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arXiv:2311.10968 · Physical review. B./Physical review. B

Two-dimensional quantum breakdown model with Krylov subspace many-body localization

Xinyu Liu, Biao Lian

Published Feb 6, 2025• 3 citations
arXivHTMLPDF

Abstract

We propose a two-dimensional (2D) quantum breakdown model of hardcore bosons interacting with disordered spins which would be classical without the bosons. It resembles particles incident into supersaturated vapor. The model exhibits a set of subsystem symmetries and has a strong fragmentation into Krylov subspaces in each symmetry sector. The Hamiltonian in each Krylov subspace maps to a single-particle problem in a Cayley-tree-like graph. At zero disorder, the Krylov subspaces exhibit either (possible) integrable features or quantum chaos with quantum scar states showing irregular energy and degeneracy patterns. At nonzero disorders, they enter a 2D many-body localization (MBL) phase beyond certain disorder strength ${W}_{*}$, as indicated by Poisson level spacing statistics and entanglement entropy growing as $logt$ with time $t$. Our theoretical arguments suggest ${W}_{*}$ is finite or zero for boson number ${N}_{b}\ensuremath{\lesssim}{L}^{\ensuremath{\gamma}}/logL$ ($1/2\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}1$) as system size $L\ensuremath{\rightarrow}\ensuremath{\infty}$. This gives a more stringent condition for MBL than that in the 1D quantum breakdown models. This model reveals the possibility of MBL in systems of quantum particles interacting with classical degrees of freedom.

DOI
10.1103/physrevb.111.054302

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We propose a two-dimensional (2D) quantum breakdown model of hardcore bosons interacting with disordered spins which would be classical without the bosons. It resembles particles incident into supersaturated vapor. The model exhibits a set of subsystem symmetries and has a strong fragmentation into Krylov subspaces in each symmetry sector. The Hamiltonian in each Krylov subspace maps to a single-particle problem in a Cayley-tree-like graph. At zero disorder, the Krylov subspaces exhibit either (possible) integrable features or quantum chaos with quantum scar states showing irregular energy and degeneracy patterns. At nonzero disorders, they enter a 2D many-body localization (MBL) phase beyond certain disorder strength W∗{W}_{*}W∗​, as indicated by Poisson level spacing statistics and entanglement entropy growing as logtlogtlogt with time ttt. Our theoretical arguments suggest W∗{W}_{*}W∗​ is finite or zero for boson number Nb\ensuremath≲L\ensuremathγ/logL{N}_{b}\ensuremath{\lesssim}{L}^{\ensuremath{\gamma}}/logLNb​\ensuremath≲L\ensuremathγ/logL (1/2\ensuremath≤\ensuremathγ\ensuremath≤11/2\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}11/2\ensuremath≤\ensuremathγ\ensuremath≤1) as system size L\ensuremath→\ensuremath∞L\ensuremath{\rightarrow}\ensuremath{\infty}L\ensuremath→\ensuremath∞. This gives a more stringent condition for MBL than that in the 1D quantum breakdown models. This model reveals the possibility of MBL in systems of quantum particles interacting with classical degrees of freedom.
We introduce a one-dimensional (1D) extended quantum breakdown model comprising a fermionic and a spin degree of freedom per site, and featuring a spatially asymmetric breakdown-type interaction between the fermions and spins. Our model resembles the breakdown process of particles incident into a cloud chamber with nonzero quantum amplitudes of both exciting and not exciting the local vapor atoms. We analytically show that, in the absence of any magnetic field for the spins, the model exhibits Hilbert space fragmentation within each symmetry sector into exponentially many Krylov subspaces and hence displays nonthermal dynamics. Here, we demonstrate that the fragmentation naturally occurs in an entangled basis and thus provides an example of ``quantum fragmentation.'' Besides establishing the nature of fragmentation analytically, we also study the long-time behavior of the entanglement entropy and its deviation from the expected Page value as a probe of ergodicity in the system. Upon introducing a magnetic field for the spins, most of the Krylov subspaces merge and the model becomes chaotic. Finally, we study the effects of strong randomness on the system and observe behavior similar to that of many-body localized systems.
We propose a quantum model of fermions simulating the electrical breakdown process of dielectrics. The model consists of MMM sites with NNN fermion modes per site and has a conserved charge QQQ. It has an on-site chemical potential $\ensuremath{\mu}$ with disorder WWW and an interaction of strength JJJ restricting each fermion to excite two more fermions when moving forward by one site. We show that the N=3N=3N=3 model with disorder W=0W=0W=0 shows a Hilbert space fragmentation and is exactly solvable except for very few Krylov subspaces. The analytical solution shows that the N=3N=3N=3 model exhibits many-body localization (MBL) as $M\ensuremath{\rightarrow}\ensuremath{\infty}$, which is stable against W>0W>0W>0 as our exact diagonalization (ED) shows. At N>3N>3N>3, our ED suggests an MBL to quantum chaos crossover at small WWW as M/NM/NM/N decreases across 1, and persistent MBL at large WWW. At W=0W=0W=0, an exactly solvable many-body scar flat band exists in many charge QQQ sectors, which has a nonzero measure in the thermodynamic limit. We further calculate the time evolution of a fermion added to the particle vacuum, which shows a breakdown (dielectric) phase when $\ensuremath{\mu}/J<1/2$ ($\ensuremath{\mu}/J>1/2$) if $W\ensuremath{\ll}J$, and no breakdown if $W\ensuremath{\gg}J$.
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.
We find that rank deficiency of the local Hamiltonian in a classically fragmented model is the key mechanism leading to quantum Hilbert space fragmentation. The rank deficiency produces local null directions that can generate entangled frozen states (EFS): entangled states embedded in mobile classical Krylov sectors that do not evolve under Hamiltonian dynamics. When the entangled frozen subspace is non-empty, the mobile classical sector splits into a mobile quantum Krylov subspace and an entangled frozen subspace, and the model exhibits quantum fragmentation. We establish this mechanism in four models of increasing symmetry structure: an asymmetric qubit projector with no symmetry, the Z2\mathbb{Z}_2Z2​-symmetric GHZ projector, a Z3\mathbb{Z}_3Z3​-symmetric cyclic qutrit projector, and the Temperley-Lieb model. For the asymmetric and GHZ projector models, we obtain closed-form expressions for irreducible Krylov dimensions, degeneracies, and sector multiplicities. The all-mobile-sector EFS in these two models exhibits a sub-volume-law bipartite entanglement entropy scaling as S∼LS \sim \sqrt{L}S∼L​. Further, we introduce the notion of weak and strong quantum fragmentation, the quantum counterpart of the weak-strong distinction in classical fragmentation. After removing the EFS, the mobile quantum Krylov subspace decomposes into irreducible blocks. In the weak case, the number of irreducible blocks remains O(1)O(1)O(1), each is individually ergodic with Gaussian Orthogonal Ensemble (GOE) level statistics, and the unresolved spectrum follows an mmmGOE distribution. In the strong case, the number of irreducible blocks grows with system size, and the gap-ratio distribution approaches Poisson as L→∞L\to\inftyL→∞.
Abstract The one-dimensional quantum breakdown model, which features spatially asymmetric fermionic interactions simulating the electrical breakdown phenomenon, exhibits an exponential U(1) symmetry and a variety of dynamical phases including many-body localization and quantum chaos with quantum scar states. We investigate the minimal quantum breakdown model with the minimal number of on-site fermion orbitals required for the interaction and identify a large number of local conserved charges in the model. We then reveal a mapping between the minimal quantum breakdown model in certain charge sectors and a quantum link model which simulates the U(1) lattice gauge theory and show that the local conserved charges map to the gauge symmetry generators. A special charge sector of the model further maps to the PXP model, which shows quantum many-body scars. This mapping unveils the rich dynamics in different Krylov subspaces characterized by different gauge configurations in the quantum breakdown model.
Recent experiments in quantum simulators have provided evidence for the many-body localized (MBL) phase in one-dimensional (1D) and two-dimensional (2D) bosonic quantum matter. The theoretical study of such a bosonic MBL phase, however, is a daunting task due to the unbounded nature of its Hilbert space. In this work, we introduce a method to compute the long-time real-time evolution of 1D and 2D bosonic systems in an MBL phase at strong disorder and weak interactions. We focus on local dynamical indicators that are able to distinguish an MBL phase from an Anderson localized one. In particular, we consider the temporal fluctuations of local observables, the spatiotemporal behavior of two-time correlators and out-of-time-correlators. We show that these few-body observables can be computed with a computational effort that depends only polynomially on the system size but is independent of the target time, by extending a recently proposed numerical method [G. De Tomasi, F. Pollmann, and M. Heyl, Phys. Rev. B 99, 241114(R) (2019)] to mixed states and bosons. Our method also allows us to surrogate our numerical study with analytical considerations of the time-dependent behavior of the studied quantities.