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arXiv:1509.01244 · Physical Review Letters

Finding Matrix Product State Representations of Highly Excited Eigenstates of Many-Body Localized Hamiltonians

Xiongjie Yu, David Pekker, Bryan K. Clark

Published Jan 3, 2017• 121 citations
arXivHTMLPDF

Abstract

A key property of many-body localized Hamiltonians is the area law entanglement of even highly excited eigenstates. Matrix product states (MPS) can be used to efficiently represent low entanglement (area law) wave functions in one dimension. An important application of MPS is the widely used density matrix renormalization group (DMRG) algorithm for finding ground states of one-dimensional Hamiltonians. Here, we develop two algorithms, the shift-and-invert MPS (SIMPS) and excited state DMRG which find highly excited eigenstates of many-body localized Hamiltonians. Excited state DMRG uses a modified sweeping procedure to identify eigenstates, whereas SIMPS applies the inverse of the shifted Hamiltonian to a MPS multiple times to project out the targeted eigenstate. To demonstrate the power of these methods, we verify the breakdown of the eigenstate thermalization hypothesis in the many-body localized phase of the random field Heisenberg model, show the saturation of entanglement in the many-body localized phase, and generate local excitations.

DOI
10.1103/physrevlett.118.017201

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A key property of many-body localized Hamiltonians is the area law entanglement of even highly excited eigenstates. Matrix product states (MPS) can be used to efficiently represent low entanglement (area law) wave functions in one dimension. An important application of MPS is the widely used density matrix renormalization group (DMRG) algorithm for finding ground states of one-dimensional Hamiltonians. Here, we develop two algorithms, the shift-and-invert MPS (SIMPS) and excited state DMRG which find highly excited eigenstates of many-body localized Hamiltonians. Excited state DMRG uses a modified sweeping procedure to identify eigenstates, whereas SIMPS applies the inverse of the shifted Hamiltonian to a MPS multiple times to project out the targeted eigenstate. To demonstrate the power of these methods, we verify the breakdown of the eigenstate thermalization hypothesis in the many-body localized phase of the random field Heisenberg model, show the saturation of entanglement in the many-body localized phase, and generate local excitations.
The eigenstates of many-body localized (MBL) Hamiltonians exhibit low entanglement. We adapt the highly successful density-matrix renormalization group method, which is usually used to find modestly entangled ground states of local Hamiltonians, to find individual highly excited eigenstates of MBL Hamiltonians. The adaptation builds on the distinctive spatial structure of such eigenstates. We benchmark our method against the well-studied random field Heisenberg model in one dimension. At moderate to large disorder, the method successfully obtains excited eigenstates with high accuracy, thereby enabling a study of MBL systems at much larger system sizes than those accessible to exact-diagonalization methods.
Many-body localization (MBL) addresses the absence of thermalization in interacting quantum systems, with nonergodic high-energy eigenstates behaving as ground states, only area-law entangled. However, computing highly excited many-body eigenstates using exact methods is very challenging. Instead, we show that one can address high-energy MBL physics using ground-state methods, which are much more amenable to many efficient algorithms. We find that a localized many-body ground state of a given interacting disordered Hamiltonian H0{\mathcal{H}}_{0}H0​ is a very good approximation for a high-energy eigenstate of a parent Hamiltonian, close to H0{\mathcal{H}}_{0}H0​ but more disordered. This construction relies on computing the covariance matrix, easily achieved using density-matrix renormalization group for disordered Heisenberg chains up to L=256L=256L=256 sites.
Anderson insulators are noninteracting disordered systems which have localized single-particle eigenstates. The interacting analog of Anderson insulators are the many-body localized (MBL) phases. The spectrum of the many-body eigenstates of an Anderson insulator is efficiently represented as a set of product states over the single-particle modes. We show that product states over matrix product operators of small bond dimension is the corresponding efficient description of the spectrum of an MBL insulator. In this language all of the many-body eigenstates are encoded by matrix product states (i.e., density matrix renormalization group wave functions) consisting of only two sets of low bond dimension matrices per site: the Gi{G}_{i}Gi​ matrices corresponding to the local ground state on site iii and the Ei{E}_{i}Ei​ matrices corresponding to the local excited state. All 2n{2}^{n}2n eigenstates can be generated from all possible combinations of these sets of matrices.
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 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.
We study the properties of excited states in one-dimensional many-body localized (MBL) systems using a matrix product state algorithm. First, the method is tested for a large disordered noninteracting system, where for comparison we compute a quasiexact reference solution via a Monte Carlo sampling of the single-particle levels. Thereafter, we present extensive data obtained for large interacting systems of $L\ensuremath{\sim}100$ sites and large bond dimensions $\ensuremath{\chi}\ensuremath{\sim}1700$, which allows us to quantitatively analyze the scaling behavior of the entanglement SSS in the system. The MBL phase is characterized by a logarithmic growth $S(L)\ensuremath{\sim}log(L)$ over a large scale separating the regimes where volume and area laws hold. We check the validity of the eigenstate thermalization hypothesis. Our results are consistent with the existence of a mobility edge.