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arXiv:1105.3014

Ground-state fidelity and tensor network states for quantum spin tubes

Aimin Chen, Qian‐Qian Shi, Jinhua Liu, Huan-Qiang Zhou

Published May 16, 2011• 0 citations
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Abstract

An efficient algorithm is developed for quantum spin tubes in the context of the tensor network representations. It allows to efficiently compute the ground-state fidelity per lattice site, which in turn enables us to identify quantum critical points, at which quantum spin tubes undergo quantum phase transitions. As an illustration, we investigate the isosceles spin 1/2 antiferromagnetic three-leg Heisenberg tube. Our simulation results suggest that two Kosterlitz-Thouless transitions occur as the degree of asymmetry of the rung interaction is tuned, thus offering an alternative route towards a resolution to the conflicting results on this issue arising from the density matrix renormalization group.

DOI
10.48550/arxiv.1105.3014

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An efficient algorithm is developed for quantum spin tubes in the context of the tensor network representations. It allows to efficiently compute the ground-state fidelity per lattice site, which in turn enables us to identify quantum critical points, at which quantum spin tubes undergo quantum phase transitions. As an illustration, we investigate the isosceles spin 1/2 antiferromagnetic three-leg Heisenberg tube. Our simulation results suggest that two Kosterlitz-Thouless transitions occur as the degree of asymmetry of the rung interaction is tuned, thus offering an alternative route towards a resolution to the conflicting results on this issue arising from the density matrix renormalization group.
We have developed an efficient tensor network algorithm for spin ladders, which generates ground-state wave functions for infinite-size quantum spin ladders. The algorithm is able to efficiently compute the ground-state fidelity per lattice site, a universal phase transition marker, thus offering a powerful tool to unveil quantum many-body physics underlying spin ladders. To illustrate our scheme, we consider the two-leg and three-leg Heisenberg spin ladders with staggering dimerization, the two-leg Heisenberg spin ladder with cyclic four-spin exchange, and the ferromagnetic frustrated two-leg ladder. The ground-state phase diagrams thus yielded are reliable, compared with the previous studies based on the the exact diagonalization and the density matrix renormalization group. Our results indicate that the ground-state fidelity per lattice site successfully captures quantum criticalities in spin ladders.
We have developed an efficient tensor network algorithm for spin ladders, which generates ground-state wave functions for infinite-size quantum spin ladders. The algorithm is able to efficiently compute the ground-state fidelity per lattice site, a universal phase transition marker, thus offering a powerful tool to unveil quantum many-body physics underlying spin ladders. To illustrate our scheme, we consider the two-leg and three-leg Heisenberg spin ladders with staggering dimerization. The ground-state phase diagram thus yielded is reliable, compared with the previous studies based on the density matrix renormalization group. Our results indicate that the ground-state fidelity per lattice site successfully captures quantum criticalities in spin ladders.
We investigate a two-leg Heisenberg spin ladder with cyclic four-spin exchange by exploiting a newly-developed tensor network algorithm. The algorithm allows to efficiently compute the ground-state fidelity per lattice site, which enables us to establish the ground-state phase diagram for quantum lattice many-body systems. The latter is based on the observation that, for an infinite-size system, any singularity on a ground-state fidelity surface characterizes a critical point, at which the system undergoes a phase transition. For the two-leg Heisenberg spin-1/2 ladder with cyclic four-spin exchange, six different phases are identified: the ferromagnetic phase, the rung singlet phase, the staggered dimer phase, the scalar chirality phase, the dominant vector chirality region, and the dominant collinear spin region. Our findings are in a good agreement with the previous studies from the exact diagonalization and the density-matrix renormalization group.
We study quantum phase transitions in the asymmetric variation of the three-leg Heisenberg tube for half-odd-integer spin, with a modulation of one of the rung exchange couplings ${J}_{\ensuremath{\perp}}^{\ensuremath{'}}$ while the other two are kept constant ${J}_{\ensuremath{\perp}}$. We focus on the strong rung-coupling regime ${J}_{\ensuremath{\perp}}\ensuremath{\gg}{J}_{\ensuremath{\parallel}}$, where ${J}_{\ensuremath{\parallel}}$ is the leg coupling, and analyze the effective spin-orbital model with a transverse crystal field in detail. Applying the Abelian bosonization to the effective model, we find that the system is in the dimer phase for the general half-odd-integer-spin cases without the rung modulation; the phase transition between the dimer and Tomonaga-Luttinger-liquid phases induced by the rung modulation is of the SU(2)-symmetric Berezinskii-Kosterlitz-Thouless type. Moreover, we perform a level spectroscopy analysis for the effective model for spin-1/2 using exact diagonalization, to determine the precise transition point $|{J}_{\ensuremath{\perp}}^{\ensuremath{'}}\ensuremath{-}{J}_{\ensuremath{\perp}}|/{J}_{\ensuremath{\parallel}}\ensuremath{\sim}0.283$ in the strong rung-coupling limit. The presence of the dimer phase in a small but finite region is also confirmed by a density-matrix renormalization group calculation on the original spin-tube model.
A systematic analysis is performed for quantum phase transitions in a two-dimensional anisotropic spin-1/2 antiferromagnetic XYX model in an external magnetic field. With the help of an innovative tensor network algorithm, we compute the fidelity per lattice site to demonstrate that the field-induced quantum phase transition is unambiguously characterized by a pinch point on the fidelity surface, marking a continuous phase transition. We also compute an entanglement estimator, defined as a ratio between the one-tangle and the sum of squared concurrences, to identify both the factorizing field and the critical point, resulting in a quantitative agreement with quantum Monte Carlo simulation. In addition, the local order parameter is "derived" from the tensor network representation of the system's ground-state wave functions.
We investigate quantum phase transitions of the S=12S=\frac{1}{2}S=21​ three-leg antiferromagnetic spin tube with asymmetric interchain (rung) exchange interactions. On the basis of the electron tube system, we propose a useful effective theory to give the global phase diagram of the asymmetric spin tube. In addition, using other effective theories we raise the reliability of the phase diagram. The density-matrix renormalization-group and the numerical diagonalization analyses show that the finite spin gap appears in a narrow region around the rung-symmetric line, in contrast to a recent paper by Nishimoto and Arikawa, [Phys. Rev. B 78, 054421 (2008)]. The numerical calculations indicate that this global phase diagram obtained by use of the effective theories is qualitatively correct. In the gapless phase on the phase diagram, the numerical data are fitted by a finite-size scaling in the c=1c=1c=1 conformal field theory. We argue that all the phase transitions between the gapful and gapless phases belong to the Berezinskii-Kosterlitz-Thouless universality class.