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arXiv:2302.11680 · Physical review. B./Physical review. B

Hilbert space fragmentation in a long-range system

Gianluca Francica, Luca Dell’Anna

Published Jul 17, 2023• 18 citations
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Abstract

We study the role of the interaction range on the Hilbert-space fragmentation and many-body scar states considering a spin-1/2 many-body Hamiltonian describing a generalized Fredkin spin chain. We show that both scar states and weak fragmentation of the Hilbert space survive for almost any range of the coupling. Moreover, when the interaction range is small enough, there are sectors with definite symmetries such that the ratio between the dimension of their largest fragment and their dimension decays algebraically with the system size. Finally we investigate the effects of such structures of the Hilbert space on the out-of-equilibrium dynamics, triggered by certain initial states, characterized by either local persistent oscillations or nonuniform stationary profile of the magnetization.

DOI
10.1103/physrevb.108.045127

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We study the role of the interaction range on the Hilbert-space fragmentation and many-body scar states considering a spin-1/2 many-body Hamiltonian describing a generalized Fredkin spin chain. We show that both scar states and weak fragmentation of the Hilbert space survive for almost any range of the coupling. Moreover, when the interaction range is small enough, there are sectors with definite symmetries such that the ratio between the dimension of their largest fragment and their dimension decays algebraically with the system size. Finally we investigate the effects of such structures of the Hilbert space on the out-of-equilibrium dynamics, triggered by certain initial states, characterized by either local persistent oscillations or nonuniform stationary profile of the magnetization.
In this work, based on the Fredkin spin chain, we introduce a family of spin-1/2 many-body Hamiltonians with a three-site interaction featuring a fragmented Hilbert space with coexisting quantum many-body scars. The fragmentation results from an emergent kinetic constraint resembling the conserved spin configuration in the one-dimensional Fermi-Hubbard model in the infinite on-site repulsion limit. To demonstrate the many-body scars, we construct an exact middle spectrum eigenstate within each fractured subsector displaying either logarithmic or area-law entanglement entropy. The interplay between fragmentation and scarring leads to rich tunable nonergodic dynamics by quenching different initial states shown through large-scale matrix product state simulations. In addition, we provide a Floquet quantum circuit that displays nonergodic dynamics due to sharing the same fragmentation structure and scarring as the Hamiltonian.
Hilbert space fragmentation (HSF) is a mechanism for generating quantum many-body scar (QMBS), which provides a route to weakly break ergodicity. Many scarred systems possess an exponentially large number of zero-energy states due to the chiral symmetry induced bipartition of the Hilbert space. In this work, we study the QMBS phenomenology under the interplay between the chiral symmetry and pseudo HSF, where the Hilbert space is approximately fragmented into different blocks. We consider a model of tilted chain of interacting spinless fermions with periodically varying tunneling strength. At small tunneling strength, we analytically derive the resonance conditions under which the system is described by an effective model with chiral symmetry and pseudo HSF. We find that the interplay between the two gives rise to a highly localized zero-energy QMBS when the particle number is even. This zero-energy QMBS induces an unusual scarred dynamical phenomenon. Specifically, the fidelity from a simple initial state oscillates around a finite fixed value without decaying, instead of showing the typical decaying collapse and revival observed when the particle number is odd and in common scarred systems. We show that the signature of the unusual scarred dynamical behaviour can also be detected in the original driven system by measuring local observables. Our findings enrich the scar phenomenon and deepen the understanding of the relation between Hilbert space structure and QMBS.
We investigate a spin-1/21/21/2 anisotropic Heisenberg model on a lattice consisting of two identical bipartite sublattices. A family of exact eigenstates generated by the restricted spectrum generating algebra (RSGA) constitutes quantum many-body scar states, characterized by subextensive entanglement entropy and supporting. These scar states are magnon-pair condensates exhibiting off-diagonal long-range order (ODLRO). At the resonance point of the inter-sublattice interaction, the model exactly maps onto a mixed spin-111 and spin-000 XY model on a bipartite lattice, which decomposes into independent sub-Hamiltonians labeled by all possible spin configurations. Each spin-000 particle is dynamically isolated from its neighbors and acts as a kinetic constraint, giving rise to emergent Hilbert space fragmentation (HSF). Our work establishes an exactly solvable platform in which quantum many-body scars, magnon-pair condensation exhibiting off-diagonal long-range order, and Hilbert space fragmentation naturally coexist.
The eigenstate thermalization hypothesis describes how isolated many-body quantum systems reach thermal equilibrium. However, quantum many-body scars and Hilbert space fragmentation violate this hypothesis and cause nonthermal behavior. We demonstrate that Hilbert space fragmentation may arise from lattice geometry in a spin-12\frac{1}{2}21​ model that conserves the number of domain walls. We generalize a known, one-dimensional, scarred model to larger dimensions and show that this model displays Hilbert space fragmentation on the Vicsek fractal lattice and the two-dimensional lattice. Using Monte Carlo methods, the model is characterized as strongly fragmented on the Vicsek fractal lattice when the number of domain walls is either small or close to the maximal value. On the two-dimensional lattice, the model is strongly fragmented when the density of domain walls is low and weakly fragmented when the density of domain walls is high. Furthermore, we show that the fragmentation persists at a finite density of domain walls in the thermodynamic limit for the Vicsek fractal lattice and the two-dimensional lattice. We also show that the model displays signatures similar to Hilbert space fragmentation on a section of the second-generation hexaflake fractal lattice and a modified two-dimensional lattice. We study the autocorrelation function of local observables and demonstrate that the model displays nonthermal dynamics.
We investigate a hierarchy of invariant Hilbert-space structures connecting Hilbert-space fragmentation, partial solvability, and quantum many-body scars. For spin chains with fragmented spin-1/21/21/2 subspaces and their local deformations, we show, using the spin-111 XXCXXCXXC chain as a representative example, how successive local deformations reduce extensive irreducible-string fragmentation through an implicitly embedded spin-1/21/21/2 invariant space and ultimately to a single scar tower, connecting the XXCXXCXXC, perturbed XXCXXCXXC, and spin-111 XYXYXY scar models. Through the intermediate embedded space, local dynamics couples states from distinct frozen irreducible-string sectors and selects a new decomposition into representations, from which smaller invariant spaces can be retained. We also establish a rigidity constraint on isolating nonsymmetric towers and a constructive correspondence that transfers a reduction between locally realizable representations of the embedded dynamics. The latter demonstrates that the reduction hierarchy can be transferred through a unitary transformation within the embedded subspace, yielding an exact scar tower built on a Néel reference state without requiring unitary equivalence of the full Hamiltonians.
Hilbert space fragmentation (HSF) is a mechanism for generating quantum many-body scar (QMBS), which provides a route to weakly break ergodicity. The zero-energy QMBSs widely exist across various systems due to the intertwining of chiral symmetry and spatial inversion symmetry. In this work, we study the phenomenology of the zero-energy QMBS under the interplay between the chiral symmetry and pseudo HSF, where the Hilbert space is approximately fragmented into different blocks. We consider a model of tilted chain of interacting spinless fermions with periodically varying tunneling strength. At small tunneling strength and under resonance condition, the system is described by an effective model with chiral symmetry and pseudo HSF. We find that the interplay between the two gives rise to a highly localized zero-energy QMBS when the particle number is even. We identify a simple product state to signalize the zero-energy QMBS, which gives rise to unusual scarred dynamics. The fidelity oscillates around a fixed value without decaying instead of showing the usual collapse and revival in common scarred systems. We show that the signature of the zero-energy QMBS can also be captured by the original Hamiltonian. Our results uncover a new scar phenomenon and provide an example that does not need the intertwining of chiral and spatial symmetries to support zero-energy QMBS.