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arXiv:1904.06273 · Physical review. B./Physical review. B

Thermal tensor renormalization group simulations of square-lattice quantum spin models

Han Li, Bin-Bin Chen, Ziyu Chen, Jan von Delft, Andreas Weichselbaum, Wei Li

Published Jul 9, 2019• 39 citations
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Abstract

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 $W=10$. 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-$T$ thermal data on $W\ensuremath{\le}8$ cylinders and $W\ensuremath{\le}10$ open strips, respectively. We extract an exponential divergence versus $T$ of the structure factor $S(M)$, 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 ${T}_{c}$.

DOI
10.1103/physrevb.100.045110

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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 \ensuremathξ\ensuremath{\xi}\ensuremathξ, such that for system sizes L\ensuremath≫\ensuremathξL\ensuremath{\gg}\ensuremath{\xi}L\ensuremath≫\ensuremathξ 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 ug∗\ensuremath≃\ensuremath−0.6694(4){u}_{g}^{*}\ensuremath{\simeq}\ensuremath{-}0.6694(4)ug∗​\ensuremath≃\ensuremath−0.6694(4) and a spontaneous magnetization mS∗\ensuremath≃0.30(1){m}_{S}^{*}\ensuremath{\simeq}0.30(1)mS∗​\ensuremath≃0.30(1) already consistent with the ground-state properties, which is obtained from extrapolated low-TTT thermal data on W\ensuremath≤8W\ensuremath{\le}8W\ensuremath≤8 cylinders and W\ensuremath≤10W\ensuremath{\le}10W\ensuremath≤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 \ensuremathξ\ensuremath{\xi}\ensuremathξ, at the ordering wave vector M=(\ensuremathπ,\ensuremathπ)M=(\ensuremath{\pi},\ensuremath{\pi})M=(\ensuremathπ,\ensuremathπ), 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​.
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.
We have extended the canonical tree tensor network (TTN) method, which was initially introduced to simulate the zero-temperature properties of quantum lattice models on the Bethe lattice, to finite temperature simulations. By representing the thermal density matrix with a canonicalized tree tensor product operator, we optimize the TTN and compute accurately the thermodynamic quantities, including the internal energy, specific heat, spontaneous magnetization, etc., at various temperatures. By varying the anisotropic coupling constant $\mathrm{\ensuremath{\Delta}}$, we obtain the phase diagram of the spin-1/2 Heisenberg XXZ model on the Bethe lattice, where three magnetic ordered phases, namely the ferromagnetic, XY, and antiferromagnetic phases, are found in low temperatures and separated from the high-TTT paramagnetic phase by a continuous thermal phase transition at Tc{T}_{c}Tc​. The XY phase is separated from the other two phases by two first-order phase transition lines at the symmetric coupling points $\mathrm{\ensuremath{\Delta}}=\ifmmode\pm\else\textpm\fi{}1$. We have also carried out a linear spin wave calculation on the Bethe lattice, showing that the low-energy magnetic excitations are always gapped, and find the obtained magnon gaps in very good agreement with those estimated from the TTN simulations. Despite the gapped excitation spectrum, Goldstone-like transverse fluctuation modes, as a manifestation of spontaneous continuous symmetry breaking, are observed in the ordered magnetic phases with $|\mathrm{\ensuremath{\Delta}}|\ensuremath{\le}1$. One remarkable feature there is that the transverse correlation length reaches ${\ensuremath{\xi}}_{c}=1/ln(z\ensuremath{-}1)$ for $T\ensuremath{\le}{T}_{c}$, the maximal value allowed on a zzz-coordinated Bethe lattice.
Simulating strongly correlated quantum many-body systems at finite temperatures is a significant challenge in computational physics. In this work, we present a scalable finite-temperature tensor network algorithm for two-dimensional quantum many-body systems. We employ the (fermionic) projected entangled pair state to represent the vectorization of the quantum thermal state, and we utilize a stochastic reconfiguration method to cool down the quantum states from infinite temperature. We validate our method by benchmarking it against the two-dimensional antiferromagnetic Heisenberg model, the ${J}_{1}\ensuremath{-}{J}_{2}$ model, and the Fermi-Hubbard model, comparing physical properties such as internal energy, specific heat, and magnetic susceptibility with results obtained from stochastic series expansion, exact diagonalization, and determinant quantum Monte Carlo.
Understanding quantum many-body states of correlated electrons is one main theme in modern condensed-matter physics. Given that the Fermi-Hubbard model, the prototype of correlated electrons, was recently realized in ultracold optical lattices, it is highly desirable to have controlled numerical methodology to provide precise finite-temperature results upon doping to directly compare with experiments. Here, we demonstrate the exponential tensor renormalization group (XTRG) algorithm [Chen et al., Phys. Rev. X 8, 031082 (2018)], complemented by independent determinant quantum Monte Carlo, offers a powerful combination of tools for this purpose. XTRG provides full and accurate access to the density matrix and thus various spin and charge correlations, down to an unprecedented low temperature of a few percent of the tunneling energy. We observe excellent agreement with ultracold fermion measurements at both half filling and finite doping, including the sign-reversal behavior in spin correlations due to formation of magnetic polarons, and the attractive hole-doublon and repulsive hole-hole pairs that are responsible for the peculiar bunching and antibunching behaviors of the antimoments.
Within the tensor network framework, the (positive) thermal density operator can be approximated by a double layer of infinite Projected Entangled Pair Operators (iPEPO) coupled via ancilla degrees of freedom.To investigate the thermal properties of the spin-1/2 Heisenberg model on the square lattice, we introduce a family of fully spin-SU(2) and lattice-C 4v symmetric on-site tensors (of bond dimensions D = 4 or D = 7) and a plaquette-based Trotter-Suzuki decomposition of the imaginary-time evolution operator.A variational optimization is performed on the plaquettes, using a full (for D = 4) or simple (for D = 7) environment obtained from the single-site Corner Transfer Matrix Renormalization Group fixed point.The method is benchmarked by a comparison to quantum Monte Carlo in the thermodynamic limit.Although the iPEPO spin correlation length starts to deviate from the exact exponential growth for inverse-temperature β 2, the behavior of various observables turns out to be quite accurate once plotted w.r.t the inverse correlation length.We also find that a direct T = 0 variational energy optimization provides results in full agreement with the β → ∞ limit of finite-temperature data, hence validating the imaginary-time evolution procedure.Extension of the method to frustrated models is described and preliminary results are shown.
Simulating strongly-correlated quantum many-body systems at finite temperatures is a significant challenge in computational physics. In this work, we present a scalable finite-temperature tensor network algorithm for two-dimensional quantum many-body systems. We employ the (fermionic) projected entangled pair state (PEPS) to represent the vectorization of the quantum thermal state and utilize a stochastic reconfiguration method to cool down the quantum states from infinite temperature. We validate our method by benchmarking it against the 2D antiferromagnetic Heisenberg model, the J1J_1J1​-J2J_2J2​ model, and the Fermi-Hubbard model, comparing physical properties such as internal energy, specific heat, and magnetic susceptibility with results obtained from stochastic series expansion (SSE), exact diagonalization, and determinant quantum Monte Carlo (DQMC).