Minimally entangled typical thermal state algorithms

Stoudenmire, E. Miles; White, Steven R.

doi:10.1088/1367-2630/12/5/055026

Cited by 274 publications

(286 citation statements)

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“…The latter is of significant interest in view of experimental realizations. Apart from a recent numerical study for local quenches [37], the spreading of signals in quenches from thermal states is basically unexplored.Our numerical simulations are based on a quench extension of a recently proposed algorithm utilizing an optimized wave function ensemble called Minimally Entangled Typical Thermal States (METTS) [38,39] product state (MPS) framework. We come back to the description of the algorithm and a discussion of its performance towards the end of this paper.…”

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confidence: 99%

“…[38,39] introduced a stochastic method in which the expectation value of a thermal density matrix is replaced by an average over an ensemble of wave functions, {|φ i }, that i) can be efficiently sampled (importance sampling) using Markov chains and ii) only hosts the minimal (small) amount of entanglement required at that temperature, thus allowing for an efficient representation in terms of MPS. This ensemble was therefore called METTS (minimally entangled typical thermal states) [38,39].We show that the METTS method, thus far only applied to static equilibrium problems, can be easily extended to study real time evolution by realizing that the expectation value of some real-time propagated operatorÂ(t) can be written aswhere the last term denotes an average over the time-evolved METTS ensemble [68]. We employ the numerical scheme illustrated in Fig.…”

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confidence: 99%

“…Our numerical simulations are based on a quench extension of a recently proposed algorithm utilizing an optimized wave function ensemble called Minimally Entangled Typical Thermal States (METTS) [38,39] product state (MPS) framework. We come back to the description of the algorithm and a discussion of its performance towards the end of this paper.…”

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confidence: 99%

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“Light-Cone” Dynamics After Quantum Quenches in Spin Chains

2014

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Signal propagation in the non equilibirum evolution after quantum quenches has recently attracted much experimental and theoretical interest. A key question arising in this context is what principles, and which of the properties of the quench, determine the characteristic propagation velocity. Here we investigate such issues for a class of quench protocols in one of the central paradigms of interacting many-particle quantum systems, the spin-1/2 Heisenberg XXZ chain. We consider quenches from a variety of initial thermal density matrices to the same final Hamiltonian using matrix product state methods. The spreading velocities are observed to vary substantially with the initial density matrix. However, we achieve a striking data collapse when the spreading velocity is considered to be a function of the excess energy. Using the fact that the XXZ chain is integrable, we present an explanation of the observed velocities in terms of "excitations" in an appropriately defined generalized Gibbs ensemble.The last few years have witnessed a number of significant advances in understanding the nonequilibirum dynamics in isolated quantum systems. Much of this activity has focussed on fundamental concepts such as thermalization [1][2][3][4][5] or the roles played by dimensionality and conservation laws [6][7][8][9][10][11][12][13][14][15][16].Another key issue concerns the spreading of correlations out of equilibrium, and in particular the "light-cone" effect after global quantum quenches. The most commonly studied protocol in this context is to prepare the system in the ground state of a given Hamiltonian, and to then suddenly change a system parameter such as a magnetic field or interaction strength. At subsequent times the spreading of correlations can then be analyzed by considering the time-dependence of two-point functions of local operators separated by a fixed distance. As shown by Lieb and Robinson [17,18], the velocity of information transfer in quantum systems is bounded. This gives rise to a causal structure in commutators of local operators at different times, although Schrödinger's equation, unlike relativistic theories, has no built-in speed limit. Recently, the Lieb-Robinson bounds have been refined [19][20][21] and extended to mixed state dynamics in open quantum systems [21,22] as well as creation of topological quantum order [23].A striking consequence of the Lieb-Robinson bound is that the equal-time correlators after a quantum quench feature a "light-cone" effect [23], which is most pronounced for quenches to conformal field theories from initial density matrices with a finite correlation length [24]: connected correlations are initially absent, but exhibit a marked increase after a time t 0 = x/2v. This observation is explained by noting [25,26] that entangled pairs of quasi-particles initially located half-way between the two points of measurement, propagate with the speed of light v and hence induce correlations after a time t 0 . These predictions have been verified numerically in several systems, se...

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“Light-Cone” Dynamics After Quantum Quenches in Spin Chains

2014

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“…The implementation of arithmetic operations with MPOs is well-known already [6,17] and is entirely independent of the specific form of the operands. In particular, the implementation can handle single-site operators as constructed in the previous section and MPOs resulting from earlier arithmetic operations on equal footing.…”

Section: Arithmetic Operations With Matrix Product Operatorsmentioning

confidence: 99%

Generic construction of efficient matrix product operators

2017

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Matrix product operators (MPOs) are at the heart of the second-generation density matrix renormalization group (DMRG) algorithm formulated in matrix product state language. We first summarize the widely known facts on MPO arithmetic and representations of single-site operators. Second, we introduce three compression methods (rescaled SVD, deparallelization, and delinearization) for MPOs and show that it is possible to construct efficient representations of arbitrary operators using MPO arithmetic and compression. As examples, we construct powers of a short-ranged spin-chain Hamiltonian, a complicated Hamiltonian of a two-dimensional system and, as proof of principle, the long-range four-body Hamiltonian from quantum chemistry.

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“…Let us note that recently a new method has been developed to treat finite temperatures, 98,99 which is very promising to reach even lower temperatures.…”

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Statics and Dynamics of Weakly Coupled Antiferromagnetic Spin-1/2 Ladders in a Magnetic Field

Bouillot

2013

Springer Theses

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We investigate weakly coupled spin-1/2 ladders in a magnetic field. The work is motivated by recent experiments on the compound (C 5 H 12 N) 2 CuBr 4 (BPCB). We use a combination of numerical and analytical methods, in particular, the density-matrix renormalization group (DMRG) technique, to explore the phase diagram and the excitation spectra of such a system. We give detailed results on the temperature dependence of the magnetization and the specific heat, and the magnetic-field dependence of the nuclear-magnetic-resonance relaxation rate of single ladders. For coupled ladders, treating the weak interladder coupling within a mean-field or quantum Monte Carlo approach, we compute the transition temperature of triplet condensation and its corresponding antiferromagnetic order parameter. Existing experimental measurements are discussed and compared to our theoretical results. Furthermore, we compute, using time-dependent DMRG, the dynamical correlations of a single spin ladder. Our results allow to describe directly the inelastic neutron scattering cross section up to high energies. We focus on the evolution of the spectra with the magnetic field and compare their behavior for different couplings. The characteristic features of the spectra are interpreted using different analytical approaches such as the mapping onto a spin chain, a Luttinger liquid or onto a t-J model. For values of parameters for which such measurements exist, we compare our results to inelastic neutron scattering experiments on the compound BPCB and find excellent agreement. We make additional predictions for the high-energy part of the spectrum that are potentially testable in future experiments.

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Minimally entangled typical thermal state algorithms

Cited by 274 publications

References 17 publications

“Light-Cone” Dynamics After Quantum Quenches in Spin Chains

“Light-Cone” Dynamics After Quantum Quenches in Spin Chains

Generic construction of efficient matrix product operators

Statics and Dynamics of Weakly Coupled Antiferromagnetic Spin-1/2 Ladders in a Magnetic Field

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