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arXiv:1910.06341 · Physical review. B./Physical review. B

Statistical localization: From strong fragmentation to strong edge modes

Tibor Rakovszky, Pablo Sala, Ruben Verresen, Michael Knap, Frank Pollmann

Published Mar 25, 2020• 175 citations
arXivHTMLPDF

Abstract

Certain disorder-free Hamiltonians are nonergodic due to a $s\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}g$ $f\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}g\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}n$ of the Hilbert space. Here, the authors introduce the notion of ``statistically localized integrals of motion'' (SLIOM) to characterize these systems. Despite SLIOMs being nonlocal operators, they become spatially localized to subextensive regions when their expectation value is taken in typical states. These can also result in statistically localized $s\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}g$ $z\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}o$ $m\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}d\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}s$, leading to topological string order for certain highly excited eigenstates as well as infinitely long-lived edge magnetization along with a thermalizing bulk.

DOI
10.1103/physrevb.101.125126

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Certain disorder-free Hamiltonians are nonergodic due to a s\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}g f\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}g\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}n of the Hilbert space. Here, the authors introduce the notion of ``statistically localized integrals of motion'' (SLIOM) to characterize these systems. Despite SLIOMs being nonlocal operators, they become spatially localized to subextensive regions when their expectation value is taken in typical states. These can also result in statistically localized s\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}g z\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}o m\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}d\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}s, leading to topological string order for certain highly excited eigenstates as well as infinitely long-lived edge magnetization along with a thermalizing bulk.
Systems exhibiting the Hilbert-space fragmentation are nonergodic, and their Hamiltonians decompose into exponentially many blocks in the computational basis. In many cases, these blocks can be labeled by eigenvalues of statistically localized integrals of motion (SLIOMs), which play a similar role in fragmented systems as local integrals of motion in integrable systems. While a nonzero perturbation eliminates all nontrivial conserved quantities from integrable models, we demonstrate for the $t\text{\ensuremath{-}}{J}_{z}$ chain that an appropriately chosen perturbation may gradually eliminate SLIOMs (one by one) by progressively merging the fragmented subspaces. This gradual recovery of ergodicity manifests as an extended critical regime characterized by multiple peaks of the fidelity susceptibility. Each peak signals a change in the number of SLIOMs and blocks, as well as an ultra-slow relaxation of local observables.
Abstract Symmetry-protected topological phases 1–4 cannot be described by any local order parameter and are beyond the conventional symmetry-breaking model 5 . They are characterized by topological boundary modes that remain stable under symmetry respecting perturbations 1–4,6–8 . In clean, gapped systems without disorder, the stability of these edge modes is restricted to the zero-temperature manifold; at finite temperatures, interactions with mobile thermal excitations lead to their decay 9–11 . Here we report the observation of a distinct type of topological edge mode 12–14 , which is protected by emergent symmetries and persists across the entire spectrum, in an array of 100 programmable superconducting qubits. Through digital quantum simulation of a one-dimensional disorder-free stabilizer Hamiltonian, we observe robust long-lived topological edge modes over up to 30 cycles for a wide range of initial states. We show that the interaction between these edge modes and bulk excitations can be suppressed by dimerizing the stabilizer strength, leading to an emergent U(1) × U(1) symmetry in the prethermal regime of the system. Furthermore, we exploit these topological edge modes as logical qubits and prepare a logical Bell state, which exhibits persistent coherence, despite the system being disorder-free and at finite temperature. Our results establish a viable digital simulation approach 15–18 to experimentally study topological matter at finite temperature and demonstrate a potential route to construct long-lived, robust boundary qubits in disorder-free systems.
Hilbert space fragmentation is an ergodicity-breaking phenomenon, in which the Hamiltonian shatters into exponentially many dynamically disconnected sectors. In many fragmented systems, these sectors can be labeled by statistically localized integrals of motion, which are nonlocal operators. We study the paradigmatic nearest-neighbor pair hopping model exhibiting the so-called strong fragmentation. We show that this model hosts local integrals of motion (LIOMs), which correspond to frozen density modes with long wavelengths. The latter modes become subdiffusive when longer-range pair hoppings are allowed. Finally, we make a connection with a tilted (Stark) chain. Contrary to the dipole-conserving effective models, the tilted chain is shown to support either a Hamiltonian or dipole moment as an LIOM. Numerical results are obtained from a numerical algorithm, in which finding LIOMs is reduced to a data compression problem.