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Annalen der Physik

Quantum Phase Transition of a Quantum Mixed Spin Chain by Employing Density Matrix Renormalization Group Method

Sheng Yang, Jing‐Bo Xu

Published Mar 31, 2021• 3 citations
Publisher

Abstract

Abstract The fidelity susceptibility and trace distance in the valence‐bond‐solid (VBS) transition of a quantum mixed spin chain is investigated, which consists of unit cells arrayed as 1/2‐1/2‐1‐1 with alternating Heisenberg antiferromagnetic exchange couplings, by using the density matrix renormalization group (DMRG) method in the matrix product state (MPS) form. It is observed that the fidelity susceptibility and the first derivative of the trace distance display explicit divergence near the quantum critical point. The information about the quantum criticality, such as the quantum critical point and correlation length critical exponent, is extracted from the finite size scaling behaviors. The obtained quantum critical point is more accurate than previous work, and the existence of the scaling function of the trace distance near the critical point is observed numerically for the first time. In addition, it is also found that the trace distance between two sites can supplement the VBS picture for describing the change of the ground state as the control parameter is varied through the quantum critical point.

DOI
10.1002/andp.202100011

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Abstract The fidelity susceptibility and trace distance in the valence‐bond‐solid (VBS) transition of a quantum mixed spin chain is investigated, which consists of unit cells arrayed as 1/2‐1/2‐1‐1 with alternating Heisenberg antiferromagnetic exchange couplings, by using the density matrix renormalization group (DMRG) method in the matrix product state (MPS) form. It is observed that the fidelity susceptibility and the first derivative of the trace distance display explicit divergence near the quantum critical point. The information about the quantum criticality, such as the quantum critical point and correlation length critical exponent, is extracted from the finite size scaling behaviors. The obtained quantum critical point is more accurate than previous work, and the existence of the scaling function of the trace distance near the critical point is observed numerically for the first time. In addition, it is also found that the trace distance between two sites can supplement the VBS picture for describing the change of the ground state as the control parameter is varied through the quantum critical point.
The deconfined quantum critical point was proposed as a second-order quantum phase transition between two broken-symmetry phases beyond the Landau-Ginzburg-Wilson paradigm. However, numerical studies cannot completely rule out a weakly-first-order transition because of strong violations of finite-size scaling. We demonstrate that the fidelity is a simple probe to study the deconfined quantum critical point. We study the ground-state fidelity susceptibility close to the deconfined quantum critical point in a spin chain using the large-scale finite-size density-matrix renormalization-group method. We find that the finite-size scaling of the fidelity susceptibility obeys the conventional scaling behavior for continuous phase transitions, supporting the idea that the deconfined quantum phase transition is continuous. We numerically determine the deconfined quantum critical point and the associated correlation length critical exponent from the finite-size-scaling theory of the fidelity susceptibility. Our results are consistent with recent results obtained directly from the matrix product states for infinite-size lattices using others observables. Our work provides a useful probe to study critical behaviors at the deconfined quantum critical point from the concept of quantum information.
= 1 andS=3/2dimers is studied in detail as a function of two considered exchange interactions ascribed to similar and dissimilar spin pairs. At zero magnetic field, the ferrimagnetic mixed spin-(1, 1, 3/2, 3/2) Heisenberg tetramer chain displays, depending on a mutual interplay between two considered exchange interactions, three distinct gapped valence-bond-solid phases separated by gap-closing quantum critical points. Using density-matrix renormalization group calculations we construct the full ground-state phase diagram as a function of the interaction ratio and magnetic field, which exhibits besides three gapped valence-bond-solid phases special Kosterlitz-Thouless and topological quantum critical points. A tangential finite-size scaling analysis is employed to obtain precise estimates of the zero-field valence-bond-solid transitions and unveil their common logarithmic correction to a power-law scaling of the correlation length.
We discuss zero‐temperature phase transititions from an antiferromagnetic ground state into a valence‐bond‐solid (VBS) in two different square‐lattice quantum spin models. In one limit these models reduce to the standard S = 1/2 XY and Heisenberg models, respectively, which have long‐range antiferromagnetic order at T = 0. Introducing particular types of four‐spin interactions, amenable to quantum Monte Carlo (QMC) simulation without negative‐sign problems, drives phase transitions into VBS states with broken Z4 symmetry. It has recently been argued that continuous quantum phase transitions of this nature represent a new class of deconfined quantum‐critical points [Senthil et al., Science 303, 1490 (2004)], which are not captured within the standard Ginzburg‐Landau framework of phase transitions. Here we discuss a QMC study aimed at accurately characterizing the transition in the XY model. We find some evidence that the transition is actually weakly first‐order, but other scenarios, such as a continuous transition violating hyperscaling, also cannot be completely ruled out. For the Heisenberg model with four‐spin interactions we have confirmed that a VBS state exists. Preliminary finite‐size scaling results show consistency with a continuous transition into the Néel state, with dynamic exponent z = 1. This model is thus a candidate for deconfined quantum criticality.
We study quantum mixed-spin chains consisting of two integer spins S1{S}^{1}S1 and two half integer spins S2{S}^{2}S2 arrayed as ${S}^{1}\ensuremath{-}{S}^{1}\ensuremath{-}{S}^{2}\ensuremath{-}{S}^{2}$ in a unit cell with antiferromagnetic nearest-neighbor couplings J1{J}_{1}J1​ (J2){(J}_{2})(J2​) between the spins of equal (different) magnitudes. By varying the ratio of two competing couplings, $\ensuremath{\alpha}{=J}_{2}{/J}_{1},$ the systems undergo successive quantum phase transitions between different valence-bond-solid (VBS) states accompanied by the vanishing of the energy gap. By a quantum Monte Carlo simulation with an improved loop cluster algorithm, we find one critical point ${\ensuremath{\alpha}}_{c}=0.762(1)$ in a $1\ensuremath{-}1\ensuremath{-}\frac{1}{2}\ensuremath{-}\frac{1}{2}$ chain and two critical points ${\ensuremath{\alpha}}_{c1}=0.479(1)$ and ${\ensuremath{\alpha}}_{c2}=1.318(1)$ in a $1\ensuremath{-}1\ensuremath{-}\frac{3}{2}\ensuremath{-}\frac{3}{2}$ chain, respectively. The calculated expectation values of the Lieb-Shultz-Mattis twist operator show the characteristic features of the single VBS transition in the $1\ensuremath{-}1\ensuremath{-}\frac{1}{2}\ensuremath{-}\frac{1}{2}$ chain and two successive VBS transitions in the $1\ensuremath{-}1\ensuremath{-}\frac{3}{2}\ensuremath{-}\frac{3}{2}$ chain at these critical points.
We study the fidelity susceptibility in an antiferromagnetic spin-1 XXZ chain numerically. By using the density-matrix renormalization group method, the effects of the alternating single-site anisotropy D on fidelity susceptibility are investigated. Its relation with the quantum phase transition is analyzed. It is found that the quantum phase transition from the Haldane spin liquid to periodic Néel spin solid can be well characterized by the fidelity. Finite size scaling of fidelity susceptibility shows a power-law divergence at criticality, which indicates the quantum phase transition is of second order. The results are confirmed by the second derivative of the ground-state energy. We also study the relationship between the entanglement entropy, the Schmidt gap and quantum phase transitions. Conclusions drawn from these quantum information observables agree well with each other.
Abstract 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.