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arXiv:2310.15759 · Physical review. B./Physical review. B

Two-dimensional excitation information by a matrix product state method on infinite helices

Xing-Yu Zhang, Runze Chi, Yang Liu, Lei Wang

Published Feb 14, 2024• 4 citations
arXivHTMLPDF

Abstract

Understanding the excitation spectrum in two-dimensional quantum many-body systems has long been a formidable challenge. In this study, we propose an innovative approach by introducing an excitation Ansatz based on an infinite matrix product state (MPS) with a helix structure. The use of infinite uniform MPS states allows us to accurately extract key properties such as energy, degeneracy, spectrum weight, and scaling behavior of low-energy excited states simultaneously. To validate the effectiveness of our method, we apply it to the critical point of the transverse-field Ising model. The scaling exponent of the energy gap extracted aligns closely with conformal bootstrap results. Subsequently, we extend our method to the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on a square lattice. Our findings reveal that the degeneracy of lowest-energy excitations serves as a reliable metric for distinguishing different phases. The phase boundaries identified by our method are consistent with previous findings. The present method provides a promising avenue for studying the excitation spectrum of two-dimensional quantum many-body systems.

DOI
10.1103/physrevb.109.075129

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Understanding the excitation spectrum in two-dimensional quantum many-body systems has long been a formidable challenge. In this study, we propose an innovative approach by introducing an excitation Ansatz based on an infinite matrix product state (MPS) with a helix structure. The use of infinite uniform MPS states allows us to accurately extract key properties such as energy, degeneracy, spectrum weight, and scaling behavior of low-energy excited states simultaneously. To validate the effectiveness of our method, we apply it to the critical point of the transverse-field Ising model. The scaling exponent of the energy gap extracted aligns closely with conformal bootstrap results. Subsequently, we extend our method to the J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ Heisenberg model on a square lattice. Our findings reveal that the degeneracy of lowest-energy excitations serves as a reliable metric for distinguishing different phases. The phase boundaries identified by our method are consistent with previous findings. The present method provides a promising avenue for studying the excitation spectrum of two-dimensional quantum many-body systems.
We show that, in certain circumstances, exact excitation energies appear as locally site-independent (or flat) modes if one records the excitation spectrum of the effective Hamiltonian while sweeping through the lattice in the variational matrix-product-state formulation of the density matrix renormalization group, a remarkable property since the effective Hamiltonian is only constructed to target the ground state. Conversely, modes that are very flat over several consecutive iterations are systematically found to correspond to faithful excitations. We suggest to use this property to extract accurate information about excited states using the standard ground-state algorithm. The results are spectacular for critical systems, for which the low-energy conformal tower of states can be obtained very accurately at essentially no additional cost, as demonstrated by confirming the predictions of boundary conformal field theory for two simple minimal models: the transverse-field Ising model and the critical three-state Potts model. This approach is also very efficient to detect the quasidegenerate low-energy excitations in topological phases and to identify localized excitations in systems with impurities. Finally, using the variance of the Hamiltonian as a criterion, we assess the accuracy of the resulting matrix-product-state representations of the excited states.
We demonstrate an efficient method that allows for simultaneous determination of the ground state, low energy excitation properties and excitation gap in quantum many body systems. To this aim we first use the \textit{ab-initio} optimization principle of tensor networks (TN), to show that the infinite density matrix renormalization group (iDMRG) in the real space is associated in a natural manner to the infinite time-evolving block decimation (iTEBD) implemented on a continuous matrix product state (MPS), and defined in imaginary time. We illustrate this association showing that the (imaginary) time matrix product state (MPS) in iTEBD reproduces accurately the properties of the two-dimensional (2D) classical Ising model, verifying in this way that the time MPS corresponds to a well-defined physical state. We apply then our scheme to the one-dimensional (1D) quantum Ising chain, where the time MPS is defined in continuous imaginary time. It is found that the time MPS at or close to the critical point is always maximally entangled, but gives essentially different correlations than those obtained from iDMRG; we argue that these correlations describe the dynamic correlations for the ground state.
Based on the matrix product state (MPS) formalism, we introduce an ansatz for capturing excited states in finite systems with open boundary conditions, providing a very efficient method for computing, e.g., the spectral gap of quantum spin chains. This method can be straightforwardly implemented on top of an existing density-matrix renormalization group or MPS ground-state code. Although this approach is built on open-boundary MPSs, we also apply it to systems with periodic boundary conditions. Despite the explicit breaking of translation symmetry by the MPS representation, we show that momentum emerges as a good quantum number and can be exploited for labeling excitations on top of MPS ground states. We apply our method to the critical Ising chain on a ring and the classical Potts model on a cylinder. Finally, we apply the same idea to compute excitation spectra for 2D quantum systems on infinite cylinders. Again, despite the explicit breaking of translation symmetry in the periodic direction, we recover momentum as a good quantum number for labeling excitations. We apply this method to the 2D transverse-field Ising model and the half-filled Hubbard model; for the latter, we obtain accurate results for, e.g., the hole dispersion for cylinder circumferences up to eight sites.
Numerical methods based on matrix product states (MPSs) are currently the standard for calculating the ground-state properties of (quasi-)one-dimensional quantum many-body systems. While the properties of the low-lying excitations in such systems are often studied in this MPS framework through by means of time-evolution simulations, we can also look at their by directly calculating eigenstates corresponding to these excitations. The so-called MPS excitation ansatz is a powerful method for finding such eigenstates with a single-particle character in the thermodynamic limit. Although this excitation ansatz has been used quite extensively, a general method for calculating expectation values for these states is lacking in the literature: We aim to fill this gap by presenting a recursive algorithm to calculate arbitrary observables expressed as matrix product operators. This method concisely encapsulates existing methods for—as well as extensions to—the excitation ansatz, such as excitations with a larger spatial support and multiparticle excitations, and is robust enough to handle further innovations. We demonstrate the versatility of our method by studying the low-lying excitations in the spin-1 Heisenberg chain and the one-dimensional Hubbard model, looking at how the excitations converge in the former, while in the latter, we present a refined method of targeting single-particle excitations inside a continuum by minimizing the energy rather than the energy itself. We hope that this technique will foster further advancements with the excitation ansatz.
Determining the low-energy eigenspectra of quantum many-body systems is a long-standing challenge in physics. In this paper, we solve this problem by introducing two algorithms to determine low-energy eigenstates based on a compact matrix product state (MPS) representation of multiple targeted eigenstates. The first algorithm utilizes a canonicalization approach that takes advantage of the imaginary-time evolution of multitarget MPSs, offering faster convergence and ease of implementation. The second algorithm employs a variational approach that optimizes local tensors on the Grassmann manifold, capable of achieving higher accuracy. These algorithms can be used independently or combined to enhance convergence speed and accuracy. We apply them to the transverse-field Ising model and demonstrate that the calculated low-energy eigenspectra agree remarkably well with the exact solution. Moreover, the eigenenergies exhibit uniform convergence in gapped phases, suggesting that the low-energy excited eigenstates have nearly the same level of accuracy as the ground state. Our results highlight the accuracy and versatility of multitarget MPS-based algorithms for determining low-energy eigenspectra and their potential applications in quantum many-body physics.
(Please refer to arXiv:1810.08050, which has completely different aims but contains all the main contents of this paper) In this work, we propose to access the information of criticality and excitations of one-dimensional quantum systems by a matrix product state (MPS) defined in the (imaginary) time direction. This state, dubbed as time MPS (tMPS), is a boundary state of tensor network (TN) that represents the ground-state simulation after Trotter-Suzuki decomposition. We show that the tMPS exhibits the structure of the continuous MPS originally proposed for the field theories. The information of excitations, e.g., dynamic correlation length and energy gap, can be accurately calculated from the tMPS. The non-universal renormalization of velocity of the excitations is given by a ratio between the correlations of the ground state and the tMPS. When the system is at the quantum critical point, the tMPS is found to show the logarithmic scaling law, where the scaling coefficient gives the central charge that characterizes the criticality of the theory. Our work also implies that the spectra of the transfer matrices defined by the ground state and tMPS provide information of the low-lying masses of the theory in the continuous limit. This could help to understand the role of the finiteness of the bond dimension as an infra red regulator. Furthermore, we show from the perspective of methodology that, the tMPS emerges from a generalized TN ab-initio optimization principle scheme, which unifies the infinite density matrix renormalization group and the infinite time-evolving block decimation algorithms.