Symmetric minimally entangled typical thermal states for canonical and grand-canonical ensembles

Binder, Moritz; Barthel, Thomas

doi:10.1103/physrevb.95.195148

Cited by 25 publications

(16 citation statements)

References 34 publications

(60 reference statements)

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“…The explicit implementation of non-Abelian symmetries has been regarded as a standard technique in ground state DMRG simulations (T = 0) [43], which has many important applications including exploring quantum spin liquids in frustrated quantum magnets [44,45], and is also shown to be useful in METTS-type thermal simulations [46,47]. However, the implementations of non-Abelian symmetries in MPO for finite-temperature simulations are still absent.…”

Section: A Symmetric Matrix Product Operatormentioning

confidence: 99%

Exponential Thermal Tensor Network Approach for Quantum Lattice Models

Chen

et al. 2018

Phys. Rev. X

101

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We speed up thermal simulations of quantum many-body systems in both one-(1D) and two-dimensional (2D) models in an exponential way by iteratively projecting the thermal density matrixρ = e −βĤ onto itself. We refer to this scheme of doubling β in each step of the imaginary time evolution as the exponential tensor renormalization group (XTRG). This approach is in stark contrast to conventional Trotter-Suzuki-type methods which evolveρ on a linear quasi-continuous grid in inverse temperature β ≡ 1/T . As an aside, the large steps in XTRG allow one to swiftly jump across finite-temperature phase transitions, i.e., without the need to resolve each singularly expensive phase transition point right away, e.g., when interested in low-energy behavior. A fine temperature resolution can be obtained, nevertheless, by using interleaved temperature grids. In general, XTRG can reach low temperatures exponentially fast, and thus not only saves computational time but also merits better accuracy due to significantly fewer truncation steps. For similar reasons, we also find that the series expansion thermal tensor network (SETTN) approach benefits in both efficiency and precision, from the logarithmic temperature scale setup. We work in an (effective) 1D setting exploiting matrix product operators (MPOs) which allows us to fully and uniquely implement non-Abelian and Abelian symmetries to greatly enhance numerical performance. We use our XTRG machinery to explore the thermal properties of Heisenberg models on 1D chains and 2D square and triangular lattices down to low temperatures approaching ground state properties. The entanglement properties, as well as the renormalization group flow of entanglement spectra in MPOs, are discussed, where logarithmic entropies (approximately ln β) are shown in both spin chains and square lattice models with gapless towers of states. We also reveal that XTRG can be employed to accurately simulate the Heisenberg XXZ model on the square lattice which undergoes a thermal phase transition. We determine its critical temperature based on thermal physical observables, as well as entanglement measures. Overall, we demonstrate that XTRG provides an elegant, versatile, and highly competitive approach to explore thermal properties, including finite temperature thermal phase transitions as well as the different ordering tendencies at various temperature scales for frustrated systems.

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Section: A Symmetric Matrix Product Operatormentioning

confidence: 99%

Exponential Thermal Tensor Network Approach for Quantum Lattice Models

Chen

et al. 2018

Phys. Rev. X

101

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“…Secondly, because METTS updates remain in a fixed quantum-number sector corresponding to the symmetries of the Hamiltonian, one cannot straightforwardly use quantum numbers to simulate the grand canonical ensemble efficiently [12,13]. On the contrary, using purification, the maximally entangled state |S allows particle numbers to fluctuate on the physical sites while keeping the total number of particles conserved in the total space (physical and auxiliary).…”

Section: Hybrid Algorithmmentioning

confidence: 99%

“…Apart from the imaginary time evolution itself, the main challenge in obtaining the Matsubara Green function is computing finite temperature versus ground state properties. Within tensor network methods, finite temperature systems can be accessed in two ways: (i) the minimally entangled typical thermal states (METTS) algorithm [11][12][13][14][15] uses a sampling technique over tensor network states to produce the thermal average; (ii) the purification technique [16][17][18][19] rewrites the thermal trace as a average in a pure state in a larger space, obtained by imaginary time evolution.…”

Section: Introductionmentioning

confidence: 99%

Minimally Entangled Typical Thermal States Algorithms for Finite Temperature Matsubara Green Functions

Bauernfeind,

Cao,

Stoudenmire

et al. 2021

Preprint

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We extend finite-temperature tensor network methods to compute Matsubara imaginary-time correlation functions, building on the minimally entangled typical thermal states (METTS) and purification algorithms. While imaginary-time correlation functions are straightforward to formulate with these methods, care is needed to avoid convergence issues that would result from naive estimators. As a benchmark, we study the single-band Anderson impurity model, even though the algorithm is quite general and applies to lattice models. The special structure of the impurity model benchmark system and our choice of basis enable techniques such as reuse of high-probability METTS for increasing algorithm efficiency. The results are competitive with state-of-the-art continuous time Monte Carlo. We discuss the behavior of computation time and error as a function of the number of purified sites in the Hamiltonian.

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“…For efficient sampling in the vast Hilbert space, the sampling approach is often combined with the importance sampling implemented by the Markov chain. However, there is a drawback in the use of the Markov chain that it can bring the autocorrelation problem [17], and thus some treatments to circumvent the problem are required in such approaches [15,16,[18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Matrix product state approach for a quantum system at finite temperatures using random phases and Trotter gates

Goto,

Kaneko,

Danshita

2021

Preprint

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We develop a numerical method based on matrix product states for simulating quantum manybody systems at finite temperatures without importance sampling and evaluate its performance in spin 1/2 systems. Our method is an extension of the random phase product state (RPPS) approach introduced recently [T. Iitaka, arXiv:2006.14459]. We show that the original RPPS approach often gives unphysical physical values for thermodynamic quantities even in the Heisenberg chain. We find that by adding the operation of Trotter gates to the RPPS, the sampling efficiency of the approach significantly increases and its results are consistent with those of the purification approach. We also apply our method to a frustrated spin 1/2 system to exemplify that it can simulate a system in which the purification approach fails.

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Symmetric minimally entangled typical thermal states for canonical and grand-canonical ensembles

Cited by 25 publications

References 34 publications

Exponential Thermal Tensor Network Approach for Quantum Lattice Models

Exponential Thermal Tensor Network Approach for Quantum Lattice Models

Minimally Entangled Typical Thermal States Algorithms for Finite Temperature Matsubara Green Functions

Matrix product state approach for a quantum system at finite temperatures using random phases and Trotter gates

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