The two-dimensional quantum $XY$ model with a transverse magnetic field was investigated with the exact diagonalization method. Upon turning on the magnetic field $h$ and the $XY$-plane anisotropy $η$, there appear a variety of phase boundaries, which meet at the multicritical point $(h,η)=(2,0)$. We devote ourselves to the Ising-universality branch, placing an emphasis on the multicritical behavior. As a probe to detect the underlying phase transitions, we adopt the fidelity susceptibility $χ_F$. The fidelity susceptibility does not rely on any presumptions as to the order parameter involved. We made a finite-size-scaling analysis of $χ_F$ for $η=1$ (Ising limit), where a number of preceding results are available. Thereby, similar analyses with $η$ scaled were carried out around the multicritical point. We found that the $χ_F$ data are described by the crossover scaling theory. A comparison with the preceding studies of the multicriticality is made.
The two-dimensional easy-plane SU$(3)$ magnet subjected to the transverse field was investigated with the exact-diagonalization method. So far, as to the $XY$ model (namely, the easy-plane SU$(2)$ magnet), the transverse-field-driven order-disorder phase boundary has been investigated with the exact-diagonalization method, and it was claimed that the end-point singularity (multicriticality) at the $XX$-symmetric point does not accord with large-$N$-theory's prediction. Aiming to reconcile the discrepancy, we extend the internal symmetry to the easy-plane SU$(3)$ with the anisotropy parameter $…
The honeycomb-lattice Ising antiferromagnet subjected to the imaginary magnetic field $H=iθT /2$ with the "topological" angle $θ$ and temperature $T$ was investigated numerically. In order to treat such a complex-valued statistical weight, we employed the transfer-matrix method. As a probe to detect the order-disorder phase transition, we resort to an extended version of the fidelity $F$, which makes sense even for such a non-hermitian transfer matrix. As a preliminary survey, for an intermediate value of $θ$, we investigated the phase transition via the fidelity susceptibility $χ_F^{(θ)}$. Th…
The square-lattice Ising antiferromagnet subjected to the imaginary magnetic field $H=i θT /2 $ with the "topological" angle $θ$ and temperature $T$ was investigated by means of the transfer-matrix method. Here, as a probe to detect the order-disorder phase transition, we adopt an extended version of the fidelity susceptibility $χ_F^{(θ)}$, which makes sense even for such a non-hermitian transfer matrix. As a preliminary survey, for an intermediate value of $θ$, we examined the finite-size-scaling behavior of $χ_F^{(θ)}$, and found a pronounced signature for the criticality; note that the magn…
We investigate quantum phase transitions in one-dimensional quantum disordered lattice models, the Anderson model and the Aubry-André 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 a…
Quantum criticality usually occurs in many-body systems. Recently it was shown that the quantum Rabi model, which describes a two-level atom coupled to a single model cavity field, presents quantum phase transitions from a normal phase to a superradiate phase when the ratio between the frequency of the two level atom and the frequency of the cavity field extends to infinity. In this work, we study quantum phase transitions in the quantum Rabi model from the fidelity susceptibility perspective. We found that the fidelity susceptibility and the generalized adiabatic susceptibility present univer…
We compare the irreversible work produced in an infinitesimal sudden quench of a quantum system at zero temperature with its ground state fidelity susceptibility, giving an explicit relation between the two quantities. We find that the former is proportional to the latter but for an extra term appearing in the irreversible work which includes also contributions from the excited states. We calculate explicitly the two quantities in the case of the quantum Ising chain, showing that at criticality they exhibit different scaling behavior. The irreversible work, rescaled by square of the quench's a…
The fidelity susceptibility measures sensitivity of eigenstates to a change of an external parameter. It has been fruitfully used to pin down quantum phase transitions when applied to ground states (with extensions to thermal states). Here we propose to use the fidelity susceptibility as a useful dimensionless measure for complex quantum systems. We find analytically the fidelity susceptibility distributions for Gaussian orthogonal and unitary universality classes for arbitrary system size. The results are verified by a comparison with numerical data.
For a field theory with a gravitational dual, we study holographic fidelity susceptibility for two states related by a deformation of a relevant operator. To do so, we study back reaction of a massive scalar field on the asymptotic behavior of the metric in an Einstein gravity coupled to a massive scalar field. Identifying the two states, holographically, with the original and back reacted geometries, the corresponding holographic fidelity susceptibility is given by the difference of the volume of an extremal time slice evaluated on the original and back reacted geometries.
We derive the exact expression for the Uhlmann fidelity between arbitrary thermal Gibbs states of the quantum XY model in a transverse field with finite system size. Using it, we conduct a thorough analysis of the fidelity susceptibility of thermal states for the Ising model in a transverse field. We compare the exact results with a common approximation that considers only the positive-parity subspace, which is shown to be valid only at high temperatures. The proper inclusion of the odd parity subspace leads to the enhancement of maximal fidelity susceptibility in the intermediate range of tem…
Automatic differentiation is an invaluable feature of machine learning and quantum machine learning software libraries. In this work it is shown how quantum automatic differentiation can be used to solve the condensed-matter problem of computing fidelity susceptibility, a quantity whose value may be indicative of a phase transition in a system. Results are presented using simulations including hardware noise for small instances of the transverse-field Ising model, and a number of optimizations that can be applied are highlighted. Error mitigation (zero-noise extrapolation) is applied within th…
The computational complexity conjecture of NP $\nsubseteq$ BQP implies that there should be an exponentially small energy gap for Quantum Annealing (QA) of NP-hard problems. We aim to verify how this computation originated gapless point could be understood based on physics, using the quantum Monte Carlo method. As a result, we found a phase transition detectable only by the divergence of fidelity susceptibility. The exponentially small gapless points of each instance are all located in the phase found in this study, which suggests that this phase transition is the physical cause of the failure…
In recent years, programmable Rydberg-atom arrays have been widely used to simulate new quantum phases and phase transitions, generating great interest among theorists and experimentalists. Based on the large-scale density matrix renormalization group method, the ground-state phase diagram of one-dimensional Rydberg chains is investigated with fidelity susceptibility as an efficient diagnostic method. We find that the competition between Rydberg blockade and external detuning produces unconventional phases and phase transitions. As the Rydberg blockade radius increases, the phase transition be…
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 $J_c$ to $J^*$ in the gapped phase by a value $|J^* - J_c| = B^2/36$, where $B^2$ is a transition width controlling the asymptotic form of the correlation length $ξ\sim \exp(-B/\sqrt{|J-J_c|})$ 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 …
We derive several closed-form expressions for the fidelity susceptibility~(FS) of the anisotropic $XY$ model in the transverse field. The basic idea lies in a partial fraction expansion of the expression so that all the terms are related to a simple fraction or its derivative. The critical points of the model are reiterated by the FS, demonstrating its validity for characterizing the phase transitions. Moreover, the critical exponents $ν$ associated with the correlation length in both critical regions are successfully extracted by the standard finite-size scaling analysis.
We study the quantum fidelity (groundstate overlap) near quantum phase transitions of the Ising universality class in one dimensional (1D) systems of finite size L. Prominent examples occur in magnetic systems (e.g. spin-Peierls, the anisotropic XY model), and in 1D topological insulators of any topologically nontrivial Altland-Zirnbauer-Kitaev universality class. The rescaled fidelity susceptibility is a function of the only dimensionless parameter LM, where 2M is the gap in the fermionic spectrum. We present analytic expressions for the fidelity susceptibility for periodic and open boundarie…
We develop the perturbation theory of the fidelity susceptibility in biorthogonal bases for arbitrary interacting non-Hermitian many-body systems with real eigenvalues. The quantum criticality in the non-Hermitian transverse field Ising chain is investigated by the second derivative of ground-state energy and the ground-state fidelity susceptibility. We show that the system undergoes a second-order phase transition with the Ising universal class by numerically computing the critical points and the critical exponents from the finite-size scaling theory. Interestingly, our results indicate that …
In this work, we explore the critical behaviors of fidelity susceptibility and trace distance susceptibility associated to the steady states of dissipative systems at continuous phase transitions. We investigate on two typical models, one is the dissipative spin-1/2 XYZ model on two-dimensional square lattice and the other is a driven-dissipative Kerr oscillator. We find that the susceptibilities of fidelity and trace distance exhabit singular behaviors near the critical points of phase transitions in both models. The critical points, in thermodynamic limit, extracted from the scalings of the …
The inner product between the ground-state eigenvectors with proximate interaction parameters, namely, the fidelity, plays a significant role in the quantum dynamics. In this paper, the critical behaviors of the transverse- and longitudinal-field fidelity susceptibilities for the d=2 quantum (transverse-field) Ising model are investigated by means of the numerical diagonalization method; the former susceptibility has been investigated rather extensively. The critical exponents for these fidelity susceptibilities are estimated as α^{(t)}_F=0.752(24) and α^{(h)}_F=1.81(13), respectively. These i…
The critical properties of the one-dimensional transverse Ising model in the presence of a longitudinal magnetic field were studied by the quantum fidelity method. We used exact diagonalization to obtain the ground-state energies and corresponding eigenvectors for lattice sizes up to 24 spins. The maximum of the fidelity susceptibility is used to locate the various phase boundaries present in the system. The type of dominant spin ordering for each phase was identified by examining the corresponding ground-state eigenvector. For a given antiferromagnetic nearest-neighbor interaction J2, we calc…
In this work we analyze the ground-state properties of the $s=1/2$ one-dimensional ANNNI model in a transverse field using the quantum fidelity approach. We numerically determined the fidelity susceptibility as a function of the transverse field $B_x$ and the strength of the next-nearest-neighbor interaction $J_2$, for systems of up to 24 spins. We also examine the ground-state vector with respect to the spatial ordering of the spins. The ground-state phase diagram shows ferromagnetic, paramagnetic, floating, $\Braket{2,2}$ phases, and we predict an infinite number of modulated phases in the t…
We analyze the scaling behavior of the fidelity, and the corresponding susceptibility, emerging in finite-size many-body systems whenever a given control parameter $λ$ is varied across a quantum phase transition. For this purpose we consider a finite-size scaling (FSS) framework. Our working hypothesis is based on a scaling assumption of the fidelity in terms of the FSS variables associated to $λ$ and to its variation $δλ$. This framework entails the FSS predictions for continuous transitions, and meanwhile enables to extend them to first-order transitions, where the FSS becomes qualitatively …
We study the behaviour of the fidelity and the Uhlmann connection in two-dimensional systems of free fermions that exhibit non-trivial topological behavior. In particular, we use the fidelity and a quantity closely related to the Uhlmann factor in order to detect phase transitions at zero and finite temperature for topological insulators and superconductors. We show that at zero temperature both quantities predict quantum phase transitions: a sudden drop of fidelity indicates an abrupt change of the spectrum of the state, while the behavior of the Uhlmann connection signals equally rapid chang…
The Harper equation arising out of a tight-binding model of electrons on a honeycomb lattice subject to a uniform magnetic field perpendicular to the plane is studied. Contrasting and complementary approaches involving von Neumann entropy, fidelity, fidelity susceptibility, multifractal analysis are employed to characterize the phase diagram. The phase diagram consists of three phases: two metallic phases and an insulating phase. A variant model where next nearest neighbor hopping is included, exhibits a mobility edge and does not allow for a simple single phase diagram characterizing all the …
A universal black-box quantum Monte Carlo approach to quantum phase transitions
Department of Physics & Astronomy, University of Southern California（南加州大学物理与天文学系）；Information Sciences Institute, University of Southern California（南加州大学信息科学研究所）
