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

Quantum phase transitions (QPTs) have attracted the intense interest of both theorists and experimentalists in condensed matter physics for decades. With variation of a non-thermal variable in the many-body Hamiltonian, the ground-state properties show abrupt changes as a result of competing ground-state phases  Sondhi et al. (1997); Sachdev (2000); Shen et al. (2020). The Landau-Ginzburg-Wilson (LGW) paradigm has provided a well-established framework, which rests on the assumption that a continuous equilibrium phase transition can be accessed by the onset of an order parameter associated with spontaneous symmetry breaking. A broad regime of order parameter fluctuations are controlled by proximity to a quantum critical point (QCP). It has been a long time to recognize that the symmetry breaking occurs spontaneously only in infinite-size quantum many-body systems, while recently they are sometimes observed in surprisingly small systems Bayha et al. (2020) and even few-body systems Cai et al. (2021); Liu et al. (2017); Hwang et al. (2015). Despite the enormous success, the breakdown of the LGW paradigm appears in a few different situations, for example, deconfined QCPs C. Wang and Senthil (2017); Shao et al. (2016) and deconfined critical universality classes Bi et al. (2020).

In recent years, an immense effort has been expended to understand QPTs in quasiperiodic systems Goldman and Kelton (1993); Iyer et al. (2013); Kohmoto et al. (1987); Lagendijk et al. (2009); hen and Ye (2019); Gonalves et al. ; Cai et al. (2013); Gong and Tong (2005); Agrawal et al. (2020); Roy et al. (2021); Goblot et al. (2020); Wang et al. (2020). The quasiperiodic system possesses a long-range periodicity, which is intermediate between that of the clean and randomly disordered cases, offering a rich playground to study quasiperiodic QCPs and unusual characteristic features, such as hierarchical energy spectra Sutherland (1986); Hiramoto and Kohmoto (1992, 1989) and localization-delocalization transitions. A paradigmatic model of the quasiperiodic system is the Aubry-André-Harper (AAH) model Liu and Gao (2017); Zeng et al. (2018); Tong et al. (2021); Liu et al. (2015); A. Purkayastha and Kulkarni (2018); Zeng et al. (2016); Wei (2019); Roy and Sharma (2019); Fraxanet et al. (2021); Xiao et al. (2021); Wang et al. (2016); Liu et al. (2015); Mondragon-Shem and Hughes (2014); Yoo et al. (2020); Zeng et al. (2017); Zhao et al. (2017); Longhi (2019); Zeng et al. (2020); Longhi (2021); Zhang et al. (2020); Longhi (2019); Liu et al. (2020), in which the quasiperiodicity is embodied in the form of a cosine modulation incommensurate with lattice spacing. With the rapid development of experimental technologies, the AAH model can be realized in optical waveguide lattices Gredeskul and Kivshar (1989); Christodoulides and Silberberg (2003); Pertsch et al. (2004), photonic crystals Negro et al. (2003); Lahini et al. (2009); Kraus et al. (2012) and cold atom systems Roati et al. (2008). For instance, the AAH Hamiltonian has been experimentally realized by cold atomic gases in a one-dimensional optical lattice perturbed by another weak incommensurate optical lattice Roati et al. (2008). These feasible platforms allow us to explore the emerging topological states of matter with additional interactions in incommensurate systems, including modulated off-diagonal hopping Liu et al. (2015), nearest neighbor $p$-wave superconductivity Yahyavi et al. (2019), a long-range $p$-wave superconducting pairing Fraxanet et al. (2021), and many-body interactions Yoo et al. (2020); Štrkalj et al. (2021).

The AAH model has gained popularity since it acts as a proxy for random potentials in the study of generic disordered system. An obstacle to comprehending the critical phenomena in the disordered systems is the undecidability of local order parameters. In close proximity to QCPs, the complex and non-local entanglement between individual constituents becomes extremely prominent at all distance scales. As such, it has been recognized that the exploration of quantum critical phenomena from the perspective of quantum information science is a great privilege, such as the von Neumann entropy Gong and Tong (2008), and quantum concurrence Werlang et al. (2010). The quantum fidelity susceptibility (QFS) has proved to be particularly useful for detecting the critical points of a symmetry-knowledge unknown system Wei and Lv (2018); Garnerone et al. (2009); Chen et al. (2008); Mao et al. (2021). It was shown that the QFS can not only identify the QCPs, but also satisfy the scaling ansatz, where the universal information can be retrieved. The most significant implication is that the finite-size scaling of such a universal order parameter dictates position of QCPs and the critical exponent of the correlation length $\nu$. For a quasi-periodic system with spatial complexity, the scaling theory of the QFS and the universality of localization transition have been partially understood. Notably, the finite-size scaling of the usual fidelity susceptibility is irrelevant to the dynamical exponent $z$. However, the critical exponents obey the scaling and hyperscaling relations, implying that there are only two independent exponents. Thus, a second independent critical exponent plays a decisive role in determining the universality class, which lies at the heart of critical phenomena. In this work, we apply the generalization of fidelity susceptibility to the one-dimensional AAH model with $p$-wave superconducting pairing, and devise a direct pathway to the determination of critical points and universal critical exponents of localization-delocalization transitions. Importantly, the theoretical predictions could be testified in state-of-the-art experiments.

The rest of the paper is organized as follows. Section II reviews the AAH model with $p$-wave superconductivity and determines its phase diagram. In Sec. III, we introduce the concept of the generalized fidelity susceptibility (GFS) and postulate its scaling hypothesis for the universal part. Section IV is devoted to the scaling behavior of the GFS in the AAH model and identification of critical exponents. Conclusions and discussions are presented in Sec. V.

The generalized AAH model with $p$-wave superconducting pairing in a quasi-periodically modulated potential is given by the following Hamiltonian:

where $c_{j}^{{\dagger}}$ $(c_{j})$ is the fermionic creation (annihilation) operator at the $j$-th site among total $N$ lattice sites, $J$ is the hopping strength between nearest-neighbor sites, $\Delta$ denotes the amplitude of $p$-wave superconducting pairing, and H.c. represents the Hermitian conjugate. The $p$-wave pairing amplitudes can be tuned by the mixture of spin-polarized fermions with a Bose-Einstein condensate Kinnunen et al. (2018), affected by an $s$-wave Feshbach resonance in a spin-polarized cold Fermi gas Liu et al. (2016), or induced by the proximity effect in stacking a superconducting wire on top of the normal metal. Here we focus on quasi-periodicity encoded in the chemical potential, keeping a constant hopping magnitude and pairing potential. The on-site potential terms are quasiperiodically varying according to the Aubry-André rule $V_{j}=V\cos(2\pi\alpha j+\phi)$, where $\alpha=(\sqrt{5}-1)/2$ is an irrational frequency and $V$ is the strength of the incommensurate potential. The parameter $\phi\in[0,2\pi)$ shifts the origin of the modulation representing a random phase. The boundary condition is imposed as $c_{N+1}$=$\sigma$$c_{1}$, where $\sigma$= $1$, $-1$, and $0$ corresponding to periodic, antiperiodic, and open boundary conditions, respectively. Without losing generality, $\Delta$ can be assumed to be real [the phase can be otherwise eliminated under global U(1) transformation] and $J=1$ is set as energy unit throughout the paper. For $\Delta=\pm 1$, the model will be equivalent to quasiperiodic Ising model Li et al. (2021); Caneva et al. (2007). When the $p$-wave pairing term is absent, i.e., $\Delta=0$, the AAH model in Eq.(1) becomes easily tractable as it can be written as $H$ = $\sum_{i,j}$ $c_{i}^{\dagger}$ ${\cal H}_{i,j}$ $c_{j}$. The eigenvectors $|\psi_{n}\rangle$ and the associated single-particle energies $\epsilon_{n}$ are obtained by diagonalizing the $N\times N$ single-particle Hamiltonian matrix ${\cal H}$. In the limit when $V/J$ $\to$ 0, Eq. (1) describes a metallic chain with all eigenstates being extended, while for $V/J$ $\to$ $\infty$ the eigenmodes are localized on one site. The Aubry-André transition from being extended to being localized is known to occur at $V/J=2$ as a consequence of Aubry-André duality between the Hamiltonian in position and momentum space.

As for a finite $p$-wave paring, i.e., $\Delta$$\neq$$0$, the Hamiltonian (1) can be diagonalized through a canonical Bogoliubov-de Gennes (BdG) transformation by introducing the new fermionic operators $\eta_{n}$ and $\eta_{n}^{{\dagger}}$,

where $u_{n,j}$ and $v_{n,j}$ denote the two components of the wave function at site $j$, and $n$ ($n=1,...,N$) is the energy band index. The eigenstates $|\psi_{n}\rangle$= $(u_{n,1}$,$u_{n,2}$,…,$v_{n,1}$,$v_{n,2}$,…$)^{T}$ can be determined by solving the Schrödinger equation $H|\psi_{n}\rangle=E_{n}|\psi_{n}\rangle$, which can be recast into a $2N$$\times$$2N$ matrix form as

where $A$ ($B$) is a $N\times N$ symmetric (antisymmetric) matrix. The nonzero elements are given by $A_{i,i}$=$V_{i}$, $A_{i,i+1}$= $A_{i+1,i}$=$-J$, and $B_{i,i+1}$=-$B_{i+1,i}$=$\Delta$. The matrix elements for the boundary terms are $A_{N,1}$=$A_{1,N}$=$-\sigma J$, and $B_{N,1}$=-$B_{1,N}$=$-\sigma\Delta$. The BdG Hamiltonian ${\cal H}$ in Eq.(9) respects an imposed particle-hole symmetry, namely, $\tau^{x}{\cal H}^{T}\tau^{x}=-{\cal H}$, where the Pauli matrix $\tau^{x}$ acts in the Nambu space. The energy levels appear in $\pm\epsilon_{n}$ conjugate pairs, with $\epsilon_{n}\geq 0$, except the zero energy mode, which is self-conjugate. As such, for finite lattices it is convenient to replace $\alpha$ with $F_{\ell-1}/F_{\ell}$, the ratio of two successive Fibonacci numbers Machida and Fujita (1986); Kohmoto (1983). Note that the irrational limit is reached as far as the numerical results are extrapolated to the scaling limit ($\ell\to\infty$). The period $F_{\ell}$ then acts like a finite length scale which controls scaling behavior. The Fibonacci-sequence quasiperiodic potential has an intimate connection with topological phase transition and Majorana modes.

While in the translational-invariant case the solution of Eq.(9) can be further reduced to the 2$\times$2 matrix form with independent momenta, in the quasidisorder case one has to diagonalize the $2N$$\times$$2N$ BdG matrix numerically, marking a qualitative difference between the disordered and the clean model. In terms of the new fermion operators, the Hamiltonian in Eq.(1) can be diagonalized as

with the single-particle eigenvalues being $\epsilon_{1}\leq$$\epsilon_{2}\leq$ $\cdots$ $\leq\epsilon_{N}$. The ground state of $H$ is the Bogoliubov vacuum state $|\psi_{g}\rangle$ annihilated by all $\eta_{n}$ for $n=1,...,N$, i.e., $\eta_{n}|\psi_{g}\rangle$=0, with an energy $E_{g}$= -$\sum_{n=1}^{N}\epsilon_{n}$. For a weak quasi-disorder strength, all the eigenstates of the system are extended, while the system becomes localized for a sufficiently strong disorder. Recently, it was found that with a nonzero superconducting pairing, a nonergodic critical phase intervenes the transition from the delocalized to localized state, and all the eigenstates are expected to be multifractal. The phase diagram shown in Fig.1 consists of the extended phase (EP), critical phase (CP), and localized phase (LP). The localized wavefunctions in the LP can be transformed into the extended ones in the EP by a Aubry-André duality occurring at $\Delta=0$ across a second-order QCP. The system undergoes a continuous QPT from the EP to the CP at $V_{c1}=2|J-\Delta|$. One finds that Eq.(1) is invariant under the transformation as $c_{j}$ $\to$ $-c_{j}^{\dagger}$ on odd $j$-th sites and $\alpha$ $\to$ $\alpha+1/2$ Wang et al. (2016). For the self-duality point ($J=\Delta$, $V=0$), the EP becomes unstable for arbitrarily weak disorder. The system displays a second-order QPT from CP to LP at $V_{c2}=2|J+\Delta|$.

In order to visually characterize the localized and extended nature of the entire energy spectrum, we evaluate the normalized inverse participation ratio (IPR) for each eigenstate $|\psi_{n}\rangle$ of the model Evers and Mirlin (2008); Licciardello and Thouless (1978); Wegner (1980); Misguich et al. (2016), given by

The IPRs can quantify the extent of distribution over the preferential bases. It should be noted that IPR is erroneously employed to describe the participation ratio in many literatures reports Gong and Tong (2005); Dukesz et al. (2009), which is the reciprocal of IPR Thouless (1974). In a specific ${\cal N}$-dimensional bases $|\varphi_{k}\rangle$, the IPR of the state $|\psi_{n}\rangle=\sum_{k=1}^{\cal N}c_{k}|\varphi_{k}\rangle$ reaches a maximal value $P_{\text{max}}=1$ when the state coincides exactly a single basis state, and attains a minimal value $P_{\rm min}=1/{\cal N}$ when the state is uniform in the selective bases. For a set of one-particle states in real space, the IPR scales inversely with the system size $N$ in the delocalized state, while appears to be independent of $N$ in the localized phase and shows intermediate behavior in the CP. Both transitions at finite strength of quasiperiodic modulation occur simultaneously for all eigenstates, as is revealed in Fig.2(a), in contrast to the presence of mobility edges in specific systems Biddle et al. (2011); Ganeshan et al. (2015); H. P. Lüschen and Bloch (2018); Guo et al. (2021). We thus further define the mean inverse participation ratio (MIPR) as

The evolution of MIPR is exhibited in Fig.2(b) on a logarithmic scale for different $p$-wave pairing strength $\Delta=0.5J$, $\Delta=1.0J$, and $\Delta=1.5J$. One finds the MIPR is capable of identifying the phase boundaries separating the extended, critical, and localized phases, which are captured by the turning points of the MIPR locating respectively at $V_{c1}=2|J-\Delta|$ and $V_{c2}=2|J+\Delta|$. The LP is gapless for periodic boundary conditions while gapped for open boundary conditions. Note that for a pure state $\rho=|\psi_{n}\rangle\langle\psi_{n}|$ of the entire system, the von Neumann entropy is zero and the IPR is inversely proportional to the participation entropy $S_{q}=\ln\sum_{k}\rho_{kk}^{q}/(1-q)$ of order $q=2$, i.e., $S_{2}=-\ln P$, which becomes the diagonal entropy for the reduced density matrix of the subsystem Polkovnikov (2011); Santos et al. (2011). As is observed in the inset of Fig.2(b), the scaling exponents extracted from the linear fittings for $V=0.2J$ and $V=0.5J$ with $\Delta=0.5J$ are approximately 0.98, which implies that the inverse of MIPR tends to scale extensively for extended states in the AAH model, resembling the volume-law scaling of the mean first-order Rényi entropy at infinite temperature, which is conjectured to be universal for translationally invariant quadratic fermionic Hamiltonians Vidmar et al. (2018), while $\bar{P}^{-1}$ declines towards a finite value close to ${\cal O}(1)$ for localized states, in analogy to the area law of the disordered averaged entanglement entropy Pastur and Slavin (2014). For critical states, the MIPR scales like $N^{-d^{*}}$ Kramer and Mackinnon (1993), where the fractal dimension $0<d^{*}<1$ depends on the fractal structure of wavefunctions. The fitting lines of $\ln(\bar{P}^{-1})$ with respect to $\ln N$ for $V=1.5J$ and $V=2.0J$ with $\Delta=0.5J$ gives rise to $d^{*}$ $\approx$ 0.75, implying that points in the whole CP belong to the same universality class. Remarkably, the fractal dimension $d^{*}$ and the associated IPR are proved to host intrinsic relation to the mean entanglement entropy Tomasi and Khaymovich (2020). One should be aware that a typical value of IPR can be used $\bar{P}_{\rm typ}$ = $\sum_{n=1}^{2N}\ln P_{n}/(2N)$, which is similar to the behavior of Eq.(12) for the AAH model (1) yet becomes more subtle when eigenstates display a single-particle mobility edge.

It is now well established that the QFS is a good measure to witness QPTs and manifest critical phenomena in translational invariant quantum systems Yang et al. (2008); Zanardi and Paunkovi (2006); You et al. (2007). The merit of the QFS in characterizing critical phenomena is the model-independent feature, which is quite suitable for quantum systems without prior knowledge of order parameters. To this end, the fidelity susceptibility is recognized as a sensitive probe of quantum criticalities in conjugate field Thesberg and Sørensen (2011), long-range interacting systems Zhu et al. (2018); Ren et al. (2020), deconfined QCP Sun et al. (2019), disordered systems, chaotic Hamiltonians Pandey et al. (2020), quantum many-body scars Surace et al. (2021), excited-state quantum phase transition Leblond et al. (2021), and holographic models Alishahiha and Astaneh (2017). Currently the investigations of the scaling of QFS in the context of quantum disordered systems are still poorly understood. In what follows, we will focus on the fidelity susceptibility and its generalization as well as the associated scaling in the AAH model.

The fidelity susceptibility provides a generic and direct approach to measure the quantum metric tensor via the transition probability of the quantum state being excited to other eigenstates during a sudden infinitesimal quench of the tuning parameter $\lambda$ You et al. (2007). The GFS of order $2r+2$ at the tuning parameter $\lambda$ associated with the state $|\psi_{n}(\lambda)\rangle$ is given by You and He (2015)

where $|\psi_{m}(\lambda)\rangle$ and $E_{m}(\lambda)$ correspond to the $m$th eigenstate and eigenvalue of this generic Hamiltonian $\hat{H}(\lambda)$, respectively. The numerator in Eq.(13) denotes the probability of exciting the system away from the state $|\psi_{n}(\lambda)\rangle$ through a relevant (or marginal) perturbation $\partial_{\lambda}\hat{H}$. The GFS of different orders is embodied by the power of the denominator. Concretely, Eq.(13) reduces respectively to the second derivative of the ground-state energy $\chi_{1}$ for $r=-1/2$ Chen et al. (2008) and the conventional quantum geometric tensor $\chi_{2}$ for $r=0$ You et al. (2007). We can anticipate that $\chi_{1}$ has a weaker divergence than $\chi_{2}$ at a critical point. The QFS can be also devised as the Riemannian metric tensor upon projecting the dynamics onto a single (non-degenerate) band Zanardi et al. (2007),

Regarding the absence of mobility edge in the energy spectrum, in the following we focus on the GFS of the lowest eigenstate $|\psi_{1}(\lambda)\rangle$. To this end, the superscript of $\chi_{2r+2}^{(1)}$ ($r$=$-1/2$, $0$, $1$) is omitted for abbreviation.

We then start the description of the finite-size scaling theory by recalling its main features that hold in the vicinity of the usual continuous QPT. The sensitivity is greatly enhanced, especially for the system at the quantum criticality compared with that away from the critical region Braun et al. (2018). Single parameter scaling posits that the correlation length $\xi$ is the only relevant length scale in the thermodynamic limit that diverges at the transition,

and the single-particle spectral gap $\epsilon_{s}$ of size $2\epsilon_{1}$ will vanish as

where $\nu$ is the correlation length exponent and $z$ is the dynamical critical exponent. For finite chains, the single parameter scaling hypothesis implies that the relevant physical quantities shall depend only of the ratio $\xi/N$, at least in the vicinity of the critical point where $\xi\gg$ 1 Fisher and Barber (1972). As $\lambda$ crosses QCPs adiabatically, the GFS shows a broad peak for a finite system size, signaling the location of pseudocritical points $\lambda_{m}$. With increasing system sizes $N$, the peaks of GFS become more pronounced and the maximal points of the GFS is expected that

where $\mu=2/\nu+2zr$ is the critical adiabatic dimension. For relevant operators $\partial_{\lambda}\hat{H}$ on sufficiently one-dimensional large-size systems, i.e., $\nu$$<$2, the pseudocritical points converge towards the critical points $\lambda_{c}$, satisfying

with $\theta$=$1/\nu$. Here one should be aware that the shift exponent $\theta$ in Eq.(18) is not necessarily equal to inverse of the correlation-length exponent $\nu$ Roncaglia et al. (2015), as it happens to the entanglement witness Osterloh et al. (2002); Ren et al. (2018). Accordingly, the GFS of a finite system with size $N$ in the neighborhood of a QCP shall obey the universal scaling form Albuquerque et al. (2010),

where $\phi_{r}(x)$ is a regular universal scaling function of the GFS of order $2r+2$, a priori unknown. Estimates for critical parameters can thus be obtained by plotting the scaled GFS $[\chi_{2r+2}(\lambda_{m})$-$\chi_{2r+2}(\lambda)]$/$\chi_{2r+2}(\lambda)$ versus $N^{1/\nu}(\lambda-\lambda_{m})$ by subtly adjusting the values of $\lambda_{m}$, $\nu$, and $z$ until data collapse is achieved. Alternatively, taking logarithm on both sides of Eqs.(17) and (18),yields

where fitting parameters $c_{2r+2}$ and $d_{2r+2}$ can determine the critical exponents $\nu$ and $z$ as

As such, we concentrate our attention on the region close to the critical point of localization-delocalization transitions, above which the eigenstates are localized within the finite localization length $\xi$ Sinha et al. (2019). Unlike the Aubry-André model with $\Delta=0$, the BdG Hamiltonian (9) for a finite $\Delta$ acts in an enlarged expanded Nambu-spinor space Altland and Zirnbauer (1997). The second-order derivative of first excitation $\chi_{1}$$\equiv$$\partial^{2}\epsilon_{1}/\partial V^{2}$ in the neighborhood of $V_{c2}$ for various odd number of system sizes $N$ with $\Delta=0.5$ is shown in Fig.3(a). One can see that $\chi_{1}$ presents a divergent peak ${\chi}_{1,\rm max}$ at $V_{m}$. The linear fittings give rise to parameters $c_{1}$ = -0.994 $\pm 0.066$ and $d_{1}$= 0.605 $\pm 0.010$. According to Eq.(III), the fitting values imply that $\nu\approx 1.006$ and $z\approx 1.384$. Furthermore, the single-particle spectral gap $2\epsilon_{1}$ with several system sizes from $N$ = 55 up to $N$ = 4181 are considered with periodic boundary conditions for $\Delta=0.5$ in Fig.3(b), indicating that $z=1.381$ $\pm 0.109$ according to Eq. (16).

We obtain all the eigenenergies and the corresponding wave functions by diagonalizing Eq.(9). The numerical results tempt us to evaluate the GFS of the AAH model through Eq.(13). First we recapitulate the finite-size scaling hypothesis of QFS in the Aubry-André model with $\Delta=0$ Mondragon-Shem and Hughes (2014). Previous work revealed the QFS near the EP-LP transition can be separately rescaled onto two different universal curves for even and odd numbers of lattice sites Wei (2019), while the retrieved critical exponents are quite close, which implies that two universal scaling functions are not necessary. The logarithm of QFS $\ln\chi_{2}$ of the Aubry-André model as a function of $V$ for $\Delta=0$ with different system sizes is exhibited in Fig.4(a). One can find that the fidelity susceptibility presents a maximum $\chi_{2,{\rm max}}$ at $V_{m}$. With increasing the system sizes, the peaks become more pronounced and $V_{m}$ gets closer to the critical point $V_{c}=2J$. The maximum value of $\ln\chi_{2}$ against the system size $N$ is displayed in the log-log scale, whose linear fit shows that $\ln\chi_{2,{\rm max}}$= $(2.000\pm 0.010)$ $\times$$\ln N$-5.400, implying $\nu=1.000\pm 0.005$ according to Eq.(17). This is consistent with the Harris criterion Harris (1974), which imposes that $\nu<2$ for phase transitions in the presence of incommensurate modulation. When the rescaled fidelity susceptibility $[\chi_{2}(V_{m})$-$\chi_{2}(V)]$/$\chi_{2}(V)$ is plotted as a function of the proper scaling variable $N^{1/\nu}(V-V_{m})$, all curves of distinct chain sizes in the vicinity of $V_{m}$ collapse into a single curve, as is shown in Fig.4(b), which corroborates the estimated critical parameter and the validity of the single-parameter scaling hypothesis (19). In particular the properly chosen value of $\phi=\pi$ much alleviates the odd-even effect.

Next, the QFS in the AAH model with respect to the strength of the incommensurate potential $V$ for odd number of lattice sizes with $\Delta=0.5$ is shown in Fig.5. The QFS exhibits an extensive scaling in the off-critical region. Therefore, the QFS per site $\chi_{2}/N$ appears to be an $N$-independent value. Instead, the QFS shows a stronger dependence on system size around $V_{c2}=3$, signaling the onset of the QCP in the AAH model. The maximum values $\chi_{2,\rm max}$ of the fidelity susceptibility near the QCP as a function of $N$ in log-log scale are plotted. The superextensive behavior at the pseudocritical point is reflected in the linear fitting $\ln\chi_{2,{\rm max}}$$\propto$(2.003 $\pm 0.033)$ $\ln N$, whose slope suggests that $\nu$$=0.999\pm 0.017$. Meanwhile, the numerical fitting in terms of Eq.(18) yields $\nu$=0.947 $\pm 0.090$. The accuracy of retrieved $\nu$ from the algebraic law (18) is generally plagued by the precision of numerical calculation.

In order to extract the dynamical exponent $z$ of CP-LP transitions, we further study the finite-size scaling of $\chi_{4}$. One can easily heed that $\chi_{4}$ displays much more divergent peaks than $\chi_{2}$ in the vicinity of QCP $V_{c2}=3$, as is disclosed in Fig.6. The linear fittings of the peak maxima $\chi_{4,{\rm max}}$ suggests $c_{4}$=-1.018 $\pm 0.1074$ and $d_{4}=4.773\pm 0.202$. According to Eq.(21) the extracted values of critical exponents $\nu$ = $0.993\pm 0.105$ and $z=1.380\pm 0.053$ for the CP-LP transition with $\Delta=0.5$ agree well with those obtained from the gap scaling Tong et al. (2021). In this vein, we continue to pick the critical exponents $\nu$ and $z$ via the scaling analysis of $\chi_{4}$ as the $p$-wave superconducting pairing $\Delta$ changes. The numerical results in Fig. 7(a) reveals that $\nu$$\simeq$ 1.000 and $z$ $\simeq$ 1.388 with little variation. It turns out that there is a discontinuity of $z$ when $\Delta$ increases from 0 to an infinitesimal value. Distinct values of $z\approx 2.375$ for $\Delta=0$ Wei (2019) and $z\approx 1.380$ for $\Delta\neq 0$ suggest that their ground states belong to different universality classes. For all $\Delta\neq 0$, the transitions across QCPs $V_{c2}=2|J+\Delta|$ belong to the same universality class as the quasiperiodic Ising chain Agrawal et al. (2020).

In this work, we investigate quantum criticality in the Aubry-André-Harper (AAH) model with $p$-wave superconducting pairing in terms of the generalized fidelity susceptibility (GFS). This quasiperiodic fermion system is Jordan-Wigner-equivalent to the quasi-periodically modulated transverse field XY chain. The interplay of spatial modulation of potential and symmetry breaking leads to quantum critical phenomena that are different from either the commensurate potential or randomly distributed potential. In the absence of $p$-wave pairing ($\Delta$=0), the AAH model hosts a phase transition from the extended state to the exponentially localized state through the self-duality point ($V=2J$). With a finite value of $\Delta$, the transition from the extended phase to the localized phase has to pass through an intermediate phase, and the critical point will develop into a critical region, which is sandwiched between the extended and localized states. Various available methods have been incorporated in identifying quantum critical points (QCPs) from numerical simulations. A useful quantity in characterizing quantum criticality of disordered systems is the inverse participation ratio (IPR), which is equivalent to the second-order participation Rényi entropy. Since there is no mobility edge in the energy spectrum, we then use the mean inverse participation ratio (MIPR) to characterize the degree of the extensivity in space of the wave function in different phases. The MIPR presents a power-law scaling $\propto N^{-d^{*}}$ in distinct phases, where $d^{*}$=1 in the extended phase, $d^{*}$=0 in the localized phase and the exponent $0<d^{*}<1$ in the intermediate critical phase.

We have developed accelerated methods for the location of critical points by the extrema of the universal order parameters. In this context, higher-order GFSs are more efficient in spotlighting the pseudo-critical points, even in the moderately large systems. The enhanced sensitivity is propitious for extracting the associated universal information from the finite-size scaling in quasiperiodic QCPs, whose system sizes are rapidly growing three-subsequence Fibonacci numbers Cookmeyer et al. (2020). This distinguishing feature becomes especially crucial in interacting many-body systems and higher-dimensional systems. One can see from Fig. 7(b) that $\chi_{4}$ has already spied on the pseudocritical point for $N=13$ via the visible peak. By performing a detailed numerical simulation, we find different orders of GFS obey power-law scaling in the vicinity of the localization transitions. The single parameter scaling of these macroscopic observables provide self-consistent results of critical exponents. Moreover, the generalization of fidelity susceptibility poses an efficient avenue to dynamic exponent $z$. The determined values of correlation-length exponent $\nu$$\simeq$ 1.000 and the dynamical exponent $z$ $\simeq$ 1.388 suggest that the quantum criticality of localization transitions in the AAH model for $\Delta\neq 0$ lies in a different universality class from the Aubry-André transition ($\Delta=0$) with $\nu$=1.000, and $z$=2.375 Wei (2019); Sinha et al. (2019), where a Aubry-André-type duality may prevent the finite energy excitations from localizing. Understanding the nature of the quasiperiodic localization transition, with and without a finite $p$-wave superconducting pairing, may thereby cut to the heart of the phenomenon. The critical properties of this fixed point are found to be intermediate to the clean and randomly disordered cases. The former case is represented by the clean transverse field Ising model in the celebrated Onsager universality class with $\nu$=$z$=1, while the latter is symbolized by the Anderson model with $\nu$=2/3, $z$=2 Cestari et al. (2011).

Last but not the least, another challenge in the study of quasiperiodic models is to separate physically measurable observables from the mathematically intriguing concepts. The quantum metric tensor has been experimentally measured with superconducting qubits Tan et al. (2019), coupled qubits in diamond Yu et al. (2020), and planar microcavity Gianfrate et al. (2020). Thus, an experimental measurement of the correlation-length exponent $\nu$ and the dynamical exponent $z$ becomes tractable. For instance, $z$ governs the low-temperature behavior of the specific heat $C_{v}\sim T^{-z}$ and can be extracted from the density of states $\rho\sim\epsilon^{1/z-1}$ or through the Kibble-Zurek mechanism Sinha et al. (2019). In this respect, our results can be explored in state-of-the-art experimental settings for moderate system sizes. Our tentative approach draws a link between quantum information science and analog quasiperiodic systems without explicit order parameters, and it would be interesting to investigate whether our results can be extended to more complex disordered models.