Abstract
We discuss a method based on sampling minimally entangled typical thermal states (METTS) that can simulate finite temperature quantum systems with a computational cost comparable to the ground state density matrix renormalization group (DMRG). Detailed implementation of each step of the method is presented, along with efficient algorithms for working with matrix product states and matrix product operators. Furthermore, we explore how the properties of METTS can reveal characteristic order and excitations of systems and discuss why METTS form an efficient basis for sampling. Finally, we explore the extent to which the average entanglement of a METTS ensemble is minimal.
Highlights
According to the elementary principles of quantum statistical mechanics, the average of an observable is found by computing A = Tr[ρ A] 1 Z Tr[e−β H A]. (1)While performing such a calculation directly is usually intractable for quantum many body systems, one can still approximate the expectation value of A by strategies based on sampling
And 8, we show the classical product states (CPS) obtained after collapsing subsequent minimally entangled typical thermal states (METTS) in simulations of the Heisenberg and AKLT models and mark points at which the spins fail to follow the diluted Néel pattern that underlies the string order in the Haldane phase
We have found that both approaches are very costly because of both the time needed to construct the matrix product operators (MPOs) and to compute the expectation values
Summary
According to the elementary principles of quantum statistical mechanics, the average of an observable is found by computing. Does each METTS provide a good characterization of the thermal ensemble, but there exists a simple algorithm, the pure state algorithm, for sampling many of them efficiently This is achieved by a random walk through the set of METTS where the state is constructed from a CPS obtained by measuring, or collapsing, the previous one. In addition to generating METTS with the correct distribution, the pure state method has other important properties that make it advantageous for performing simulations It may be defined in a completely general way, allowing one to choose a specific implementation based on the problem of interest. Since the computational cost of METTS is comparable to that of ground state DMRG, we expect that it can treat comparably large ladders and two-dimensional (2D) systems This promises to be of great value for studying frustrated and fermionic models beyond one dimension—especially models with nontrivial phases at finite temperature. We will discuss how to produce and sample METTS in detail, and we will demonstrate that there is great flexibility within the basic algorithm that can allow us to optimize it for our system of interest
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