Fermionic Projected Entangled Pair States at Finite Temperature

An algorithm for imaginary time evolution of a fermionic projected entangled pair state (PEPS) with ancillas from infinite temperature down to a finite temperature state is presented. As a benchmark application, it is applied to spinless fermions hopping on a square lattice subject to $p$-wave pairing interactions. With a tiny bias it allows to evolve the system across a high-temperature continuous symmetry-breaking phase transition.Comment: 7 pages, 11 figures; new results on a high-Tc phase transition were adde

Time Evolution of an Infinite Projected Entangled Pair State: an Algorithm from First Principles

A typical quantum state obeying the area law for entanglement on an infinite 2D lattice can be represented by a tensor network ansatz -- known as an infinite projected entangled pair state (iPEPS) -- with a finite bond dimension $D$. Its real/imaginary time evolution can be split into small time steps. An application of a time step generates a new iPEPS with a bond dimension $k$ times the original one. The new iPEPS does not make optimal use of its enlarged bond dimension $kD$, hence in principle it can be represented accurately by a more compact ansatz, favourably with the original $D$. In this work we show how the more compact iPEPS can be optimized variationally to maximize its overlap with the new iPEPS. To compute the overlap we use the corner transfer matrix renormalization group (CTMRG). By simulating sudden quench of the transverse field in the 2D quantum Ising model with the proposed algorithm, we provide a proof of principle that real time evolution can be simulated with iPEPS. A similar proof is provided in the same model for imaginary time evolution of purification of its thermal states.Comment: 9 pages, 10 figures, replaced with the published versio

Variational Approach to Projected Entangled Pair States at Finite Temperature

The projected entangled pair state (PEPS) ansatz can represent a thermal state in a strongly correlated system. We introduce a novel variational algorithm to optimize this tensor network. Since full tensor environment is taken into account, then with increasing bond dimension the optimized PEPS becomes the exact Gibbs state. Our presentation opens with a 1D version for a matrix product state (MPS) and then generalizes to PEPS in 2D. Benchmark results in the quantum Ising model are presented.Comment: 9 pages, 12 figures; an extended version with new numerical result

Striped critical spin liquid in a spin-orbital entangled RVB state in a projected entangled-pair state representation

We introduce a spin-orbital entangled (SOE) resonating valence bond (RVB) state on a square lattice of spins-$\frac12$ and orbitals represented by pseudospins-$\frac12$. Like the standard RVB state, it is a superposition of nearest-neighbor hard-core coverings of the lattice by spin singlets, but adjacent singlets are favoured to have perpendicular orientations and, more importantly, an orientation of each singlet is entangled with orbitals' state on its two lattice sites. The SOE-RVB state can be represented by a projected entangled pair state (PEPS) with a bond dimension $D=4$. This representation helps to reveal that the state is a superposition of striped coverings conserving a topological quantum number. The stripes are a critical quantum spin liquid. We propose a spin-orbital Hamiltonian supporting a SOE-RVB ground state.Comment: 8 pages, 10 figure

Projected Entangled Pair States at Finite Temperature: Iterative Self-Consistent Bond Renormalization for Exact Imaginary Time Evolution

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 2D 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 (S-T) decomposition, the renormalization has in principle to take into account full environment made of the new tensors with the bond dimension $k\times D$. Here we propose a self-consistent renormalization procedure operating with the original bond dimension $D$, but without compromising the accuracy of the S-T 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.Comment: 9 pages, 14 figures, improved presentation. arXiv admin note: text overlap with arXiv:1311.727

Projected Entangled Pair States at Finite Temperature: Imaginary Time Evolution with Ancillas

A projected entangled pair state (PEPS) with ancillas is evolved in imaginary time. This tensor network represents a thermal state of a 2D lattice quantum system. A finite temperature phase diagram of the 2D quantum Ising model in a transverse field is obtained as a benchmark application.Comment: 7 pages, 9 figures; version accepted to publication in Phys. Rev. B; added new numerical results and reference

Time Evolution of an Infinite Projected Entangled Pair State: an Efficient Algorithm

An infinite projected entangled pair state (iPEPS) is a tensor network ansatz to represent a quantum state on an infinite 2D lattice whose accuracy is controlled by the bond dimension $D$. Its real, Lindbladian or imaginary time evolution can be split into small time steps. Every time step generates a new iPEPS with an enlarged bond dimension $D' > D$, which is approximated by an iPEPS with the original $D$. In Phys. Rev. B 98, 045110 (2018) an algorithm was introduced to optimize the approximate iPEPS by maximizing directly its fidelity to the one with the enlarged bond dimension $D'$. In this work we implement a more efficient optimization employing a local estimator of the fidelity. For imaginary time evolution of a thermal state's purification, we also consider using unitary disentangling gates acting on ancillas to reduce the required $D$. We test the algorithm simulating Lindbladian evolution and unitary evolution after a sudden quench of transverse field $h_x$ in the 2D quantum Ising model. Furthermore, we simulate thermal states of this model and estimate the critical temperature with good accuracy: $0.1\%$ for $h_x=2.5$ and $0.5\%$ for the more challenging case of $h_x=2.9$ close to the quantum critical point at $h_x=3.04438(2)$.Comment: published version, presentation improve

Overcoming the Sign Problem at Finite Temperature: Quantum Tensor Network for the Orbital ege_g Model on an Infinite Square Lattice

The variational tensor network renormalization approach to two-dimensional (2D) quantum systems at finite temperature is applied for the first time to a model suffering the notorious quantum Monte Carlo sign problem --- the orbital $e_g$ model with spatially highly anisotropic orbital interactions. Coarse-graining of the tensor network along the inverse temperature $\beta$ yields a numerically tractable 2D tensor network representing the Gibbs state. Its bond dimension $D$ --- limiting the amount of entanglement --- is a natural refinement parameter. Increasing $D$ we obtain a converged order parameter and its linear susceptibility close to the critical point. They confirm the existence of finite order parameter below the critical temperature $T_c$, provide a numerically exact estimate of~$T_c$, and give the critical exponents within $1\%$ of the 2D Ising universality class.Comment: 8 pages, 8 figure

Variational tensor network renormalization in imaginary time: Two-dimensional quantum compass model at finite temperature

Progress in describing thermodynamic phase transitions in quantum systems is obtained by noticing that the Gibbs operator $e^{-\beta H}$ for a two-dimensional (2D) lattice system with a Hamiltonian $H$ can be represented by a three-dimensional tensor network, the third dimension being the imaginary time (inverse temperature) $\beta$. Coarse-graining the network along $\beta$ results in a 2D projected entangled-pair operator (PEPO) with a finite bond dimension $D$. The coarse-graining is performed by a tree tensor network of isometries. The isometries are optimized variationally --- taking into account full tensor environment --- to maximize the accuracy of the PEPO. The algorithm is applied to the isotropic quantum compass model on an infinite square lattice near a symmetry-breaking phase transition at finite temperature. From the linear susceptibility in the symmetric phase and the order parameter in the symmetry-broken phase the critical temperature is estimated at ${\cal T}_c=0.0606(4)J$, where $J$ is the isotropic coupling constant between $S=1/2$ pseudospins.Comment: 12 pages, 15 figures, slightly revised after referees' report

Order in quantum compass and orbital e_{g} models

We investigate thermodynamic phase transitions in the compass model and in $e_g$ orbital model on an infinite square lattice by variational tensor network renormalization (VTNR) in imaginary time. The onset of nematic order in the quantum compass model is estimated at ${\cal T}_c/J=0.0606(4)$. For~the $e_g$ orbital model one finds: ($i$) a very accurate estimate of ${\cal T}_c/J=0.3566\pm 0.0001$ and ($ii$)~the~critical exponents in the Ising universality class. Remarkably large difference in frustration results in so distinct values of ${\cal T}_c$, while entanglement influences the quality of ${\cal T}_c$ estimation.Comment: 4 pages, 2 figures, accepted by Acta Physica Polonica