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.
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.
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 Tc.
Within the Projected Entangled Pair State (PEPS) tensor network formalism, a simple update (SU) method has been used to investigate the time evolution of a two-dimensional U(1) critical spin-1/2 spin liquid under Hamiltonian quench [Phys. Rev. B 106, 195132 (2022)]. Here we introduce two different variational frameworks to describe the time dynamics of SU(2)-symmetric translationally-invariant PEPS, aiming to improve the accuracy. In one approach, after using a Trotter-Suzuki decomposition of the time evolution operator in term of two-site elementary gates, one considers a single bond embedded in an environment approximated by a Corner Transfer Matrix Renormalization Group (CTMRG). A variational update of the two tensors on the bond is performed under the application of the elementary gate and then, after symmetrization of the site tensors, the environment is updated. In the second approach, a cluster optimization is performed on a finite (periodic) cluster, maximizing the overlap of the exact time-evolved state with a symmetric finite-size PEPS ansatz. Observables are then computed on the infinite lattice contracting the infinite-PEPS (iPEPS) by CTMRG. We show that the variational schemes outperform the SU method and remain accurate over a significant time interval before hitting the entanglement barrier. Studying the spectrum of the transfer matrix, we find that the asymptotic correlations are very well preserved under time evolution, including the critical nature of the singlet correlations, as expected from the Lieb-Robinson (LR) bound theorem. Consistently, the system (asymptotic) boundary is found to be described by the same Conformal Field Theory of central charge c = 1 c = 1 during time evolution. We also compute the time-evolution of the short distance spin-spin correlations and estimate the LR velocity.
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.
A projected entangled pair state (PEPS) with ancillas can be evolved in imaginary time to obtain thermal states of a strongly correlated quantum system on a two-dimensional lattice. Every application of a Suzuki-Trotter gate multiplies the PEPS bond dimension D by a factor k. It has to be renormalized back to the original D. In order to preserve the accuracy of the Suzuki-Trotter (ST) decomposition, the renormalization in principle has to take into account full environment made of the new tensors with the bond dimension $k\ifmmode\times\else\texttimes\fi{}D$. Here, we propose a self-consistent renormalization procedure operating with the original bond dimension D, but without compromising the accuracy of the ST decomposition. The iterative procedure renormalizes the bond using full environment made of renormalized tensors with the bond dimension D. After every renormalization, the new renormalized tensors are used to update the environment, and then the renormalization is repeated again and again until convergence. As a benchmark application, we obtain thermal states of the transverse field quantum Ising model on a square lattice, both infinite and finite, evolving the system across a second-order phase transition at finite temperature.
The U(1)-symmetric iPEPS study of the spin-1/2 square J1-J2 Heisenberg model
We develop an improved variant of U(1)-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground state phase diagram of the spin-1/2 square J1-J2 Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the U(1)-symmetric iPEPS ansatz (for a fixed bond dimension D) set a better upper bound, improving previous tensor-network-based results. By studying N\'{e}el order parameter and correlation function in the large-D limit, we find that a deconfined quantum-critical point occurs at J2c1=0.530(5), where the correlation length of the system diverges as ξ∼D1.2. The estimated critical anomalous exponents ηs=0.65(5) and ηd=1.9(1) for spin and dimer correlations respectively show strong deviation from those of the well-known J-Q model, suggesting possibly a different universality class. The pattern of local bond energy for the intermediate phase is compatible with the columnar valence bond solid (VBS) state, with a quite competitive variational ground-state energy in comparison with plaquette VBS one predicted by the density matrix renormalization group. We show that the associated VBS order parameter first monotonically increases with J2 and finally a first-order quantum phase transition takes place at J2c2=0.610(3) to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.
A Gibbs operator ${e}^{\ensuremath{-}\ensuremath{\beta}H}$ for a two-dimensional (2D) lattice system with a Hamiltonian H can be represented by a 3D tensor network, with the third dimension being the imaginary time (inverse temperature) $\ensuremath{\beta}$. Coarse graining the network along $\ensuremath{\beta}$ results in a 2D projected entangled-pair operator (PEPO) with a finite bond dimension. The coarse graining is performed by a tree tensor network of isometries. They are optimized variationally to maximize the accuracy of the PEPO as a representation of the 2D thermal state ${e}^{\ensuremath{-}\ensuremath{\beta}H}$. The algorithm is applied to the two-dimensional Hubbard model on an infinite square lattice. Benchmark results at finite temperature are obtained that are consistent with the best cluster dynamical mean-field theory and power-series expansion in the regime of parameters where they yield mutually consistent results.