An integrable spin chain with Hilbert space fragmentation and solvable real time dynamics

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 T and hard rod deformations known from Quantum Field Theories.

Fragmentation and emergent integrable transport in the weakly tilted Ising chain

“…Domain wall conservation enforces a peculiar form of the nearest-neighbor exchange term: it involves a projector which ensures that the two outer spins (surrounding the hopping pair) point in the same direction. Similar type of hopping have been recently discussed in [33][34][35][36]. Since only isolated magnons (i.e.…”

“…in the constrained PXXP model [31], in the strongly-coupled regime of a Z 2 lattice gauge theory coupled to fermions [32] and in interacting correlated hopping models [33]. At the non-interacting point, that is non-trivial due to the constraint, it has been shown to emerge in the strong coupling limit of the canonical XXZ spin chain [34][35][36]. The leitmotif of some of these studies is the Hilbert space fragmentation [37,38] due to imposed or emergent constraints, which make the constrained XXZ chain a natural candidate to describe integrable sectors, if present.…”

Fractionalized holes in one-dimensional $\mathbb{Z}_2$ gauge theory coupled to fermion matter -- deconfined dynamics and emergent integrability

“…We study in detail the dynamics of such holes and provide several independent manifestations of their deconfined nature. In the limit of strong string tension, we find that holes become heavy and undergo a peculiar correlated hopping of the type studied previously in [36][37][38][39][40][41]. We uncover emergent integrability of the slow hole dynamics in that regime and demonstrate its equivalence to the constrained XXZ model of Alcaraz and Bariev [42].…”

“…The factor (1 − n h ) inhibits hopping between next-nearest sites if the intermediate site is already occupied with a hole. This type of correlated hopping was first investigated by Bariev [36] who demonstrated its integrable nature, for related recent studies see [38,39,41]. In addition to the Bariev's hopping, a nearest neighbour repulsion U eff is induced between the holes by the second-order perturbation theory.…”

Subdiffusive hydrodynamics of nearly-integrable anisotropic spin chains

“…Because of integrability, even in a hightemperature thermal state, these strings remain stable, and their characteristic velocity scale does not change appreciably, although their other properties are highly renormalized, as we now discuss. ∆ = ∞ limit.-It is instructive to consider the ∆ = ∞ limit first; this limit is sometimes called the "folded XXZ model" [68][69][70][71]. Here, the quasiparticle picture simplifies: all strings with s > 1 are frozen, and the only dynamics is due to the s = 1 strings, or magnons, moving in a static background of spin domains with velocity v = O (1) [see also Ref.…”