NTQMP@京大 （2014/12/16）

ロシアより I をこめて
(From Russia with “I”)

桂 法称（東大理・物理学専攻）
 H. Katsura and I. Maruyama, JPA 43, 175003 (2010).
 S. Tanaka, R. Tamura, and H. Katsura, PRA 86, 032326 (2012).
 H. Katsura, arXiv:1407.4267 (to be published in J. Stat. Mech.).

Outline
1. Introduction and Motivation•

Quantum integrable models: history & present status

2. MPS, MPO and Tensor Networks•

How to express many-body states? MPS & MPO•

Tensor networks ~ stat. mech. rephrasing

3. Integrability in the Age of MPS & MPO•

Yang-Baxter eq., commuting transfer matrices•

Algebraic Bethe ansatz in a nutshell•

Bethe states as MPS, Factorizing F-matrix

4. Entanglement Meets Integrability• How to cook up integrable density matrix?
• Integrable MPO with D=4

Crude History of Quantum Integrable Models
 Lattice models
Bethe ansatz
1931: Heisenberg XXX chain (Bethe)
1944: 2d Ising model (Onsager, Kaufman) Free fermions
1958-1966: XXZ chain (Orbach, Walker, Yang-Yang)
1961-1962: XY chains (Lieb-Schultz-Mattis, Katsura)
1968: Hubbard chain (Lieb-Wu)
TQ relation
1972: XYZ chain (Baxter)
2d stat. mech.: 6-, 8-, 16-vertex models, RSOS models
 Continuum models & field theories
1963: Quantum NLS (Lieb-Liniger)
1979: Massive Thirring (Bergknoff-Thacker)
Sine-Gordon (Sklyanin-Takhtajan-Faddeev)
Kondo problem: Andrei, Wiegmann, Kawakami, …
 Mathematical sophistication Ancestor of MPS/MPO?
1979-: QISM (Leningrad group) or ABA
1985-: Quantum group (Drinfeld, Jimbo), Yangian, …

Present Status
 Applications to cond-mat. and atomic physics
1. Quantum magnetism: KCuF3, Sr2CuO3, …
Inverse scattering meets neutron scattering!
2. Cold atoms in optical lattices
Realization & manipulation of 1d systems
e.g. Kinoshita et al., Science 305, 1125 (‘04).

Lake et al., PRL 111 (‘13)

 Classical & Quantum nonequilibrium models
1. Solvable stochastic processes
ASEP, TASEP, KPZ eq.⇔ XXZ, quantum NLS

From Prosen’s slide.

See short review by Spohn, arXiv:1204.2657.

2. Solvable dissipative dynamics
Exact NESS of boundary-driven Lindblad eq.
Prosen, PRL 106, 107 (‘11), Prosen, Ilievski & Popkov, NJP 15 (‘13).

 AdS/CFT integrability
Duality between N = 4 super Yang-Mills & IIB superstrings
Bethe ansatz computation of scaling dimensions
See e.g. Beisert et al., LMP 99 (’11), arXiv:1012.3982.

Today’s subject
 Integrability in the age of MPS, MPO & tensor network
Quantum inverse scattering method (QISM) & algebraic Betheansatz (ABA) borrowed the ideas from classical soliton theory.
(See e.g. Faddeev’s lecture notes, Korepin’s textbook.)
But they can be reformulated in the language of MPS & MPO.
1. H. Katsura & I. Maruyama, JPA 43, 175003 (2010).
2. V. Murg, V. E. Korepin & F. Verstraete, PRB 86, 045125 (2012).
QISM & ABA inthe age of solitons?

Conference “Quantum Solitons” in the early 1980s (from Reshetikhin’s webpage)Outline
1. Introduction and Motivation•

Quantum integrable models: history & present status

2. MPS, MPO and Tensor Networks•

How to express many-body states? MPS & MPO•

Tensor networks ~ stat. mech. rephrasing

3. Integrability in the Age of MPS & MPO•

Yang-Baxter eq., commuting transfer matrices•

Algebraic Bethe ansatz in a nutshell•

Bethe states as MPS, Factorizing F-matrix

4. Entanglement Meets Integrability• How to cook up integrable density matrix?
• Integrable MPO with D=4

How to express many-body states Preliminary
Consider a finite “spin” chain of length N.
Local Hilbert space:spanned by
Total Hilbert space:
A general many-body state

We need to store

can be expressed as

numbers! Too hard to deal with…

 Product statesEx.)

Classical states with no entanglement. Too boring to study…

Auxiliary Space & Matrix Product States (MPS)
Consider an auxiliary vector spacematrices
act. Any stateon which D×D
can be expressed with

Matrix-productstates (MPS)
We can handle states with small D.
States with D ~ dL are still hard to deal with…
Ex.1) GHZ state (d=2, D=2)

Ex.2) Valence-bond-solid (VBS) state (d=3, D=2)
Affleck-Kennedy-Lieb-Tasaki (AKLT)  “Haldane gap”
Finitely correlated states: Fannes et al., Kluemper et al.

Matrix Product Operators
Any operator on
Any operator on

We need to store

can be expressed as a sum of
can be expressed in the form:

numbers! Too hard to deal with…

 Matrix-product operators (MPO)
Again consider an auxiliary vector spaceon which D×Dmatrices
act. Any operator on
can be expressed as

or more generally, a sum ofBoundary matrix.

Tensor Networks ~ Stat. Mech. Rephrasing
 MPS (uniform case)
Building block(Local weight)

Stat. mech. model on a comb!

 MPO (uniform case)
Building block(Local weight)
Transfer matrix of a 2d vertex model!

MPO naturally act on MPS. But tensor
products of aux. space should appear.

Outline
1. Introduction and Motivation•Quantum integrable models: history & present status

2. MPS, MPO and Tensor Networks•

How to express many-body states? MPS & MPO•

Tensor networks ~ stat. mech. rephrasing

3. Integrability in the Age of MPS & MPO•

Yang-Baxter eq., commuting transfer matrices•

Algebraic Bethe ansatz in a nutshell•

Bethe states as MPS, Factorizing F-matrix

4. Entanglement Meets Integrability• How to cook up integrable density matrix?
• Integrable MPO with D=4

Integrability in the age of MPS & MPO (1) Local Lax matrix (L-operator)
Consider a finite “spin” chain of length N.
: auxiliary space,: quantum space of j-th site.
L-operators acting on
is defined by
λ: spectral parameter

This is not an MPO! (Trace is not taken over V0.)Graphical rules:

Integrability in the age of MPS & MPO (2) Monodromy matrix (acting on)

This is still not an MPO… (Trace is not taken over V0.)
 Transfer matrix

This is an MPO! (Trace is taken over V0.) T(λ) can be thoughtof as a transfer matrix of 2d vertex models (d2D2 configurations).
So far, everything is quite general.
How does integrability come into play??

Integrability in the age of MPS & MPO (3) Yang-Baxter relation: tensor product,: R-matrix: matrix product

RLL = LLRrelation
Sufficient condition(Infinitely) many solutions have been found. Many important
examples have the difference property:

Integrability in the age of MPS & MPO (4) From local to global
Suppose R-matrix is invertible.
Noting[T(λ), T(μ)]=0 for any λ, μ.

and the cyclic rule

Commutingtransfer matrices!

 Infinitely many conserved charges
implies
Ik’s are mutually commuting! They are spatially local andsimultaneously diagonalizable via Algebraic Bethe ansatz.

Integrability in the age of MPS & MPO (4) From local to global
Suppose R-matrix is invertible.
Noting[T(λ), T(μ)]=0 for any λ, μ.

and the cyclic rule

Commutingtransfer matrices!

 Infinitely many conserved charges
implies
Ik’s are mutually commuting! They are spatially local andsimultaneously diagonalizable via Algebraic Bethe ansatz.

Integrability in the age of MPS & MPO (4) From local to global
Suppose R-matrix is invertible.
Noting[T(λ), T(μ)]=0 for any λ, μ.

and the cyclic rule

Commutingtransfer matrices!

 Infinitely many conserved charges
implies
Ik’s are mutually commuting! They are spatially local andsimultaneously diagonalizable via Algebraic Bethe ansatz.

Integrability in the age of MPS & MPO (4) From local to global
Suppose R-matrix is invertible.
Noting[T(λ), T(μ)]=0 for any λ, μ.

and the cyclic rule

Commutingtransfer matrices!

 Infinitely many conserved charges
implies
Ik’s are mutually commuting! They are spatially local andsimultaneously diagonalizable via Algebraic Bethe ansatz.

Integrability in the age of MPS & MPO (4) From local to global
Suppose R-matrix is invertible.
Noting[T(λ), T(μ)]=0 for any λ, μ.

and the cyclic rule

Commutingtransfer matrices!

 Infinitely many conserved charges
implies
Ik’s are mutually commuting! They are spatially local andsimultaneously diagonalizable via Algebraic Bethe ansatz.

Algebraic Bethe Anstaz in a Nutshell (1) The simplest case (d=D=2, difference property)
Pauli matrices

L-operators

R-matrixRLL=LLR can be checked easily (with Mathematica).
 Six-vertex model ~ stat. mech. rephrasing
Quantum space:
Auxiliary space:

Only six configurations have nonzero weights.
Ice rule: (# of incoming arrows)=(# of outgoing arrows)

Algebraic Bethe ansatz in a Nutshell (2) Quantum Hamiltonian
Pauli matrix on j-th site:

Heisenberg XXX Hamiltonian

H and T(λ) are commuting.  They share the same eigenstates.
 ABCD of ABA

in the basis of auxiliary space

From Yang-Baxter eq., we get, which gives
algebraic relations among A,B,C & D, e.g.,

Algebraic Bethe ansatz in a Nutshell (3) Diagonalization of T
is an eigenstate of T(λ)=A(λ)+D(λ).

because there must be
is a new state! (1-magnon state)
Demanding thatis an eigenstate ofT(λ), we get the condition on λ’s. Bethe equation:

Now the hard work begins. Yang-Yang, string hypothesis …
Generalized Pauli principle: λ’s are distinct.

.

Bethe states as MPS (1) Domain wall boundary condition (DWBC) in aux. space
Bethe states:
Partition function of6-vertex with DWBC!

 Construction of A↑ and A↓

Bethe states as MPS (2) Factorizing F-matrix (Drinfeld twist)
There is a similarity tr. which makes A↑ diagonal:
One can prove this by induction on n. In addition,
Bethe states & DWBC remain unchanged!!

We never have the vertex.
A↓(j) satisfy an analogue of theZamolodchikov-Faddeev algebra.
For explicit expressions, see Katsura & Maruyama, JPA 43 (‘10).
This gives a proof of ansatz raised by Alcaraz & Lazo, JPA 37 (‘04).

Outline
1. Introduction and Motivation•Quantum integrable models: history & present status

2. MPS, MPO and Tensor Networks•

How to express many-body states? MPS & MPO•

Tensor networks ~ stat. mech. rephrasing

3. Integrability in the Age of MPS & MPO•

Yang-Baxter eq., commuting transfer matrices•

Algebraic Bethe ansatz in a nutshell•

Bethe states as MPS, Factorizing F-matrix

4. Entanglement Meets Integrability• How to cook up integrable density matrix?
• Integrable MPO with D=4

How to cook up Integrable Density Matrix?
 Vertex/Face correspondence
Transfer matrix

Yang-Baxter relation
This correspondence is not always
one-to-one. Face-type models
include Ising, hard-hexagon, IRF, …

 MPO/ladder state correspondence
AB

Bijection, leads to

Reduced density matrix for A:.
.

Quantum hard-hexagon model Frustration-free model
Parent Hamiltonian:
For a triangular ladder,G.S.:
Face weights (n.n. exclusion) Entanglement spectrum (ES)Double-row
transfer matrixof hard-hexagons!
Entanglement Hamiltonian

The 2d classical model is integrable(Baxter, 1980). Critical at zc=11.09…

Spectrum of HE at z=zc
is described by c=4/5 CFT.
(3-state Potts universality)
See Tanaka, Tamura & Katsura,
PRA 86, 032326 (2012).

kIntegrable MPO with D=4 (1) Deformation of hexagonal VBS density matrix: orthonormal basis in(aux. space),
Consider the following MPO:

4×4matrices

Hexagonal VBS on 2-leg ladder

AB

Entanglement spectrum
S=3/2 and S=1
spins are mixed.

Reduced density matrix
ES resembles the spectrumof the ferromagnetic XXX!
Lou, Tanaka, Katsura & Kawashima, PRB 84, 245128 (2011).

Integrable MPO with D=4 (2) Miraculous properties
M(x,y) related to VBS does not exhibit any integrability…
However, M(cosθ,sinθ) have the following properties:
Property 1. For any N and arbitrary λ, θ∈R, the operator
M(cosθ,sinθ) commutes with the XXX transfer matrix T(λ).
Property 2. For any N and arbitrary θ1, θ2∈R, the operator
M(cosθ1,sinθ1) and M(cosθ2,sinθ2) commute with each other.
They strongly suggest the integrabilty of M(cosθ,sinθ).
In fact, they can be proved as corollaries ofTheorem. For any N and arbitrary θ ∈R, the operator
M(cosθ,sinθ) is written in terms of transfer matrices as

Proof is based on a similarity transformationand the Yang-Baxter eq. For details, see H.Katsura, arXiv:1407.4267.

Summary• Modern formulation of QISM & ABA• Bethe states can be expressed as MPS• Factorizing F-matrix simplifies the MPS• Construction of integrable density matrix• Integrable MPO and hexagonal VBS state Prospects• ABA with OBC: reflection eq.: RKRK=KRKR
V.Murg, V.E.Korepin & F.Verstraete, PRB 86 (2012).

• Integrable density matrix + Suzuki-Trotter
Exact computation of Renyi entropies for ladders?

• Application to many-body localization? (Cirac, Huse, Hastings, …)• 3(=2+1) D generalizations
Tetrahedron relation: RRRR=RRRR (Zamolodchikov, Baxter-Bazhanov)Topological invariants of 3-manifolds (Witten, Reshetikhin, Turaev-Viro)