Improved energy extrapolation with infinite projected entangled-pair states applied to the 2D Hubbard model

An infinite projected entangled-pair state (iPEPS) is a variational tensor network ansatz for 2D wave functions in the thermodynamic limit where the accuracy can be systematically controlled by the bond dimension $D$. We show that for the doped Hubbard model in the strongly correlated regime ($U/t=8$, $n=0.875$) iPEPS yields lower variational energies than state-of-the-art variational methods in the large 2D limit, which demonstrates the competitiveness of the method. In order to obtain an accurate estimate of the energy in the exact infinite $D$ limit we introduce and test an extrapolation technique based on a truncation error computed in the iPEPS imaginary time evolution algorithm. The extrapolated energies are compared with accurate quantum Monte Carlo results at half filling and with various other methods in the doped, strongly correlated regime.Comment: 8 pages, 6 figure

Variational optimization with infinite projected entangled-pair states

We present a scheme to perform an iterative variational optimization with infinite projected entangled-pair states (iPEPS), a tensor network ansatz for a two-dimensional wave function in the thermodynamic limit, to compute the ground state of a local Hamiltonian. The method is based on a systematic summation of Hamiltonian contributions using the corner transfer-matrix method. Benchmark results for challenging problems are presented, including the 2D Heisenberg model, the Shastry-Sutherland model, and the t-J model, which show that the variational scheme yields considerably more accurate results than the previously best imaginary time evolution algorithm, with a similar computational cost and with a faster convergence towards the ground state.Comment: 11 pages, 9 figures, revised (published) version, correction in Fig.

Emergent Haldane phase in the S=1S=1 bilinear-biquadratic Heisenberg model on the square lattice

Infinite projected entangled pair states simulations of the $S=1$ bilinear-biquadratic Heisenberg model on the square lattice reveal an emergent Haldane phase in between the previously predicted antiferromagnetic and 3-sublattice 120$^\circ$ magnetically ordered phases. This intermediate phase preserves SU(2) spin and translational symmetry but breaks lattice rotational symmetry, and it can be adiabatically connected to the Haldane phase of decoupled $S=1$ chains. Our results contradict previous studies which found a direct transition between the two magnetically ordered states.Comment: 5 pages, 4 figures, plus supplemental materia

A tensor network study of the complete ground state phase diagram of the spin-1 bilinear-biquadratic Heisenberg model on the square lattice

Using infinite projected entangled pair states, we study the ground state phase diagram of the spin-1 bilinear-biquadratic Heisenberg model on the square lattice directly in the thermodynamic limit. We find an unexpected partially nematic partially magnetic phase in between the antiferroquadrupolar and ferromagnetic regions. Furthermore, we describe all observed phases and discuss the nature of the phase transitions involved.Comment: 27 pages, 15 figures; v3: adjusted sections 1 and 3, and added a paragraph to section 5.2.

A ground state study of the spin-1 bilinear-biquadratic Heisenberg model on the triangular lattice using tensor networks

Making use of infinite projected entangled pair states, we investigate the ground state phase diagram of the nearest-neighbor spin-1 bilinear-biquadratic Heisenberg model on the triangular lattice. In agreement with previous studies, we find the ferromagnetic, 120 degree magnetically ordered, ferroquadrupolar and antiferroquadrupolar phases, and confirm that all corresponding phase transitions are first order. Moreover, we provide an accurate estimate of the location of the ferroquadrupolar to 120 degree magnetically ordered phase transition, thereby fully establishing the phase diagram. Also, we do not encounter any signs of the existence of a quantum paramagnetic phase. In particular, contrary to the equivalent square lattice model, we demonstrate that on the triangular lattice the one-dimensional Haldane phase does not reach all the way up to the two-dimensional limit. Finally, we investigate the possibility of an intermediate partially-magnetic partially-quadrupolar phase close to $\theta = \pi/2$, and we show that, also contrary to the square lattice case, this phase is not present on the triangular lattice.Comment: 14 pages, 15 figures; v2: shortened section II.B and added a paragraph to section IV.

Fermionic Quantum Critical Point of Spinless Fermions on a Honeycomb Lattice

Spinless fermions on a honeycomb lattice provide a minimal realization of lattice Dirac fermions. Repulsive interactions between nearest neighbors drive a quantum phase transition from a Dirac semimetal to a charge-density-wave state through a fermionic quantum critical point, where the coupling of Ising order parameter to the Dirac fermions at low energy drastically affects the quantum critical behavior. Encouraged by a recently discovery of absence of the fermion sign problem in this model, we study the fermionic quantum critical point using the continuous time quantum Monte Carlo method with worm sampling technique. We estimate the transition point $V/t= 1.356(1)$ with the critical exponents $\nu =0.80(3)$ and $\eta =0.302(7)$. Compatible results for the transition point are also obtained with infinite projected entangled-pair states.Comment: Single column, 21 pages, appendix on worm updates and \pi-flux lattic

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

Infinite Matrix Product States vs Infinite Projected Entangled-Pair States on the Cylinder: a comparative study

In spite of their intrinsic one-dimensional nature matrix product states have been systematically used to obtain remarkably accurate results for two-dimensional systems. Motivated by basic entropic arguments favoring projected entangled-pair states as the method of choice, we assess the relative performance of infinite matrix product states and infinite projected entangled-pair states on cylindrical geometries. By considering the Heisenberg and half-filled Hubbard models on the square lattice as our benchmark cases, we evaluate their variational energies as a function of both bond dimension as well as cylinder width. In both examples we find crossovers at moderate cylinder widths, i.e. for the largest bond dimensions considered we find an improvement on the variational energies for the Heisenberg model by using projected entangled-pair states at a width of about 11 sites, whereas for the half-filled Hubbard model this crossover occurs at about 7 sites.Comment: 11 pages, 9 figure

Probing the stability of the spin liquid phases in the Kitaev-Heisenberg model using tensor network algorithms

We study the extent of the spin liquid phases in the Kitaev-Heisenberg model using infinite Projected Entangled-Pair States tensor network ansatz wave functions directly in the thermodynamic limit. To assess the accuracy of the ansatz wave functions we perform benchmarks against exact results for the Kitaev model and find very good agreement for various observables. In the case of the Kitaev-Heisenberg model we confirm the existence of 6 different phases: N\'eel, stripy, ferromagnetic, zigzag and two spin liquid phases. We find finite extents for both spin liquid phases and discontinuous phase transitions connecting them to symmetry-broken phases.Comment: 9 pages, 7 figures. Adjusted notation in equations 4-8. Added bond labeling to lower panel in figure 1. Included missing acknowledgement