Infinite Time-evolving Block Decimation 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.

Two-dimensional coherent spectrum of interacting spinons from matrix product states

We compute numerically the second- and third-order nonlinear magnetic susceptibilities of an Ising ladder model in the context of two-dimensional coherent spectroscopy by using the infinite time-evolving block decimation method. The Ising ladder model couples a quantum Ising chain to a bath of polarized spins, thereby effecting the decay of spinon excitations. We show that its third-order susceptibility contains a robust spinon echo signal in the weak-coupling regime, which appears in the two-dimensional coherent spectrum as a highly anisotropic peak on the frequency plane. The spinon echo peak reveals the dynamical properties of the spinons. In particular, the spectral peak corresponding to the high-energy spinons, which couple to the bath, is suppressed with increasing coupling, whereas those corresponding to low-energy spinons do not show any significant changes.

Statistics of the number ofdefects after quantum annealing in a thermal environment.

We study the statistics of the kink number generated by quantum annealing ina one-dimensional transverse Ising model coupled to a bosonic thermal bath. Using the freezing ansatz for quantum annealing in the thermal environment, we show the relation between the ratio of the second to the first cumulant of the kink number distribution and the average kink density. The theoretical result is confirmed thoroughly by numerical simulation using the non-Markovian infinite time-evolving block decimation which we proposed recently. The simulation using D-Wave's quantum annealer is also discussed. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

Frustrated mixed-spin ladders: Evidence for a bond-order wave phase between rung-singlet and Haldane phases

In frustrated spin ladders the interplay of frustration and correlations leads to the familiar Haldane (H) and rung-singlet (RS) phases. The nature of the transition between these two phases is still under debate. In this paper we tackle this issue using tools of quantum information theory. We consider frustrated mixed-spin-(1, 1/2) ladders with antiferromagnetic leg, rung and diagonal couplings, and calculate various quantities, such as the entanglement entropy (EE), the Schmidt gap, and the level degeneracy of the entanglement spectrum (ES). We use two numerical techniques, the infinite time-evolving block decimation (iTEBD) and the density matrix renormalization group (DMRG). We demonstrate that there exists an intermediate phase in which the ES levels do not exhibit the characteristic degeneracies of the H and RS phases. To understand the underlying physics in this phase, we investigate short-range spin correlations along legs, rungs and diagonals and show that in this intermediate phase long-wavelength modulations occur, akin to bond order waves.

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.

Excitonic effects on high-harmonic generation in Mott insulators

To study excitonic effects on high-harmonic generation (HHG) in Mott insulators, we investigate pumped nonequilibrium dynamics in the one-dimensional extended Hubbard model. By employing time-dependent calculations based on the exact diagonalization and infinite time-evolving block decimation methods, we find the strong enhancement of the HHG intensity around the exciton energy. The subcycle analysis in the sub-Mott-gap regime shows that the intensity region of the time-resolved spectrum around the exciton energy splits into two levels and oscillates following the driving electric field. This excitonic dynamics is qualitatively different from the dynamics of free doublon and holon but favorably contributes to HHG in the Mott insulator.

Classical Simulation and Theory of Quantum Annealing in a Thermal Environment.

We study quantum annealing in the quantum Ising model coupled to a thermal environment. When the speed of quantum annealing is sufficiently slow, the system evolves following the instantaneous thermal equilibrium. This quasistatic and isothermal evolution, however, fails near the end of annealing because the relaxation time grows infinitely, therefore yielding excess energy from the thermal equilibrium. We develop a phenomenological theory based on this picture and derive a scaling relation of the excess energy after annealing. The theoretical results are numerically confirmed using a novel non-Markovian method that we recently proposed based on a path-integral representation of the reduced density matrix and the infinite time evolving block decimation. In addition, we discuss crossovers from weak to strong coupling as well as from the adiabatic to quasistatic regime, and propose experiments on the D-Wave quantum annealer.

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.

Quantum criticality in spin-1/2 anisotropic XY model with staggered Dzyaloshinskii–Moriya interaction

By utilizing the infinite time evolving block decimation method in infinite matrix product state representation, the quantum criticality and critical exponents varying are investigated in the spin-1/2 anisotropic XY chain with staggered Dzyaloshinskii–Moriya interaction. The phase diagram is obtained from the entanglement measurement, where a XY phase line δ=0 separates the Néel phase. Along this critical line, the central charge c=1 is extracted from the finite entanglement and the finite correlation length. In addition, the characteristic critical exponents are obtained from the local transverse magnetization, nonlocal transverse Néel order, and the correlation length, respectively. It is found that all the critical exponents are varying continuously along the phase transition line δ=0, and the ratios of critical exponents imply that the phase transition is in conformity with the weak universality. The linear relations of the critical exponents are able to illustrate the dependence between the critical exponents and the Dzyaloshinskii–Moriya interaction.

Quantum phases and transitions of spin-1/2 quantum compass chain

Quantum phases (QPs) and quantum phase transitions (QPTs) are very important parts of the strongly correlated quantum many-body systems in condensed matter. To study the QPs and QPTs, the systems should include rich quantum phase diagram. In this sense, the corresponding quantum spin models should have strong quantum fluctuation, strong geometric frustration, complicated spin-spin exchange or orbital degrees of freedom, which induces a variety of spontaneous symmetry breaking (SSB) or hidden spontaneous symmetry breaking. The QPs induced by the SSB can be characterized by local order parameters, a concept that originates from Landau-Ginzburg-Wilson paradigm (LGW). However, there is also a novel class of topological QPs beyond LGW, which has aroused one’s great interest since the Haldane phase was found. Such QPs can be characterized only by topological long-range nonlocal string correlation order parameters instead of local order parameters. In this paper, we investigate a spin-1/2 quantum compass chain model (QCC) with orbital degrees of freedom in <i>x</i>, <i>y</i> and <i>z</i> components. The prototype of QCC is the quantum compass model including novel topological QPs beyond LGW, and consequently one can also anticipate the existence of novel topological QPs in QCC. However, very little attention has been paid to the QPs and QPTs for QCC, which deserves to be further investigated. By using the infinite time evolving block decimation in the presentation of matrix product states, we study the QPs and QPTs of QCC. To characterize QPs and QPTs of QCC, the ground state energy, local order parameter, topological long-range nonlocal string correlation order parameters, critical exponent, correlation length and central charge are calculated. The results show the phase diagram of QCC including local antiferromagnetic phase, local stripe antiferromagnetic phase, oscillatory odd Haldane phase and monotonic odd Haldane phase. The QPTs from oscillatory odd Haldane phase to local stripe antiferromagnetic phase and from local antiferromagnetic phase to monotonic odd Haldane phase are continuous; on the contrary, QPTs from local stripe antiferromagnetic phase to local antiferromagnetic phase and from oscillatory odd Haldane phase to monotonic odd Haldane phase are discontinuous. The crossing point where the line of continuous QPTs meets with the line of discontinuous QPTs is the multiple critical point. The critical exponents <i>β</i> of local antiferromagnetic order parameter, local stripe antiferromagnetic order parameter, topological long-range nonlocal oscillatory odd string correlation order parameter, and topological long-range nonlocal monotonic odd string correlation order parameter are all equal to 1/8. Moreover, <inline-formula><tex-math id="M3">\begin{document}$\beta =1/8$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211433_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211433_M3.png"/></alternatives></inline-formula> and the central charges <inline-formula><tex-math id="M4">\begin{document}$c = 1/2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211433_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211433_M4.png"/></alternatives></inline-formula> at the critical points show that the QPTs from local phases to nonlocal phases belong to the Ising-type universality class.

Solitonic excitations in the Ising anisotropic chain BaCo2V2O8 under large transverse magnetic field

We study the dynamics of the quasi-one-dimensional Ising-Heisenberg antiferromagnet ${\mathrm{BaCo}}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{8}$ under a transverse magnetic field. Combining inelastic neutron scattering experiments and theoretical analyses by field theories and numerical simulations, we mainly elucidate the structure of the spin excitation spectrum in the high-field phase, appearing above the quantum phase transition point ${\ensuremath{\mu}}_{0}{H}_{c}\ensuremath{\approx}10\phantom{\rule{0.28em}{0ex}}\mathrm{T}$. We find that it is characterized by collective solitonic excitations superimposed on a continuum. These solitons are strongly bound in pairs due to the effective staggered field induced by the nondiagonal $g$ tensor of the compound and are topologically different from the fractionalized spinons in the weak-field region. The dynamical susceptibility numerically calculated with the infinite time-evolving block decimation method shows an excellent agreement with the measured spectra, which enables us to identify the dispersion branches with elementary excitations. The lowest-energy dispersion has an incommensurate nature and has a local minimum at an irrational wave number due to the applied transverse field.

Directed percolation in nonunitary quantum cellular automata

Probabilistic cellular automata (CA) provide a classic framework for studying nonequilibrium statistical physics on lattices. A notable example is the Domany-Kinzel CA, which has been used to investigate the process of directed percolation and the critical dynamics of the nonequilibrium phase transition between absorbing and percolating phases. In this work, we construct a nonunitary quantum CA that generalizes the Domany-Kinzel CA and study the resulting dynamical evolution using numerical simulations using the tensor network infinite time-evolving block decimation (iTEBD) algorithm. We demonstrate that the system undergoes the absorbing/percolating phase transition and that the addition of the Hamiltonian generates coherences, which are a distinct feature of quantum dynamics. A proposal for the implementation of the model with Rydberg array is put forward, which does not require local addressing of individual sites.

Simulated quantum annealing as a simulator of nonequilibrium quantum dynamics

Simulated quantum annealing based on the path-integral Monte Carlo is one of the most common tools to simulate quantum annealing on classical hardware. Nevertheless, it is in principle highly non-trivial whether or not this classical algorithm can correctly reproduce the quantum dynamics of quantum annealing, particularly in the diabatic regime. We study this problem numerically through the generalized Kibble-Zurek mechanism of defect distribution in the simplest ferromagnetic one-dimensional transverse-field Ising model with and without coupling to the environment. We find that,in the absence of coupling to the environment, simulated quantum annealing correctly describes the annealing-time dependence of the average number of defects, but a detailed analysis of the defect distribution shows clear deviations from the theoretical prediction. When the system is open (coupled to the environment), the average number of defects does not follow the theoretical prediction but is qualitatively compatible with the numerical result by the infinite time-evolving block decimation combined with the quasi-adiabatic propagator path integral, which is valid in a very short time region. The distribution of defects in the open system turns out to be not far from the theoretical prediction. It is surprising that the classical stochastic dynamics of simulated quantum annealing ostensibly reproduce some aspects of the quantum dynamics. However, a serious problem is that it is hard to predict for which physical quantities in which system it is reliable. Those results suggest the necessity to exert a good amount of caution in using simulated quantum annealing to study the detailed quantitative aspects of the dynamics of quantum annealing.

High-harmonic generation in one-dimensional Mott insulators

We study high-harmonic generation (HHG) in the one-dimensional Hubbard model in order to understand its relation to elementary excitations as well as the similarities and differences to semiconductors. The simulations are based on the infinite time-evolving block decimation (iTEBD) method and exact diagonalization. We clarify that the HHG originates from the doublon-holon recombination, and the scaling of the cutoff frequency is consistent with a linear dependence on the external field. We demonstrate that the subcycle features of the HHG can be reasonably described by a phenomenological three step model for a doublon-holon pair. We argue that the HHG in the one-dimensional Mott insulator is closely related to the dispersion of the doublon-holon pair with respect to its relative momentum, which is not necessarily captured by the single-particle spectrum due to the many-body nature of the elementary excitations. For the comparison to semiconductors, we introduce effective models obtained from the Schrieffer-Wolff transformation, i.e. a strong-coupling expansion, which allows us to disentangle the different processes involved in the Hubbard model: intraband dynamics of doublons and holons, interband dipole excitations, and spin exchanges. These demonstrate the formal similarity of the Mott system to the semiconductor models in the dipole gauge, and reveal that the spin dynamics, which does not directly affect the charge dynamics, can reduce the HHG intensity. We also show that the long-range component of the intraband dipole moment has a substantial effect on the HHG intensity, while the correlated hopping terms for the doublons and holons essentially determine the shape of the HHG spectrum. A new numerical method to evaluate single-particle spectra within the iTEBD method is also introduced.

Bond-Alternation-Induced Topological Quantum Gaussian Transition and Topological Quantum Crossover in Ising Chains with the Dzyaloshinskii-Moriya Interaction

Non-local orders, entanglement entropy, and quantum fidelity are investigated in an infinite-size bond-alternating Ising chain with the Dzyaloshinskii-Moriya interaction by employing the infinite matrix product state representation with the infinite time evolving block decimation method. Directly computing two distinct types of finite string correlations for very large lattice distances, in contrast to an extrapolated extreme value for finite size chains, reveals two topologically ordered phases. As the bond alternation varies, a topological quantum phase transition with continuously variable critical exponents along the phase boundary occurs between the two Haldane phases for a Dzyaloshinskii-Moriya interaction stronger than the Ising interaction while a topological quantum crossover between them happens through an intermediate antiferromagentic phase demonstrated with the quantum fidelity for a Dzyaloshinskii-Moriya interaction weaker than the Ising interaction. The critical exponents of the order parameters and the central charges from the entanglement entropy quantify the universality classes of the phase transition points. Anisotropic Heisenberg types of spin chains with bond alternations are finally discussed to share the same criticality.

Kibble–Zurek Mechanism in a Dissipative Transverse Ising Chain

We study the Kibble-Zurek mechanism in the transverse Ising chain coupled to a dissipative boson bath, making use of a new numerical method with the infinite time evolving block decimation combined with the discrete-time path integral. We first show the ground-state phase diagram and confirm that a quantum phase transition takes place in the presence of the system-bath coupling. Then we present the time dependence of the energy expectation value of the spin Hamiltonian and the scaling of the kink density with respect to the time period over which the spin Hamiltonian crosses a quantum phase transition. The energy of spins starts to grow from the energy at the ground state of the full system near a quantum phase transition. The kink density decays as a power law with respect to the time period. These results confirm that the Kibble-Zurek mechanism happens. We discuss the exponent for the decay of the kink density in comparison with a theoretical result with the quantum Monte-Carlo simulation. A comparison to an experimental study is also briefly mentioned.

Photoinduced η -pairing at finite temperatures

We numerically prove photoinduced $\eta$-pairing in a half-filled fermionic Hubbard chain at both zero and finite temperature. The result, obtained by combining the matrix-product-state based infinite time-evolving block decimation technique and the purification method, applies to the thermodynamic limit. Exciting the Mott insulator by a laser electric field docked on via the Peierls phase, we track the time-evolution of the correlated many-body system and determine the optimal parameter set for which the nonlocal part of the $\eta$-pair correlation function becomes dominant during the laser pump at zero and low temperatures. These correlations vanish at higher temperatures and long times after pulse irradiation. In the high laser frequency strong Coulomb coupling regime we observe a remnant enhancement of the Brillouin-zone boundary pair-correlation function also at high temperatures, if the Hubbard interaction is about a multiple of the laser frequency, which can be attributed to an enhanced double occupancy in the virtual Floquet state.

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.

Enhancement of magnetization plateaus in low-dimensional spin systems

We study the low-energy properties and, in particular, the magnetization process of a spin-1/2 Heisenberg $J_1-J_2$ sawtooth and frustrated chain (also known as zig-zag ladder) with a spatially anisotropic $g$-factor. We treat the problem both analytically and numerically while keeping the $J_2/J_1$ ratio generic. Numerically, we use complete and Lanczos diagonalization as well as the infinite time-evolving block decimation (iTEBD) method. Analytically we employ (non-)Abelian bosonization. Additionally for the sawtooth chain, we provide an analytical description in terms of flat bands and localized magnons. By considering a specific pattern for the $g$-factor anisotropy for both models, we show that a small anisotropy significantly enhances a magnetization plateau at half saturation. For the magnetization of the frustrated chain, we show the destruction of the $1/3$ of the full saturation plateau in favor of the creation of a plateau at half-saturation. For large anisotropies, the existence of an additional plateau at zero magnetization is possible. Here and at higher magnetic fields, the system is locked in the half-saturation plateau, never reaching full saturation.