[Paper Review] An integrable spin chain with Hilbert space fragmentation and solvable real time dynamics

Balázs Pozsgay, Tamás Gombor+4 more|arXiv (Cornell University)|May 5, 2021

Quantum many-body systemsPhysics and Astronomy102 references83 citations

TL;DR

This paper introduces the folded XXZ spin chain as a minimal quantum integrable model with Hilbert space fragmentation, where particle dynamics are constrained by domain walls leading to exponentially degenerate energy levels. Despite its simplicity, the model allows exact real-time dynamics via a non-local mapping to the SU(3) XX (Maassarani-Mathieu) chain, enabling solvable quantum quen protocols and revealing persistent oscillations due to degeneracy-induced equilibration breakdown.

ABSTRACT

We revisit the so-called folded XXZ model, which was treated earlier by two independent research groups. We argue that this spin-1/2 chain is one of the simplest quantum integrable models, yet it has quite remarkable physical properties. The particles have constant scattering lengths, which leads to a simple treatment of the exact spectrum and the dynamics of the system. The Hilbert space of the model is fragmented, leading to exponentially large degeneracies in the spectrum, such that the exponent depends on the particle content of a given state. We provide an alternative derivation of the Hamiltonian and the conserved charges of the model, including a new interpretation of the so-called "dual model" considered earlier. We also construct a non-local map that connects the model with the Maassarani-Mathieu spin chain, also known as the SU(3) XX model. We consider the exact solution of the model with periodic and open boundary conditions, and also derive multiple descriptions of the exact thermodynamics of the model. We consider quantum quenches of different types. In one class of problems the dynamics can be treated relatively easily: we compute an example for the real time dependence of a local observable. In another class of quenches the degeneracies of the model lead to the breakdown of equilibration, and we argue that they can lead to persistent oscillations. We also discuss connections with the $T\bar T$- and hard rod deformations known from Quantum Field Theories.

Motivation & Objective

• To identify the simplest quantum integrable model with genuine interactions beyond free theories.
• To analyze Hilbert space fragmentation in the folded XXZ model, where domain walls cause exponentially large energy degeneracies dependent on particle content.
• To derive exact real-time dynamics for quantum quen protocols using a non-local mapping to the SU(3) XX model.
• To explore connections between the model's structure and deformations in quantum field theory, such as T¯T and hard rod deformations.
• To establish a framework for exact non-equilibrium dynamics in integrable systems with broken equilibration due to degeneracy.

Proposed method

• Derives the Hamiltonian and conserved charges of the folded XXZ model via an alternative construction, clarifying its integrability structure.
• Introduces a non-local unitary map connecting the folded XXZ model to the Maassarani-Mathieu (SU(3) XX) spin chain, enabling exact dynamics.
• Applies Bethe Ansatz techniques to solve the spectrum and dynamics under periodic and open boundary conditions.
• Uses generalized Gibbs ensemble (GGE) and generalized hydrodynamics (GHD) frameworks to analyze non-equilibrium relaxation.
• Analyzes quantum quen protocols, computing exact time evolution of local observables in specific initial states.
• Connects the model to T¯T and hard rod deformations via its underlying scattering and geometric structure.

Experimental results

Research questions

• RQ1 Can a minimal quantum integrable model with genuine interactions and Hilbert space fragmentation be exactly solvable in real time?
• RQ2 How does the presence of domain walls lead to exponential degeneracy and breakdown of equilibration in quantum quen dynamics?
• RQ3 What is the role of non-local mappings in connecting different integrable spin chains, such as folded XXZ and SU(3) XX?
• RQ4 To what extent can exact real-time dynamics be computed in fragmented Hilbert spaces, and what are the signatures of non-thermal steady states?
• RQ5 How are the model's properties related to known deformations in QFT, such as T¯T and hard rod models?

Key findings

• The folded XXZ model exhibits Hilbert space fragmentation due to non-dynamical domain walls, leading to exponentially large degeneracies that scale with particle number.
• Exact real-time dynamics of local observables after quantum quen are computed analytically, showing non-trivial time dependence and persistent oscillations due to degeneracy.
• A non-local unitary map is constructed that maps the folded XXZ model to the Maassarani-Mathieu (SU(3) XX) spin chain, enabling exact solvability.
• The model is shown to be related to T¯T and hard rod deformations, suggesting a deeper connection between integrable spin chains and QFT deformations.
• In certain quench protocols, equilibration fails due to degeneracy, resulting in persistent oscillations instead of thermalization.
• The model provides a solvable benchmark for testing GGE and GHD predictions, with exact agreement in asymptotic limits.