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 more than $50$ 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. But they also present distinct scalings with 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 study the dynamics of an interacting quantum spin chain under the application of a linearly increasing field. This model exhibits a type of localization known as Stark many-body localization. The dynamics shows a strong dependence on the initial conditions, indicating that the system violates the conventional ("strong") eigenstate thermalization hypothesis at any finite gradient of the field. This is contrary to reports of a numerically observed ergodic phase. Therefore, the localization is crucially distinct from disorder-driven many-body localization, in agreement with recent predictions …
Investigating many-body localization (MBL) using exact numerical methods is limited by the exponentialgrowth of the Hilbert space. However, localized eigenstates display multifractality and only extend over a vanishing fraction of the Hilbert space. Here, building on this remarkable property, we develop a simple yet efficient decimation scheme to discard the irrelevant parts of the Hilbert space of the random-field Heisenberg chain. This leads to an Hilbert space fragmentation in small clusters, allowing to access larger systems at strong disorder. The MBL transition is quantitatively predicte…
Emerging dynamical constraints resulting from inter-site interactions severely limit particle mobility in polar lattice gases. Whereas in absence of disorder hard-core Hubbard models with only strong nearest-neighbor interactions present Hilbert space fragmentation but no many-body localization for typical states, the $1/r^3$ tail of the dipolar interaction results in Hilbert space shattering, as well as in a dramatically slowed down dynamics and eventual disorder-free localization. Our results show that the study of the intriguing interplay between disorder- and interaction-induced many-body …
We study Hilbert space fragmentation in the extended Fermi-Hubbard model with nearest and next-nearest-neighbor interactions. Using a generalized spin/mover picture and saddle point methods, we derive lower bounds for the scaling of the number of frozen states and for the size of the largest block preserved under the dynamics. We find fragmentation for strong nearest- and next-nearest-neighbor repulsions as well as for the combined case. Our results suggest that the involvement of next-nearest-neighbor repulsions leads to an increased tendency for localization. We then model the dynamics for l…
We present a large scale exact diagonalization study of the one dimensional spin $1/2$ Heisenberg model in a random magnetic field. In order to access properties at varying energy densities across the entire spectrum for system sizes up to $L=22$ spins, we use a spectral transformation which can be applied in a massively parallel fashion. Our results allow for an energy-resolved interpretation of the many body localization transition including the existence of an extensive many-body mobility edge. The ergodic phase is well characterized by Gaussian orthogonal ensemble statistics, volume-law en…
The 1-D Anderson model possesses a completely localized spectrum of eigenstates for all values of the disorder. We consider the effect of projecting the Hamiltonian to a truncated Hilbert space, destroying time reversal symmetry. We analyze the ensuing eigenstates using different measures such as inverse participation ratio and sample-averaged moments of the position operator. In addition, we examine amplitude fluctuations in detail to detect the possibility of multifractal behavior (characteristic of mobility edges) that may arise as a result of the truncation procedure.
We show how a finite number of conservation laws can globally `shatter' Hilbert space into exponentially many dynamically disconnected subsectors, leading to an unexpected dynamics with features reminiscent of both many body localization and quantum scars. A crisp example of this phenomenon is provided by a `fractonic' model of quantum dynamics constrained to conserve both charge and dipole moment. We show how the Hilbert space of the fractonic model dynamically fractures into disconnected emergent subsectors within a particular charge and dipole symmetry sector. This shattering can occur in a…
In contrast with Anderson localization where a genuine localization is observed in real space, the many-body localization (MBL) problem is much less understood in the Hilbert space, support of the eigenstates. In this work, using exact diagonalization techniques we address the ergodicity properties in the underlying ${\cal{N}}$-dimensional complex networks spanned by various computational bases for up to $L=24$ spin-1/2 particles (i.e. Hilbert space of size ${\cal{N}}\simeq 2.7\,10^6$). We report fully ergodic eigenstates in the delocalized phase (irrespective of the computational basis), whil…
In this paper we propose a new perspective to analyze the many-body localization (MBL) transition when recast in terms of a single-particle tight-binding model in the space of many-body configurations. We compute the distribution of tunneling rates between many-body states separated by an extensive number of spin flips at the lowest order in perturbation theory starting from the insulator, and determine the scaling of their typical amplitude with the number of accessible states in the Hilbert space. By using an analogy with the Rosenzweig-Porter random matrix ensemble, we propose an ergodicity…
We study the localization in the Hilbert space of a modified Tomonaga-Luttinger model. For the standard version of this model, the states are found to be extended in the basis of Slater determinants, representing the eigenstates of the non-interacting system. The linear dispersion which leads to the fact that these eigenstates are extended in the modified model is replaced by one with random level spacings modeling the complicated one-particle spectra of realistic models. The localization properties of the eigenstates are studied. The interactions are simplified and an effective one-dimensiona…
We construct an analytic theory of many-body localization (MBL) in random spin chains. The approach is based on a first quantized perspective in which MBL is understood as a localization phenomenon on the high dimensional lattice defined by the discrete Hilbert space of the clean system. We construct a field theory on that lattice and apply it to discuss the stability of a weak disorder (`Wigner-Dyson') and a strong disorder (`Poisson') phase.
Two electrons move in a quasi one--dimensional wire under the influence of a short--range interaction. We restrict Hilbert space to those states where the two electrons are close to each other. Using supersymmetry, we present a complete analytical solution to this problem. The two--body interaction affects the density of states and, thereby, the localization length. We derive a criterion for the onset of changes of the localization length due to the two--body interaction.
Disordered systems provide paradigmatic instances of ergodicity breaking and localization phenomena. Here we explore the dynamics of excitations in a system of Rydberg atoms held in optical tweezers. The finite temperature produces an intrinsic uncertainty in the atomic positions, which translates into quenched correlated disorder in the interatomic interaction strengths. In a simple approach, the dynamics in the many-body Hilbert space can be understood in terms of a one-dimensional Anderson-like model with disorder on every other site, featuring both localized and delocalized states. We cond…
Ergodicity in quantum many-body systems is - despite its fundamental importance - still an open problem. Many-body localization provides a general framework for quantum ergodicity, and may therefore offer important insights. However, the characterization of many-body localization through simple observables is a difficult task. In this article, we introduce a measure for distances in Hilbert space for spin-1/2 systems that can be interpreted as a generalization of the Anderson localization length to the many-body Hilbert space. We show that this many-body localization length is equivalent to a …
We study the breaking of ergodicity measured in terms of return probability in the evolution of a quantum state of a spin chain. In the non ergodic phase a quantum state evolves in a much smaller fraction of the Hilbert space than would be allowed by the conservation of extensive observables. By the anomalous scaling of the participation ratios with system size we are led to consider the distribution of the wave function coefficients, a standard observable in modern studies of Anderson localization. We finally present a criterion for the identification of the ergodicity breaking (many-body loc…
Many-body localization (MBL) is a novel prototype of ergodicity breaking due to the emergence of local integrals of motion (LIOMs) in a disordered interacting quantum system. To better understand the role played by the existence of such macroscopically LIOMs, we explore and study some of their structural properties across the MBL transition. We first, consider a one-dimensional XXZ spin chain in a disordered magnetic field and introduce and implement a non-perturbative, fast, and accurate method of constructing LIOMs. In contrast to already existing methods, our scheme allows obtaining LIOMs n…
Interplay of localization and topology in disordered dimerized array of Rydberg atoms
Szkoła Doktorska Nauk Ścisłych i Przyrodniczych, Uniwersytet Jagielloński（雅盖隆大学自然科学博士学院）；Instytut Fizyki Teoretycznej, Wydział Fizyki, Astronomii i Informatyki Stosowanej, Uniwersytet Jagielloński（雅盖隆大学物理、天文学与应用信息学系理论物理研究所）；Mark Kac Complex Systems Research Center, Jagiellonian University in Kraków（克拉科夫雅盖隆大学马克·卡克复杂系统研究中心）
We report the results of the numerical study of the non-dissipative quantum Josephson junction chain with the focus on the statistics of many-body wave functions and local energy spectra. The disorder in this chain is due to the random offset charges. This chain is one of the simplest physical systems to study many-body localization. We show that the system may exhibit three distinct regimes: insulating, characterized by the full localization of many-body wavefunctions, fully delocalized (metallic) one characterized by the wavefunctions that take all the available phase volume and the intermed…
We study the transport properties of two electrons in a quasi one-dimensional disordered wire. The electrons are subject to both, a disorder potential and a short range two-body interaction. Using the approach developed by Iida et al. [ Ann. Phys. (N.Y.) 200 (1990) 219 ], the supersymmetry technique, and a suitable truncation of Hilbert space, we work out the two-point correlation function in the framework of a non-linear sigma model. We study the loop corrections to arbitrary order. We obtain a remarkably simple and physically transparent expression for the change of the localization length c…
We explore correlations of eigenstates around the many-body localization (MBL) transition in their dependence on the energy difference (frequency) $ω$ and disorder $W$. In addition to the genuine many-body problem, XXZ spin chain in random field, we consider localization on random regular graphs (RRG) that serves as a toy model of the MBL transition. Both models show a very similar behavior. On the localized side of the transition, the eigenstate correlation function $β(ω)$ shows a power-law enhancement of correlations with lowering $ω$; the corresponding exponent depends on $W$. The correlati…
We generalize the notion of "localization landscape," introduced by M. Filoche and S. Mayboroda [Proc. Natl. Acad. Sci. USA 109, 14761 (2012)] for the single-particle Schrodinger operator, to a wide class of interacting many-body Hamiltonians. The many-body localization landscape (MBLL) is defined on a graph in the Fock space, whose nodes represent the basis vectors in the Fock space and edges correspond to transitions between the nodes connected by the hopping term in the Hamiltonian. It is shown that in analogy to the single-particle case, the inverse MBLL plays the role of an effective pote…
We consider a model arising in facilitated Rydberg chains with positional disorder which features a Hilbert space with the topology of a $d$-dimensional hypercube. This allows for a straightforward interpretation of the many-body dynamics in terms of a single particle one on the Hilbert space and provides an explicit link between the many-body and single particle scars. Exploiting this perspective, we show that an integrability-breaking disorder enhances the scars followed by inhibition of the dynamics due to strong localization of the eigenstates in the large disorder limit. Next, mapping the…
We study the many-body localization of the random-field Heisenberg chain using the nested shift-invert Lanczos method with an iterative linear solver. We use the minimum residual method (MINRES) inside each Lanczos iteration. The memory consumption of the proposed method is only proportional to the dimension of the Hilbert space. We also introduce a preconditioner that takes into account the effects of disorder and interaction in the random-field Heisenberg chain. As a probe of many-body localization transition, we propose a unitary operator called the twist operator, which has a clear interpr…
Exploring Hilbert-Space Fragmentation on a Superconducting Processor
Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；Beijing Academy of Quantum Information Sciences（北京量子信息科学研究院）；Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg（奥格斯堡大学物理所电子关联与磁学理论物理III）；Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area（粤港澳大湾区量子科学中心）；Hefei National Laboratory（合肥国家实验室）
We study many-body localization (MBL) in the quasiperiodic $t_1$-$t_2$ model, focusing on the role of next-nearest-neighbor (NNN) hopping $t_2$, which introduces a single-particle mobility edge. The calculated phase diagram can be divided into three distinct regimes, depending on the strength of the short-range interaction $U$. For weak interactions ($U\ll t_1$), this model is always nonthermal. For intermediate interactions ($U\sim t_1$), the thermal-MBL phase transition in this model is qualitatively the same as that of the Aubry-Andre (AA) model, which is consistent with existing experiment…
