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arXiv:1708.08212 · Physical Review A

Fidelity susceptibility study of quantum long-range antiferromagnetic Ising chain

Gaoyong Sun

Published Oct 20, 2017• 62 citations
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Abstract

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 ${h}_{c}$ 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.

DOI
10.1103/physreva.96.043621

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We study the fidelity susceptibility of a quantum antiferromagnetic Ising chain with a long-range power-law interaction 1/r\ensuremathα1/{r}^{\ensuremath{\alpha}}1/r\ensuremathα using the large-scale density matrix renormalization group method. We find that the critical adiabatic dimension \ensuremathμ=2\ensuremath{\mu}=2\ensuremathμ=2 and the critical exponent of the correlation length \ensuremathν=1\ensuremath{\nu}=1\ensuremathν=1 for arbitrary \ensuremathα>0\ensuremath{\alpha}>0\ensuremathα>0, indicating all quantum phase transitions are second-order Ising transitions. In addition, we numerically determine the complete phase diagram for 0<\ensuremathα\ensuremath≤30<\ensuremath{\alpha}\ensuremath{\le}30<\ensuremathα\ensuremath≤3 from the data collapse of the fidelity susceptibility and show that the critical point hc{h}_{c}hc​ changes monotonously with respect to \ensuremathα\ensuremath{\alpha}\ensuremathα. 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.
We study the phase diagram of a spin-1 antiferromagnetic chain with isotropic antiferromagnetic interactions decaying with a power law $\ensuremath{\propto}{r}^{\ensuremath{-}\ensuremath{\alpha}}$ $(\ensuremath{\alpha}\ensuremath{\ge}1)$ accompanied by modulated single-ion anisotropy. Employing the techniques of the density-matrix renormalization group, the effects of long-range interactions and single-ion anisotropy on a variety of correlations are investigated. In order to check the consistency, the fidelity susceptibilities are evaluated across quantum phase transitions. The quantum critical points are faithfully detected and orders of phase transitions are determined. The correlation-length critical exponent is extracted from scaling functions of the fidelity susceptibility. The presence of long-range interactions leads to a quantitative change of the phase boundaries and reduces the order of the phase transition under certain conditions. A direct first-order transition between the periodic N\'eel phase and the large-DDD phase occurs for slowly decaying antiferromagnetic interactions.
Antiferromagnetic Hamiltonians with short range, non-frustrating interactions are well known to exhibit long range magnetic order in dimensions d ≥2 but exhibit only quasi-long-range order, with power-law decay of correlations, in d = 1 (for half-integer spin). On the other hand, non-frustrating long range interactions can induce long range order in d = 1. We study Hamiltonians in which the long range interactions have an adjustable amplitude λ, as well as an adjustable power law 1/| x | α , using a combination of quantum Monte Carlo and analytic methods: spin-wave, large N non-linear σ model, and renormalization group methods. We map out the phase diagram in the λ–α plane and study the nature of the critical line separating the phases with long range and quasi-long-range order. We find that this corresponds to a novel line of critical points with continuously varying critical exponents and a dynamical exponent, z <1.
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.
We study quantum phase transitions in Heisenberg antiferromagnetic chains with a staggered power-law decaying long-range interactions. Employing the density-matrix renormalization group (DMRG) algorithm and the fidelity susceptibility as the criticality measure, we establish more accurate values of quantum critical points than the results obtained from the spin-wave approximation, quantum Monte Carlo, and DMRG in literatures. The deviation is especially evident for strong long-range interactions. We extend isotropic long-range interactions to the anisotropic cases and find that kaleidoscope of quantum phases emerge from the interplay of anisotropy of the long-range exchange interaction and symmetry breaking. We demonstrate nonfrustrating long-range interactions induce the true long-range order in Heisenberg antiferromagnetic chains with a continuous symmetry breaking, lifting the restrictions imposed by the Mermin-Wagner theorem.
Abstract We present the zero temperature phase diagram of the bond alternating Ising chain in the presence of Dzyaloshinskii–Moriya interaction. An abrupt change in ground state fidelity is a signature of quantum phase transition. We obtain the renormalization of fidelity in terms of quantum renormalization group without the need to know the ground state. We calculate the fidelity susceptibility and its scaling behavior close to quantum critical point (QCP) to find the critical exponent which governs the divergence of correlation length. The model consists of a long range antiferromagnetic order with nonzero staggered magnetization which is separated from a helical ordered phase at QCP. Our results state that the critical exponent is independent of the bond alternation parameter (λ) while the maximum attainable helical order depends on λ.