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arXiv:2102.06111 · Physical review. B./Physical review. B

Hilbert space fragmentation and exact scars of generalized Fredkin spin chains

Christopher M. Langlett, Shenglong Xu

Published Jun 25, 2021• 68 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.103.l220304

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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.
We study the dynamics and thermalization of the Fredkin spin chain, a system with local three-body interactions, particle conservation, and explicit kinetic constraints. We consider deformations away from its stochastic point in order to tune between regimes where kinetic energy dominates and those where potential energy does. By means of exact diagonalization, perturbation theory, and variational matrix product states, we show there is a sudden change of behavior in the dynamics that occurs, from one of fast thermalization to one of slow metastable (prethermal) dynamics near the stochastic point. This change in relaxation is connected to the emergence of additional kinetic constraints, which lead to the fragmentation of Hilbert space in the limit of a large potential energy. We also show that this change can lead to thermalization being evaded for special initial conditions because of nonthermal eigenstates (akin to quantum many-body scars). We provide clear evidence for the existence of these nonthermal states for large system sizes even when far from the large-potential-energy limit, and explain their connection to the emergent kinetic constraints. Published by the American Physical Society 2024
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.
We study the periodic dynamics of a spin-one chain driven using a square-pulse protocol with amplitude Q0Q_0Q0​ and frequency ωDω_DωD​. The Hamiltonian of the spin chain hosts a thermodynamically large number of Z2Z_2Z2​-valued conserved quantities WℓW_{\ell}Wℓ​ on the links ℓ\ellℓ. This allows us to study the Floquet dynamics of this chain within a given sector with fixed values of WℓW_{\ell}Wℓ​. For the sector with all Wℓ=1W_{\ell}=1Wℓ​=1, we find signatures of quantum many-body scar states for ℏωD≫Q0\hbar ω_D \gg Q_0ℏωD​≫Q0​; they lead to oscillatory dynamics and fidelity revival for specific initial states. Upon lowering ωDω_DωD​, we find an ergodic regime exhibiting fast thermalization consistent with the prediction of the (Floquet) eigenstate thermalization hypothesis. In addition, we identify special drive frequencies ωn∗=Q0/(2nℏ)ω_n^{\ast}= Q_0/(2n \hbar)ωn∗​=Q0​/(2nℏ) (where n=1,2,3,⋯n = 1, 2, 3, \cdotsn=1,2,3,⋯) at which the Floquet Hamiltonian exhibits prethermal strong Hilbert space fragmentation (HSF) with the largest fragment being ergodic; in contrast, a weak HSF is found at ωn′=Q0/[ℏ(2n+1)]ω'_n= Q_0/[\hbar(2n+1)]ωn′​=Q0​/[ℏ(2n+1)] (where n=0,1,2,⋯n = 0, 1, 2, \cdotsn=0,1,2,⋯). We also study the sector with Wℓ={⋯1,1,−1,1,1,−1⋯ }W_{\ell} =\{\cdots 1,1,-1,1,1,-1 \cdots \}Wℓ​={⋯1,1,−1,1,1,−1⋯} which shows strong HSF at ωn∗ω_n^{\ast}ωn∗​ but no fragmentation at ωn′ω'_nωn′​. Our analysis indicates that the strong HSF in this sector harbors an integrable largest fragment. We provide numerical support for our analytical and perturbative results using exact-diagonalization (ED) studies on finite chains of length L≤24L\le 24L≤24. Our numerical results for entanglement entropy, fidelity, and correlation functions of the driven chain provide definitive signatures of prethermal strong HSF for both sectors.
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.
Motivated by recent observations of ergodicity breaking due to Hilbert space fragmentation in 1D Fermi-Hubbard chains with a tilted potential [Scherg et al., arXiv:2010.12965], we show that the same system also hosts quantum many-body scars in a regime U≈Δ≫J at electronic filling factor ν=1. We numerically demonstrate that the scarring phenomenology in this model is similar to other known realizations such as Rydberg atom chains, including persistent dynamical revivals and ergodicity-breaking many-body eigenstates. At the same time, we show that the mechanism of scarring in the Fermi-Hubbard model is different from other examples in the literature: the scars originate from a subgraph, representing a free spin-1 paramagnet, which is weakly connected to the rest of the Hamiltonian's adjacency graph. Our work demonstrates that correlated fermions in tilted optical lattices provide a platform for understanding the interplay of many-body scarring and other forms of ergodicity breaking, such as localization and Hilbert space fragmentation.
To provide a physical example of quantum scars, we study the many-body scars in the spin-1 Kitaev chain where the so-called PXP Hamiltonian is exactly embedded in the spectra. Regarding the conserved quantities, the Hilbert space is fragmented into disconnected subspaces and we explore the associated constrained dynamics. The continuous revivals of the fidelity and the entanglement entropy when the initial state is prepared in $|{\mathbb{Z}}_{k}\ensuremath{\rangle}$ (k=2,3k=2,3k=2,3) state illustrate the essential physics of the PXP model. We study the quantum phase transitions in the one-dimensional spin-1 Kitaev-Heisenberg model using the density-matrix renormalization group and Lanczos exact diagonalization methods, and determine the phase diagram. We parametrize the two terms in the Hamiltonian by the angle $\ensuremath{\phi}$, where the Kitaev term is $K\ensuremath{\equiv}sin(\ensuremath{\phi})$ and competes with the Heisenberg $J\ensuremath{\equiv}cos(\ensuremath{\phi})$ term. One finds a rich ground-state phase diagram as a function of the angle $\ensuremath{\phi}$. Depending on the ratio $K/J\ensuremath{\equiv}tan(\ensuremath{\phi})$, the system either breaks the symmetry to one of distinct symmetry broken phases, or preserves the symmetry in a quantum spin-liquid phase with frustrated interactions. We find that the scarred state is stable for perturbations which obey Z2{\mathbb{Z}}_{2}Z2​-symmetry, while it becomes unstable against Heisenberg-type perturbations.