Fidelity Susceptibility Research Articles

The Quantum Geometric Tensor in a Parameter-Dependent Curved Space.

We introduce a quantum geometric tensor in a curved space with a parameter-dependent metric, which contains the quantum metric tensor as the symmetric part and the Berry curvature corresponding to the antisymmetric part. This parameter-dependent metric modifies the usual inner product, which induces modifications in the quantum metric tensor and Berry curvature by adding terms proportional to the derivatives with respect to the parameters of the determinant of the metric. The quantum metric tensor is obtained in two ways: By using the definition of the infinitesimal distance between two states in the parameter-dependent curved space and via the fidelity susceptibility approach. The usual Berry connection acquires an additional term with which the curved inner product converts the Berry connection into an object that transforms as a connection and density of weight one. Finally, we provide three examples in one dimension with a nontrivial metric: an anharmonic oscillator, a Morse-like potential, and a generalized anharmonic oscillator; and one in two dimensions: the coupled anharmonic oscillator in a curved space.

Sensitivity of (multi)fractal eigenstates to a perturbation of the Hamiltonian

We study the response of an isolated quantum system governed by the Hamiltonian drawn from the Gaussian Rosenzweig-Porter random matrix ensemble to a perturbation controlled by a small parameter. We focus on the density of states, local density of states and the eigenfunction amplitude overlap correlation functions which are calculated exactly using the mapping to the supersymmetric nonlinear sigma model. We show that the susceptibility of eigenfunction fidelity to the parameter of perturbation can be expressed in terms of these correlation functions and is strongly peaked at the localization transition: It is independent of the effective disorder strength in the ergodic phase, grows exponentially with increasing disorder in the fractal phase and decreases exponentially in the localized phase. As a function of the matrix size, the fidelity susceptibility remains constant in the ergodic phase and increases in the fractal and in the localized phases at modestly strong disorder. We show that there is a critical disorder strength inside the insulating phase such that for disorder stronger than the critical the fidelity susceptibility decreases with increasing the system size. The overall behavior is very similar to the one observed numerically in a recent work by Sels and Polkovnikov [Phys. Rev. E 104, 054105 (2021)] for the normalized fidelity susceptibility in a disordered XXZ spin chain.

Resonant energy scales and local observables in the many-body localized phase

We formulate a theory for resonances in the many-body localized (MBL) phase of disordered quantum spin chains in terms of local observables. A key result is to show that there are universal correlations between the matrix elements of local observables and the many-body level spectrum. This reveals how the matrix elements encode the energy scales associated with resonance, thereby allowing us to show that these energies are power-law-distributed. Using these results, we calculate analytically the distributions of local polarizations and of eigenstate fidelity susceptibilities. The first of these quantities characterizes the proximity of MBL systems to noninteracting ones, while the second highlights their extreme sensitivity to local perturbations. Our theoretical approach is to consider the effect of varying a local field, which induces a parametric dynamics of spectral properties. We corroborate our results numerically using exact diagonalization in finite systems.

Signatures of topological phase transitions in higher Landau levels of HgTe/CdTe quantum wells from an information theory perspective

We analyze the structure of low energy Hamiltonian eigenstates in zincblende heterostructures (like HgTe/CdTe quantum wells) near the gamma point, under magnetic fields, to characterize topological phase transitions (TPT) under an information-theoretic perspective. Using information markers like entanglement, quantum fluctuations, fidelity susceptibility, participation ration, area in phase space, etc., we realize that higher Landau levels (LL) feel the topological phase transition slightly displaced with regard to the edge state, thus leading to the concept of “higher Landau level |n|>0 TPT”, as “echoes” of the standard edge state n=0 TPT. We compute the critical magnetic field and the critical HgTe layer thickness at which these information measures of higher Landau levels undergo a structural change.

Thermal topological phase transition in SnTe from ab initio calculations

One of the key issues in the physics of topological insulators is whether the topologically non-trivial properties survive at finite temperatures and, if so, whether they disappear only at the temperature of topological gap closing. Here, we study this problem, using quantum fidelity as a measure, by means of \emph{ab-initio} methods supplemented by an effective dissipative theory built on the top of the \emph{ab-initio} electron and phonon band structures. In the case of SnTe, the prototypical crystal topological insulator, we reveal the presence of a characteristic temperature, much lower than the gap-closing one, that marks a loss of coherence of the topological state. The transition is not present in a purely electronic system but it appears once we invoke coupling with a dissipative bosonic bath. Features in the dependence with temperature of the fidelity susceptibility can be related to changes in the band curvature, but signatures of a topological phase transition appear in the fidelity only though the non-adiabatic coupling with soft phonons. Our argument is valid for valley topological insulators, but in principle can be generalized to the broader class of topological insulators which host any symmetry-breaking boson.

Quantum criticality driven by the cavity coupling in the Rabi-dimer model

The superradiant phase transition (SPT) controlled by the interacting strength between the two-level atom and the photons has been a hot topic in the Rabi model and the Rabi-dimer model. The latter describes two Rabi cavities coupled with an inter-cavity hopping parameter. Moreover, the SPT in the Rabi-dimer model is found to be the same universal class that in the Rabi model by investigating the correlation-length critical exponent. In this paper, we are concerned about whether the inter-cavity hopping parameter between two Rabi cavities (i.e., the Rabi-dimer model) will induce the SPT and to which the universal class of the phase transition belongs. We analytically derive the phase boundary of the SPT and investigate the ground-state properties of the system. We uncover that the inter-cavity induced SPT can be apparently understood from the ground-state energy and the ground-state photon population, as well as the ground-state expectation value of the squared anti-symmetric mode. From the scaling analysis of the fidelity susceptibility, we numerically verify that the SPT driven by the cavity coupling belongs to the same universal class as the one driven by the atom–cavity interaction. Our work enriches the studies on the SPT and its critical behaviors in the Rabi-dimer model.

Steady-state susceptibility in continuous phase transitions of dissipative systems

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 critical controlling parameters to the system size or nonlinearity agree well with the existed results.

Anharmonicity-induced excited-state quantum phase transition in the symmetric phase of the two-dimensional limit of the vibron model

In most cases, excited state quantum phase transitions can be associated with the existence of critical points (local extrema or saddle points) in a system's classical limit energy functional. However, an excited-state quantum phase transition might also stem from the lowering of the asymptotic energy of the corresponding energy functional. One such example occurs in the 2D limit of the vibron model, once an anharmonic term in the form of a quadratic bosonic number operator is added to the Hamiltonian. The study of this case in the broken-symmetry phase was presented in Phys. Rev. A. 81 050101 (2010). In the present work, we delve further into the nature of this excited-state quantum phase transition and we characterize it in the, previously overlooked, symmetric phase of the model making use of quantities such as the effective frequency, the expected value of the quantum number operator, the participation ratio, the density of states, and the quantum fidelity susceptibility. In addition to this, we extend the usage of the quasilinearity parameter, introduced in molecular physics, to characterize the phases in the spectrum of the anharmonic 2D limit of the vibron model and a down-to-earth analysis has been included with the characterization of the critical energies for the linear isomers HCN/HNC.

Critical crossover phenomena driven by symmetry-breaking defects at quantum transitions.

We study the effects of symmetry-breaking defects at continuous quantum transitions (CQTs) of homogeneous systems, which may arise from localized external fields coupled to the order-parameter operator. The problem is addressed within renormalization-group (RG) and finite-size scaling frameworks. We consider the paradigmatic one-dimensional quantum Ising models at their CQT, in the presence of defects which break the global Z_{2} symmetry. We show that such defects can give rise to notable critical crossover regimes where the ground-state properties experience substantial and rapid changes, from symmetric conditions to characterization of these crossover phenomena driven by defects. In particular, this is demonstrated by analyzing the ground-state fidelity associated with small changes of the defect strength. Within the critical crossover regime, the fidelity susceptibility shows a power-law divergence when increasing the system size, related to the RG dimension of the defect strength; in contrast, outside the critical defect regime, it remains finite. We support the RG scaling arguments with numerical results.

Long-range order and quantum criticality in antiferromagnetic chains with long-range staggered interactions.

We study quantum phase transitions in Heisenberg antiferromagnetic chains with a staggered power-law decaying long-range interactions. Employing the density-matrix renormalization group (DMRG) algorithm and the fidelity susceptibility as the criticality measure, we establish more accurate values of quantum critical points than the results obtained from the spin-wave approximation, quantum Monte Carlo, and DMRG in literatures. The deviation is especially evident for strong long-range interactions. We extend isotropic long-range interactions to the anisotropic cases and find that kaleidoscope of quantum phases emerge from the interplay of anisotropy of the long-range exchange interaction and symmetry breaking. We demonstrate nonfrustrating long-range interactions induce the true long-range order in Heisenberg antiferromagnetic chains with a continuous symmetry breaking, lifting the restrictions imposed by the Mermin-Wagner theorem.

Quantum criticality and universality in the p -wave-paired Aubry-André-Harper model

We investigate the quantum criticality and universality in Aubry-Andr\'{e}-Harper (AAH) model with $p$-wave superconducting pairing $\Delta$ in terms of the generalized fidelity susceptibility (GFS). We show that the higher-order GFS is more efficient in spotlighting the critical points than lower-order ones, and thus the enhanced sensitivity is propitious for extracting the associated universal information from the finite-size scaling in quasiperiodic systems. The GFS obeys power-law scaling for localization transitions and thus scaling properties of the GFS provide compelling values of critical exponents. Specifically, we demonstrate that the fixed modulation phase $\phi=\pi$ alleviates the odd-even effect of scaling functions across the Aubry-Andr\'{e} transition with $\Delta=0$, while the scaling functions for odd and even numbers of system sizes with a finite $\Delta$ cannot coincide irrespective of the value of $\phi$. A thorough numerical analysis with odd number of system sizes reveals the correlation-length exponent $\nu \simeq 1.000$ and the dynamical exponent $z$ $\simeq$ 1.388 for transitions from the critical phase to the localized phase,suggesting the unusual universality class of localization transitions in the AAH model with a finite $p$-wave superconducting pairing lies in a different universality class from the Aubry-Andr\'{e} transition. The results may be testified in near term state-of-the-art experimental settings.

Quantum fidelity susceptibility in excited state quantum phase transitions: application to the bending spectra of nonrigid molecules

We characterize excited state quantum phase transitions in the two dimensional limit of the vibron model with the quantum fidelity susceptibility, comparing the obtained results with the information provided by the participation ratio. As an application, we locate the eigenstate closest to the barrier to linearity and determine the linear or bent character of the different overtones for particular bending modes of six molecular species. We perform a fit and use the optimized eigenvalues and eigenstates in three cases and make use of recently published results for the other three cases.

Disorder-driven phase transitions in bosonic fractional quantum Hall liquids

We investigate the disorder-driven phase transitions in bosonic fractional quantum Hall liquids at filling factors $f=1/2$ and $f=1$ in the lowest Landau level. We use the evolution of ground-state entanglement entropy, fidelity susceptibility, and Hall conductance with increasing disorder strength to identify the underlying phase transitions. The critical disorder strengths obtained from these different quantities are consistent with each other, validating the reliability of our numerical calculations based on exact diagonalization. At $f=1/2$, we observe a clear transition from the bosonic Laughlin state to a trivial insulating phase. At $f=1$, we identify a direct phase transition from the non-Abelian bosonic Moore-Read state to a trivial insulating phase, although some signs of a disorder-induced intermediate fractional quantum Hall phase were recently reported for the $f=5/2$ fermionic cousin.

Quantum Criticality Driven by the Cavity Coupling in the Rabi-Dimer Model

In the previous researches, the superradiant phase transition (SPT) controlled by the interacting strength between the two-level atom and the photons has been a hot topic in the Rabi model and the Rabi-dimer model which describes two Rabi cavities coupled by an inter-cavity hopping parameter. Moreover, the SPT in the Rabi-dimer model is found to be the same universal class as that in the Rabi model by investigating the correlation-length critical exponent. In our research, we concern about that whether the inner-cavity hopping parameter between two Rabi cavities (i.e., the Rabi-dimer model) will induce the SPT and which universal class of the phase transition belongs to. By analyzing the ground-state energy and the ground-state photon population, we verify that the SPT can be induced by the inter-cavity hopping parameter. From the scaling analysis of the fidelity susceptibility, we verify that the SPT driven by the cavity coupling belongs to the same universal class as the one driven by the atom-cavity interaction. Our work enriches the studies on the SPT and its critical behaviors in the Rabi-dimer model.

Biorthogonal quantum criticality in non-Hermitian many-body systems

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 the biorthogonal quantum phase transitions are described by the biorthogonal fidelity susceptibility instead of the conventional fidelity susceptibility.

Fidelity and entanglement entropy for infinite-order phase transitions

We study the fidelity and the entanglement entropy for the ground states of quantum systems that have infinite-order quantum phase transitions. In particular, we consider the quantum O(2) model with a spin-$S$ truncation, where there is an infinite-order Gaussian (IOG) transition for $S = 1$ and there are Berezinskii-Kosterlitz-Thouless (BKT) transitions for $S \ge 2$. We show that the height of the peak in the fidelity susceptibility ($\chi_F$) converges to a finite thermodynamic value as a power law of $1/L$ for the IOG transition and as $1/\ln(L)$ for BKT transitions. The peak position of $\chi_F$ resides inside the gapped phase for both the IOG transition and BKT transitions. On the other hand, the derivative of the block entanglement entropy with respect to the coupling constant ($S^{\prime}_{vN}$) has a peak height that diverges as $\ln^{2}(L)$ [$\ln^{3}(L)$] for $S = 1$ ($S \ge 2$) and can be used to locate both kinds of transitions accurately. We include higher-order corrections for finite-size scalings and crosscheck the results with the value of the central charge $c = 1$. The crossing point of $\chi_F$ between different system sizes is at the IOG point for $S = 1$ but is inside the gapped phase for $S \ge 2$, while those of $S^{\prime}_{vN}$ are at the phase-transition points for all $S$ truncations. Our work elaborates how to use the finite-size scaling of $\chi_F$ or $S^{\prime}_{vN}$ to detect infinite-order quantum phase transitions and discusses the efficiency and accuracy of the two methods.

Dynamical obstruction to localization in a disordered spin chain.

We analyze a one-dimensional XXZ spin chain in a disordered magnetic field. As the main probes of the system's behavior, we use the sensitivity of eigenstates to adiabatic transformations, as expressed through the fidelity susceptibility, in conjunction with the low-frequency asymptotes of the spectral function. We identify a region of maximal chaos-with exponentially enhanced susceptibility-which separates the many-body localized phase from the diffusive ergodic phase. This regime is characterized by slow transport, and we argue that the presence of such slow dynamics highly constrains any possible localization transition in the thermodynamic limit. Rather, the results are more consistent with absence of the localized phase.

Multipartite entanglement and quantum criticality of Rydberg atoms trapped in a two-dimensional optical lattice.

We investigate the multipartite entanglement and quantum criticality of neutral atoms on a two-dimensional square lattice, interacting via laser excitation to atom Rydberg states. It is found that the first derivative of residual entanglement with respect to detuning has peaks near the critical point, and corresponding critical behaviors are shown to obey conventional finite-sized scaling, from which we numerically determine the quantum critical point and the critical exponent of the associated correlation length. We also show that there is a sharp peak in the fidelity susceptibility near the critical point, and the critical exponent of the associated correlation length is obtained based on the finite size analysis.

Uhlmann fidelity and fidelity susceptibility for integrable spin chains at finite temperature: exact results

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 temperatures. We show that this enhancement persists in the thermodynamic limit and scales quadratically with the system size. The correct low-temperature behavior is captured by an approximation involving the two lowest many-body energy eigenstates, from which simple expressions are obtained for the thermal susceptibility and specific heat.

Quantum criticality and correlations in the Ising-Gamma chain

In this paper, we study quantum phase transitions (QPTs) and critical phenomena of the one-dimensional Ising-Gamma model, which describes the competition between the Ising interactions and the off-diagonal Gamma exchange interactions. The Hamiltonian is analytically solved in virtue of Jordan–Wigner, Fourier and Bogoliubov transformations. The phase diagram consists of three phases, including antiferromagnetic phase, paramagnetic phase and gapless spiral phase. A detailed study of information measures including the Wigner–Yanase skew information (WYSI) and quantum fidelity susceptibility (QFS) is conducted. The second-order QPTs are witnessed both by WYSI and QFS. When the QPT occurs between distinct gapped phases, a logarithmic scaling behavior of local measures and a superextensive behavior of fidelity susceptibility are identified, while they exhibit size-dependent behaviors for the gapped-to-gapless transitions. The correlation-length exponent extracted from various order parameters agrees well with each other.