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arXiv:0706.2495 · Physical Review B

Fidelity susceptibility, scaling, and universality in quantum critical phenomena

Shi-Jian Gu, Ho-Man Kwok, Wen-Qiang Ning, Hai-Qing Lin

Published Jun 9, 2008• 211 citations
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Abstract

We study fidelity susceptibility in the one-dimensional asymmetric Hubbard model and show that the fidelity susceptibility can be used to identify the universality class of the quantum phase transitions in this model. The Kosterlitz--Thouless-type transition occurred at half-filling and the Landau transition away from half-filling can be discriminated from distinct critical exponents of the fidelity susceptibility.

DOI
10.1103/physrevb.77.245109

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We study fidelity susceptibility in the one-dimensional asymmetric Hubbard model and show that the fidelity susceptibility can be used to identify the universality class of the quantum phase transitions in this model. The Kosterlitz--Thouless-type transition occurred at half-filling and the Landau transition away from half-filling can be discriminated from distinct critical exponents of the fidelity susceptibility.
We investigated the fidelity susceptibility in the one-dimension (1D) Hubbard model and the extended Hubbard model at half-filling via the density matrix renormalization group. From the numerical results, we argue that in the 1D Hubbard model, the fidelity susceptibility shows a divergence at two points which is infinitesimally close to the critical point while it is always extensive exactly at the critical point. For the extended Hubbard model, we found that for a properly chosen driving parameter, the fidelity susceptibility is able to reveal the quantum phase transitions between the PS (phase separation)-superconducting, superconducting-CDW (charge-density wave), CDW-SDW(spin-density wave), SDW-PS, CDW-BOW (bond-order wave), and the BOW-SDW phases.
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 investigate quantum phase transitions in one-dimensional quantum disordered lattice models, the Anderson model and the Aubry-Andr\'e model, from the fidelity susceptibility approach. First, we find that the fidelity susceptibility and the generalized adiabatic susceptibility are maximum at the quantum critical points of the disordered models, through which one can locate the quantum critical point in disordered lattice models. Second, finite-size scaling analysis of the fidelity susceptibility and of the generalized adiabatic susceptibility show that the correlation length critical exponent and the dynamical critical exponent at the quantum critical point of the one-dimensional Anderson model are respectively 2/3 and 2 and of the Aubry-Andr\'e model are respectively 1 and 2.375. Thus the quantum phase transitions in the Anderson model and in the Aubry-Andr\'e model are of different universality classes. Because the fidelity susceptibility and the generalized adiabatic susceptibility are directly connected to the dynamical structure factor which are experimentally accessible in the linear response regime, the fidelity susceptibility in quantum disordered systems may be observed experimentally in the near future.
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 notion of fidelity susceptibility, introduced within the context of quantum metric tensor, has been an important quantity to characterize the criticality near quantum phase transitions. We demonstrate that for topological phase transitions in Dirac models, provided the momentum space is treated as the manifold of the quantum metric, the fidelity susceptibility coincides with the curvature function whose integration gives the topological invariant. Thus the quantum criticality of the curvature function near a topological phase transition also describes the criticality of the fidelity susceptibility, and the correlation length extracted from the curvature function also gives a momentum scale over which the fidelity susceptibility decays. To map out the profile and criticality of the fidelity susceptibility, we turn to quantum walks that simulate one-dimensional class BDI and two- dimensional class D Dirac models and demonstrate their accuracy in capturing the critical exponents and scaling laws near topological phase transitions.
We show that the peak which can be observed in fidelity susceptibility around the Berezinskii-Kosterlitz-Thouless transition is shifted from the quantum critical point (QCP) at Jc{J}_{c}Jc​ to J∗{J}^{*}J∗ in the gapped phase by a value $|{J}^{*}\ensuremath{-}{J}_{c}|={B}^{2}/36$, where B2{B}^{2}B2 is a transition width controlling the asymptotic form of the correlation length $\ensuremath{\xi}\ensuremath{\sim}exp(\ensuremath{-}B/\sqrt{|J\ensuremath{-}{J}_{c}|})$ in that phase. This is in contrast to the conventional continuous QCP where the maximum is an indicator of the position of the critical point. The shape of the peak is universal, emphasizing the close connection between fidelity susceptibility and the correlation length. We support those arguments with numerical matrix product state simulations of the one-dimensional Bose-Hubbard model in the thermodynamic limit, where the broad peak is located at J∗=0.212{J}^{*}=0.212J∗=0.212 that is significantly different from Jc=0.3048(3){J}_{c}=0.3048(3)Jc​=0.3048(3). In the spin-3/23/23/2 XXZ model the shift from Jc=1{J}_{c}=1Jc​=1 to J∗=1.0021{J}^{*}=1.0021J∗=1.0021 is small but the narrow universal peak is much more pronounced over the nonuniversal background.