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arXiv:2210.08019

Integer characteristic polynomial factorization and Hilbert space fragmentation

Nicolas Regnault, Liu, Shuo, Bogdan Andrei Bernevig

Published Oct 14, 2022• 0 citations
arXivHTMLPDF

Abstract

Models with Hilbert space fragmentation are characterized by (exponentially) many dynamically disconnected subspaces, not associated with conventional symmetries but captured by nontrivial Krylov subspaces. These subspaces usually exhibit a whole range of thermalization properties, from chaotic to integrable, to quantum many-body scars. However, so far, they have not been properly defined, nor can they be easily found, given a Hamiltonian. In this work, we consider Hamiltonians that have integer representations, a common feature of many (most) celebrated models in condensed matter. We show the equivalence of the integer characteristic polynomial factorization and the existence of Krylov subspaces generated from integer vectors. Considering the pair-hopping model, we illustrate how the factorization property can be used as a method to unveil Hilbert space fragmentation. We discuss the generalization over other rings of integers, for example those based on the cyclotomic field which are relevant when working in a given ($\ne 0, π$) momentum sector.

DOI
10.48550/arxiv.2210.08019

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Models with Hilbert space fragmentation are characterized by (exponentially) many dynamically disconnected subspaces, not associated with conventional symmetries but captured by nontrivial Krylov subspaces. These subspaces usually exhibit a whole range of thermalization properties, from chaotic to integrable, to quantum many-body scars. However, so far, they have not been properly defined, nor can they be easily found, given a Hamiltonian. In this work, we consider Hamiltonians that have integer representations, a common feature of many (most) celebrated models in condensed matter. We show the equivalence of the integer characteristic polynomial factorization and the existence of Krylov subspaces generated from integer vectors. Considering the pair-hopping model, we illustrate how the factorization property can be used as a method to unveil Hilbert space fragmentation. We discuss the generalization over other rings of integers, for example those based on the cyclotomic field which are relevant when working in a given (≠0,π\ne 0, π=0,π) momentum sector.
A quantum model exhibits Hilbert space fragmentation (HSF) if its Hilbert space decomposes into exponentially many dynamically disconnected subspaces, known as Krylov subspaces. A model may however have different HSFs depending on the method for identifying them. Here we establish a connection between two vastly distinct methods recently proposed for identifying HSF: the commutant algebra (CA) method and integer characteristic polynomial factorization (ICPF) method. For a Hamiltonian consisting of operators admitting rational number matrix representations, we prove a theorem that, if its center of commutant algebra have all eigenvalues being rational, the HSF from the ICPF method must be equal to or finer than that from the CA method. We show that this condition is satisfied by most known models exhibiting HSF, for which we demonstrate the validity of our theorem. We further discuss representative models for which ICPF and CA methods yield different HSFs. Our results may facilitate the exploration of a unified definition of HSF.
Hilbert space fragmentation refers to exponential growth in the number of dynamically disconnected Krylov sectors with system size. It is taken as evidence of ergodicity breaking, since conventional symmetries generate at most a polynomial number of sectors. However, we demonstrate that generalized symmetries can fragment the Hilbert space. Models with higher-form, subsystem, and gauge symmetries can have exponentially many symmetry sectors. We further prove that non-invertible symmetries can induce additional fragmentation within individual symmetry sectors. Fragmentation in several known models arises from generalized symmetries, and the presence of exponentially many Krylov sectors therefore does not by itself imply ergodicity breaking. Finally, we show that disorder free localization arises naturally from Krylov-restricted thermalization when sectors lack translation invariance, requiring neither ergodicity breaking nor gauge symmetry.
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 is a novel type of ergodicity breaking in closed quantum systems. Recently, an algebraic approach was utilized to provide a definition of Hilbert space fragmentation characterizing families of Hamiltonian systems based on their (generalized) symmetries. In this paper we reveal a simple connection between the aforementioned classification of physical systems and their coherence generation properties, quantified by the coherence generating power (CGP). The maximum CGP (in the basis associated with the algebra of each family of Hamiltonians) is exactly related to the number of independent Krylov subspaces KKK, which is precisely the characteristic used in the classification of the system. In order to gain further insight, we numerically simulate paradigmatic models with both ordinary symmetries and Hilbert space fragmentation, comparing the behavior of the CGP in each case with the system dimension. More generally, allowing the time evolution to be any unitary channel in a specified algebra, we show analytically that the scaling of the Haar averaged value of the CGP depends only on KKK. These results illustrate the intuitive relationship between coherence generation and symmetry algebras.
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→∞.
The Hamiltonian formulation of lattice gauge theories plays a central role in quantum simulations of gauge theories, and understanding their spectrum and other properties is expected to become crucial in the upcoming years. The relevant Hamiltonians in this framework possess local symmetry at each lattice site and may exhibit higher-form symmetries. There are then an exponentially large number of dynamically disconnected symmetry sectors, most of which are not translation-invariant. An exponential number of dynamically disconnected sectors, i.e., Hilbert space fragmentation, can also occur in systems in which no such symmetries have been identified. In this contribution, we describe an emergent gauge symmetry that is valid only in a subset of sectors of the fragmented S=1S=1S=1 dipole-conserving spin chain. These non-invertible symmetries can label exponentially many of the model's sectors. Simulating this Hamiltonian, which is not gauge-invariant, yields an exact quantum simulation of a gauge theory.