Energy gap at first-order quantum phase transitions: An anomalous case

We show for the one-dimensional quantum $XY$ model with $s=1/2$ that the energy gap between the ground and first excited states behaves anomalously as a function of the system size at the first-order quantum phase transition. Although it is generally the case that the energy gap closes exponentially at a quantum first-order transition, the gap in the present model behaves non-monotonically as a function of the system size, apparently very irregularly in some cases. This property of the gap is similar to that of the infinite-range quantum $XY$ model, in which the gap closes polynomially, exponentially, or even factorially fast depending on the choice of the series of system sizes toward the thermodynamic limit. This observation is surprising in consideration of the apparent maturity of our understanding of the one-dimensional quantum $XY$ model. Our result is also important from the viewpoint of quantum annealing, where the rate of gap closing determines the efficiency of computation.We show for the one-dimensional quantum $XY$ model with $s=1/2$ that the energy gap between the ground and first excited states behaves anomalously as a function of the system size at the first-order quantum phase transition. Although it is generally the case that the energy gap closes exponentially at a quantum first-order transition, the gap in the present model behaves non-monotonically as a function of the system size, apparently very irregularly in some cases. This property of the gap is similar to that of the infinite-range quantum $XY$ model, in which the gap closes polynomially, exponentially, or even factorially fast depending on the choice of the series of system sizes toward the thermodynamic limit. This observation is surprising in consideration of the apparent maturity of our understanding of the one-dimensional quantum $XY$ model. Our result is also important from the viewpoint of quantum annealing, where the rate of gap closing determines the efficiency of computation.

arXiv:1501.02926Jan 13, 20150 citations

Anomalous behavior of the energy gap in the one-dimensional quantum XY model

Manaka Okuyama, Yuuki Yamanaka, Hidetoshi Nishimori +1 more

We reexamine the well-studied one-dimensional spin-1/2 XY model to reveal its nontrivial energy spectrum, in particular the energy gap between the ground state and the first excited state. In the case of the isotropic XY model, the XX model, the gap behaves very irregularly as a function of the system size at a second order transition point. This is in stark contrast to the usual power-law decay of the gap and is reminiscent of the similar behavior at the first order phase transition in the infinite-range quantum XY model. The gap also shows nontrivial oscillatory behavior for the phase transitions in the anisotropic model in the incommensurate phase. We observe a close relation between this anomalous behavior of the gap and the correlation functions. These results, those for the isotropic case in particular, are important from the viewpoint of quantum annealing where the efficiency of computation is strongly affected by the size dependence of the energy gap.We reexamine the well-studied one-dimensional spin-1/2 XY model to reveal its nontrivial energy spectrum, in particular the energy gap between the ground state and the first excited state. In the case of the isotropic XY model, the XX model, the gap behaves very irregularly as a function of the system size at a second order transition point. This is in stark contrast to the usual power-law decay of the gap and is reminiscent of the similar behavior at the first order phase transition in the infinite-range quantum XY model. The gap also shows nontrivial oscillatory behavior for the phase transitions in the anisotropic model in the incommensurate phase. We observe a close relation between this anomalous behavior of the gap and the correlation functions. These results, those for the isotropic case in particular, are important from the viewpoint of quantum annealing where the efficiency of computation is strongly affected by the size dependence of the energy gap.

arXiv:1508.02814Physical Review ENov 13, 201516 citations

First-order quantum phase transitions

M. A. Contínentino, André S. Ferreira

arXiv:cond-mat/0610624Journal of Magnetism and Magnetic MaterialsNov 17, 20069 citations

Anomalous behavior of the energy gap in the one-dimensional quantum XY model

秀稔西森, Hidetoshi Nishimori, 真佳奥山 +4 more

We reexamine the well-studied one-dimensional spin-1/2 XY model to reveal its nontrivial energy spectrum, in particular the energy gap between the ground state and the first excited state. In the case of the isotropic XY model, the XX model, the gap behaves very irregularly as a function of the system size at a second order transition point. This is in stark contrast to the usual power-law decay of the gap and is reminiscent of the similar behavior at the first order phase transition in the infinite-range quantum XY model. The gap also shows nontrivial oscillatory behavior for the phase transitions in the anisotropic model in the incommensurate phase. We observe a close relation between this anomalous behavior of the gap and the correlation functions. These results, those for the isotropic case in particular, are important from the viewpoint of quantum annealing where the efficiency of computation is strongly affected by the size dependence of the energy gapWe reexamine the well-studied one-dimensional spin-1/2 XY model to reveal its nontrivial energy spectrum, in particular the energy gap between the ground state and the first excited state. In the case of the isotropic XY model, the XX model, the gap behaves very irregularly as a function of the system size at a second order transition point. This is in stark contrast to the usual power-law decay of the gap and is reminiscent of the similar behavior at the first order phase transition in the infinite-range quantum XY model. The gap also shows nontrivial oscillatory behavior for the phase transitions in the anisotropic model in the incommensurate phase. We observe a close relation between this anomalous behavior of the gap and the correlation functions. These results, those for the isotropic case in particular, are important from the viewpoint of quantum annealing where the efficiency of computation is strongly affected by the size dependence of the energy gap

Nov 1, 20150 citations

We show for the one-dimensional quantum

XY

model with

s=1/2

that the energy gap between the ground and first excited states behaves anomalously as a function of the system size at the first-order quantum phase transition. Although it is generally the case that the energy gap closes exponentially at a quantum first-order transition, the gap in the present model behaves non-monotonically as a function of the system size, apparently very irregularly in some cases. This property of the gap is similar to that of the infinite-range quantum

XY

model, in which the gap closes polynomially, exponentially, or even factorially fast depending on the choice of the series of system sizes toward the thermodynamic limit. This observation is surprising in consideration of the apparent maturity of our understanding of the one-dimensional quantum

XY

model. Our result is also important from the viewpoint of quantum annealing, where the rate of gap closing determines the efficiency of computation.