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arXiv:0807.3491 · Europhysics Letters (EPL)

Scaling dimension of fidelity susceptibility in quantum phase transitions

Shi-Jian Gu, Hai-Qing Lin

Published Jul 1, 2009• 59 citations
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Abstract

We analyze ground-state behaviors of fidelity susceptibility (FS) and show that the FS has its own distinct dimension instead of real system's dimension in general quantum phases. The scaling relation of the FS in quantum phase transitions (QPTs) is then established on more general grounds. Depending on whether the FS's dimensions of two neighboring quantum phases are the same or not, we are able to classify QPTs into two distinct types. For the latter type, the change in the FS's dimension is a characteristic that separates two phases. As a non-trivial application to the Kitaev honeycomb model, we find that the FS is proportional to L 2 ln L in the gapless phase, while L 2 in the gapped phase. Therefore, the extra dimension of ln L can be used as a characteristic of the gapless phase.

DOI
10.1209/0295-5075/87/10003

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We analyze ground-state behaviors of fidelity susceptibility (FS) and show that the FS has its own distinct dimension instead of real system's dimension in general quantum phases. The scaling relation of the FS in quantum phase transitions (QPTs) is then established on more general grounds. Depending on whether the FS's dimensions of two neighboring quantum phases are the same or not, we are able to classify QPTs into two distinct types. For the latter type, the change in the FS's dimension is a characteristic that separates two phases. As a non-trivial application to the Kitaev honeycomb model, we find that the FS is proportional to L 2 ln L in the gapless phase, while L 2 in the gapped phase. Therefore, the extra dimension of ln L can be used as a characteristic of the gapless phase.
We study exactly both the ground-state fidelity susceptibility and bond-bond correlation function in the Kitaev honeycomb model. Our results show that the fidelity susceptibility can be used to identify the topological phase transition from a gapped A phase with Abelian anyon excitations to a gapless B phase with non-Abelian anyon excitations. We also find that the bond-bond correlation function decays exponentially in the gapped phase, but algebraically in the gapless phase. For the former case, the correlation length is found to be 1/ξ=2sinh⁡−1[2Jz−1/(1−Jz)]1/\xi=2\sinh^{-1}[\sqrt{2J_z -1}/(1-J_z)]1/ξ=2sinh−1[2Jz​−1​/(1−Jz​)], which diverges around the critical point Jz=(1/2)+J_z=(1/2)^+Jz​=(1/2)+.
A general relation between quantum phase transitions and the second derivative of the fidelity (or the ``fidelity susceptibility'') is proposed. The validity and the limitation of the fidelity susceptibility in characterizing quantum phase transitions are thus established. Moreover, based on the bosonization method, general formulas of the fidelity and the fidelity susceptibility are obtained for a class of one-dimensional gapless systems known as the Tomonaga-Luttinger liquid. Applying these formulas to the one-dimensional spin-1∕21∕21∕2 XXZXXZXXZ model, we find that quantum phase transitions, even of the Beresinskii-Kosterlitz-Thouless type, can be signaled by the fidelity susceptibility.
It has been found via numerical simulations that the geometric phase (GP) and fidelity susceptibility (FS) across the quantum critical points exhibit some universal scaling laws. Here we propose a singular function expansion method to find their exact singular forms and the related coefficients across the critical points. For models where the gaps are closed and reopened at special points (${k}_{0}=0,\ensuremath{\pi}$), scaling laws can be found as a function of the system length NNN and parameter deviation $\ensuremath{\lambda}\ensuremath{-}{\ensuremath{\lambda}}_{c}$, where ${\ensuremath{\lambda}}_{c}$ refers to one of the critical parameters. Although the GP and FS are defined in totally different ways, we find that these two measurements are essentially determined by the same physics, and as a consequence, their coefficients are closely related. Some of these exact relations are found in the anisotropic XY model and extended Ising models. We also show that the constant term in FS may be accompanied by a discontinuous jump across the critical points and, thus, does not have a universal scaling form. These findings should be in contrast to the cases where the gaps are not closed and reopened at the special points, in which some of the above scaling laws may break down as a function of the system length. Finally, we investigate the second-order derivative of GP, which may also exhibit some scaling laws across the critical point. These exact results can greatly enrich our understanding of GP and FS in the characterization of quantum phase transitions and may even find important applications in related physical quantities, such as entanglement, discord, correlation, and quantum Euler numbers, which may also exhibit scaling laws across the critical points.
We study the quantum phase transitions in the two-dimensional spin-orbit models in terms of fidelity susceptibility and reduced fidelity susceptibility. An order-to-order phase transition is identified by fidelity susceptibility in the two-dimensional Heisenberg XXZ model with Dzyaloshinsky-Moriya interaction on a square lattice. The finite size scaling of fidelity susceptibility shows a power-law divergence at criticality, which indicates the quantum phase transition is of second order. Two distinct types of quantum phase transitions are witnessed by fidelity susceptibility in Kitaev-Heisenberg model on a hexagonal lattice. We exploit the symmetry of two-dimensional quantum compass model, and obtain a simple analytic expression of reduced fidelity susceptibility. Compared with the derivative of ground-state energy, the fidelity susceptibility is a bit more sensitive to phase transition. The violation of power-law behavior for the scaling of reduced fidelity susceptibility at criticality suggests that the quantum phase transition belongs to a first-order transition. We conclude that fidelity susceptibility and reduced fidelity susceptibility show great advantage to characterize diverse quantum phase transitions in spin-orbit models.
Quantum phase transition is one of the most important issues in condensed matter physics,and quantum fidelity is an important concept emerging from quantum information theory.In this paper,we describe as simply as possible the role of fidelity and its leading term,i.e.the so-called fidelity susceptibility,in quantum phase transitions.As a purely geometric quantity,the key advantage of the fidelity approach is that no a priori knowledge of the order parameter is assumed.Specifically,we discuss definitions of the fidelity,the fidelity susceptibility,and the corresponding quantum adiabatic dimension,as well as their role in quantum phase transitions.We also use the Lipkin-Meshkov-Glick model and the Kitaev honeycomb model as examples to clarify these issues.
The behavior of the ground-state fidelity susceptibility in the vicinity of a quantum critical point is investigated. We derive scaling relations describing its singular behavior in the quantum critical regime. Unlike in previous studies, these relations are solely expressed in terms of conventional critical exponents. We also describe in detail a quantum Monte Carlo scheme that allows for the evaluation of the fidelity susceptibility for a large class of many-body systems and apply it in the study of the quantum phase transition for the transverse-field Ising model on the square lattice. Finite-size analysis applied to the so-obtained numerical results confirms the validity of our scaling relations. Furthermore, we analyze the properties of a closely related quantity, the ground-state energy's second derivative, which can be numerically evaluated in a particularly efficient way. The usefulness of both quantities as alternative indicators of quantum criticality is examined.