Fidelity and criticality in the nonreciprocal Aubry-Andr{é}-Harper model
Chen-Chang Zeng Zhen Cai Guang-Heng Wang Gaoyong Sun



College of Physics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China


College of Physics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China


College of Physics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China


Corresponding author: gysun@nuaa.edu.cn
College of Physics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China
Key Laboratory of Aerospace Information Materials and Physics (NUAA), MIIT, Nanjing 211106, China


We study the critical behaviors of the ground and first excited states in the one-dimensional nonreciprocal Aubry-André-Harper model using both the self-normal and biorthogonal fidelity susceptibilities. We demonstrate that fidelity susceptibilities serve as a probe for the phase transition in the nonreciprocal AAH model. For ground states, characterized by real eigenenergies across the entire regime, both fidelity susceptibilities near the critical points scale as N^2, akin to the Hermitian AAH model. However, for the first-excited states, where 𝒫𝒯 transitions occur, the fidelity susceptibilities exhibit distinct scaling laws, contingent upon whether the lattice consists of even or odd sites. For even lattices, the self-normal fidelity susceptibilities near the critical points continue to scale as N^2. For odd lattices, the biorthogonal fidelity susceptibilities diverge, while the self-normal fidelity susceptibilities exhibit linear behavior, indicating a novel scaling law.





Fidelity and criticality in the nonreciprocal Aubry-André-Harper model
    Gaoyong Sun
    April 29, 2024
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§ INTRODUCTION

Non-Hermitian systems have become a prominent focus of physics research in recent years, thanks to their distinctive physical properties without the counterparts in Hermitian systems <cit.>.
The well-known fascinating phenomena are non-Hermitian skin effects <cit.>, 
exceptional points <cit.>,
and continuous quantum phase transition without gap closing <cit.>.
Recent findings have demonstrated that the eigenstates of non-Hermitian systems exhibit distinct behaviors <cit.>,
highlighting the crucial importance of understanding phases and phase transitions within this realm.

The Aubry-André-Harper (AAH) model is an intriguing simple model employed for studying the phase transition from the extended to localized phases <cit.> or topological phases <cit.> in Hermitian systems.
Recently, the AAH model has been extended to incorporate non-Hermitian systems <cit.>, providing a platform for exploring the interplay between non-Hermiticity and quasiperiodicity, where the phase transition between delocalization and localization <cit.>as well as the topological phases <cit.> persist. 
Moreover, a 𝒫𝒯 transition is expected to take place within the non-Hermitian AAH model <cit.>.
Nevertheless, it has been observed that certain eigenstates, such as the ground state, can remain real <cit.> even as they cross the critical points (exceptional points) of the 𝒫𝒯 transitions. 
A fundamental question arises: Can the eigenstate, characterized by the real eigenenergies across the entire regime of a parameter, adequately describe the 𝒫𝒯 transition? 










The exploration of phase transitions in non-Hermitian systems has emerged as a significant focus in recent years within non-Hermitian research. Consequently, characterizing these transitions is critically important within this domain.
A simple method for studying quantum phase transitions in Hermitian systems involves analyzing the fidelity susceptibility <cit.>, 
facilitated by the advancements in quantum information science.
Fidelity susceptibility, a proven valuable tool for studying quantum phase transitions in Hermitian systems, has recently been extended to non-Hermitian systems as well <cit.>.
In particular, fidelity susceptibility has been employed to characterize transitions from delocalization to localization <cit.>, wherein the system is argued to segregate into three subsequences <cit.> contingent upon the odd/even lattice structure.
An intriguing question arises: Can fidelity susceptibility effectively characterize phase transitions of the non-Hermitian AAH model? Moreover, should fidelity susceptibility be segmented into three subsequences?

We will study the quantum criticality, particularly at the exceptional points, of non-Hermitian AAH model 
using both the self-normal and biorthogonal fidelity susceptibilities of the ground state and the first excited state.
We find that the fidelity susceptibility exhibits smooth changes when the energies are real, yet undergoes significant alterations when the energy of the system transitions from real to complex.
Consequently, for ground states, it is found that both fidelity susceptibilities near the critical points scale as N^2, indicating a second-order phase transition. For first-excited states, where 𝒫𝒯 transitions occur, the fidelity susceptibilities exhibit distinct scaling laws. The self-normal fidelity susceptibilities scale as N^2 for even lattices, but exhibit linear behaviors for odd lattices, indicating a new scaling law. 




This paper is organized as follows. 
In Sec. <ref>, we introduce the one-dimensional nonreciprocal AAH model and provide a brief discussion of its characteristics.
In Sec. <ref>, we present the definitions of the self-normal and biorthogonal fidelity susceptibility of non-Hermitian systems.
In Sec. <ref>, we study the ground-state and first-excited phase transitions using the self-normal and biorthogonal fidelity susceptibility.
In Sec. <ref>, we summarize our results.









§ MODEL


In the following, we consider the one-dimensional nonreciprocal AAH model as shown in Fig.<ref>(a).
The Hamiltonian is given by,

    H=∑_j=1^N -(J_L c_j^† c_j+1+J_R c_j+1^† c_j)+2 λcos (2 πα j)c_j^† c_j,

where c_j^† and c_j represent the fermionic creation and annihilation operators at the jth lattice site respectively. 
Here, N denotes the length of the chain;
J_L=J e^-g, J_R=J e^g are the left and right hopping integrals between two adjacent lattices, 
which are nonreciprocal hopping terms introducing a non-Hermiticity;
λ is the amplitude of the chemical potential. The quasiperiodicity is imposed by the irrational number α, referred to as the golden rate (√(5)-1)/2. 
In a finite lattice, the ratio of fibonacci series F_n-1/F_n can substitute for the golden rate α when the lattice size is N = F_n. 
When g=0, the system in Eq.(<ref>) is the conventional Hermitian AAH model, which undergoes a phase transition from the extended phase to the localized phase at λ=1.
Interestingly, the system can be divided into three subsequences, F_3n = 8,34,144,610,2584,..., F_3n+1 = 13,55,233,987,... and F_3n+2 = 21,89,377,1597,...,
dependent on the odd/even properties of the denominators F_n in the irrational number α.
The subsequence F_3n is even, while the other two subsequences are odd. 
When g ≠ 0, the system in Eq.(<ref>) represents the non-Hermitian AAH model due to the nonreciprocal hopping, 
undergoing a phase transition from the extended phase to the localized phase at λ=e^g.
Throughout the paper, we investigate the characteristics of the nonreciprocal AAH model by employing fidelity susceptibility with J=1 under periodic boundary conditions (PBC).








§ FIDELITY SUSCEPTIBILITY


In a quantum system, two states can exhibit similarity within the same phase, even when the interval of the controlled parameter between them is large.
Around the critical point of the phase transition, despite a small parameter gap between the two states, they can exhibit an increased overlap enhanced by the quantum criticality.
For a non-Hermitian system, quantum fidelity is characterized by two definitions: the self-normal fidelity and the biorthogonal fidelity, depending on the usage of the eigenstates.

The self-normal fidelity is defined as the overlap between two right eigenstates corresponding to distinct parameters,

    F^S_n(λ, δλ) =⟨ψ_n^R(λ) |ψ_n^R(λ+δλ)⟩,

where |ψ_n^R(λ) ⟩ is the nth right eigenstate of the AAH model at λ.
The biorthogonal fidelity is defined as the inner product of the left and right eigenstates corresponding to two distinct parameters,

    F^B_n(λ, δλ) =⟨ψ_n^L(λ) |ψ_n^R(λ+δλ)⟩⟨ψ_n^L(λ+δλ) |ψ_n^R(λ)⟩,

where |ψ_n^L(λ) ⟩ is the nth left eigenstate of the AAH model at λ.
The values of the fidelities F^S_n(λ, δλ) and F^B_n(λ, δλ) dependent on the controlled parameters λ and δλ. 
In the Taylor expansion of fidelity, the main contribution originates from the second derivative term, and the coefficient associated with this term is termed the fidelity susceptibility, defined as:

    χ_F(λ)=lim _δλ→ 0-2 ln |F(λ, δλ)|/(δλ)^2

For second-order transitions, the fidelity susceptibility scales as,

    χ_F(λ) = L^2/νΦ((λ -λ_m)L^1/ν),

from the finite-size scaling theory, where ν is the correlation length critical exponent and λ denotes the peak position of the fidelity susceptibility.
Moreover, the fidelity susceptibility also exhibits a scaling law,

    χ_F(λ_m) ∝ N^2/ν,

near the critical point.
Thus one can simply determine the correlation length critical exponent ν from Eq.(<ref>).
However, the scaling laws of fidelity susceptibility are less known for the 𝒫𝒯 transitions.

In the following, we primarily employ the self-normal fidelity susceptibility χ^S_F_n and the biorthogonal fidelity susceptibility χ^B_F_n of the ground state and the first excited state 
to investigate phase transitions, especially the 𝒫𝒯 transitions, of the one-dimensional nonreciprocal AAH model as described in Eq.(<ref>).









§ PHASE TRANSITIONS


We employ exact diagonalization to explore the nonreciprocal AAH model, unveiling the phase diagram depicted in Fig.<ref>(b).
The phase diagram is derived from the real-complex transition [c.f. Fig.<ref>(c)(d)], thereby characterizing it as a 𝒫𝒯 transition.
As illustrated in Fig.<ref>, the system demonstrates non-zero imaginary components within the energy spectrum around λ≈ 2.71, 
slightly deviating from the analytical value λ = e^g due to the finite-size effect.
As the system size increases, the critical point tends toward λ = e^g.

Interestingly, we find that the ground-state energy of the AAH model changes smoothly and maintains its real value despite the 𝒫𝒯 transition occurring.
To examine whether the ground state can accurately depict the 𝒫𝒯 transition, 
we calculate both the self-normal fidelity susceptibility and the biorthogonal fidelity susceptibility χ_F_0^S and χ_F_0^B of the the ground state, as depicted in Fig.<ref>.
As can be seen from Fig.<ref>, both the self-normal fidelity susceptibility and the biorthogonal fidelity susceptibility per site increase [c.f. Fig.<ref>(a)(b)], 
indicating a phase transition in the ground state.
Interestingly, we observe that both the self-normal fidelity susceptibility and the biorthogonal fidelity susceptibility collapse into two distinct curves [c.f. Fig.<ref>(c)(d)] depending on the odd/even lattice.
The correlation length critical exponent ν≈ 1 is determined, consistent with the Hermitian AAH model.
Additionally, the correlation length critical exponent is doubly verified by fitting the maximum values of both χ_F_0^S and χ_F_0^B [c.f. Fig.<ref>(d)(e)].
Our findings suggest that the ground-state fidelity susceptibility serves as an excellent tool for detecting the phase transition. 
However, it appears that the ground-state phase transition is a second-order phase transition rather than a 𝒫𝒯 transition.

To explore the properties of the fidelity susceptibility around the 𝒫𝒯 transition, we focus on studying the first-excited state.
For the first-excited states, we observe that the fidelity susceptibilities display distinct scaling laws depending on whether the lattice comprises even or odd sites. 
For even lattices, the self-normal fidelity susceptibilities near the critical points persist in scaling as N^2 [c.f. Fig.<ref>].
While, for odd lattices, we find that the real part of the biorthogonal fidelity become 1/2 at the critical point [c.f. Fig.<ref>(a)], 
indicating the biorthogonal fidelity susceptibilities would diverge as δλ→ 0. 
In contrast, the self-normal fidelity susceptibilities demonstrate linear scaling instead [c.f. Fig.<ref>(b)], with a slope of 0.9744.
Our findings also indicate that the fidelity susceptibility serves as a reliable tool for detecting the 𝒫𝒯 transition. 
However, it exhibits a distinct behavior compared to the ground-state fidelity susceptibility, suggesting the presence of a novel scaling law for 𝒫𝒯 transition.








§ CONCLUSION


In summary, our investigation focuses on exploring the quantum criticality and fidelity susceptibility of both the ground states and first excited states within the one-dimensional nonreciprocal AAH model. We show that the scaling laws of the AAH model can be divided into two subsequences of the fidelity susceptibility depending on the odd/even properties of the denominators in the irrational number.
We therefore demonstrate that the fidelity susceptibility serves as a reliable tool for characterizing phase transitions in non-Hermitian systems.

In the ground state of the AAH model, characterized by real eigenenergies, we observe that the scaling behavior of the fidelity susceptibilities is perfectly consistent with that of the Hermitian AAH model, exhibiting an equivalent correlation length critical exponent of ν=2. 
In the first-excited state, where the system has a 𝒫𝒯 transition, the fidelity susceptibilities exhibit different scaling laws for the odd/even sites. For even lattices, the self-normal fidelity susceptibilities scale as N^2 as the ground state fidelity susceptibilities. However, the self-normal fidelity susceptibilities for odd lattices exhibit linear scaling law χ_F_1^S(λ_m) ∝ N, and the biorthogonal fidelity susceptibilities instead diverge. Exploring whether these scaling laws persist in the many-body AAH model would be an intriguing avenue for future investigation.









G.S. is appreciative of support from "the Fundamental Research Funds for the Central Universities, under the Grant No. NS2023055", the NSFC under the Grants No. 11704186, and the High Performance Computing Platform of Nanjing University of Aeronautics and Astronautics. C.Z., Z.C. and G.W. are appreciative of support from the Innovation Training Fund for College Students under Grant No .202310287211Y.










 






