Classification and dichotomy theorems of integrability and non-integrability of quantum spin chains

Naoto Shiraishi (University of Tokyo)

N. Shiraishi, Europhys. Lett. 128 17002 (2019)

M. Yamaguchi, Y. Chiba, N. Shiraishi, arXiv:2411.02162

M. Yamaguchi, Y. Chiba, N. Shiraishi, arXiv:2411.02163

N. Shiraishi, arXiv:2501.15506

N. Shiraishi and M. Yamaguchi, arXiv:2504.14315 (updated!)

Outline

Background

S=1/2 spin chain with NN and NNN interaction

Isotropic spin chain

Outline

Background

S=1/2 spin chain with NN and NNN interaction

Isotropic spin chain

Quantum integrability/non-integrability

local = a sum of (spatially) local quantities.

nontrivial = Its support is larger than Hamiltonian’s.

However, no established definition of quantum integrability/non-integrability exists…

Our subject: integrability/non-integrability

(J.-S. Caux, J. Mossel, J. Stat. Mech. P02023 (2011),

C. Gogolin and J. Eisert, Rep. Prog. Phys. 79, 056001 (2016).)

Focus on nontrivial local conserved quantities (CQ) as characterization of systems with local interaction.

integrability/non-integrability and local conserved quantities (CQ)

Expected correspondences are

(and one of k-support operator has nonzero coefficient.)

Of course, these two have a gap, but…

Dichotomy conjecture

(Related conjectures: M. P. Grabowski and P. Mathieu, J. Phys. A: Math. Gen. 28, 4777 (1995), T. Gombor and B. Pozsgay, Phys. Rev. E 104, 054123(2021))

Dichotomy conjecture

A locally-interacting system possesses

1. k-local CQ for all k, or
2. no nontrivial local CQ.

# of nontrivial k-local CQ

absent

non-integrable

integrable

(partially-integrable)

No model sits in this region

…all existing models satisfy this dichotomy:

Our goal of this talk

Dichotomy conjecture is highly plausible

(All known models satisfy this conjecture).

But, can we show this conjecture rigorously?

→Yes! We prove the dichotomy in

• S=1/2 chains with NN & NNN interaction
• isotropic spin chains.

Outline

Background

S=1/2 spin chain with NN and NNN interaction

Isotropic spin chain

Classification of NN and NNN symmetric S=1/2 chain

with is nonintegrable (no local CQ) except for known integrable models.

(NNN case: N. Shiraishi, arXiv:2501.15506)

(NN case: M. Yamaguchi, Y. Chiba, N. Shiraishi, arXiv:2411.02162 / arXiv:2411.02163)

List of known integrable models

Classical:

Bethe solvable:

XY model: (+z field)

XXZ model: (+z field)

XYZ model:

Implications

This is the first complete classification of

integrable/non-integrable for a wide class of systems.

• No missing integrable systems in this class.

​

• No intermediate (partially-integrable) systems. →Dichotomy is proven in these classes.

Proof of non-integrability: basic idea

Take XYZ chain with z magnetic field as an example.

Proof of non-integrability: outline 1

Proof of non-integrability: outline 2

We have

Outline

Background

S=1/2 spin chain with NN and NNN interaction

Isotropic spin chain

Motivation: dichotomy as a guidepost

However, dichotomy (in particular, conjecture by Grabowski-Mathieu) originally aims to serve as a guidepost for studying unexplored models.

Can we show the dichotomy for a broad class of models including unexplored ones?

Dichotomy of isotropic spin chain

(N. Shiraishi and M. Yamaguchi, arXiv:2504.14315)

Remarks and comments

Comments

• Our criterion is a slightly weaker condition than the Reshetikhin condition (= lowest-order Yang-Baxter equation)

Proof outline: non-integrability 1

(To show this, we use the fuzzy (non-commutative) spherical harmonics as basis matrices.)

In particular, a candidate of 3-local CQ is

Proof outline: non-integrability 2

is also SU(2) sym.

time anti-sym.

time sym.

(A. Hokkyo, arXiv:2501.18400)

Proof outline: integrability

(Z. Zhang, arXiv:2504.17773, but we found a logical gap in this proof.

Hokkyo (in prep.) showed an alternative proof)

Summary

• We establish dichotomy of no nontrivial local CQ or all k-local conserved CQ in
1. S=1/2 chains with NN and NNN interaction
2. NN isotropic spin chain.

​

• We completely classify int./non-int. for ①.

​

• The criterion for ② is very simple.

END

Grabowski-Mathieu conjecture

Conjecture 1:

Absence of 3-local CQ → non-integrable

Comment below conjecture 1:

Presence of 3-local CQ → integrable

(M. P. Grabowski and P. Mathieu, J. Phys. A: Math. Gen. 28, 4777 (1995)

Reshetikin condition

Reshetikin condition is

Fuzzy spherical harmonics

Fuzzy spherical harmonics

How to show step 1?

Since the local Hamiltonian is expanded as

Known integrable models of isotropic spin chain

Four infinite sequences:

Further extensions?

One may expect the same dichotomy theorem for less symmetries.

Dichotomy might be true, but the criterion will be a more complicated one.

Is dichotomy universal?

The dichotomy (no partially-integrable system) has been strongly expected to general systems.

However, surprisingly, for non-Hermite bosonic system we have a model only with 3-local CQ!

(see M. Yamaguchi’s poster)

Whether dichotomy holds for Hermite spin systems is an open problem.

Non-integrability in high dimension

is non-integrable if . ( can be zero)

(This model includes XX model and Heisenberg model)

(N. Shiraishi and H. Tasaki, arXiv:2412.18504)

is non-integrable if . ( can be zero)

(This model includes transverse Ising model)

(Y. Chiba, arXiv:2412.18903)

Classification of next-nearest-neighbor interacting symmetric S=1/2 chain

with is nonintegrable except (i) classical case, and (ii) Bethe solvable case.

Classical:

Bethe solvable:

(N. Shiraishi, arXiv:2501.15506)

Proof of classification of NNN chain

By global spin rotation, the system is reduced to

Proof of classification of NNN chain:�case division

Rank 3:

Rank 2:

Rank 1:

Non-integrability in S=1 chain

is nonintegrable except for the following three known integrable models:

• (Barber-Batchelor)
• (Takhtajan-Babujian)
• (Lai-Sutherland)

​

(H.-K. Park and S.-B. Lee, arXiv:2410.23286,

A. Hokkyo, M. Yamaguchi, and Y. Chiba, arXiv:2411.04945)

Liouville integrability

Classical integrability is established

(http://elib.mi.sanu.ac.rs/files/journals/tm/26/tm1411.pdf)

(https://www.sciencedirect.com/science/article/abs/pii/S0307904X19305116?dgcid=rss_sd_all)

Integrable

Non-integrable

Sufficiently many conserved quantity (Lioubille integrability).

Orbit: composition of oscillators

No conserved quantity.

Orbit: highly complicated

These criteria do not work for quantum systems!

k-local conserved quantity

k-support operator: Operators supported by k contiguous sites (and their sum).

contiguous support

k-local conserved quantity: Conserved quantity which consists of at most k-support operators.

(and one of k-support operator has nonzero coefficient.)

Big difference between classical and quantum integrability

Classical case

• conserved quantity
• composition of oscillators

Integrability

Quantum case

→All quantum systems are integrable…?

Distinction between integrable and non-integrable in quantum system

(M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, PRL 98, 050405 (2007))

(L. F. Santos and M. Rigol, PRE 81, 036206 (2010))

Thermalization (mixing property)

Integrable : Not thermalize

Non-integrable : Thermalize

Level statistics

Integrable : Poisson dist.

Non-integrable : Wigner-Dyson dist.

Other differences: transport property, …

There must exist quantum integrability and non-integrability!

Necessity to restrict class of conserved quantity

Let’s re-consider the following statement:

Good reasons to restrict conserved quantities to local ones

• System with LCQ does not thermalize.

• Known exactly solvable models (and quantum inverse scattering method) accompany LCQ.

• Nonlocal conserved quantities appear not to matter macro physics.

(W. Beugeling, R. Moessner, and M. Haque, PRE 89, 042112 (2014))

Some observations

Main result

(N. Shiraishi, Europhys. Lett. 128 17002 (2019) )

This is the first rigorous proof of non-integrability!

Grabowski-Mathieu conjecture is solved in the affirmative in this model.

Symbols in this talk (1)

We denote

Ex)

A candidate of 3-local conserved quantity is

at most 64 terms!

Symbols in this talk (2)

Symbols in this talk (2)

Consequence from consideration of 4-support operators

Coefficients which might be nonzero are only

It suffices to prove one of them is zero!

Analysis on 3-support operators

→

Usually, 4 types of commutators “generate” a single 3-support operator

Operators generated by only 3 types of commutators

Operators generated by only 3 types of commutators

Relations between coefficients

The obtained three equalities

Its 3-local conserved quantity is expressed as

which is consistent with the condition in step 1:

Why these two are connected?

Case of 3-support operators (revisit)

k-support

k-1-support

Same terms are canceled!

What we seek for case of k-support

k-support

k-1-support

Same terms are canceled!

Final result (k-support operator)

Preliminary process: �global spin rotation

Standard form

Our Hamiltonian is reduced to

Classification theorem (standard)

rank 1:

integrable (classical)

integrable (transverse-field Ising)

otherwise

non-integrable

rank 2:

integrable (XY model)

otherwise

non-integrable

Classification theorem (standard)

rank 3:

integrable (Heisenberg model +field)

non-integrable

integrable (XXZ model +z field)

integrable (XYZ model)

non-integrable

High-rank case does not include low-rank case

However, this guess is not true!

Low-rank non-integrability does not imply high-rank non-integrability

One may feel that if is non-integrable, then a perturbed one is trivially non-integrable.

rank 1

rank 3

→integrable (Heisenberg chain with a magnetic field)

→non-integrable (mixed-field Ising model)

Proof (rank 3 and rank 2)

The proofs for rank 3 and for rank 2 are similar, while that for rank 1 is very different from them.

Proof for rank 3

Essentially the same as that for XYZ+h model.

Proof for rank 2

We shall provide the proof outline in next slides.

Proof outline for rank 2: Step 1

They are essentially

Proof outline for rank 2: Step 2

We consider a sequence as

…

These coefficients have been computed.

Proof outline for rank 1: strategy

This system is treated by Chiba.

( Y. Chiba, Phys. Rev. B 109, 035123 (2024))

Proof outline for rank 1: Step A-1

Proof outline for rank 1: Step A-2

…

We consider a sequence as

These coefficients have been computed.

Proof outline for rank 1: Step B-1

Remark on Step B-1

Operators with different support sizes are connected.

Proof outline for rank 1: Step B-2

We consider a sequence as

…

These coefficients have been computed.

Quasi-local conserved quantity

In other words, quasi-local operator has not small support, but its long-range effect decays extremely quickly.

Almost doubling-product operator

In rank 2, almost doubling-product but non-doubling-product operator remains in step 1:

non doubling-product

doubling-product

Why almost doubling-product operator remains

=

=

Due to this, we cannot derive contradiction at this stage.

two-dimension case (1)

Step 1

remove

add

possible support

All possible k-support operators are doubling-product in one dimension.

two-dimension case (2)

Step 2

…

We consider