Infinite Time-evolving Block Decimation Algorithm Research Articles

Two-dimensional isometric tensor networks on an infinite strip

The exact contraction of a generic two-dimensional (2D) tensor network state (TNS) is known to be exponentially hard, making simulation of 2D systems difficult. The recently introduced class of isometric TNS (isoTNS) represents a subset of TNS that allows for efficient simulation of such systems on finite square lattices. The isoTNS ansatz requires the identification of an "orthogonality column" of tensors, within which one-dimensional matrix product state (MPS) methods can be used for calculation of observables and optimization of tensors. Here we extend isoTNS to infinitely long strip geometries and introduce an infinite version of the Moses Move algorithm for moving the orthogonality column around the network. Using this algorithm, we iteratively transform an infinite MPS representation of a 2D quantum state into a strip isoTNS and investigate the entanglement properties of the resulting state. In addition, we demonstrate that the local observables can be evaluated efficiently. Finally, we introduce an infinite time-evolving block decimation algorithm (iTEBD\textsuperscript{2}) and use it to approximate the ground state of the 2D transverse field Ising model on lattices of infinite strip geometry.

Characteristic quantum phase in Heisenberg antiferromagnetic chain with exchange and single-ion anisotropies.

We investigate the ground-state phase diagram for a spin-one quantum Heisenberg antiferromagnetic chain with exchange and single-ion anisotropies in an external magnetic field by using the infinite time-evolving block decimation algorithm to compute the ground-state fidelity per lattice site. We detect all phase boundaries solely by computing the ground-state fidelity per lattice site, with the prescription that a phase transition point is attributed to a pinch point on the ground-state fidelity surface. Furthermore, the results indicate that a magnetization plateau corresponds to a fidelity plateau on the ground-state fidelity surface, thus offering an alternative route for investigating the magnetization processes of quantum many-body spin systems. We characterize all phases by using the local-order parameter, the spin correlation, the momentum distribution of the spin correlation structure factor, and mutual information as a function of the lattice distance. The commensurate and incommensurate phases are distinguished by the mutual information. In addition, the central charges at criticalities are identified by performing a finite-entanglement scaling analysis. The results show that all phase transitions between spin liquids and magnetization plateaus belong to the Pokrovsky-Talapov universality class.

An optimized infinite time-evolving block decimation algorithm for lattice systems

The infinite time-evolving block decimation algorithm (iTEBD) provides an efficient way to determine the ground state and dynamics of the quantum lattice systems in the thermodynamic limit. In this paper we suggest an optimized way to take the iTEBD calculation, which takes advantage of additional reduced decompositions to speed up the calculation. The numerical calculations show that for a comparable computation time our method provides more accurate results than the traditional iTEBD, especially for lattice systems with large on-site degrees of freedom.

Dynamical phase transitions in the two-dimensional transverse-field Ising model

We investigate two separate notions of dynamical phase transitions in the two-dimensional nearest-neighbor transverse-field Ising model on a square lattice using matrix product states and a new \textit{hybrid} infinite time-evolving block decimation algorithm, where the model is implemented on an infinitely long cylinder with a finite diameter along which periodic boundary conditions are employed. Starting in an ordered initial state, our numerical results suggest that quenches below the dynamical critical point give rise to a ferromagnetic long-time steady state with the Loschmidt return rate exhibiting \textit{anomalous} cusps even when the order parameter never crosses zero. Within the accessible timescales of our numerics, quenches above the dynamical critical point suggest a paramagnetic long-time steady state with the return rate exhibiting \textit{regular} cusps connected to zero crossings of the order parameter. Additionally, our simulations indicate that quenching slightly above the dynamical critical point leads to a coexistence region where both anomalous and regular cusps appear in the return rate. Quenches from the disordered phase further confirm our main conclusions. Our work supports the recent finding that anomalous cusps arise only when local spin excitations are the energetically dominant quasiparticles. Our results are accessible in modern Rydberg experiments.

Hybrid infinite time-evolving block decimation algorithm for long-range multidimensional quantum many-body systems

In recent years, the infinite time-evolution block decimation (iTEBD) method has been demonstrated to be one of the most efficient and powerful numerical schemes for time-evolution in one-dimensional quantum many-body systems. However, a major shortcoming of the method, along with other state-of-the-art algorithms for many-body dynamics, has been their restriction to one spatial dimension. We present an algorithm based on a \textit{hybrid} extension of iTEBD where finite blocks of a chain are first locally time-evolved before an iTEBD-like method combines these processes globally. This in turn permits simulating the dynamics of many-body systems in the thermodynamic limit in $d\geq1$ dimensions including in the presence of long-range interactions. Our work paves the way for simulating the dynamics of many-body phenomena that occur exclusively in higher dimensions, and whose numerical treatments have hitherto been limited to exact diagonalization of small systems, which fundamentally limits a proper investigation of dynamical criticality. We expect the algorithm presented here to be of significant importance to validating and guiding investigations in state-of-the-art ion-trap and ultracold-atom experiments.

Photoinduced charge-order melting dynamics in a one-dimensional interacting Holstein model

Transient quantum dynamics in an interacting fermion-phonon system are investigated. In particular, a charge order (CO) melting after a short optical-pulse irradiation and roles of the quantum phonons on the transient dynamics are focused on. A spinless-fermion model in a one-dimensional chain coupled with local phonons is analyzed numerically. The infinite time-evolving block decimation algorithm is adopted as a reliable numerical method for one-dimensional quantum many-body systems. Numerical results for the photoinduced CO melting dynamics without phonons are well interpreted by the soliton picture for the CO domains. This interpretation is confirmed by the numerical simulation for an artificial local excitation and the classical soliton model. In the case of the large phonon frequency corresponding to the antiadiabatic condition, the CO melting is induced by propagations of the polaronic solitons with the renormalized soliton velocity. On the other hand, in the case of the small phonon frequency corresponding to the adiabatic condition, the first stage of the CO melting dynamics occurs due to the energy transfer from the fermionic to phononic systems, and the second stage is brought about by the soliton motions around the bottom of the soliton band. Present analyses provide a standard reference for the photoinduced CO melting dynamics in low-dimensional many-body quantum systems.

Pulse excitation to continuous-wave excitation in a low-dimensional interacting quantum system

Real-time dynamics in one-dimensional transverse Ising model coupled with the time-dependent oscillating field is analyzed by using the infinite time-evolving block decimation algorithm and the Floquet theory. In particular, the transient dynamics induced by the pulse field and their connections to the dynamics by a continuous-wave field are focused on. During the pulse-field irradiation, the order parameter shows a characteristic oscillation, in which the frequency shifts from the pulse-field frequency. This is considered as a kind of the Rabi oscillation, but the frequency strongly depends on the intersite Ising interaction. After turning off the pulse field, the oscillation remains with a frequency $\mathit{\Omega}$ and a damping constant $\gamma$. In the case of low fluence, both $\mathit{\Omega}$ and $\gamma$ are scaled by the pulse amplitude in a wide range of the parameter values of the model. In the case of high fluence, $\mathit{\Omega}$ and $\gamma$ are arranged by a product of the pulse amplitude and the pulse width. This implies that the dynamics after turning off the pulse field are decided by a population of the excited state when the pulse field is turned off.

Electronic properties and 1/3 magnetization plateau of the S=1/2 magnetism Cu3 (P2 O6 OH)2

First, the geometrical structure, energy band structure, density of states, and overlap population of the copper hydroxydiphosphate ( ) are investigated systematically using the first-principles density functional method. The generalized gradient approximations (GGA)+U method is adopted to consider the on-site coulomb repulsion on Cu 3d orbits. The theoretical structural results obtained by the geometry optimization agree well with the experimental data. Energy band structures show that the belongs to a semiconductor. It is found that the on-site Coulomb repulsion plays a key role in the opening of the energy gap. The uppest valence band is found to be mainly contributed from O 2p and Cu 3d states, but the lowest conduction band is characterized by Cu 3d state. Furthermore, a strong hybridization between O-2p and Cu-3d orbits is observed. Then, the magnetization, bipartite entanglement, and the nearest-neighbor correlations in such a cuprate are calculated by the infinite time-evolving block decimation algorithm. Distinctive magnetization plateaus accompanied with some interesting bipartite entanglement and correlation plateaus are observed at T=0 k in the thermodynamic limit. © 2014 Wiley Periodicals, Inc.

Reentrance and entanglement in the one-dimensional Bose-Hubbard model

Re-entrance is a novel feature where the phase boundaries of a system exhibit a succession of transitions between two phases A and B, like A-B-A-B, when just one parameter is varied monotonically. This type of re-entrance is displayed by the 1D Bose Hubbard model between its Mott insulator (MI) and superfluid phase as the hopping amplitude is increased from zero. Here we analyse this counter-intuitive phenomenon directly in the thermodynamic limit by utilizing the infinite time-evolving block decimation algorithm to variationally minimize an infinite matrix product state (MPS) parameterized by a matrix size chi. Exploiting the direct restriction on the half-chain entanglement imposed by fixing chi, we determined that re-entrance in the MI lobes only emerges in this approximate when chi >= 8. This entanglement threshold is found to be coincident with the ability an infinite MPS to be simultaneously particle-number symmetric and capture the kinetic energy carried by particle-hole excitations above the MI. Focussing on the tip of the MI lobe we then applied, for the first time, a general finite-entanglement scaling analysis of the infinite order Kosterlitz-Thouless critical point located there. By analysing chi's up to a very moderate chi = 70 we obtained an estimate of the KT transition as t_KT = 0.30 +/- 0.01, demonstrating the how a finite-entanglement approach can provide not only qualitative insight but also quantitatively accurate predictions.

Ground-state fidelity and Kosterlitz–Thouless phase transition for the spin-1/2 Heisenberg chain with next-to-the-nearest-neighbor interaction

The Kosterlitz–Thouless transition for the spin-1/2 Heisenberg chain with the next-to-the-nearest-neighbor interaction is investigated in the context of an infinite matrix product state algorithm, which is a generalization of the infinite time-evolving block decimation algorithm (Vidal 2007 Phys. Rev. Lett. 98 070201) to accommodate both the next-to-the-nearest-neighbor interaction and spontaneous dimerization. It is found that, in the critical regime, the algorithm automatically leads to infinite degenerate ground-state wavefunctions, due to the finiteness of the truncation dimension. This results in pseudo-symmetry spontaneous breakdown, which is reflected as a catastrophe point in the ground-state fidelity per lattice site. In addition, this allows the introduction of a pseudo-order parameter to characterize the Kosterlitz–Thouless transition. Our work demonstrates that the ground-state fidelity per lattice site is able to capture the Kosterlitz–Thouless transition, which is in sharp contrast to the fidelity susceptibility that fails to detect it.

Bifurcation in ground-state fidelity for a one-dimensional spin model with competing two-spin and three-spin interactions

A one-dimensional quantum spin model with the competing two-spin and three-spin interactions is investigated in the context of a tensor network algorithm based on the infinite matrix product state representation. The algorithm is an adaptation of Vidalʼs infinite time-evolving block decimation algorithm to a translation-invariant one-dimensional lattice spin system involving three-spin interactions. The ground-state fidelity per lattice site is computed, and its bifurcation is unveiled, for a few selected values of the coupling constants. We succeed in identifying critical points and deriving local order parameters to characterize different phases in the conventional Ginzburg–Landau–Wilson paradigm.

Classical Simulation of Infinite-Size Quantum Lattice Systems in Two Spatial Dimensions

We present an algorithm to simulate two-dimensional quantum lattice systems in the thermodynamic limit. Our approach builds on the projected entangled-pair state algorithm for finite lattice systems [F. Verstraete and J. I. Cirac, arxiv:cond-mat/0407066] and the infinite time-evolving block decimation algorithm for infinite one-dimensional lattice systems [G. Vidal, Phys. Rev. Lett. 98, 070201 (2007)10.1103/PhysRevLett.98.070201]. The present algorithm allows for the computation of the ground state and the simulation of time evolution in infinite two-dimensional systems that are invariant under translations. We demonstrate its performance by obtaining the ground state of the quantum Ising model and analyzing its second order quantum phase transition.

Infinite time-evolving block decimation algorithm beyond unitary evolution

The infinite time-evolving block decimation algorithm [G. Vidal, Phys. Rev. Lett. 98, 070201 (2007)] allows to simulate unitary evolution and to compute the ground state of one-dimensional (1D) quantum lattice systems in the thermodynamic limit. Here we extend the algorithm to tackle a much broader class of problems, namely, the simulation of arbitrary one-dimensional evolution operators that can be expressed as a (translationally invariant) tensor network. Relatedly, we also address the problem of finding the dominant eigenvalue and eigenvector of a one-dimensional transfer matrix that can be expressed in the same way. New applications include the simulation, in the thermodynamic limit, of open (i.e., master equation) dynamics and thermal states in 1D quantum systems, as well as calculations with partition functions in two-dimensional (2D) classical systems, on which we elaborate. The present extension of the algorithm also plays a prominent role in the infinite projected entangled-pair states approach to infinite 2D quantum lattice systems.