Spin singlet• VBS state
in Majumdar-Ghosh model or Shastry-Sutherland model
In AKLT model
0
F. C. Alcaraz and M. Lazo, J. Phys. A 37 (‘04).
Coordinate BA
algebraic BA
Factorizing F Matrices
Matrix-product BA
Bethe Ansatz(BA)
J. M. Maillet and J. S. de Santos: q-alg/9612012.
H. Bethe: Z. Phys. 71 (1931) 205.
Our workDMRG @ Kyoto
Verstraete, Cirac, Phys. Rev. Lett. 104, 190405 (2010)
Matrix Product State (MPS)
DMRG PWFRG TEBD
Ostlund, Rommer PRL.75.3537 (1995)
•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
F. C. Alcaraz and M. Lazo, J. Phys. A 37 (‘04).
Coordinate BA
algebraic BA
Factorizing F Matrices
Matrix-product BA
Bethe Ansatz(BA)
J. M. Maillet and J. S. de Santos: q-alg/9612012.
H. Bethe: Z. Phys. 71 (1931) 205.
Our workDMRG @ Kyoto
Verstraete, Cirac, Phys. Rev. Lett. 104, 190405 (2010)
Matrix Product State (MPS)
DMRG PWFRG TEBD
Ostlund, Rommer PRL.75.3537 (1995)
•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
In Heisenberg chainJ. Phys.A.43.175003
0
Matrix Product BAAlcaraz and Lazo’s work.
DMRG @ Kyoto
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.
DMRG @ Kyoto
F. C. Alcaraz and M. Lazo, J. Phys. A 37 (‘04).
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)
DMRG @ Kyoto
( Others =0 )
Questions and Motivations:1. Why the dimension of matrices is 2n? Not 2L?2. What is an explicit form of Ω?
Answers:2n due to the six vertex model / Ω is the DWBC
XXZ model = 6-vertex model (2D statistical model)
Bethe state( L sites, n down spins )
DMRG @ Kyoto
( Others =0 )
Answers:2n due to the six vertex model / Ω is the DWBC
XXZ model = 6-vertex model (2D statistical model)
Bethe state( L sites, n down spins )
DMRG @ Kyoto
( Others =0 )
Answers:2n due to the six vertex model / Ω is the DWBC
XXZ model = 6-vertex model (2D statistical model)
Bethe state( L sites, n down spins )
DMRG @ Kyoto
( Others =0 )
Answers:2n due to the six vertex model / Ω is the DWBC
XXZ model = 6-vertex model (2D statistical model)
Bethe state( L sites, n down spins )
DMRG @ Kyoto
( Others =0 )
Answers:2n due to the six vertex model / Ω is the DWBC
XXZ model = 6-vertex model (2D statistical model)
Bethe state( L sites, n down spins )
DMRG @ Kyoto
( Others =0 )
Answers:2n due to the six vertex model / Ω is the DWBC
XXZ model = 6-vertex model (2D statistical model)
Bethe state( L sites, n down spins )
DMRG @ Kyoto
( Others =0 )
MPS is
Answers:2n due to the six vertex model / Ω is the DWBC
XXZ model = 6-vertex model (2D statistical model)
Bethe state( L sites, n down spins )
DMRG @ Kyoto
( Others =0 )
MPS is
The dimension of matrix is 2n
Answer!
Answers:2n due to the six vertex model / Ω is the DWBC
XXZ model = 6-vertex model (2D statistical model)
Bethe state( L sites, n down spins )
DMRG @ Kyoto
( Others =0 )
MPS is
Answers:2n due to the six vertex model / Ω is the DWBC
XXZ model = 6-vertex model (2D statistical model)
Bethe state( L sites, n down spins )
DMRG @ Kyoto
( Others =0 )
MPS is
Domain Wall Boundary Condition(DWBC)
Answer!
Answers:2n due to the six vertex model / Ω is the DWBC
XXZ model = 6-vertex model (2D statistical model)
Bethe state( L sites, n down spins )
DMRG @ Kyoto
( 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
DMRG @ Kyoto
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 !
DMRG @ Kyoto
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
DMRG @ Kyoto
the boundary matrix Ω
Ω
Ω
Uniform MPS with the Boundary matrix for the Bethe ansatz
with H. Katsura (Gakusyuin University, Japan)
with H. Ueda (Osaka Univ.), K. Okunishi (Niigata Univ.)
with M. Orii(Osaka Univ.), H. Ueda
DMRG @ Kyoto
• 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''
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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the boundary matrix Ω
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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