Infinite Matrix Product State Representation Research Articles

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.

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.

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.

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.

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.

Quantum ground-state entanglement for spin-1/2 alternating Heisenberg model in one spatial dimension

Quantum entanglement for a one-dimensional quantum antiferromagnetic-ferromagnetic spin-1/2 alternating Heisenberg chain is investigated by employing the infinite matrix product state representation. The entanglement entropies for the model are shown and the peaks of the entropies tell us the quantum phase transitions happen when the control parameter crosses its critical value. Besides, spontaneous magnetization is given and the critical exponent is extracted from the data. Finally, a finite-entanglement scaling of the von Neumann entropy and finite-entanglement scaling exponent are given with different truncation dimension, from which the type of the transition and its conformal anomaly parameter c are obtained at the critical point.

Long-range string orders and topological quantum phase transitions in the one-dimensional quantum compass model

In order to investigate the quantum phase transition in the one-dimensional quantum compass model, we numerically calculate non-local string correlations, entanglement entropy and fidelity per lattice site by using the infinite matrix product state representation with the infinite time evolving block decimation method. In the whole range of the interaction parameters, we find that four distinct string orders characterize the four different Haldane phases and the topological quantum phase transition occurs between the Haldane phases. The critical exponents of the string order parameters β = 1/8 and the cental charges c = 1/2 at the critical points show that the topological phase transitions between the phases belong to an Ising type of universality classes. In addition to the string order parameters, the singularities of the second derivative of the ground state energies per site, the continuous and singular behaviors of the Von Neumann entropy and the pinch points of the fidelity per lattice site manifest that the phase transitions between the phases are of the second-order, in contrast to the first-order transition suggested in previous studies.

Symmetry fractionalization: symmetry-protected topological phases of the bond-alternating spin-1/2 Heisenberg chain

We study different phases of the one-dimensional bond-alternating spin-1/2 Heisenberg model by using the symmetry fractionalization mechanism. We employ the infinite matrix-product state representation of the ground state (through the infinite-size density matrix renormalization group algorithm) to obtain inequivalent projective representations and commutation relations of the (unbroken) symmetry groups of the model, which are used to identify the different phases. We find that the model exhibits trivial as well as symmetry-protected topological phases. The symmetry-protected topological phases are Haldane phases on even/odd bonds, which are protected by the time-reversal (acting on the spin as σ → −σ), parity (permutation of the chain about a specific bond), and dihedral (π-rotations about a pair of orthogonal axes) symmetries. Additionally, we investigate the phases of the most general two-body bond-alternating spin-1/2 model, which respects the time-reversal, parity, and dihedral symmetries, and obtain its corresponding twelve different types of the symmetry-protected topological phases.

The finite scaling for S = 1 XXZ chains with uniaxial single-ion-type anisotropy

The scaling behavior of criticality for spin-1 XXZ chains with uniaxial single-ion-type anisotropy is investigated by employing the infinite matrix product state representation with the infinite time evolving block decimation method. At criticality, the accuracy of the ground state of a system is limited by the truncation dimension χ of the local Hilbert space. We present four evidences for the scaling of the entanglement entropy, the largest eigenvalue of the Schmidt decomposition, the correlation length, and the connection between the actual correlation length ξ and the energy. The result shows that the finite scalings are governed by the central charge of the critical system. Also, it demonstrates that the infinite time evolving block decimation algorithm by the infinite matrix product state representation can be a quite accurate method to simulate the critical properties at criticality.

Topological quantum phase transition in bond-alternating spin-12Heisenberg chains

We investigate string correlations in an infinite-size spin-1/2 bond-alternating Heisenberg chain. By employing the infinite matrix product state representation with the infinite time evolving block decimation method, a finite string correlation for extremely large lattice distances is directly observed, contrast to an extrapolated extreme value for finite size chains. We find that a topological quantum phase transition occurs between two different phases separated and characterized by two different long-range string orders in the space of bond-alternating interactions. Also, the critical exponent $\beta$ from the long-range string orders is obtained as $\beta=1/12$ and the central charge at the critical point is obtained as $c \simeq 1$, which shows that the topological phase transition belongs to the Gussian universality class. In addition, it is shown that, for the topological quantum phase transition, the phase boundary can be captured by the singular behavior of the von Neumann entropy and the pinch point of the fidelity per site.

Non-local Correlations in the Haldane Phase for an XXZ Spin-1 Chain: A Perspective from Infinite Matrix Product State Representation

String correlations are investigated in an infinite-size XXZ spin-1 chain. By using the infinite matrix product state representation, we calculate a long-range string order. In the XY phase, the string correlations decay within a relatively very large lattice distance, which makes a finite-size study difficult to verify the non-existence of the string order. Thus, in the Haldane phase, the non-vanishing string correlations in the limit of a very large distance allow to characterize the phase boundaries to the XY phase and the Neel phase, which implies that the transverse long-range string order is the order parameter for the Haldane phase. In addition, the singular behaviors of the von Neumann entropy and the fidelity per lattice site are shown to capture clearly the phase transition points that are consistent with the results from the string order. The estimated critical points including a BKT transition from the XY phase to the Haldane phase agree well with the previous results: $\Delta_{c2} = 0$ for the XY-Haldane phase transition and $\Delta_{c3} = 1.185$ for the Haldane-N'eel phase transition from the density renormalization group. From a finite-entanglement scaling of the von Neumann entropy with respect to the truncation dimension, the central charges are found to be $c \simeq 1.0$ at $\Delta_{c2} = 0$ and $c \simeq 0.5$ at $\Delta_{c3} = 1.185$, respectively, which shows that the XY-Haldane phase transition at $\Delta_{c2} = 0$ belongs to the Heisenberg universality class, while the Haldane-Neel phase transition at $\Delta_{c2} = 1.185$ belongs to the two-dimensional classical Ising universality class. It is also shown that, the long-range order parameters and the von Neumann entropy, as well as the fidelity per site approach, can be applied to characterize quantum phase transitions as a universal phase transition indicator for one-dimensional lattice many-body systems.

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.