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Continuous-Time Thermal Tensor Network Approach for Quantum Lattice Models

Bin-Bin Chen, Yun-Jing Liu, Ziyu Chen, Wei Li

Published Sep 5, 2016• 0 citations
OpenAlex

Abstract

In this Letter we propose a continuous-time thermal tensor network (TTN) approach for efficient simulations of quantum lattice models. The continuous-time TTN method is based on numerically exact Taylor series expansions of equilibrium density operators $e^{-\beta H}$ (with $H$ the total Hamiltonian and $\beta$ the imaginary time), as well as unitary time-evolution operators $e^{-iHt}$, and is thus Trotter-error free. We discover, through simulating a typical XXZ spin chain, that not only the Hamiltonian $H$, but also its powers $H^n$, can be efficiently expressed as matrix product operators, which enables us to calculate with high precision the equilibrium and dynamical properties of quantum lattice models at finite temperatures. Our series-expansion TTN method provides an alternative to conventional Trotter-Suzuki renormalization group (RG) approaches, and achieves an unprecedented standard of thermal RG simulations in terms of accuracy and flexibility.

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Series-expansion thermal tensor network approach for quantum lattice models

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In this Letter we propose a continuous-time thermal tensor network (TTN) approach for efficient simulations of quantum lattice models. The continuous-time TTN method is based on numerically exact Taylor series expansions of equilibrium density operators e−βHe^{-\beta H}e−βH (with HHH the total Hamiltonian and β\betaβ the imaginary time), as well as unitary time-evolution operators e−iHte^{-iHt}e−iHt, and is thus Trotter-error free. We discover, through simulating a typical XXZ spin chain, that not only the Hamiltonian HHH, but also its powers HnH^nHn, can be efficiently expressed as matrix product operators, which enables us to calculate with high precision the equilibrium and dynamical properties of quantum lattice models at finite temperatures. Our series-expansion TTN method provides an alternative to conventional Trotter-Suzuki renormalization group (RG) approaches, and achieves an unprecedented standard of thermal RG simulations in terms of accuracy and flexibility.
We propose a series-expansion thermal tensor network (SETTN) approach for efficient simulations of quantum lattice models. This continuous-time SETTN method is based on the numerically exact Taylor series expansion of the equilibrium density operator ${e}^{\ensuremath{-}\ensuremath{\beta}H}$ (with HHH the total Hamiltonian and $\ensuremath{\beta}$ the imaginary time), and is thus Trotter-error free. We discover, through simulating XXZ spin chain and square-lattice quantum Ising models, that not only the Hamiltonian HHH, but also its powers Hn{H}^{n}Hn, can be efficiently expressed as matrix product operators, which enables us to calculate with high precision the equilibrium and dynamical properties of quantum lattice models at finite temperatures. Our SETTN method provides an alternative to conventional Trotter-Suzuki renormalization-group (RG) approaches, and achieves a very high standard of thermal RG simulations in terms of accuracy and flexibility.
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.
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.
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.
In this work, we benchmark the well-controlled and numerically accurate exponential thermal tensor renormalization group (XTRG) in the simulation of interacting spin models in two dimensions. Finite temperature introduces a finite thermal correlation length $\ensuremath{\xi}$, such that for system sizes $L\ensuremath{\gg}\ensuremath{\xi}$ finite-size calculations actually simulate the thermodynamic limit. In this paper, we focus on the square lattice Heisenberg antiferromagnet (SLH) and quantum Ising models (QIM) on open and cylindrical geometries up to width W=10W=10W=10. We explore various one-dimensional mapping paths in the matrix product operator (MPO) representation, whose performance is clearly shown to be geometry dependent. We benchmark against quantum Monte Carlo (QMC) data, yet also the series-expansion thermal tensor network results. Thermal properties including the internal energy, specific heat, and spin structure factors, etc. are computed with high precision, obtaining excellent agreement with QMC results. XTRG also allows us to reach remarkably low temperatures. For SLH, we obtain an energy per site ${u}_{g}^{*}\ensuremath{\simeq}\ensuremath{-}0.6694(4)$ and a spontaneous magnetization ${m}_{S}^{*}\ensuremath{\simeq}0.30(1)$ already consistent with the ground-state properties, which is obtained from extrapolated low-TTT thermal data on $W\ensuremath{\le}8$ cylinders and $W\ensuremath{\le}10$ open strips, respectively. We extract an exponential divergence versus TTT of the structure factor S(M)S(M)S(M), as well as the correlation length $\ensuremath{\xi}$, at the ordering wave vector $M=(\ensuremath{\pi},\ensuremath{\pi})$, which represents the renormalized classical behavior and can be observed over a narrow but appreciable temperature window, by analyzing the finite-size data by XTRG simulations. For the QIM with a finite-temperature phase transition, we employ several thermal quantities, including the specific heat, Binder ratio, as well as the MPO entanglement to determine the critical temperature Tc{T}_{c}Tc​.
Low-dimensional quantum magnets, particularly those with strong spin frustration, are characterized by their notable spin fluctuations. Nuclear magnetic resonance (NMR) serves as a sensitive probe of low-energy fluctuations that offers valuable insight into rich magnetic phases and emergent phenomena in quantum magnets. Although experimentally accessible, the numerical simulation of NMR relaxation rates, specifically the spin-lattice relaxation rate 1/T11/T_11/T1​, remains a significant challenge. Analytical continuation based on Monte Carlo calculations are hampered by the notorious negative sign for frustrated systems, and the real-time simulations incur significant costs to capture low-energy fluctuations. Here we propose computing the relaxation rate using thermal tensor networks (TTNs), which provides a streamlined approach by calculating its imaginary-time proxy. We showcase the accuracy and versatility of our methodology by applying it to one-dimensional spin chains and two-dimensional lattices, where we find that the critical exponents ηηη and zνzνzν can be extracted from the low-temperature scalings of the simulated 1/T11/T_11/T1​ near quantum critical points. Our results also provide insights into the low-dimensional and frustrated magnetic materials, elucidating universal scaling behaviors in the Ising chain compound CoNb2_22​O6_66​ and revealing the renormalized classical behaviors in the triangular-lattice antiferromagnet Ba8_88​CoNb6_66​O24_{24}24​. We apply the approach to effective model of the family of frustrated magnets AYbCh2_22​ (A = Na, K, Cs, and Ch = O, S, Se), and find dramatic changes from spin ordered to the proposed quantum spin liquid phase. Overall, with high reliability and accuracy, the TTN methodology offers a systematic strategy for studying the intricate dynamics observed across a broad spectrum of quantum magnets and related fields.