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arXiv:2201.07314 · Physical review. B./Physical review. B

Momentum-resolved time evolution with matrix product states

Maarten Van Damme, Laurens Vanderstraeten

Published May 23, 2022• 7 citations
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Abstract

We introduce a method based on matrix product states (MPS) for computing spectral functions of (quasi-) one-dimensional spin chains, working directly in momentum space in the thermodynamic limit. We simulate the time evolution after applying a momentum operator to an MPS ground state by working with the momentum superposition of a window MPS. We show explicitly for the spin-1 Heisenberg chain that the growth of entanglement is smaller in momentum space, even inside a two-particle continuum, such that we can attain very accurate spectral functions with relatively small bond dimension. We apply our method to compute spectral lineshapes of the gapless XXZ chain and the square-lattice ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on a six-leg cylinder.

DOI
10.1103/physrevb.105.205130

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We introduce a method based on matrix product states (MPS) for computing spectral functions of (quasi-) one-dimensional spin chains, working directly in momentum space in the thermodynamic limit. We simulate the time evolution after applying a momentum operator to an MPS ground state by working with the momentum superposition of a window MPS. We show explicitly for the spin-1 Heisenberg chain that the growth of entanglement is smaller in momentum space, even inside a two-particle continuum, such that we can attain very accurate spectral functions with relatively small bond dimension. We apply our method to compute spectral lineshapes of the gapless XXZ chain and the square-lattice J1\ensuremath−J2{J}_{1}\ensuremath{-}{J}_{2}J1​\ensuremath−J2​ Heisenberg model on a six-leg cylinder.
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.
A variational ansatz for momentum eigenstates of translation-invariant quantum spin chains is formulated. The matrix product state ansatz works directly in the thermodynamic limit and allows for an efficient implementation (cubic scaling in the bond dimension) of the variational principle. Unlike previous approaches, the ansatz includes topologically nontrivial states (kinks, domain walls) for systems with symmetry breaking. The method is benchmarked using the spin-\textonehalf{} XXZXXZXXZ antiferromagnet and the spin-1 Heisenberg antiferromagnet, and we obtain surprisingly accurate results.
We propose a method to simulate the real time evolution of one-dimensional quantum many-body systems at finite temperature by expressing both the density matrices and the observables as matrix product states. This allows the calculation of expectation values and correlation functions as scalar products in operator space. The simulations of density matrices in inverse temperature and the local operators in the Heisenberg picture are independent and result in a grid of expectation values for all intermediate temperatures and times. Simulations can be performed using real arithmetics with only polynomial growth of computational resources in inverse temperature and time for integrable systems. The method is illustrated for the XXZ model and the single impurity Anderson model.
We propose a method to simulate the real time evolution of one-dimensional quantum many-body systems at finite temperature by expressing both the density matrices and the observables as matrix product states. This allows the calculation of expectation values and correlation functions as scalar products in operator space. The simulations of density matrices in inverse temperature and the local operators in the Heisenberg picture are independent and result in a grid of expectation values for all intermediate temperatures and times. Simulations can be performed using real arithmetics with only polynomial growth of computational resources in inverse temperature and time for integrable systems. The method is illustrated for the XXZ model and the single impurity Anderson model.
We present an algorithm for studying quantum systems at finite temperature using continuous matrix product operator representation. The approach handles both short-range and long-range interactions in the thermodynamic limit without incurring any time discretization error. Moreover, the approach provides direct access to physical observables including the specific heat, local susceptibility, and local spectral functions. After verifying the method using the prototypical quantum XXZ chains, we apply it to quantum Ising models with power-law decaying interactions and on the infinite cylinder, respectively. The approach offers predictions that are relevant to experiments in quantum simulators and the nuclear magnetic resonance spin-lattice relaxation rate.