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arXiv:0804.0537 · Physical Review A

Fidelity approach to Gaussian transitions

Yu-Chin Tzeng, Hsiang-Hsuan Hung, Yung‐Chung Chen, Min-Fong Yang

Published Jun 13, 2008• 46 citations
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Abstract

The fidelity approach to the Gaussian transitions in spin-one $\mathit{XXZ}$ spin chains with three different values of Ising-type anisotropy $\ensuremath{\lambda}$ is analyzed by means of the density matrix renormalization group technique for systems of large sizes. We find that, despite the success in the cases of $\ensuremath{\lambda}=2.59$ and 1, the fidelity susceptibility fails to detect the Gaussian transition for $\ensuremath{\lambda}=0.5$. Thus our results demonstrate the limitation of the fidelity susceptibility in characterizing quantum phase transitions, which was proposed recently in general frameworks.

DOI
10.1103/physreva.77.062321

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The fidelity approach to the Gaussian transitions in spin-one XXZ\mathit{XXZ}XXZ spin chains with three different values of Ising-type anisotropy \ensuremathλ\ensuremath{\lambda}\ensuremathλ is analyzed by means of the density matrix renormalization group technique for systems of large sizes. We find that, despite the success in the cases of \ensuremathλ=2.59\ensuremath{\lambda}=2.59\ensuremathλ=2.59 and 1, the fidelity susceptibility fails to detect the Gaussian transition for \ensuremathλ=0.5\ensuremath{\lambda}=0.5\ensuremathλ=0.5. Thus our results demonstrate the limitation of the fidelity susceptibility in characterizing quantum phase transitions, which was proposed recently in general frameworks.
By means of the density matrix renormalization group technique, the scaling relation of the fidelity susceptibility proposed recently is verified for the spin-one XXZXXZXXZ spin chain with an on-site anisotropic term. Moreover, from the results of both the fidelity susceptibility and the entanglement entropy, the critical points and some of the corresponding critical exponents are determined through a proper finite-size scaling analysis, and these values agree with the findings in the literature. Thus our work provides a numerical support of the use of the fidelity in detecting quantum phase transitions.
The phase diagram of a quantum XYXYXY spin chain with Gaussian-distributed random anisotropies and transverse fields is investigated, with focus on the fidelity susceptibility, a recently introduced quantum information theoretical measure. Monitoring the finite-size scaling of the probability distribution of this quantity as well as its average and typical values, we detect a disorder-induced disappearance of criticality and the emergence of Griffiths phases in this model. It is found that the fidelity susceptibility is not self-averaging near the disorder-free quantum-critical lines. At the Ising critical point the fidelity susceptibility scales as a disorder-strength independent stretched exponential of the system size, in contrast with the quadratic scaling at the corresponding point in the disorder-free XYXYXY chain. Along the line where the average anisotropy vanishes the fidelity susceptibility appears to scale extensively, whereas in the disorder-free case this point is quantum critical with quadratic finite-size scaling.
We study the fidelity susceptibility of a quantum antiferromagnetic Ising chain with a long-range power-law interaction $1/{r}^{\ensuremath{\alpha}}$ using the large-scale density matrix renormalization group method. We find that the critical adiabatic dimension $\ensuremath{\mu}=2$ and the critical exponent of the correlation length $\ensuremath{\nu}=1$ for arbitrary $\ensuremath{\alpha}>0$, indicating all quantum phase transitions are second-order Ising transitions. In addition, we numerically determine the complete phase diagram for $0<\ensuremath{\alpha}\ensuremath{\le}3$ from the data collapse of the fidelity susceptibility and show that the critical point hc{h}_{c}hc​ changes monotonously with respect to $\ensuremath{\alpha}$. This work will shed light on the nature of phase transitions in the quantum long-range antiferromagnetic Ising chain from a quantum information perspective.
We study the criticality of a long-range quantum ferromagnetic Ising chain with algebraically decaying interactions $1/{r}^{\ensuremath{\alpha}}$ via the fidelity susceptibility based on the exact diagonalization and the density-matrix renormalization-group techniques. We find that critical exponents change monotonically from the mean-field universality class to the short-range Ising universality class for intermediate $\ensuremath{\alpha}$, which are consistent with recent results obtained from renormalization-group techniques. In addition, we determine the critical values for $1.8\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}3$ from the finite-size scaling of the fidelity susceptibility. Our work provides very nice numerical data from the fidelity susceptibility for the quantum long-range ferromagnetic Ising chain.
The Gaussian transition in the spin-1 Heisenberg chain with single-ion anisotropy is extremely difficult to treat, both analytically and numerically. We introduce an improved density-matrix renormalization group procedure with strict error control, which we use to access very large systems. By considering the bulk entropy, we determine the Gaussian transition point to four-digit accuracy, Dc/J=0.96845(8){D}_{c}/J=0.968\phantom{\rule{0.16em}{0ex}}45(8)Dc​/J=0.96845(8), resolving a long-standing debate in quantum magnetism. With this value, we obtain high-precision data for the critical behavior of quantities, including the ground-state energy, gap, and transverse string-order parameter, and for the critical exponent $\ensuremath{\nu}=1.472(2)$. Applying our improved technique at Jz=0.5{J}_{z}=0.5Jz​=0.5 highlights essential differences in critical behavior along the Gaussian transition line.
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.