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arXiv:0707.4574 · Physical Review B

Ground-state fidelity in one-dimensional gapless models

Min-Fong Yang

Published Nov 8, 2007• 144 citations
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Abstract

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-$1∕2$ $XXZ$ model, we find that quantum phase transitions, even of the Beresinskii-Kosterlitz-Thouless type, can be signaled by the fidelity susceptibility.

DOI
10.1103/physrevb.76.180403

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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.
We clarify the long-standing controversy concerning the behavior of the ground-state fidelity in the vicinity of a quantum phase transition of the Berezinskii-Kosterlitz-Thouless type in one-dimensional systems. Contrary to the prediction based on the Gaussian approximation of the Luttinger-liquid approach, it is shown that the fidelity susceptibility does not diverge at the transition but has a cusplike peak ${\ensuremath{\chi}}_{c}\ensuremath{-}\ensuremath{\chi}(\ensuremath{\lambda})\ensuremath{\sim}\sqrt{|{\ensuremath{\lambda}}_{c}\ensuremath{-}\ensuremath{\lambda}|}$, where $\ensuremath{\lambda}$ is a parameter driving the transition and ${\ensuremath{\chi}}_{c}$ is the peak value at the transition point $\ensuremath{\lambda}={\ensuremath{\lambda}}_{c}$. Numerical claims of the logarithmic divergence of fidelity susceptibility with the system size (or temperature) are explained by logarithmic corrections due to marginal operators, which is supported by numerical calculations for large systems.
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 slightly generalized quantum fidelity susceptibilities where the differential change in the fidelity is measured with respect to a different term than the one used for driving the system toward a quantum phase transition. As a model system we use the spin-1/21/21/2 J1{J}_{1}J1​-J2{J}_{2}J2​ antiferromagnetic Heisenberg chain. For this model, we study three fidelity susceptibilities, ${\ensuremath{\chi}}_{\ensuremath{\rho}}$, ${\ensuremath{\chi}}_{D}$, and ${\ensuremath{\chi}}_{\mathrm{AF}}$, which are related to the spin stiffness, the dimer order, and antiferromagnetic order, respectively. All these ground-state fidelity susceptibilities are sensitive to the phase diagram of the J1{J}_{1}J1​-J2{J}_{2}J2​ model. We show that they all can accurately identify a quantum critical point in this model occurring at ${J}_{2}^{c}\ensuremath{\sim}0.241{J}_{1}$ between a gapless Heisenberg phase for J2<J2c{J}_{2}<{J}_{2}^{c}J2​<J2c​ and a dimerized phase for J2>J2c{J}_{2}>{J}_{2}^{c}J2​>J2c​. This phase transition, in the Berezinskii-Kosterlitz-Thouless universality class, is controlled by a marginal operator and is therefore particularly difficult to observe.
The potential of fidelity in accurately detecting unconventional quantum phase transitions (QPTs) remains an open question. Specific attention is directed towards high-order phase transitions, including the infinite-order Berezinskii-Kosterlitz-Thouless (BKT) transition and the Gaussian transition. In this study, we extend the concept of fidelity susceptibility to fixed-point fidelity susceptibility in the thermodynamic limit and assess this measure using tensor-network algorithms. Our results demonstrate that the fixed-point fidelity susceptibility effectively identifies the BKT and Gaussian phase transitions, a capability lacking in the traditional finite-fidelity approach. Furthermore, we establish a correlation between the fixed-point fidelity method and the entanglement method, demonstrating their equivalence. This research not only confirms the efficacy of fidelity in detecting QPTs but also enhances the understanding of fidelity-related methodologies.
We show a general connection between the reduced-fidelity susceptibility and quantum phase transitions, and derive an explicit expression of the reduced-fidelity susceptibility for the one-dimensional spin-1/2 dimerized Heisenberg chain, which has both SU(2) and translational symmetries. We find that the reduced-fidelity susceptibility is directly related to the square of the second derivative of ground-state energy, which means that it is an effective indicator of the second-order quantum phase transitions. In terms of this indicator, we explicitly examine the critical behavior of the spin-1/2 dimerized Heisenberg chain. Moreover, we give another two exemplifications to show that the results may also be extended to high-spin systems.