Time-evolving Block Decimation Method Research Articles

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.

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.

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.

Nonthermalized dynamics of flat-band many-body localization

We find that a flat-band fermion system with interactions and without disorders exhibits non-thermalized ergodicity-breaking dynamics, an analog of many-body localization (MBL). In the previous works, we observed flat-band many-body localization (FMBL) in the Creutz ladder model. The origin of FMBL is a compact localized state governed by local integrals of motion (LIOMs), which are to be obtained explicitly. In this work, we clarify the dynamical aspects of FMBL. We first study dynamics of two-particles, and find that the states are not substantially modified by weak interactions, but the periodic time evolution of entanglement entropy emerges as a result of a specific mechanism inherent in the system. On the other hand, as the strength of the interactions is increased, the modification of the states takes place with inducing instability of the LIOMs. Furthermore, many-body dynamics of the system at finite fillings is numerically investigated by time-evolving block decimation (TEBD) method. For a suitable choice of the filling, non-thermal and low entangled dynamics appears. This behavior is a typical example of the disorder-free FMBL.

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.

Entanglement Entropy on the Boundary of the Square-Lattice ±J Ising Model

A classical analogue of the entanglement entropy is calculated on the system boundary of the two-dimensional Edwards-Anderson model, where the nearest-neighbor interaction is stochastically chosen from +J and -J. The boundary spin distribution is obtained by means of the time-evolving block decimation (TEBD) method, where the random ensemble is created from the successive multiplications of position-dependent transfer matrices, whose width is up to N = 300. The random average of the entanglement entropy is calculated on the Nishimori line, and it is confirmed that the entanglement entropy shows critical singularity at the Nishimori point. The central charge of the boundary state is estimated.

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.

Ultrafast nonequilibrium evolution of excitonic modes in semiconductors

We study the time evolution of excitonic states after photo-excitation in the one-dimensional spin-less extended Falicov-Kimball model. Several numerical methods are employed and benchmarked against each other: time-dependent mean-field simulations, the second-Born approximation (2BA) within the Kadanoff-Baym formalism, the generalized Kadanoff-Baym Ansatz (GKBA) implemented with the 2BA and the infinite time-evolving block decimation (iTEBD) method. It is found that the GKBA gives the best agreement with iTEBD and captures the relevant physics. We find that excitations to the particle-hole continuum and resonant excitations of the equilibrium exciton result in a qualitatively different dynamics. In the former case, the exciton binding energy remains positive and the frequency of the corresponding coherent oscillations is smaller than the band gap. On the other hand, resonant excitations trigger a collective mode whose frequency is larger than the band gap. We discuss the origin of these different behaviors by evaluating the nonequilibrium susceptibility using the nonthermal distribution and a random phase approximation. The peculiar mode with frequency larger than the band gap is associated with a partial population inversion with a sharp energy cutoff. We also discuss the effects of the cooling by a phonon bath. We demonstrate the real-time development of coherence in the polarization, which indicates excitonic condensation out of equilibrium.

Universal scrambling in gapless quantum spin chains

Information scrambling, characterized by the out-of-time-ordered correlator (OTOC), has attracted much attention, as it sheds new light on chaotic dynamics in quantum many-body systems. The scale invariance, which appears near the quantum critical region in condensed matter physics, is considered to be important for the fast decay of the OTOC. In this paper, we focus on the one-dimensional spin-1/2 XXZ model, which exhibits quantum criticality in a certain parameter region, and investigate the relationship between scrambling and the scale invariance. We quantify scrambling by the averaged OTOC over the Pauli operator basis, which is related to the operator space entanglement entropy (OSEE). Using the infinite time-evolving block decimation method, we numerically calculate time dependence of the OSEE in the early-time region in the thermodynamic limit. We show that the averaged OTOC decays faster in the gapless region than in the gapped region. In the gapless region, the averaged OTOC behaves in the same manner regardless of the anisotropy parameter. This result is consistent with the fact that the low-energy excitations of the gapless region belong to the same universality class as the Tomonaga-Luttinger liquid with the central charge $c=1$. Furthermore, we estimate $c$ by fitting the numerical data of the OSEE with an analytical result of the two-dimensional conformal field theory, and confirm that $c$ is close to unity. Thus, our numerical results suggest that the scale invariance leads to a universal behavior of the OTOC that is independent of the anisotropic parameter, which reflects the universality of the two-dimensional conformal field theory at low temperatures. Although the one-dimensional XXZ model is integrable, our results suggest such a universal behavior of generic nonintegrable systems, because the Tomonaga-Luttinger liquid serves as a low-energy effective theory for many nonintegrable and integrable systems. On the other hand, the OTOC in our numerical result does not exhibit the exponential decay, as our parameter regime is far from the semiclassical limit.

Tensor-network approach to compute genuine multisite entanglement in infinite quantum spin chains

We devise a method based on the tensor-network formalism to calculate genuine multisite entanglement in ground states of infinite spin chains containing spin-1/2 or spin-1 quantum particles. The ground state is obtained by employing an infinite time-evolving block decimation method acting upon an initial matrix product state for the infinite spin system. We explicitly show how such infinite matrix product states with translational invariance provide a natural framework to derive the generalized geometric measure, a computable measure of genuine multisite entanglement, in the thermodynamic limit of quantum many-body systems with both spin-1/2 and higher-spin particles.

Quantum phase transition in the one-dimensional quantum Heisenberg XYZ model with Dzyaloshinskii–Moriya interaction

We investigated the quantum phase transition occurred in one-dimensional quantum Heisenberg XYZ model with Dzyaloshinskii–Moriya interaction via the infinite matrix product state representation with the infinite time evolving block decimation method. Entanglement entropy and local order parameter in and near the transition point are given. Scaling relation plays crucial roles on identifying a quantum system with a physically different phase. The scaling relation of the entanglement entropy, local order parameter and finite correlation length with the truncation dimension are also obtained. All the interesting results give a theoretical justification for the high accuracy of infinite time evolved block decimation algorithm which works in the thermodynamical limit.

Hybrid Semiclassical Theory of Quantum Quenches in One-Dimensional Systems.

We develop a hybrid semiclassical method to study the time evolution of one-dimensional quantum systems in and out of equilibrium. Our method handles internal degrees of freedom completely quantum mechanically by a modified time-evolving block decimation method while treating orbital quasiparticle motion classically. We can follow dynamics up to time scales well beyond the reach of standard numerical methods to observe the crossover between preequilibrated and locally phase equilibrated states. As an application, we investigate the quench dynamics and phase fluctuations of a pair of tunnel-coupled one-dimensional Bose condensates. We demonstrate the emergence of soliton-collision-induced phase propagation, soliton-entropy production, and multistep thermalization. Our method can be applied to a wide range of gapped one-dimensional systems.

Impurity and soliton dynamics in a Fermi gas with nearest-neighbor interactions

We study spinless fermions with repulsive nearest-neighbor interactions perturbed by an impurity particle or a local potential quench. Using the numerical time-evolving block decimation method and a simplified analytic model, we show that the pertubations create a soliton-antisoliton pair. If solitons are already present in the bath, the two excitations have a drastically different dynamics: The antisoliton does not annihilate with the solitons and is therefore confined close to its origin while the soliton excitation propagates. We discuss the consequences for experiments with ultracold gases.

Quantum entanglement and critical exponents in one-dimensional spin-1 bond-alternating XXZ chains

The characterization of the quantum phase transition in a lowdimensional system has attracted a considerable amount of attention in quantum manybody systems. As one of the fundamental models in quantum magnetism, spin-1 models have richer phase diagrams and show more complex physical phenomena. In the spin-1 antiferromagnetic XXZ model, the Haldane phase and the Nel phase are the gapped topologic phases which cannot be characterized by the local order parameters. To characterize the nature in such phases, one has to calculate the non-local long range order parameters. Normally, the non-local order parameter in the topological phase is obtained from the extrapolation of finite-sized system in numerical study. However, it is difficult to extract the critical exponents with such an extrapolated non-local order parameter due to the numerical accuracy. In a recently developed tensor network representation, i.e., the infinite matrix product state (iMPS) algorithm, it was shown that the non-local order can be directly calculated from a very large lattice distance in an infinite-sized system rather than an extrapolated order parameter in a finite-sized system. Therefore, it is worthwhile using this convenient technique to study the non-local orders in the topological phases and characterize the quantum criticalities in the topological quantum phase transitions. In this paper, by utilizing the infinite matrix product state algorithm based on the tensor network representation and infinite time evolving block decimation method, the quantum entanglement, fidelity, and critical exponents of the topological phase transition are investigated in the one-dimensional infinite spin-1 bond-alternating XXZ Heisenberg model. It is found that there is always a local dimerization order existing in the whole parameter range when the bond-alternative strength parameter changes from 0 to 1. Also, due to the effect of the bond-alternating, there appears a quantum phase transition from the long-rang ordering topological Nel phase to the local ordering dimerization phase. The von Neumann entropy, fidelity per lattice site, and order parameters all give the same phase transition point at c = 0.638. To identify the type of quantum phase transition, the central charge c 0.5 is extracted from the ground state von Neumann entropy and the finite correlation length, which indicates that the phase transition belongs to the two-dimensional Ising universality class. Furthermore, it is found that the Nel order and the susceptibility of Nel order have power-law relations to |-c|. From the numerical fitting of the Nel order and its susceptibility, we obtain the characteristic critical exponents ' = 0.236 and ' = 0.838. It indicates that such critical exponents from our method characterize the nature of the quantum phase transition. Our critical exponents from the iMPS method can provide guidance for studying the properties of the phase transition in quantum spin systems.

Decoherence of an impurity in a one-dimensional fermionic bath with mass imbalance

We study the transport, decoherence and dissipation of an impurity interacting with a bath of free fermions in a one-dimensional lattice. Numerical simulations are made with the time-evolving block decimation method. We introduce a mass imbalance between the impurity and bath particles and find that the fastest decoherence occurs for a light impurity in a bath of heavy particles. By contrast, the fastest dissipation of energy occurs when the masses are equal. We present a simple model for decoherence in the heavy bath limit, and a linear density response description of the interaction which predicts maximum dissipation for equal masses.

The scaling behavior for one-dimensional quantum two-spin and three-spin model

The scaling behavior of physical observable in the phase transition point for one-dimensional quantum two-spin and three-spin model is investigated by employing the infinite matrix product state representation with the infinite time evolving block decimation method. The analytical results, which are checked numerically finding excellent agreement for the model with central charge c = 4/5 , demonstrate that the scaling behavior with finite truncation dimension $ \chi$ in local Hilbert space can be employed to characterize the one-dimensional quantum spin model. Meanwhile, the infinite time evolving block decimation algorithm by the infinite matrix product state representation can be a quite accurate method to obtain the critical properties at continuous phase transition point.

Symmetry-protected intermediate trivial phases in quantum spin chains

Symmetry-protected trivial (SPt) phases of matter are the product-state analogue of symmetry-protected topological (SPT) phases. This means, SPt phases can be adiabatically connected to a product state by some path that preserves the protecting symmetry. Moreover, SPt and SPT phases can be adiabatically connected to each other when interaction terms that break the symmetries protecting the SPT order are added in the Hamiltonian. It is also known that spin-1 SPT phases in quantum spin chains can emerge as effective intermediate phases of spin-2 Hamiltonians. In this paper we show that a similar scenario is also valid for SPt phases. More precisely, we show that for a given spin-2 quantum chain, effective intermediate spin-1 SPt phases emerge in some regions of the phase diagram, these also being adiabatically connected to non-trivial intermediate SPT phases. We characterize the phase diagram of our model by studying quantities such as the entanglement entropy, symmetry-related order parameters, and 1-site fidelities. Our numerical analysis uses Matrix Product States (MPS) and the infinite Time-Evolving Block Decimation (iTEBD) method to approximate ground states of the system in the thermodynamic limit. Moreover, we provide a field theory description of the possible quantum phase transitions between the SPt phases. Together with the numerical results, such a description shows that the transitions may be described by Conformal Field Theories (CFT) with central charge c=1. Our results are in agreement, and further generalize, those in [Y. Fuji, F. Pollmann, M. Oshikawa, Phys. Rev. Lett. 114, 177204 (2015)].

Flux quench in a system of interacting spinless fermions in one dimension

We study a quantum quench in a one-dimensional spinless fermion model (equivalent to the XXZ spin chain), where a magnetic flux is suddenly switched off. This quench is equivalent to imposing a pulse of electric field and therefore generates an initial particle current. This current is not a conserved quantity in presence of a lattice and interactions and we investigate numerically its time-evolution after the quench, using the infinite time-evolving block decimation method. For repulsive interactions or large initial flux, we find oscillations that are governed by excitations deep inside the Fermi sea. At long times we observe that the current remains non-vanishing in the gapless cases, whereas it decays to zero in the gapped cases. Although the linear response theory (valid for a weak flux) predicts the same long-time limit of the current for repulsive and attractive interactions (relation with the zero-temperature Drude weight), larger nonlinearities are observed in the case of repulsive interactions compared with that of the attractive case.

Topological phase boundary in a generalized Kitaev model**Project supported by the National Basic Research Program of China (Grant No. 2012CB921704).

We study the effects of the next-nearest-neighbor hopping and nearest-neighbor interactions on topological phases in a one-dimensional generalized Kitaev model. In the noninteracting case, we define a topological number and calculate exactly the phase diagram of the system. With addition of the next-nearest-neighbor hopping, the change of phase boundary between the topological and trivial regions can be described by an effective shift of the chemical potential. In the interacting case, we obtain the entanglement spectrum, the degeneracies of which correspond to the topological edge modes, by using the infinite time-evolving block decimation method. The results show that the interactions change the phase boundary as adding an effective chemical potential which can be explained by the change of the average number of particles.