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arXiv:1701.02758 · Physical review. E

Numerical linked cluster expansions for quantum quenches in one-dimensional lattices

Krishnanand Mallayya, Marcos Rigol

Published Mar 7, 2017• 20 citations
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Abstract

We discuss the application of numerical linked cluster expansions (NLCEs) to study one dimensional lattice systems in thermal equilibrium and after quantum quenches from thermal equilibrium states. For the former, we calculate observables in the grand canonical ensemble, and for the latter we calculate observables in the diagonal ensemble. When converged, NLCEs provide results in the thermodynamic limit. We use two different NLCEs: a maximally connected expansion introduced in previous works and a site-based expansion. We compare the effectiveness of both NLCEs. The site-based NLCE is found to work best for systems in thermal equilibrium. However, in thermal equilibrium and after quantum quenches, the site-based NLCE can diverge when the maximally connected one converges. We relate this divergence to the exponentially large number of clusters in the site-based NLCE and the behavior of the weights of observables in those clusters. We discuss the effectiveness of resummations to cure the divergence. Our NLCE calculations are compared to exact diagonalization ones in lattices with periodic boundary conditions. NLCEs are found to outperform exact diagonalization in periodic systems for all quantities studied.

DOI
10.1103/physreve.95.033302

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We discuss the application of numerical linked cluster expansions (NLCEs) to study one dimensional lattice systems in thermal equilibrium and after quantum quenches from thermal equilibrium states. For the former, we calculate observables in the grand canonical ensemble, and for the latter we calculate observables in the diagonal ensemble. When converged, NLCEs provide results in the thermodynamic limit. We use two different NLCEs: a maximally connected expansion introduced in previous works and a site-based expansion. We compare the effectiveness of both NLCEs. The site-based NLCE is found to work best for systems in thermal equilibrium. However, in thermal equilibrium and after quantum quenches, the site-based NLCE can diverge when the maximally connected one converges. We relate this divergence to the exponentially large number of clusters in the site-based NLCE and the behavior of the weights of observables in those clusters. We discuss the effectiveness of resummations to cure the divergence. Our NLCE calculations are compared to exact diagonalization ones in lattices with periodic boundary conditions. NLCEs are found to outperform exact diagonalization in periodic systems for all quantities studied.
We implement numerical linked cluster expansions (NLCEs) to study dynamics of lattice systems following quantum quenches, and focus on a hard-core boson model in one-dimensional lattices. We find that, in the nonintegrable regime and within the accessible times, local observables exhibit exponential relaxation. We determine the relaxation rate as one departs from the integrable point and show that it scales quadratically with the strength of the integrability breaking perturbation. We compare the NLCE results with those from exact diagonalization calculations on finite chains with periodic boundary conditions, and show that NLCEs are far more accurate.
We use numerical linked cluster expansions (NLCEs) to study correlated lattice systems in one and two dimensions.For fermions in the honeycomb lattice, we study its finite-temperature properties and short-range spin correlations using NLCEs and determinantal quantum Monte Carlo (DQMC) simulations.For the homogeneous system, we analyze a number of thermodynamic quantities, including the entropy, the specific heat, uniform and staggered spin susceptibilities, short-range spin correlations, and the double occupancy at and away from half filling.Employing a local density approximation (LDA), we examine the viability of adiabatic cooling by increasing the interaction strength for homogeneous as well as for trapped systems.We also use NLCEs to study thermodynamic properties of the two-dimensional spin-1/2 Ising, XY, and Heisenberg models with bimodal random-bond disorder on the square and honeycomb lattices.In all cases, the nearest-neighbor coupling between the spins takes values ±J with equal probability.We obtain the disorder averaged (over all disorder configurations) energy, entropy, specific heat, and uniform magnetic susceptibility in each case.These results are compared with the corresponding ones in the clean models.Analytic expressions are obtained for low orders in the expansion of these thermodynamic quantities in inverse temperature.For many-body localization in disordered isolated systems, we study quantum quenches in the thermodynamic limit.By a quantum quench it is meant that the initial state is stationary with respect iii to an initial Hamiltonian, which is suddenly changed to a new (time-independent) Hamiltonian.The latter then drives the (unitary) dynamics of the system.We are interested in the time average of observables after relaxation following the quench, which can be computed under the so-called diagonal ensemble.
Disordered quantum systems undergoing a many-body localization (MBL) transition fail to reach thermal equilibrium under their own dynamics. Distinguishing between asymptotically localized or delocalized dynamics based on numerical results is, however, nontrivial due to finite-size effects. Numerical linked cluster expansions (NLCE) provide a means to tackle quantum systems directly in the thermodynamic limit but are challenging for models without translational invariance. Here we demonstrate that NLCE provides a powerful tool to explore MBL by simulating quench dynamics in disordered spin-1/21/21/2 two-leg ladders and Fermi-Hubbard chains. Combining NLCE with an efficient real-time evolution of pure states, we obtain converged results for the decay of the imbalance on long time scales and show that, especially for intermediate disorder below the putative MBL transition, NLCE outperforms direct simulations of finite systems with open or periodic boundaries. Furthermore, while spin is delocalized even in strongly disordered Hubbard chains with frozen charge, we unveil that an additional tilted potential leads to a drastic slowdown of the spin imbalance and nonergodic behavior on accessible times. Our work sheds light on MBL in systems beyond the well-studied disordered Heisenberg chain and emphasizes the usefulness of NLCE for this purpose.
We show that numerical linked cluster expansions (NLCEs) based on sufficiently large building blocks allow one to obtain accurate low-temperature results for the thermodynamic properties of spin lattice models with continuous disorder distributions. Specifically, we show that such results can be obtained computing the disorder averages in the NLCE clusters before calculating their weights. We provide a proof of concept using three different NLCEs based on L, square, and rectangle building blocks. We consider both classical (Ising) and quantum (Heisenberg) spin-1/2 models and show that convergence can be achieved down to temperatures that are up to two orders of magnitude lower than the relevant energy scale in the model. Additionally, we provide evidence that in one dimension one can obtain accurate results for observables such as the energy down to their ground-state values.
We develop a numerical linked cluster expansion (NLCE) method that can be applied directly to inhomogeneous systems, for example, Hamiltonians with disorder and dynamics initiated from inhomogeneous initial states. We demonstrate the method by calculating dynamics for single-spin expectations and spin correlations in two-dimensional spin models on a square lattice, starting from a checkerboard state. We show that NLCE can give moderate to dramatic improvement over an exact diagonalization of comparable computational cost and that the advantage in computational resources grows exponentially as the size of the clusters included grows. Although the method applies to any type of NLCE, our explicit benchmarks use the rectangle expansion. Besides showing the capability to treat inhomogeneous systems, these benchmarks demonstrate the rectangle expansion's utility out of equilibrium.
We study quantum quenches in the transverse-field Ising model defined on different lattice geometries such as chains, two- and three-leg ladders, and two-dimensional square lattices. Starting from fully polarized initial states, we consider the dynamics of the transverse and the longitudinal magnetization for quenches to weak, strong, and critical values of the transverse field. To this end, we rely on an efficient combination of numerical linked cluster expansions (NLCEs) and a forward propagation of pure states in real time. As a main result, we demonstrate that NLCEs comprising solely rectangular clusters provide a promising approach to study the real-time dynamics of two-dimensional quantum many-body systems directly in the thermodynamic limit. By comparing to existing data from the literature, we unveil that NLCEs yield converged results on time scales which are competitive to other state-of-the-art numerical methods.