Local Integrals Of Motion Research Articles

Local Integrals of Motion in Dipole-Conserving Models with Hilbert Space Fragmentation.

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.

Local integrals of motion and the stability of many-body localization in Wannier-Stark potentials

Local integrals of motion and the stability of many-body localization in Wannier-Stark potentials

Disorder-induced spin-charge separation in the one-dimensional Hubbard model

Many-body localization is believed to be generically unstable in quantum systems with continuous non-Abelian symmetries, even in the presence of strong disorder. Breaking these symmetries can stabilize the localized phase, leading to the emergence of an extensive number of quasilocally conserved quantities known as local integrals of motion, or $l$ bits. Using a sophisticated nonperturbative technique based on continuous unitary transforms, we investigate the one-dimensional Hubbard model subject to both spin and charge disorder, compute the associated $l$ bits and demonstrate that the disorder gives rise to a novel form of spin-charge separation. We examine the role of symmetries in delocalizing the spin and charge degrees of freedom, and show that while symmetries generally lead to delocalization through multiparticle resonant processes, certain subsets of states appear stable.

Multibody expansion of the local integrals of motion: how many pairs of particle-hole do we really need to describe the quasiparticles in the many-body localized phase?

The emergent integrability in a many-body localized (MBL) system can be well characterized by the existence of the complete set of local integrals of motion (LIOMs). Such exactly conserved and exponentially localized operators are often understood as quasiparticle operators which can be expanded in terms of single-particle operators dressed with different numbers of particle-hole pairs. Here, we consider a one-dimensional XXZ spin- Heisenberg chain in the presence of a random field and try to quantify the corrections needed to be considered in the picture of quasiparticles associated with LIOMs due to the presence of particle-hole excitations. To this end, we explicitly present the multibody expansion of LIOM creation operators of the system in the MBL regime. We analytically obtain the coefficients of this expansion and discuss the effect of higher-order corrections associated with different numbers of particle-hole excitations. Our analysis shows that depending on the localization length of the system, there exist a regime in which the contributions that come from higher-order terms can break down the effective one-particle description of the LIOMs and such quasiparticles become essentially many-body-like.

Dynamics of entanglement generation in a many-body localized system using the local integrals of motion

The study of many-body quantum systems, that fail to thermalize in the presence of disorder, has recently attracted lots of interests. This is due to the appearance of the many-body localized phase and breakdown of eigenstate thermalization hypothesis in such systems which can be described by the local integrals of motion. In this paper, we consider a disordered spin chain in the many-body localized phase and try to study the dynamics of entanglement generation in this system using the local integrals of motion. To this end, we, first, solve the non-interacting system analytically to describe the mechanism of entanglement generation for different kinds of initial states, exactly. Then, we generalize this approach to the interacting system to learn the dynamics of entanglement generation. Finally, we discuss the physical meaning of different behaviors in the dynamics of entanglement generation in the presence and absence of interaction.‎

Structural properties of local integrals of motion across the many-body localization transition via a fast and efficient method for their construction

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 not only in the deep MBL phase but rather, near the transition point too. Then, we take the matrix representation of LIOM operators as an adjacency matrix of a directed graph whose elements describe the connectivity of ordered eigenbasis in the Hilbert-space. Our cluster size analysis for this graph shows that the MBL transition coincides with a percolation transition in the Hilbert-space. By performing finite-size scaling, we compare the critical disorder and correlation exponent $\nu$ both in the presence and absence of interaction. Finally, we also discuss how the distribution of diagonal elements of LIOM operators in a typical cluster signals the transition.

Localization and slow-thermalization in a cluster spin model

Novel cluster spin model with interactions and disorder is introduced and studied. In specific type of interactions, we find an extensive number of local integrals of motions (LIOMs), which are a modified version of the stabilizers in quantum information, i.e., mutually commuting operators specifying all quantum states in the system. These LIOMs can be defined for any strength of the interactions and disorder, and are of compact-support instead of exponentially-decaying tail. Hence, even under the presence of interactions, integrability is held, and all energy eigenstates are labeled by these LIOMs and can be explicitly obtained. Integrable dynamics is, then, expected to occur. The compact-support nature of the LIOMs crucially prevents the thermalization and entanglement spreading. We numerically investigate dynamics of the system governed by the existence of the compact-support LIOMs, and clarify the effects of additional interactions, which break the compact-support nature of the LIOMs. There, we find that the ordinary many-body localization behaviors emerge, such as the logarithmic growth of the entanglement entropy in the time evolution. Besides the ergodicity breaking dynamic, we find that symmetry protected topological order preserves for specific states even in the presence of the interactions.

Deformation of localized states and state transitions in systems of randomly hopping interacting fermions

We numerically study the random-hopping fermions (the Cruetz ladder) with repulsion and investigate how the interactions deform localized eigenstates by means of the one particle-density matrix (OPDM). The ground state exhibits resurgence of localization from the compact localized state to strong-repulsion-induced localization. On the other hand, excited states in the middle of the spectrum tend to extend by the repulsion. The transition property obtained by numerical calculations of the OPDM is deeply understood by studying a solvable model in which local integrals of motion (LIOMs) are obtained explicitly. The present work clarifies the utility of the OPDM and also how compact-support LIOMs in non-interacting limit are deformed by the repulsion.

Local resonances and parametric level dynamics in the many-body localized phase

By varying the disorder realisation in the many-body localised (MBL) phase, we investigate the influence of resonances on spectral properties. The standard theory of the MBL phase is based on the existence of local integrals of motion (LIOM), and eigenstates of the time evolution operator can be described as LIOM configurations. We show that smooth variations of the disorder give rise to avoided level crossings, and we identify these with resonances between LIOM configurations. Through this parametric approach, we develop a theory for resonances in terms of standard properties of non-resonant LIOM. This framework describes resonances that are locally pairwise, and is appropriate in arbitrarily large systems deep within the MBL phase. We show that resonances are associated with large level curvatures on paths through the ensemble of disorder realisations, and we determine the curvature distribution. By considering the level repulsion associated with resonances we calculate the two-point correlator of the level density. We also find the distributions of matrix elements of local observables and discuss implications for low-frequency dynamics.

Local integrals of motion and the quasiperiodic many-body localization transition

We study the many-body localization (MBL) transition for interacting fermions subject to quasiperiodic potentials by constructing the local integrals of motion (LIOMs) in the MBL phase as time-averaged local operators. We study numerically how these time-averaged operators evolve across the MBL transition. We find that the norm of such time-averaged operators drops discontinuously to zero across the transition; as we discuss, this implies that LIOMs abruptly become unstable at some critical localization length of order unity. We analyze the LIOMs using hydrodynamic projections and isolating the part of the operator that is associated with interactions. Equipped with these data we perform a finite-size scaling analysis of the quasiperiodic MBL transition. Our results suggest that the quasiperiodic MBL transition occurs at considerably stronger quasiperiodic modulations and has a larger correlation-length critical exponent than previous studies had found.

Flat-band full localization and symmetry-protected topological phase on bilayer lattice systems

In this work, we present bilayer flat-band Hamiltonians, in which all bulk states are localized and specified by extensive local integrals of motion (LIOMs). The present systems are bilayer extension of the Creutz ladder, which is studied previously. In order to construct models, we employ building blocks, cube operators, which are linear combinations of fermions defined in each cube of the bilayer lattice. There are eight cubic operators, and the Hamiltonians are composed of the number operators of them, the LIOMs. A suitable arrangement of locations of the cube operators is needed to have exact projective Hamiltonians. The projective Hamiltonians belong to a topological classification class, the BDI class. With the open boundary condition, the constructed Hamiltonians have gapless edge modes, which commute with each other as well as the Hamiltonian. This result comes from a symmetry analogous to the one-dimensional chiral symmetry of the BDI class. These results indicate that the projective Hamiltonians describe a kind of symmetry-protected topological phase matter. Careful investigation of topological indexes, such as the Berry phase and string operator, is given. We also show that, by using the gapless edge modes, a generalized Sachdev-Ye-Kitaev model is constructed.

Spectral Lyapunov exponents in chaotic and localized many-body quantum systems

We consider the spectral statistics of the Floquet operator for disordered, periodically driven spin chains in their quantum chaotic and many-body localized phases (MBL). The spectral statistics are characterized by the traces of powers $t$ of the Floquet operator, and our approach hinges on the fact that, for integer $t$ in systems with local interactions, these traces can be re-expressed in terms of products of dual transfer matrices, each representing a spatial slice of the system. We focus on properties of the dual transfer matrix products as represented by a spectrum of Lyapunov exponents, which we call \textit{spectral Lyapunov exponents}. In particular, we examine the features of this spectrum that distinguish chaotic and MBL phases. The transfer matrices can be block-diagonalized using time-translation symmetry, and so the spectral Lyapunov exponents are classified according to a momentum in the time direction. For large $t$ we argue that the leading Lyapunov exponents in each momentum sector tend to zero in the chaotic phase, while they remain finite in the MBL phase. These conclusions are based on results from three complementary types of calculation. We find exact results for the chaotic phase by considering a Floquet random quantum circuit with on-site Hilbert space dimension $q$ in the large-$q$ limit. In the MBL phase, we show that the spectral Lyapunov exponents remain finite by systematically analyzing models of non-interacting systems, weakly coupled systems, and local integrals of motion. Numerically, we compute the Lyapunov exponents for a Floquet random quantum circuit and for the kicked Ising model in the two phases. As an additional result, we calculate exactly the higher point spectral form factors (hpSFF) in the large-$q$ limit, and show that the generalized Thouless time scales logarithmically in system size for all hpSFF in the large-$q$ chaotic phase.

Nonthermalized dynamics of flat-band many-body localization

We find that a flat-band fermion system with interactions and without disorders exhibits non-thermalized ergodicity-breaking dynamics, an analog of many-body localization (MBL). In the previous works, we observed flat-band many-body localization (FMBL) in the Creutz ladder model. The origin of FMBL is a compact localized state governed by local integrals of motion (LIOMs), which are to be obtained explicitly. In this work, we clarify the dynamical aspects of FMBL. We first study dynamics of two-particles, and find that the states are not substantially modified by weak interactions, but the periodic time evolution of entanglement entropy emerges as a result of a specific mechanism inherent in the system. On the other hand, as the strength of the interactions is increased, the modification of the states takes place with inducing instability of the LIOMs. Furthermore, many-body dynamics of the system at finite fillings is numerically investigated by time-evolving block decimation (TEBD) method. For a suitable choice of the filling, non-thermal and low entangled dynamics appears. This behavior is a typical example of the disorder-free FMBL.

Slow Propagation in Some Disordered Quantum Spin Chains

We introduce the notion of transmission time to study the dynamics of disordered quantum spin chains and prove results relating its behavior to many-body localization properties. We also study two versions of the so-called Local Integrals of Motion (LIOM) representation of spin chain Hamiltonians and their relation to dynamical many-body localization. We prove that uniform-in-time dynamical localization expressed by a zero-velocity Lieb-Robinson bound implies the existence of a LIOM representation of the dynamics as well as a weak converse of this statement. We also prove that for a class of spin chains satisfying a form of exponential dynamical localization, sparse perturbations result in a dynamics in which transmission times diverge at least as a power law of distance, with a power for which we provide lower bound that diverges with increasing sparseness of the perturbation.

Construction of many-body-localized models where all the eigenstates are matrix-product-states

The inverse problem of ‘eigenstates-to-Hamiltonian’ is considered for an open chain of N quantum spins in the context of many-body-localization. We first construct the simplest basis of the Hilbert space made of 2N orthonormal matrix-product-states (MPS), that will thus automatically satisfy the entanglement area-law. We then analyze the corresponding N local integrals of motions (LIOMs) that can be considered as the local building blocks of these 2N MPS, in order to construct the parent Hamiltonians that have these 2N MPS as eigenstates. Finally we study the matrix-product-operator form of the diagonal ensemble density matrix that allows to compute long-time-averaged observables of the unitary dynamics. Explicit results are given for the memory of local observables and for the entanglement properties in operator-space, via the generalized notion of Schmidt decomposition for density matrices describing mixed states.

Exact projector Hamiltonian, local integrals of motion, and many-body localization with symmetry-protected topological order

In this work, we construct an exact projector Hamiltonian with interactions, which is given by a sum of mutually commuting operators called stabilizers. The model is based on the recently studied Creutz-ladder of fermions, in which flat-band structure and strong localization are realized. These stabilizers are local integrals of motion from which many-body localization (MBL) is realized. All energy eigenstates are explicitly obtained even in the presence of local disorders. All states are MBL states, that is, this system is a full many-body localized (FMBL) system. We show that this system has a topological order and stable gapless edge modes exist under the open boundary condition. By the numerical study, we investigate stability of the FMBL and topological order.

Quantitative Impact of Integrals of Motion on the Eigenstate Thermalization Hypothesis.

Even though the eigenstate thermalization hypothesis (ETH) may be introduced as an extension of the random matrix theory, physical Hamiltonians and observables differ from random operators. One of the challenges is to embed local integrals of motion (LIOMs) within the ETH. Here we make steps towards a unified treatment of the ETH in integrable and nonintegrable models with translational invariance. Specifically, we focus on the impact of LIOMs on the fluctuations and structure of the diagonal matrix elements of local observables. We first show that nonvanishing fluctuations entail the presence of LIOMs. Then we introduce a generic protocol to construct observables, subtracted by their projections on LIOMs as well as products of LIOMs. The protocol systematically reduces fluctuations and/or the structure of the diagonal matrix elements. We verify our arguments by numerical results for integrable and nonintegrable models.

Comparing many-body localization lengths via nonperturbative construction of local integrals of motion

Many-body localization (MBL), characterized by the absence of thermalization and the violation of conventional thermodynamics, has elicited much interest both as a fundamental physical phenomenon and for practical applications in quantum information. A phenomenological model, which describes the system using a complete set of local integrals of motion (LIOMs), provides a powerful tool to understand MBL, but can be usually only computed approximately. Here we explicitly compute a complete set of LIOMs with a non-perturbative approach, by maximizing the overlap between LIOMs and physical spin operators in real space. The set of LIOMs satisfies the desired exponential decay of weight of LIOMs in real-space. This LIOM construction enables a direct mapping from the real space Hamiltonian to the phenomenological model and thus enables studying the localized Hamiltonian and the system dynamics. We can thus study and compare the localization lengths extracted from the LIOM weights, their interactions, and dephasing dynamics, revealing interesting aspects of many-body localization. Our scheme is immune to accidental resonances and can be applied even at phase transition point, providing a novel tool to study the microscopic features of the phenomenological model of MBL.

Strong disorder renormalization for the dynamics of many-body-localized systems: iterative elimination of the fastest degree of freedom via the Floquet expansion

The Vosk–Altman strong disorder renormalization for the unitary dynamics of various random quantum spin chains is reformulated as follows: the local degree of freedom characterized by the highest eigenfrequency Ω can be considered as a high-frequency-Floquet-periodic-driving for the neighboring slower degrees of freedom. Then the two first orders of the high-frequency expansion for the effective Floquet Hamiltonian can be used to generate the emergent local integrals of motion (LIOMs) and to derive the renormalization rules for the effective dynamics of the remaining degrees of freedom. The flow for this effective Floquet Hamiltonian is equivalent to the RSRG-X procedure to construct the whole set of eigenstates that generalizes the Fisher RSRG procedure constructing the ground state. This general framework is applied to the random-transverse-field XXZ spin chain in its many-body-localized phase, in order to derive the renormalization rules associated to the elimination of the biggest transverse field and to the elimination of the biggest coupling respectively.

Even and odd normalized zero modes in random interacting Majorana models respecting the parity P and the time-reversal-symmetry T

For random interacting Majorana models where the only symmetries are the parity P and the time-reversal-symmetry T, various approaches are compared to construct exact even and odd normalized zero modes Γ in finite size, i.e. Hermitian operators that commute with the Hamiltonian, that square to the identity, and that commute (even) or anticommute (odd) with the parity P. Even normalized zero-modes are well known under the name of ‘pseudo-spins’ in the field of many-body-localization or more precisely ‘local integrals of motion’ (LIOMs) in the many-body-localized-phase where the pseudo-spins happens to be spatially localized. Odd normalized zero-modes are popular under the name of ‘Majorana zero modes’ or ‘strong zero modes’. Explicit examples for small systems are described in detail. Applications to real-space renormalization procedures based on blocks containing an odd number of Majorana fermions are also discussed.