This thesis is focused on many-body localization (MBL) and the development of algorithms using the tensor networks representation for many-body localized systems. Chapter 1 is a detailed discussion on the MBL phenomenon. Chapter 2 gives an introduction to the numerical techniques that are frequently used in studying many-body quantum systems. Chapter 3 studies the shift-and-invert matrix product state method for obtaining highly excited states of MBL systems. Chapter 4 studies an efficient full diagonalization method using Wegner unitary gates. Chapter 5 studies the distribution of the entanglement entropy in the MBL-ergodic transition region. Chapter 6 studies the the influence of pseudo-spin SU (2) symmetry on the MBL phase. Chapter 7 is a side project that studies the influence of the duality-twisted boundary conditions on the critical twist defect chains. The MBL-ergodic transition is a dynamical phase transition which happens at finite energy density for a disordered and isolated many-body interacting system. It is a natural extension of Anderson localization where the localized phase survives the inter-particle interaction. In Chapter 1, we describe its relation to quantum thermalization, typical lattice models that harbor a MBL phase, MBL system’s phenomenological description, the characteristics of the MBL phase, experimental studies of MBL phenomenon in 1D and higher dimensions, the latest developments and the open questions on the nature of the MBL-ergodic transition. In Chapter 2, we provide a short but self-contained discussion of basic numerical methods that are frequently used in studying MBL physics. We mainly focus on the exact diagonalization methods and the matrix product state (MPS) representation, and explain their advantages and disadvantages. In Chapter 3, because obtaining the interior eigenstates of MBL Hamiltonians is a key step to studying the MBL phases and the transitions, we develop algorithms that can capture individual excited states to high fidelity. Using the fact that eigenstates of many-body localized systems have area-law entanglement and can therefore be efficiently represented as a matrix product state (MPS), we designed two algorithms that can generate excited states in the MPS representation, and use them to test the basic properties of MBL in the regime of large one-dimensional random field Heisenberg chains that were previously inaccessible due to the limitations of exact diagonalization (ED). In Chapter 4, we investigate an efficient way to diagonlize a fully many-body localized (FMBL) Hamiltonian. Using the Wegner flow technique for subsystems, we constructively build unitary gates that can greedily transform a local part of the FMBL Hamiltonian with the aim of lowering its average energy variance. Applying the Wegner unitary successively can make the system “flow” towards the diagonalized form. We find that the performance of this method is mainly controlled by the length of each individual Wegner unitary gate, and gets better with increasing length of the gate. We compare the performance of this constructive method with previous methods based on quasi-Newtonian optimization methods, and find that they have similar performance, while our constructive method is far more efficient. We also comment the possibility of constructing a tensor network state that resembles a multi-scale entanglement renormalization ansatz (MERA) using this constructive method. In Chapter 5, using strong subadditivity (SSA) theorem, we develop two order parameters – cut-averaged entanglement entropy (CAEE) and its slope (SCAEE), and show that they can be used to directly identify the volume or area law scaling in single eigenstates. We study the distribution of the SCAEE over disorder realizations. This distribution appears Gaussian at weak disorder, while on the MBL side, the distribution of the SCAEE is peaked at zero slope and has an exponential tail. In the critical regime we find that the distribution of the SCAEE is bimodal both over multiple disorder realizations as well as for single disorder realizations. The variance of the SCAEE distribution in the transition region seems to grow with system size when considered over disorder realizations. Our system sizes are too small to pin down its maximal value, but they are consistent with (among other possibilities) the maximal variance possible which would lead to half the states having zero SCAEE and half having maximal SCAEE. This scenario would lead to an entanglement entropy at the transition which scales as a volume law with half its thermal value. In Chapter 6, we consider the strongly disordered one-dimensional Hubbard model with spin disorder, which, as we argue, cannot be categorized as a conventional MBL phase. Though the spin rotation SU (2) symmetry is broken by the spin disorder, this model still preserves the pseudo-spin SU (2) symmetry under periodic boundary conditions. On the theoretical side, using the pseudo-spin algebra we show that a significant number of the excited states at any disorder strength have logarithmic correction to their von Neumann entanglement entropy, which violates the area-law entanglement for a typical MBL phase. A common feature of this group of excited states is that they usually have far more double occupancies than single occupancies. On the numerical side, we studied the time evolution after quantum quench from two extreme cases of product states – all single occupancies at quarter filling, and all double occupancies at half filling. We find in the former case entanglement entropy behaves as in a typical MBL system, while the latter is clearly delocalized, which suggests the existence of incomplete set of local integrals of motion. With the above evidence, it is convincing that this model provides a playground for studying a non-ergodic, non-MBL phase, and its relation to continuous SU (2) symmetry. Chapter 7 deviates from the common scene of MBL physics. In this chapter, we consider a series of one-dimensional critical twist defect models with various boundary conditions. These twist defects are embedded in the background of a generalization of Kitaev Z 2 toric code model, and can also be understood as a discrete Z k gauge theory in its deconfined phase. We analytically construct the duality transformation of these critical models, and use numerical methods to extract information on the central charges, sound velocities, and conformal dimensions. With the numerical data, we show the relation between the CFT contents of the critical chains with periodic boundary conditions and the critical chains with duality-twisted boundary conditions.
This thesis is focused on many-body localization (MBL) and the development of algorithms using the tensor networks representation for many-body localized systems. Chapter 1 is a detailed discussion on the MBL phenomenon. Chapter 2 gives an introduction to the numerical techniques that are frequently used in studying many-body quantum systems. Chapter 3 studies the shift-and-invert matrix product state method for obtaining highly excited states of MBL systems. Chapter 4 studies an efficient full diagonalization method using Wegner unitary gates. Chapter 5 studies the distribution of the entanglement entropy in the MBL-ergodic transition region. Chapter 6 studies the the influence of pseudo-spin SU (2) symmetry on the MBL phase. Chapter 7 is a side project that studies the influence of the duality-twisted boundary conditions on the critical twist defect chains. The MBL-ergodic transition is a dynamical phase transition which happens at finite energy density for a disordered and isolated many-body interacting system. It is a natural extension of Anderson localization where the localized phase survives the inter-particle interaction. In Chapter 1, we describe its relation to quantum thermalization, typical lattice models that harbor a MBL phase, MBL system’s phenomenological description, the characteristics of the MBL phase, experimental studies of MBL phenomenon in 1D and higher dimensions, the latest developments and the open questions on the nature of the MBL-ergodic transition. In Chapter 2, we provide a short but self-contained discussion of basic numerical methods that are frequently used in studying MBL physics. We mainly focus on the exact diagonalization methods and the matrix product state (MPS) representation, and explain their advantages and disadvantages. In Chapter 3, because obtaining the interior eigenstates of MBL Hamiltonians is a key step to studying the MBL phases and the transitions, we develop algorithms that can capture individual excited states to high fidelity. Using the fact that eigenstates of many-body localized systems have area-law entanglement and can therefore be efficiently represented as a matrix product state (MPS), we designed two algorithms that can generate excited states in the MPS representation, and use them to test the basic properties of MBL in the regime of large one-dimensional random field Heisenberg chains that were previously inaccessible due to the limitations of exact diagonalization (ED). In Chapter 4, we investigate an efficient way to diagonlize a fully many-body localized (FMBL) Hamiltonian. Using the Wegner flow technique for subsystems, we constructively build unitary gates that can greedily transform a local part of the FMBL Hamiltonian with the aim of lowering its average energy variance. Applying the Wegner unitary successively can make the system “flow” towards the diagonalized form. We find that the performance of this method is mainly controlled by the length of each individual Wegner unitary gate, and gets better with increasing length of the gate. We compare the performance of this constructive method with previous methods based on quasi-Newtonian optimization methods, and find that they have similar performance, while our constructive method is far more efficient. We also comment the possibility of constructing a tensor network state that resembles a multi-scale entanglement renormalization ansatz (MERA) using this constructive method. In Chapter 5, using strong subadditivity (SSA) theorem, we develop two order parameters – cut-averaged entanglement entropy (CAEE) and its slope (SCAEE), and show that they can be used to directly identify the volume or area law scaling in single eigenstates. We study the distribution of the SCAEE over disorder realizations. This distribution appears Gaussian at weak disorder, while on the MBL side, the distribution of the SCAEE is peaked at zero slope and has an exponential tail. In the critical regime we find that the distribution of the SCAEE is bimodal both over multiple disorder realizations as well as for single disorder realizations. The variance of the SCAEE distribution in the transition region seems to grow with system size when considered over disorder realizations. Our system sizes are too small to pin down its maximal value, but they are consistent with (among other possibilities) the maximal variance possible which would lead to half the states having zero SCAEE and half having maximal SCAEE. This scenario would lead to an entanglement entropy at the transition which scales as a volume law with half its thermal value. In Chapter 6, we consider the strongly disordered one-dimensional Hubbard model with spin disorder, which, as we argue, cannot be categorized as a conventional MBL phase. Though the spin rotation SU (2) symmetry is broken by the spin disorder, this model still preserves the pseudo-spin SU (2) symmetry under periodic boundary conditions. On the theoretical side, using the pseudo-spin algebra we show that a significant number of the excited states at any disorder strength have logarithmic correction to their von Neumann entanglement entropy, which violates the area-law entanglement for a typical MBL phase. A common feature of this group of excited states is that they usually have far more double occupancies than single occupancies. On the numerical side, we studied the time evolution after quantum quench from two extreme cases of product states – all single occupancies at quarter filling, and all double occupancies at half filling. We find in the former case entanglement entropy behaves as in a typical MBL system, while the latter is clearly delocalized, which suggests the existence of incomplete set of local integrals of motion. With the above evidence, it is convincing that this model provides a playground for studying a non-ergodic, non-MBL phase, and its relation to continuous SU (2) symmetry. Chapter 7 deviates from the common scene of MBL physics. In this chapter, we consider a series of one-dimensional critical twist defect models with various boundary conditions. These twist defects are embedded in the background of a generalization of Kitaev Z 2 toric code model, and can also be understood as a discrete Z k gauge theory in its deconfined phase. We analytically construct the duality transformation of these critical models, and use numerical methods to extract information on the central charges, sound velocities, and conformal dimensions. With the numerical data, we show the relation between the CFT contents of the critical chains with periodic boundary conditions and the critical chains with duality-twisted boundary conditions.
"Under certain conditions, an interacting system can defy the concept of thermalization, a keystone in our understanding of physical processes. Many-body localization (MBL) is a phase of matter in which thermalization does not apply and ergodicity is broken. This striking behavior, which can appears on closed, interacting, quantum systems subject to strong disorder, has been the focus of a large body of theoretical, numerical, and experimental work in recent years. In this thesis we numerically study several aspects of MBL and the ergodic-MBL transition. Chapter 1 introduces the concept of thermalization in quantum systems, which relies on the idea of the eigenstate thermalization hypothesis (ETH). I then present localization as a phenomenon that breaks ETH, both in its single-particle as in its many-body versions. I discuss some of the main aspects that are known about MBL, as well as some of its open questions, some of which we tackle in later chapters. Chapter 2 discusses the main numerical methods used in this thesis for the study of MBL. I provide both theoretical background as well as discuss some more practical matters, such as their advantages and disadvantages, or details on the software developed/used in my studies. In Chapter 3 I present our work on MBL from the point of view of single-particle orbitals. In this work we access a complete set of approximate integrals of motion of a one-dimensional system by computing the one-particle orbitals (OPO) of highly-excited MBL energy eigenstates, which are obtained through the shift-and-invert matrix product state (SIMPS) algorithm. We then study the properties of the OPOs over large systems, up to L = 64. We find that the OPOs drawn from eigenstates at different energy densities have high overlap and their occupations are correlated with the energy of the eigenstates. Moreover, the standard deviation of the inverse participation ratio of these orbitals is maximal at the nose of the mobility edge. Also, the OPOs decay exponentially in real space, with a correlation length that increases at low disorder. In addition, we find that the probability distribution of the strength of the large-range coupling constants of the number operators generated by the OPOs approach a log-uniform distribution at strong disorder. In Chapter 4 I present our work on the hybridization of eigenstates in either the MBL or the ergodic phase, as well as at the transition. We do so by adiabatially evolving highly excited eigenstates of the Hamiltonian and measuring their their hybridization process with other eigenstates of the system. This hybridization, which dresses the eigenstate and has the potential of bringing it out of the MBL phase through the transition, is a consequence of the ""collisions"" of eigenstates in energy, which avoid level crossings every time their energy gap is small. The hybridization of eigenstates with each other involves only local regions of the system in the MBL phase, ignores locality in the ergodic phase, and is range-independent at the transition. This range independence suggests the proliferation of long-range resonances at the transition, as well as the divergence of a localization length. In Chapter 5 I present our studies on the typical and extreme (atypically strong) correlations across a one-dimensional system in the ergodic-MBL phase diagram. While typical correlations decay exponentially with range in the MBL phase, in the ergodic phase they are constant and independent of the range. Surprisingly, we identify a moderate region of the phase in which typical correlations decay as a stretched exponential with range r, and in particular as exp[-r^(1/2)] at the transition, a decay that is reminiscent of the random singlet phase. Moreover, at the transition the distribution of the logarithm of the correlations show vanishing even excess moments and non-zero range-invariant odd excess moments. This distinct behavior at the transition is in contrast with ergodic and MBL phenomenologies. In addition, we study the extreme correlations in the system. Our results suggest that strong long-range correlations proliferate at the transition, in contrast with a decay with range in the MBL phase and the lack of strong long-range correlations in the ergodic phase. Finally, we analyze the probability that a single bit of information is shared across two halves of a system, which has been proposed as a robust order parameter in the ergodic-MBL phase diagram. We find that this probability is non-zero deep in the MBL phase, but vanishes at moderate disorder, well above the transition, thus not providing a proper order parameter. Chapter 6 I summarize the work presented with an emphasis on context and perspective."
The research work in this thesis is based on strongly interacting quantum lattice systems. The biggest part of research was conducted using state-of-the-art tensor network simulations. Tensor networks provide efficient and highly accurate representations of quantum states when any simply connected patch of the quantum state is slightly entangled. Matrix Product States (MPS) is a tensor network representation for quantum states which is quasi-exact for one-dimensional systems when entanglement entropy of any bipartition of the state follows ``area-law". Projected Entangled Pair States (PEPS) is the extension of MPS for higher dimensional systems, where entanglement entropy area-law is non-trivial. These formalisms and their relevant ground state optimization techniques, Density Matrix Renormalization Group (DMRG) for MPS and Simple Update (SU) with Tensor Renormalization Group (TRG) for PEPS are thoroughly analysed. Entanglement entropy area-law is usually obeyed by the ground state of local Hamiltonians, while generic highly excited states follow a volume-law (they span a finite part of the Hilbert space). Recently, a class of interacting system was shown to undergo a dynamical phase transition (Many Body Localization) where the entropy of every eigenstate follows an area-law. This transition is achieved when the system is highly disordered and the quantum many body state becomes localized similarly to the free particle Anderson localization. The area-law property makes highly excited eigenstates of Many Body Localized (MBL) systems efficiently represented by the MPS ansatz. We develop a highly optimized algorithm (eDMRG) which goes beyond ground state optimization and successfully target eigenstates in any part of the spectrum. This algorithm is used together with analytical calculations, based on local integrability of MBL systems, to identify the universal behaviour of the entanglement spectrum of highly excited eigenstates in MBL systems. In the second part we study interacting bosons in 2D optical lattices. Strongly interacting bosons are simulated using the Bose-Hubbard (BH) model when the interactions are strictly local and the Extended Bose-Hubbard (EBH) model, when additional dipolar interactions are present. The ground states of BH/EBH Hamiltonians in a hexagonal lattice are studied. The phase spectrum includes various insulating and critical phases which are studied in detail using infinite-PEPS ansatz. Additional results on entanglement entropy scaling with respect to the filling of the lattice are presented. Finally, a strongly interacting Harper-Hofstadter Hamiltonian is realized by combining synthetic fields with the BH model. The homogeneity of the system is then broken by a parabolic trapping potential, similar to the ones used in cold atom experiments. Using time-dependent Gutzwiller ansatz (GA) the expansion dynamics of the cloud in large square lattices are studied. In contrast to the expansion dynamics of the BH model, it is found that the synthetic fields generate a self-trapping effect. Using phenomenology and simulations the dynamics are studied in the hard-core and soft-core limit.
This paper focuses on developing a new algorithm (En-DMRG) for obtaining highly excited states of disordered and strongly interacting spin systems. This line of research is stimulated by the growing interest in understanding the many-body localized (MBL) state, which emerges in disordered interacting systems at high energy densities through a disorder-driven dynamic phase transition. By studying the one-dimensional Heisenberg spin chain (with up to 72 spins) in a random field, the authors demonstrate the accuracy of the method in obtaining energy eigenstates and the corresponding statistical results of the quantum system in the MBL phase. The method presented in this work has the potential to be applicable in a variety of quantum statistical physics problems, possibly including the MBL phase in higher dimensions.
Despite the exponentially large amount of information required in the quantum description of many-body systems, finite size numerical simulations have paved the way for recent progress in the understanding of quantum phases of matter both in and out-of-equilibrium. \n \nIn this spirit, using exact diagonalization for a small number of spins ~ 20, first, we investigate the fate of both ergodic quantum many-body systems that thermalize at long times and systems with quenched disorder that exhibit many-body localization (MBL) and fail to thermalize (due to an extensive set of local conserved degrees of freedom) under periodic driving. Ergodic systems always delocalize in energy space and heat up to infinite temperature, for both local driving, in which the time dependence of the Hamiltonian is restricted to a few contiguous degrees of freedom, and global driving. On the other hand, MBL systems remain localized at finite energy density for local driving, while in the case of global driving these systems can either heat up to infinite temperature or remain localized depending on the parameters of the drive. Underlying the latter is the emergence of an effective time independent MBL Hamiltonian describing the long time dynamics of the system. These numerical predictions have recently been verified experimentally in a one dimensional quasi-disordered optical lattice of interacting fermions wherein unitary time evolution can be experimentally probed. \n \nSubsequently, we study the stability of MBL systems with respect to another perturbation, in this case by thermal inclusions, which remains poorly understood. We consider a simple model to address this question: a two level system interacting with 𝑁 localized bits subject to random fields. In the large 𝑁 limit, the localized bits are thermalized for any nonzero ratio of interactions to disorder strength. Moreover, at finite 𝑁, in the transition region, the single-site eigenstate entanglement entropies exhibit bimodal distributions, so that localized bits are either "on" (strongly entangled) or "off" (weakly entangled) in eigenstates. The clusters of "on" bits vary significantly between eigenstates of the same sample, which provides evidence for a heterogenous discontinuous transition out of the localized phase in single site observables. These results also imply the MBL phase is unstable in systems with short-range interactions and quenched randomness in dimensions 𝑑 that are high but finite. \n \nThe second part of this thesis is motivated by the recent breakthroughs of artificial intelligence in domains such as computer vision involving highly dimensional data of complexity not unlike the one found in the description of many-body systems. Using Monte Carlo simulations to generate synthetic data, we find that kernel methods, a class of machine learning algorithms, can learn the mathematical \nform of physical discriminators associated with phase transitions, such as order parameters and Hamiltonian constraints, for a set of two dimensional spin models: the ferromagnetic Ising model, a conserved-order-parameter Ising model, and the Ising gauge theory. \n \nOur work shows that the combination of numerical simulations with the ability of machine learning algorithms for automatic feature detection hold promise to become standard tools in the exploration and characterization of different phases of matter in condensed matter and other many-body systems.
Abstract Statistical mechanics provides a framework for describing the physics of large, complex many-body systems using only a few macroscopic parameters to determine the state of the system. For isolated quantum many-body systems, such a description is achieved via the eigenstate thermalization hypothesis (ETH), which links thermalization, ergodicity and quantum chaotic behavior. However, tendency towards thermalization is not observed at finite system sizes and evolution times in a robust many-body localization (MBL) regime found numerically and experimentally in the dynamics of interacting many-body systems at strong disorder. Although the phenomenology of the MBL regime is well-established, the central question remains unanswered: under what conditions does the MBL regime give rise to an MBL phase , in which the thermalization does not occur even in the asymptotic limit of infinite system size and evolution time? This review focuses on recent numerical investigations aiming to clarify the status of the MBL phase, and it establishes the critical open questions about the dynamics of disordered many-body systems. The last decades of research have brought an unprecedented new variety of tools and indicators to study the breakdown of ergodicity, ranging from spectral and wave function measures, matrix elements of observables, through quantities probing unitary quantum dynamics, to transport and quantum information measures. We give a comprehensive overview of these approaches and attempt to provide a unified understanding of their main features. We emphasize general trends towards ergodicity with increasing length and time scales, which exclude naive single-parameter scaling hypothesis, necessitate the use of more refined scaling procedures, and prevent unambiguous extrapolations of numerical results to the asymptotic limit. Providing a concise description of numerical methods for studying ETH and MBL, we explore various approaches to tackle the question of the MBL phase. Persistent finite size drifts towards ergodicity consistently emerge in quantities derived from eigenvalues and eigenvectors of disordered many-body systems. The drifts are related to continuous inching towards ergodicity and non-vanishing transport observed in the dynamics of many-body systems, even at strong disorder. These phenomena impede the understanding of microscopic processes at the ETH-MBL crossover. Nevertheless, the abrupt slowdown of dynamics with increasing disorder strength provides premises suggesting the proximity of the MBL phase. This review concludes that the questions about thermalization and its failure in disordered many-body systems remain a captivating area open for further explorations.
Classical simulation of quantum many-body systems is in general a challenging problem for the simple reason that the dimension of the Hilbert space grows exponentially with the system size. In particular, merely encoding a generic quantum many-body state requires an exponential number of bits. However, condensed matter physicists are mostly interested in local Hamiltonians and especially their ground states, which are highly non-generic. Thus, we might hope that at least some physical systems allow efficient classical simulation.Starting with one-dimensional (1D) quantum systems (i.e., the simplest nontrivial case), the first basic question is: Which classes of states have efficient classical representations? It turns out that this question is quantitatively related to the amount of entanglement in the state, for states with ``little entanglement'' are well approximated by matrix product states (a data structure that can be manipulated efficiently on a classical computer). At a technical level, the mathematical notion for ``little entanglement'' is area law, which has been proved for unique ground states in 1D gapped systems. We establish an area law for constant-fold degenerate ground states in 1D gapped systems and thus explain the effectiveness of matrix-product-state methods in (e.g.) symmetry breaking phases. This result might not be intuitively trivial as degenerate ground states in gapped systems can be long-range correlated.Suppose an efficient classical representation exists. How can one find it efficiently? The density matrix renormalization group is the leading numerical method for computing ground states in 1D quantum systems. However, it is a heuristic algorithm and the possibility that it may fail in some cases cannot be completely ruled out. Recently, a provably efficient variant of the density matrix renormalization group has been developed for frustration-free 1D gapped systems. We generalize this algorithm to all (i.e., possibly frustrated) 1D gapped systems. Note that the ground-state energy of 1D gapless Hamiltonians is computationally intractable even in the presence of translational invariance.It is tempting to extend methods and tools in 1D to two and higher dimensions (2+D), e.g., matrix product states are generalized to tensor network states. Since an area law for entanglement (if formulated properly) implies efficient matrix product state representations in 1D, an interesting question is whether a similar implication holds in 2+D. Roughly speaking, we show that an area law for entanglement (in any reasonable formulation) does not always imply efficient tensor network representations of the ground states of 2+D local Hamiltonians even in the presence of translational invariance. It should be emphasized that this result does not contradict with the common sense that in practice quantum states with more entanglement usually require more space to be stored classically; rather, it demonstrates that the relationship between entanglement and efficient classical representations is still far from being well understood.Excited eigenstates participate in the dynamics of quantum systems and are particularly relevant to the phenomenon of many-body localization (absence of transport at finite temperature in strongly correlated systems). We study the entanglement of excited eigenstates in random spin chains and expect that its singularities coincide with dynamical quantum phase transitions. This expectation is confirmed in the disordered quantum Ising chain using both analytical and numerical methods.Finally, we study the problem of generating ground states (possibly with topological order) in 1D gapped systems using quantum circuits. This is an interesting problem both in theory and in practice. It not only characterizes the essential difference between the entanglement patterns that give rise to trivial and nontrivial topological order, but also quantifies the difficulty of preparing quantum states with a quantum computer (in experiments).