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arXiv:2403.14314 · Physical review. B./Physical review. B

Subspace-restricted thermalization in a correlated-hopping model with strong Hilbert space fragmentation characterized by irreducible strings

Sreemayee Aditya, Deepak Dhar, Diptiman Sen

Published Jul 9, 2024• 10 citations
arXivHTMLPDF

Abstract

We introduce a one-dimensional correlated-hopping model of spinless fermions in which a particle can hop between two neighboring sites only if the sites to the left and right of those two sites have different particle numbers. Using a bond-to-site mapping, this model involving four-site terms can be mapped to an assisted pair-flipping model involving only three-site terms. This model shows strong Hilbert space fragmentation. We define irreducible strings (ISs) to label the different fragments, determine the number of fragments, and the sizes of fragments corresponding to some special ISs. In some classes of fragments, the Hamiltonian can be diagonalized completely, and in others it can be seen to have a structure characteristic of models which are not fully integrable. In the largest fragment in our model, the number of states grows exponentially with the system size, but the ratio of this number to the total Hilbert space size tends to zero exponentially in the thermodynamic limit. Within this fragment, we provide numerical evidence that only a weak version of the eigenstate thermalization hypothesis (ETH) remains valid; we call this subspace-restricted ETH. To understand the out-of-equilibrium dynamics of the model, we study the infinite-temperature time-dependent autocorrelation functions starting from a random initial state; we find that these exhibit a different behavior near the boundary compared to the bulk. Finally, we propose an experimental setup to realize our correlated-hopping model.

DOI
10.1103/physrevb.110.045418

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We introduce a one-dimensional correlated-hopping model of spinless fermions in which a particle can hop between two neighboring sites only if the sites to the left and right of those two sites have different particle numbers. Using a bond-to-site mapping, this model involving four-site terms can be mapped to an assisted pair-flipping model involving only three-site terms. This model shows strong Hilbert space fragmentation. We define irreducible strings (ISs) to label the different fragments, determine the number of fragments, and the sizes of fragments corresponding to some special ISs. In some classes of fragments, the Hamiltonian can be diagonalized completely, and in others it can be seen to have a structure characteristic of models which are not fully integrable. In the largest fragment in our model, the number of states grows exponentially with the system size, but the ratio of this number to the total Hilbert space size tends to zero exponentially in the thermodynamic limit. Within this fragment, we provide numerical evidence that only a weak version of the eigenstate thermalization hypothesis (ETH) remains valid; we call this subspace-restricted ETH. To understand the out-of-equilibrium dynamics of the model, we study the infinite-temperature time-dependent autocorrelation functions starting from a random initial state; we find that these exhibit a different behavior near the boundary compared to the bulk. Finally, we propose an experimental setup to realize our correlated-hopping model.
We investigate a one-dimensional correlated-hopping model of spinless fermions with an East constraint. We first analytically unravel the complete fragmentation structure of this model by labeling each fragment by a unique root configuration and utilizing the transfer matrix method. We show that the growth of the size of each fragment of the model follows the widely studied Dyck sequence and is therefore analytically tractable with the help of Catalan triangles. While the eigenstate thermalization hypothesis (ETH) does not hold within the full Hilbert space which exhibits Poisson statistics of the energy level spacing, an examination of various quantities restricted to the largest fragments shows that a weaker version of the subspace-restricted thermalization holds. This weaker violation of the ETH within the largest fragments is supported by the presence of subspace-restricted quantum many-body scars due to quantum fragmentation. Our analysis further reveals that the filling fraction at which the model has the largest fragment shifts with increasing system sizes from half-filling due to the distinct fragmentation structure. Next, we show that the inclusion of a nearest-neighbor density-density interaction with strength VVV induces a spectral transition within the largest fragment from a weakly ETH-violating phase containing scars to a statistical bubble localized phase as VVV increases. In particular, the $V\ensuremath{\rightarrow}\ensuremath{\infty}$ limit produces an integrable model. We find that the addition of finite-VVV stabilizes the ground state near half-filling while keeping intact the fragmentation structure of the East model. However, this behavior abruptly changes exactly at $V=\ensuremath{\infty}$ due to the emergence of a distinct fragmentation structure; we analytically explore this in a comprehensive manner. The infinite-VVV model has many interesting properties, among which the appearance of the ground state and the largest fragment at two different filling fractions is specially noteworthy. Finally, we propose an experimental setup to realize the infinite-VVV model as a particular limit of a special kind of $t\text{\ensuremath{-}}V$ model with an on-site potential.
We investigate thermalization in finite quantum systems with strong long-range interactions. Nearly conserved quantities inherited from the fully connected limit organize the Hilbert space into weakly coupled sectors and split the many-body spectrum into energy bands. Despite the resulting breakdown of global ergodicity, quantum chaos develops within individual energy bands, enabling the definition of microcanonical ensembles within the bands. This supports a band-resolved formulation of thermalization, which we term fragmented eigenstate thermalization hypothesis (fETH). Unlike conventional ETH, finite-size scaling in fETH obeys a symmetry-imposed selection rule that restricts which system sizes can be compared. This band-resolved description has consequences for equilibrium statistical mechanics. While microcanonical ensembles remain confined to a single band, canonical ensembles mix different bands. This mismatch explains ensemble inequivalence without invoking equilibrium phase transitions. Our results apply broadly to Hamiltonians near a fully permutation-symmetric limit.
We study the quantum dynamics of a simple translation invariant, center-of-mass (CoM) preserving model of interacting fermions in one dimension (1D), which arises in multiple experimentally realizable contexts. We show that this model naturally displays the phenomenology associated with fractonic systems, wherein single charges can only move by emitting dipoles. This allows us to demonstrate the rich Krylov fractured structure of this model, whose Hilbert space shatters into exponentially many dynamically disconnected subspaces. Focusing on exponentially large Krylov subspaces, we show that these can be either be integrable or non-integrable, thereby establishing the notion of Krylov-restricted thermalization. We analytically find a tower of integrable Krylov subspaces of this Hamiltonian, all of which map onto spin-1/2 XX models of various system sizes. We also discuss the physics of the non-integrable subspaces, where we show evidence for weak Eigenstate Thermalization Hypothesis (ETH) restricted to each non-integrable Krylov subspace. Further, we show that constraints in some of the thermal Krylov subspaces cause the long-time expectation values of local operators to deviate from behavior typically expected from translation-invariant systems. Finally, we show using a Schrieffer-Wolff transformation that such models naturally appear as effective Hamiltonians in the large electric field limit of the interacting Wannier-Stark problem, and comment on connections of our work with the phenomenon of Bloch many-body localization.
This paper explores the effect of the strong-to-weak fragmentation transition, namely, the freezing transition, and its rich characteristics in a family of one-dimensional spinless fermionic models with facilitated hoppings involving short-to-long-range East constraints. Focusing on this family of models with range-qqq constraints, our investigation furnishes an exhaustive understanding of the fragmented Hilbert space utilizing the enumerative combinatorics and transfer-matrix methods. This further allows us to get insight into the freezing transition in this family of models with the help of the generalization of Catalan numbers introduced by Frey and Sellers for q>1q>1q>1, further revealing that increasing the range of constraints drives the transition to transpire at lower filling fractions as nc=1/(q+1){n}_{c}=1/(q+1)nc​=1/(q+1). This distinct fragmentation structure also yields the emergence of ground states at multiple fillings, which can be explained from the root structure of the largest fragment within the full Hilbert space. Furthermore, the ground state exhibits signatures of criticality with logarithmic scaling of entanglement entropy. Thereafter, our investigation exemplifies that the above transition has a profound impact on the thermalization of bulk and boundary autocorrelators at long times, which includes intricate filling-dependent inhomogeneous long-time autocorrelation profiles across the chain in open boundary conditions (OBCs). Finally, we probe the effect of the strong-to-weak fragmentation transition on the transport at intermediate times in periodic boundary conditions (PBCs), restricting ourselves to models up to range-3 constraints. This investigation discloses a vast range of anomalous transport possibilities, ranging from size-stretched exponential relaxation through superdiffusive to subdiffusive behaviors, akin to the fragmentation structure supported by the filling fraction and range of constraints. In brevity, our paper reveals intriguing possibilities conspired by constraints with varying ranges, and those-induced Hilbert-space fragmentation and freezing transition, thus offering exciting avenues for future explorations.
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 study many-body localization (MBL) in a pair-hopping model exhibiting strong fragmentation of the Hilbert space. We show that several Krylov subspaces have both ergodic statistics in the thermodynamic limit and a dimension that scales much slower than the full Hilbert space but still exponentially. Such a property allows us to study the MBL phase transition in systems including up to 64 spins. The different Krylov spaces that we consider show clear signatures of a many-body localization transition, both in the Kullback-Leibler divergence of the distribution of their level spacing ratio and their entanglement properties. However, they also present distinct scalings with the system size. Depending on the subspace, the critical disorder strength can be nearly independent of the system size or conversely show an approximately linear increase with the number of spins.