Infinite Matrix Product State Research Articles

Physical observable for one-dimensional the spin-1 Blume–Capel model

We carry out an extensive study, within the infinite matrix product state algorithm, one-dimensional spin-1 Blume–Capel model on the spin-chain and subject to a changeable external magnetic field. The phase transition points are obtained by investigating the behavior of physical observable and as well as exacting the critical exponent from the scaling relationship. The final results are achieved for the physical observable-quantum entanglement entropy, local order parameter and its scaling: the scaling relationship between quantum entanglement entropy and truncation dimension, the scaling of Schmidt decomposition coefficient λmax and truncation dimension, the critical exponent of local order parameter near the phase transition point is extracted with the scaling behavior for the model with exchange coupling γ=0.6. These techniques are an easy way to exact the critical exponent for quantum phase transitions.

Quantum phase transition and topological order scaling in spin-1 bond-alternating Heisenberg model with Dzyaloshinskii-Moriya interaction

<sec>Quantum phase transitions are driven by quantum fluctuations due to the uncertainty principle in many-body physics. In quantum phase transitions, the ground-state changes dramatically. The quantum entanglement, specific heat, magnetization and other physical quantities diverge according to certain functions, and show specific scaling behaviors. In addition, there is a topological quantum phase transition beyond the conventional Landau-Ginzburg-Wilson paradigm, which is relevant to emergent phenomena in strongly correlated electron systems, with topological nonlocal order parameters as a salient feature. Thus, topological order is a new paradigm in the study of topological quantum phase transitions.</sec><sec>To investigate competition mechanism of the different quantum spin interactions, in this paper, the one-dimensional spin-1 bond-alternating Heisenberg model with Dzyaloshinskii-Moriya (DM) interaction is considered. The DM interaction drives the quantum fluctuations resulting in a phase transition. By using the one-dimensional infinite matrix product state algorithm in tensor network representation, the quantum entanglement entropy and order parameters are calculated from the ground-state function. The numerical result shows that with the change of bond alternating strength, there is a quantum phase transition from the topological ordered Haldane phase to the local ordered dimer phase. Based on the von Neumann entropy and order parameter, the phase diagram of this model is obtained. There is a critical line that separates the Haldane and the dimer phase. The DM interaction inhibits the dimerization of the quantum spin system and finally breaks the fully dimerization. Due to the fact that the structurally symmetry of system is broken, the local dimer order exists in the whole parameter range when the bond-alternative strength parameter changes. The first derivative of the local dimer order behaves as a peak corresponding to the critical point. Furthermore, from the numerical scaling of the first derivative of dimer order and the non-local string order near the phase transition point, the characteristic critical exponents <i>α</i> and <i>β</i> are obtained, respectively. It shows that the characteristic critical exponent <i>α</i> decreases, and <i>β</i> increases gradually with the interaction strength of DM increasing. The resulting state i.e. the anti-symmetric anisotropic DM interaction produced by spin-orbit coupling, affects the critical properties of the system in the phase transition. This reveals that the competition mechanism of the quantum spin interaction, also provides some guidance for the future study of the critical behavior in topological quantum phase transition with the DM interaction.</sec>

Cumulants and scaling functions of infinite matrix product states

The order parameter cumulants of infinite matrix product ground states are evaluated across a quantum phase transition. A scheme using the Binder cumulant, finite-entanglement scaling and scaling functions to obtain the critical point and exponents of the correlation length and cumulants is presented. Analogous to the scaling relations that relate the exponents of various thermodynamic quantities, a cumulant exponent relation is derived and used to check the consistency and relationship between the cumulant exponents. This scheme gives a numerically economical way of accurately obtaining the critical exponents. Examples of this scheme are shown for four one-dimensional models - the transverse field Ising model, the topological Kondo insulator, the S = 1 Heisenberg chain with single-ion anisotropy and the Bose-Hubbard model. A two-dimensional model is also exemplified in the square lattice transverse field Ising model on an infinite cylinder. These exemplary systems portray a variety of local and string order parameters as well as phase transition classes that can be studied with the scaling functions and infinite matrix product states.

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.

Precise Extrapolation of the Correlation Function Asymptotics in Uniform Tensor Network States with Application to the Bose-Hubbard and XXZ Models

We analyze the problem of extracting the correlation length from infinite matrix product states (MPS) and corner transfer matrix (CTM) simulations. When the correlation length is calculated directly from the transfer matrix, it is typically significantly underestimated for finite bond dimensions used in numerical simulation. This is true even when one considers ground states at a distance from the critical point. We introduce extrapolation procedure to overcome this problem. To that end we quantify how much the dominant part of the MPS/CTM transfer matrix spectrum deviates from being continuous. The latter is necessary to capture the exact asymptotics of the correlation function where the exponential decay is typically modified by an additional algebraic term. By extrapolating such a refinement parameter to zero, we show that we are able to recover the exact value of the correlation length with high accuracy. In a generic setting, our method reduces the error by a factor of $\sim 100$ as compared to the results without extrapolation and a factor of $\sim 10$ as compared to simple extrapolation schemes using bond dimension. We test our approach in a number of solvable models both in 1d and 2d. Subsequently, we apply it to 1d XXZ spin-$\frac32$ and the Bose-Hubbard models in a massive regime in the vicinity of BKT critical point. We then fit the scaling form of the correlation length and extract the position of the critical point and obtain results comparable or better than those of other state-of-the-art numerical methods. Finally, we show how the algebraic part of the correlation function asymptotics can be directly recovered from the scaling of the dominant form factor within our approach. Our method provides the means for detailed studies of phase diagrams of quantum models in 1d and, through the finite correlation length scaling of projected entangled pair states, also in 2d.

Finite Correlation Length Scaling with Infinite Projected Entangled-Pair States

We show how to accurately study 2D quantum critical phenomena using infinite projected entangled-pair states (iPEPS). We identify the presence of a finite correlation length in the optimal iPEPS approximation to Lorentz-invariant critical states which we use to perform a finite correlation-length scaling (FCLS) analysis to determine critical exponents. This is analogous to the one-dimensional (1D) finite entanglement scaling with infinite matrix product states. We provide arguments why this approach is also valid in 2D by identifying a class of states that despite obeying the area law of entanglement seems hard to describe with iPEPS. We apply these ideas to interacting spinless fermions on a honeycomb lattice and obtain critical exponents which are in agreement with Quantum Monte Carlo results. Furthermore, we introduce a new scheme to locate the critical point without the need of computing higher order moments of the order parameter. Finally, we also show how to obtain an improved estimate of the order parameter in gapless systems, with the 2D Heisenberg model as an example.

Topological phase transition and the effect of Hubbard interactions on the one-dimensional topological Kondo insulator

The effect of a local Kondo coupling and Hubbard interaction on the topological phase of the one-dimensional topological Kondo insulator (TKI) is numerically investigated using the infinite matrix-product state density-matrix renormalization group algorithm. The groundstate of the TKI is a symmetry-protected topological (SPT) phase protected by inversion symmetry. It is found that on its own, the Hubbard interaction that tends to force fermions into a one-charge per site order is insufficient to destroy the SPT phase. However when the local Kondo Hamiltonian term that favors a topologically trivial groundstate with a one-charge per site order is introduced, the Hubbard interaction assists in the destruction of the SPT phase. This topological phase transition occurs in the charge sector where the correlation length of the charge excitation diverges while the correlation length of the spin excitation remains finite. The critical exponents, central charge and the phase diagram separating the SPT phase from the topologically trivial phase are presented.

Topological orders and quantum phase transitions in a one-dimensional extended quantum compass model

By using the infinite time evolving block decimation in the presentation of infinite matrix product states, we study an extended quantum compass model (EQCM). This model does not only include extremely rich phase diagrams due to competitions of orbital degrees of freedom and anisotropic couplings between pseudospin-1/2 operators but also have the capacity to describe property of protected qubits for quantum computation which leads to lots of attentions paid to the phase boundaries of the EQCM. However, few attentions are paid to long-range topological string correlation order parameters of the EQCM. To study order parameters, one should understand spontaneous symmetry breaking which relates to Landau quantum phase transitions theory. Once spontaneous symmetry breaking happens, there should exist some local order which can be described by a local order parameter. This order parameter can be used to distinguish the phase from others. For continuous quantum phase transitions, in the critical regime, critical exponents can be extracted. Unfortunately, the long-range topological string correlation orders are beyond Landau quantum phase transitions theory, one can not directly use two paradigms of Landau-Ginzburg-Wilson. Usually, one can define a local order parameter by local magnetization. Naturally, one can also refer to this way to define the long-range topological string correlation order parameters by long-range topological string correlations on the following conditions, i.e. the quantum system undergoes a hidden spontaneous symmetry breaking; the long-range topological string correlation order parameter can be used to distinguish the phase from others; for continuous quantum phase transitions, the long-range topological string correlation order parameter satisfies scaling law when control parameter getting close to critical point. Based on above idea, in order to characterize the topological ordered phases and quantum phase transitions in the EQCM, even/odd long-range topological string correlations are introduced based on even/odd bonds. Hereafter, fidelity per lattice site, even/odd long-range topological string correlations, the saturation behavior of odd long-range topological string correlations and order parameters are calculated. The long-range topological string correlations show three distinguished behaviors which include decaying to zero, monotonic saturation and oscillatory saturation. By the above characterizations, oscillatory/monotonic odd long-range topological string correlation order parameter is derived. Then ground-state phase diagram of order parameters is computed which includes oscillatory/monotonic odd long-range topological string correlation order phase and antiferromagnetic phase. In the critical regime, critical exponent β=1/8 extracted from monotonic odd long-range topological string correlation order parameter and local magnetization shows the phase transition belongs to Ising universality. In addition, the phase transition points, the order of the phase transitions of fidelity show consistent with the results of order parameters.

Probing the anomalous dynamical phase in long-range quantum spin chains through Fisher-zero lines.

Using the framework of infinite matrix product states, the existence of an anomalous dynamical phase for the transverse-field Ising chain with sufficiently long-range interactions was first reported in J. C. Halimeh and V. Zauner-Stauber [Phys. Rev. B 96, 134427 (2017)10.1103/PhysRevB.96.134427], where it was shown that anomalous cusps arise in the Loschmidt-echo return rate for sufficiently small quenches within the ferromagnetic phase. In this work we further probe the nature of the anomalous phase through calculating the corresponding Fisher-zero lines in the complex time plane. We find that these Fisher-zero lines exhibit a qualitative difference in their behavior, where, unlike in the case of the regular phase, some of them terminate before intersecting the imaginary axis, indicating the existence of smooth peaks in the return rate preceding the cusps. Additionally, we discuss in detail the infinite matrix product state time-evolution method used to calculate Fisher zeros and the Loschmidt-echo return rate using the matrix product state transfer matrix. Our work sheds further light on the nature of the anomalous phase in the long-range transverse-field Ising chain, while the numerical treatment presented can be applied to more general quantum spin chains.

Dynamical phase diagram of quantum spin chains with long-range interactions

Using an infinite Matrix Product State (iMPS) technique based on the time-dependent variational principle (TDVP), we study two major types of dynamical phase transitions (DPT) in the one-dimensional transverse-field Ising model (TFIM) with long-range power-law ($\propto1/r^{\alpha}$ with $r$ inter-spin distance) interactions out of equilibrium in the thermodynamic limit -- \textit{DPT-I}: based on an order parameter in a (quasi-)steady state, and \textit{DPT-II}: based on non-analyticities (cusps) in the Loschmidt-echo return rate. We construct the corresponding rich dynamical phase diagram, whilst considering different quench initial conditions. We find a nontrivial connection between both types of DPT based on their critical lines. Moreover, and very interestingly, we detect a new DPT-II dynamical phase in a certain range of interaction exponent $\alpha$, characterized by what we call \textit{anomalous cusps} that are distinct from the \textit{regular cusps} usually associated with DPT-II. Our results provide the characterization of experimentally accessible signatures of the dynamical phases studied in this work.

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.

Infinite matrix product states versus 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.

Critical phenomena and Kibble–Zurek scaling in the long-range quantum Ising chain

We investigate an extension of the quantum Ising (QI) model in one spatial dimension including long-range interactions in its statics and dynamics with possible applications from heteronuclear polar molecules in optical lattices to trapped ions described by two-state spin systems. We introduce the statics of the system via both numerical techniques with finite size and infinite size matrix product states (MPSs) and a theoretical approaches using a truncated Jordan–Wigner transformation for the ferromagnetic and antiferromagnetic case and show that finite size effects have a crucial role shifting the quantum critical point of the external field by fifteen percent between thirty-two and around five-hundred spins. We numerically study the Kibble–Zurek hypothesis in the long-range QI model with MPSs. A linear quench of the external field through the quantum critical point yields a power-law scaling of the defect density as a function of the total quench time. For example, the increase of the defect density is slower for longer-range models and the critical exponent changes by twenty-five percent. Our study emphasizes the importance of such long-range interactions in statics and dynamics that could point to similar phenomena in a different setup of dynamical systems or for other models.

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.

Gaussian phase transition and critical exponents in spin-1 bond-alternative Heisenberg chains

The quantum Gaussian phase transition is investigated for the infinite spin-1 bond-alternative Heisenberg model in one spatial dimension. By using a tensor network representation with an infinite matrix product state approach, the ground state energy, bipartite entanglement entropy, non-local string order, and fidelity per lattice site are calculated to characterize the phase transition. At the quantum phase transition point, the scaling behavior of various physical observables with respect to the finite truncation dimension are discussed for the ground state wavefunctions. In addition, the central charge is extracted from the finite entanglement entropies and the finite correlation lengths. Furthermore, the various critical exponents of the string order are calculated. The characteristic critical exponents and the central charge determine the universality class of the phase transition.

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.

Integrable open spin chains related to infinite matrix product states

In this paper we study an su$(m)$-invariant open version of the Haldane-Shastry spin chain whose ground state can be obtained from the chiral correlator of the $c=m-1$ free boson boundary conformal field theory. We show that this model is integrable for a suitable choice of the chain sites depending on the roots of the Jacobi polynomial $P_N^{\beta-1,\beta'-1}$, where $N$ is the number of sites and $\beta,\beta'$ are two positive parameters. We also compute in closed form the first few nontrivial conserved charges arising from the twisted Yangian invariance of the model. We evaluate the chain's partition function, determine the ground state energy and deduce a complete description of the spectrum in terms of Haldane's motifs and a related classical vertex model. In particular, this description entails that the chain's level density is normally distributed in the thermodynamic limit. We also analyze the spectrum's degeneracy, proving that it is much higher than for a typical Yangian-invariant model.

Characterization of topological phases in the compass ladder model

The phase diagram of the quantum compass ladder model is investigated through numerical density matrix renormalization group based on infinite matrix product state algorithm and analytic effective perturbation theory. For this model we obtain two symmetry-protected topological phases, protected by a symmetry, and a topologically-trivial Z2-symmetry-breaking phase. The symmetry-protected topological phases—labeled by symmetry fractionalization—belong to different topological classes, where the complex-conjugate symmetry uniquely distinguishes them. An important result of this classification is that, as revealed by the nature of the Z2-symmetry-breaking phase, the associated quantum phase transitions are accompanied by an explicit symmetry breaking, and thus a local-order parameter conclusively identifies the phase diagram of the underlying model. This is in stark contrast to previous studies which require a non-local string order parameter to distinguish the corresponding quantum phase transitions. We numerically examine our results and show that the local-order parameter is related to the magnetization exponent .

Dimerized phase and entanglement in the one-dimensional spin-1 bilinear biquadratic model

Dimerized phase and quantum entanglement are investigated in the one-dimensional spin-1 bilinear biquadratic model. Employing the infinite matrix product state representation, groundstate wavefunctions are numerically obtained by using the infinite time evolving block decimation method in the infinite lattice system. From a bipartite entanglement measure of the groundstates, i.e., von Neumann entropy, the phase transition points can be clearly extracted. Moreover, the even-bond and odd-bond von Neumann entropies show two different values in the spontaneous dimerized phase. It implies that the quantum entanglement can distinguish the two degenerate groundstates. Then, we define a dimer entropy in the spontaneous dimerized phase. Comparing to the dimer order parameter, the dimer entropy can play a role of a local order parameter to characterize the spontaneous dimerized phase.

Quantum fidelity, string order parameter, and topological quantum phase transition in a spin-1/2 dimerized and frustrated Heisenberg chain

Topological quantum phase transitions are numerically investigated in a spin-1/2 dimerized and frustrated Heisenberg chain by using infinite matrix product state representation with the infinite time evolving block decimation method. Quantum fidelity approach is employed to detect the degenerate ground states and quantum phase transitions. By calculating the long-range string order parameters, we find two topological Haldane phases characterized by two long-range string orders. Also, continuous and discontinuous behaviors of von Neumann entropy show that phase transitions between two topological Haldane phases are topologically continuous and discontinuous quantum phase transitions. For the topologically continuous phase transition, the central charge at the critical point is obtained as c = 1, which means that the topologically continuous quantum phase transition belongs to the Gaussian universality class.