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arXiv:2202.00244 · Physical review. B./Physical review. B

Efficient simulation of quantum many-body thermodynamics by tailoring a zero-temperature tensor network

Ding-Zu Wang, Guo‐Feng Zhang, Maciej Lewenstein, Shi-Ju Ran

Published Apr 27, 2022• 1 citation
arXivHTMLPDF

Abstract

Various successful approaches using numerical annealing and renormalization groups have been conceived to study the thermodynamics of strongly correlated systems where perturbation or expansion theories fail to work. As the process of lowering the temperatures is usually involved in different manners, these approaches in general become much less efficient or accurate at the low temperatures. In this work, we propose to access the finite-temperature properties from the tensor network (TN) representing the zero-temperature partition function. We propose to ``scissor'' a finite part from such an infinite-size TN and ``stitch'' it to possess the periodic boundary condition along the imaginary-time direction. We dub this approach ``TN tailoring.'' Exceptional accuracy is achieved with a fine-tuning process, surpassing the previous methods including the linearized tensor renormalization group [W. Li, S.-J. Ran, S.-S. Gong, Y. Zhao, B. Xi, F. Ye, and G. Su, Phys. Rev. Lett. 106, 127202 (2011)], the continuous matrix product operator [W. Tang, H.-H. Tu, and L. Wang, Phys. Rev. Lett. 125, 170604 (2020)], etc. High efficiency is demonstrated, where the time cost is nearly independent of the target temperature, including the extremely low temperatures. The proposed idea can be extended to higher-dimensional systems of bosons and fermions.

DOI
10.1103/physrevb.105.155155

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Various successful approaches using numerical annealing and renormalization groups have been conceived to study the thermodynamics of strongly correlated systems where perturbation or expansion theories fail to work. As the process of lowering the temperatures is usually involved in different manners, these approaches in general become much less efficient or accurate at the low temperatures. In this work, we propose to access the finite-temperature properties from the tensor network (TN) representing the zero-temperature partition function. We propose to scissor'' a finite part from such an infinite-size TN and stitch'' it to possess the periodic boundary condition along the imaginary-time direction. We dub this approach ``TN tailoring.'' Exceptional accuracy is achieved with a fine-tuning process, surpassing the previous methods including the linearized tensor renormalization group [W. Li, S.-J. Ran, S.-S. Gong, Y. Zhao, B. Xi, F. Ye, and G. Su, Phys. Rev. Lett. 106, 127202 (2011)], the continuous matrix product operator [W. Tang, H.-H. Tu, and L. Wang, Phys. Rev. Lett. 125, 170604 (2020)], etc. High efficiency is demonstrated, where the time cost is nearly independent of the target temperature, including the extremely low temperatures. The proposed idea can be extended to higher-dimensional systems of bosons and fermions.
Numerical simulations of strongly correlated fermions at finite temperature are essential for studying high-temperature superconductivity and other quantum many-body phenomena. The recently developed tangent-space tensor renormalization group (tanTRG) provides an efficient and accurate framework by representing thermal density operators as matrix product operators. However, the particle number generally varies during the cooling process. The conventional strategy of fine-tuning chemical potentials to reach a target filling is computationally demanding. Here we propose a fixed-NNN tanTRG algorithm that stabilizes the average particle number by adaptively tuning the chemical potential within the imaginary-time evolution. We benchmark its accuracy on exactly solvable free fermions, and further apply it to the square-lattice Hubbard model. For hole-doped cases, we study the temperature evolution of charge and spin correlations, identifying several characteristic temperature scales for stripe formation. Our results establish fixed-NNN tanTRG as an efficient and reliable tool for finite-temperature studies of correlated fermion systems.
We introduce a tensor network method for approximating thermal equilibrium states of quantum many-body systems at low temperatures. Whereas the usual approach starts from infinite temperature and evolves the state in imaginary time (toward lower temperature), our ansatz is constructed from the zero-temperature limit, the ground state, which can be found with a standard tensor network approach. Motivated by properties of the ground state for conformal field theories, our ansatz is especially well suited near criticality. Moreover, it allows an efficient computation of thermodynamic quantities and entanglement properties. We demonstrate the performance of our approach with a tree tensor network ansatz, although it can be extended to other tensor networks, and present results illustrating its effectiveness in capturing the finite-temperature properties in one- and two-dimensional scenarios. In particular, in the critical one-dimensional case we show how the ansatz reproduces the finite temperature scaling of entanglement in a conformal field theory.
Thermal tensor networks constitute an efficient and versatile representation for quantum lattice models at finite temperatures. By Trotter-Suzuki decomposition, one obtains a DDD+1 dimensional TTN for the DDD-dimensional quantum system and then employs efficient renormalizaton group (RG) contractions to obtain the thermodynamic properties with high precision. The linearized tensor renormalization group (LTRG) method, which can be used to contract TTN efficiently and calculate the thermodynamics, is briefly reviewed and then generalized to a bilayer form. We dub this bilayer algorithm as LTRG++ and explore its performance in both finite- and infinite-size systems, finding the numerical accuracy significantly improved compared to single-layer algorithm. Moreover, we show that the LTRG++ algorithm in an infinite-size system is in essence equivalent to transfer-matrix renormalization group method, while reformulated in a tensor network language. As an application of LTRG++, we simulate an extended fermionic Hubbard model numerically, where the phase separation phenomenon, ground-state phase diagram, as well as quantum criticality-enhanced magnetocaloric effects, are investigated.
Constructing quantum Hamiltonians for simulating and controlling the exotic physics of many-body systems belongs to the most important topics of condensed matter physics and quantum technologies. The main challenge that hinders the future investigations is the extremely high complexity for either their numerical simulations or experimental realizations. In this work, we propose the temperature-adaptive entanglement simulator (TAES) that mimics and tunes the thermodynamics of the one-dimensional (1D) many-body system by embedding a small-size model in an entanglement bath. The entanglement bath is described by the interactions located at the boundaries of the small-size model, whose coupling constants are optimized by means of differentiable tensor network at target temperatures. With the benchmark on 1D spin chains, TAES surpasses the state-of-the-art accuracy compared with the existing finite-temperature approaches such as linearized and differential tensor renormalization group algorithms. By tuning the couplings of the entanglement bath with the temperature fixed, the bulk entropy exhibits similar behavior compared to that obtained by tuning the temperature. Our work provides novel opportunities of engineering the distribution of fluctuations and mimicking the non-equilibrium phenomena in a uniform temperature within the canonical ensemble framework using the optimized boundary interactions.
We speed up thermal simulations of quantum many-body systems in both one-(1D) and twodimensional (2D) models in an exponential way by iteratively projecting the thermal density matrix ρ ¼ e -β Ĥ 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 evolve ρ on a linear quasicontinuous 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 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 squarelattice 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.
Tensor network (TN), a young mathematical tool of high vitality and great potential, has been undergoing extremely rapid developments in the last two decades, gaining tremendous success in condensed matter physics, atomic physics, quantum information science, statistical physics, and so on. In this lecture notes, we focus on the contraction algorithms of TN as well as some of the applications to the simulations of quantum many-body systems. Starting from basic concepts and definitions, we first explain the relations between TN and physical problems, including the TN representations of classical partition functions, quantum many-body states (by matrix product state, tree TN, and projected entangled pair state), time evolution simulations, etc. These problems, which are challenging to solve, can be transformed to TN contraction problems. We present then several paradigm algorithms based on the ideas of the numerical renormalization group and/or boundary states, including density matrix renormalization group, time-evolving block decimation, coarse-graining/corner tensor renormalization group, and several distinguished variational algorithms. Finally, we revisit the TN approaches from the perspective of multi-linear algebra (also known as tensor algebra or tensor decompositions) and quantum simulation. Despite the apparent differences in the ideas and strategies of different TN algorithms, we aim at revealing the underlying relations and resemblances in order to present a systematic picture to understand the TN contraction approaches.