Bi-Orthogonal Floquet Fidelity Susceptibility

• The paper introduces a framework defining bi-orthogonal Floquet fidelity susceptibility using left/right Floquet eigenstates to diagnose exceptional points.
• It contrasts overlap-based biorthogonal methods with conventional self-normal fidelity, revealing enhanced singular behavior near PT transitions.
• The study bridges static non-Hermitian fidelity theory with Floquet diagnostics, outlining momentum-resolved scaling laws and divergence at exceptional points.

Searching arXiv for recent and foundational papers on biorthogonal Floquet fidelity susceptibility, non-Hermitian fidelity susceptibility, and Floquet exceptional points. Bi-orthogonal Floquet fidelity susceptibility is a non-Hermitian extension of fidelity susceptibility defined for periodically driven systems in which the relevant eigenstates are the left and right Floquet eigenstates of a non-unitary Floquet operator or of an effective non-Hermitian Floquet Hamiltonian. In this setting, ordinary same-side overlaps are not the natural geometric objects, because left and right eigenvectors are distinct and can coalesce at exceptional points. The directly Floquet-specific formulation is introduced as a diagnostic for Floquet exceptional points (FEPs) in a periodically quenched non-Hermitian two-band system (Dash et al., 2 Sep 2025), while the broader conceptual and perturbative foundations come from static non-Hermitian fidelity theory in biorthogonal bases (Sun et al., 2020, Tzeng et al., 2020, Tu et al., 2022) and from explicit overlap-based implementations of biorthogonal fidelity susceptibility in a non-Hermitian lattice model (Zeng et al., 2024). Across these works, the central theme is consistent: biorthogonal fidelity is the natural state-sensitivity probe in non-Hermitian spectral problems, and its susceptibility can display stronger singular behavior than self-normal constructions near $\mathcal{PT}$ transitions and exceptional points.

1. Definition and geometric setting

In non-Hermitian systems one must distinguish right and left eigenstates. For a parameter-dependent Hamiltonian $H(\lambda)$, the right and left eigenstates satisfy

$$H(\lambda)|\psi_i^R(\lambda)\rangle = E_i(\lambda)|\psi_i^R(\lambda)\rangle, \qquad H^\dagger(\lambda)|\psi_i^L(\lambda)\rangle = E_i^*(\lambda)|\psi_i^L(\lambda)\rangle.$$

A standard biorthogonal normalization is

$$\langle \psi_i^L(\lambda)|\psi_j^R(\lambda)\rangle=\delta_{ij}, \qquad \sum_i |\psi_i^R(\lambda)\rangle \langle \psi_i^L(\lambda)| = 1,$$

and in this basis the Hamiltonian is diagonal as

$$H(\lambda)=\sum_i E_i(\lambda)\,|\psi_i^R(\lambda)\rangle\langle \psi_i^L(\lambda)|.$$

This structure is the basis for the claim that quantities intended to diagnose non-Hermitian criticality should be defined biorthogonally rather than through ordinary same-side overlaps (Sun et al., 2020).

A closely related metricized formulation introduces a positive-definite Hermitian metric operator $G$ satisfying

$\frac{\partial}{\partial t}G=i(GH-H^\dagger G),$

with the gauge choice

$G=\sum_i \ket{L_i}\bra{L_i}$

away from an EP, so that the physical inner product becomes $\langle\psi|G|\varphi\rangle$ and the fidelity reduces to a biorthogonal overlap formula (Tzeng et al., 2020). In the $\mathcal{PT}$-symmetric setting, the same metricized fidelity is written as

$H(\lambda)$0

with quadratic expansion

$H(\lambda)$1

The derivative form is

$H(\lambda)$2

(Tu et al., 2022).

These static definitions supply the formal background for the Floquet generalization. A plausible implication is that bi-orthogonal Floquet fidelity susceptibility should be understood as the periodically driven counterpart of these left-right overlap and perturbative constructions, with the relevant states taken from the Floquet spectrum.

2. Floquet formulation and direct definition

The directly Floquet-specific construction appears in a periodically quenched non-Hermitian two-band system with step Hamiltonians

$H(\lambda)$3

and

$H(\lambda)$4

The one-period propagator over time $H(\lambda)$5 is

$H(\lambda)$6

and the Floquet eigenvalue problem is written as

$H(\lambda)$7

The effective Floquet Hamiltonian is defined by

$H(\lambda)$8

Because the propagator is non-unitary, the quasienergies are generally complex, and the left and right Floquet eigenstates $H(\lambda)$9 and $$H(\lambda)|\psi_i^R(\lambda)\rangle = E_i(\lambda)|\psi_i^R(\lambda)\rangle, \qquad H^\dagger(\lambda)|\psi_i^L(\lambda)\rangle = E_i^*(\lambda)|\psi_i^L(\lambda)\rangle.$$0 must be distinguished (Dash et al., 2 Sep 2025).

The bi-orthogonal Floquet fidelity susceptibility is then defined in spectral form as

$$H(\lambda)|\psi_i^R(\lambda)\rangle = E_i(\lambda)|\psi_i^R(\lambda)\rangle, \qquad H^\dagger(\lambda)|\psi_i^L(\lambda)\rangle = E_i^*(\lambda)|\psi_i^L(\lambda)\rangle.$$1

with

$$H(\lambda)|\psi_i^R(\lambda)\rangle = E_i(\lambda)|\psi_i^R(\lambda)\rangle, \qquad H^\dagger(\lambda)|\psi_i^L(\lambda)\rangle = E_i^*(\lambda)|\psi_i^L(\lambda)\rangle.$$2

and

$$H(\lambda)|\psi_i^R(\lambda)\rangle = E_i(\lambda)|\psi_i^R(\lambda)\rangle, \qquad H^\dagger(\lambda)|\psi_i^L(\lambda)\rangle = E_i^*(\lambda)|\psi_i^L(\lambda)\rangle.$$3

The paper explicitly states that this quantity is, in general, a tensor quantity, while focusing on the cases $$H(\lambda)|\psi_i^R(\lambda)\rangle = E_i(\lambda)|\psi_i^R(\lambda)\rangle, \qquad H^\dagger(\lambda)|\psi_i^L(\lambda)\rangle = E_i^*(\lambda)|\psi_i^L(\lambda)\rangle.$$4 and $$H(\lambda)|\psi_i^R(\lambda)\rangle = E_i(\lambda)|\psi_i^R(\lambda)\rangle, \qquad H^\dagger(\lambda)|\psi_i^L(\lambda)\rangle = E_i^*(\lambda)|\psi_i^L(\lambda)\rangle.$$5 (Dash et al., 2 Sep 2025).

The same work also gives an overlap-based reconstruction: $$H(\lambda)|\psi_i^R(\lambda)\rangle = E_i(\lambda)|\psi_i^R(\lambda)\rangle, \qquad H^\dagger(\lambda)|\psi_i^L(\lambda)\rangle = E_i^*(\lambda)|\psi_i^L(\lambda)\rangle.$$6 with the small-$$H(\lambda)|\psi_i^R(\lambda)\rangle = E_i(\lambda)|\psi_i^R(\lambda)\rangle, \qquad H^\dagger(\lambda)|\psi_i^L(\lambda)\rangle = E_i^*(\lambda)|\psi_i^L(\lambda)\rangle.$$7 expansion

$$H(\lambda)|\psi_i^R(\lambda)\rangle = E_i(\lambda)|\psi_i^R(\lambda)\rangle, \qquad H^\dagger(\lambda)|\psi_i^L(\lambda)\rangle = E_i^*(\lambda)|\psi_i^L(\lambda)\rangle.$$8

The authors note that this reconstruction is not written explicitly in the paper, but it encodes the perturbative content of the matrix-element formula they use (Dash et al., 2 Sep 2025). By contrast, the overlap-first static implementation in the nonreciprocal Aubry-André-Harper model defines

$$H(\lambda)|\psi_i^R(\lambda)\rangle = E_i(\lambda)|\psi_i^R(\lambda)\rangle, \qquad H^\dagger(\lambda)|\psi_i^L(\lambda)\rangle = E_i^*(\lambda)|\psi_i^L(\lambda)\rangle.$$9

and

$$\langle \psi_i^L(\lambda)|\psi_j^R(\lambda)\rangle=\delta_{ij}, \qquad \sum_i |\psi_i^R(\lambda)\rangle \langle \psi_i^L(\lambda)| = 1,$$0

(Zeng et al., 2024). This suggests a direct structural correspondence between static and Floquet biorthogonal fidelity constructions.

3. Relation to self-normal fidelity and why biorthogonality matters

Several of the cited works contrast biorthogonal fidelity with same-side or self-normal fidelity. In the many-body framework, the self-normal density matrix is

$$\langle \psi_i^L(\lambda)|\psi_j^R(\lambda)\rangle=\delta_{ij}, \qquad \sum_i |\psi_i^R(\lambda)\rangle \langle \psi_i^L(\lambda)| = 1,$$1

whereas the biorthogonal density matrix is

$$\langle \psi_i^L(\lambda)|\psi_j^R(\lambda)\rangle=\delta_{ij}, \qquad \sum_i |\psi_i^R(\lambda)\rangle \langle \psi_i^L(\lambda)| = 1,$$2

The associated pure-state fidelities are

$$\langle \psi_i^L(\lambda)|\psi_j^R(\lambda)\rangle=\delta_{ij}, \qquad \sum_i |\psi_i^R(\lambda)\rangle \langle \psi_i^L(\lambda)| = 1,$$3

and

$$\langle \psi_i^L(\lambda)|\psi_j^R(\lambda)\rangle=\delta_{ij}, \qquad \sum_i |\psi_i^R(\lambda)\rangle \langle \psi_i^L(\lambda)| = 1,$$4

The central claim is that biorthogonal quantum phase transitions are described by the biorthogonal fidelity susceptibility rather than the conventional self-normal fidelity susceptibility (Sun et al., 2020).

The overlap-based nonreciprocal AAH study uses a different self-normal convention,

$$\langle \psi_i^L(\lambda)|\psi_j^R(\lambda)\rangle=\delta_{ij}, \qquad \sum_i |\psi_i^R(\lambda)\rangle \langle \psi_i^L(\lambda)| = 1,$$5

and explicitly remarks that this quantity uses only right states and therefore does not directly encode the left-right geometry that is natural in non-Hermitian spectral theory (Zeng et al., 2024).

The reason biorthogonality matters is stated in closely aligned ways across the literature. In a non-Hermitian problem, right eigenstates are generally not orthogonal to each other under the ordinary inner product, and left and right eigenvectors must both be used to define physically meaningful overlaps (Dash et al., 2 Sep 2025). Near an exceptional point the eigenvectors themselves coalesce, the operator becomes defective, and this induces strong singular behavior in any quantity sensitive to parametric changes of the eigenstates (Dash et al., 2 Sep 2025). The static metricized analysis frames the same issue geometrically: the standard inner product is no longer the correct one, and fidelity must be evaluated in a parameter-dependent Hilbert-space geometry (Tzeng et al., 2020).

A common misconception is that non-Hermitian fidelity can be obtained by simply normalizing right eigenvectors and computing ordinary overlaps. The cited works do not support that equivalence. Instead, they repeatedly distinguish self-normal and biorthogonal constructions and show that the latter is the faithful probe of non-Hermitian critical geometry (Sun et al., 2020, Zeng et al., 2024).

4. Exceptional points, $$\langle \psi_i^L(\lambda)|\psi_j^R(\lambda)\rangle=\delta_{ij}, \qquad \sum_i |\psi_i^R(\lambda)\rangle \langle \psi_i^L(\lambda)| = 1,$$6 transitions, and singular behavior

In the Floquet setting, the quasienergy degeneracy conditions are

$$\langle \psi_i^L(\lambda)|\psi_j^R(\lambda)\rangle=\delta_{ij}, \qquad \sum_i |\psi_i^R(\lambda)\rangle \langle \psi_i^L(\lambda)| = 1,$$7

and at an FEP one additionally has

$$\langle \psi_i^L(\lambda)|\psi_j^R(\lambda)\rangle=\delta_{ij}, \qquad \sum_i |\psi_i^R(\lambda)\rangle \langle \psi_i^L(\lambda)| = 1,$$8

so the Floquet propagator becomes defective (Dash et al., 2 Sep 2025). The paper emphasizes that this is the genuine exceptional-point condition: not just quasienergy degeneracy, but coalescence of Floquet eigenvectors and a Jordan-block structure of $$\langle \psi_i^L(\lambda)|\psi_j^R(\lambda)\rangle=\delta_{ij}, \qquad \sum_i |\psi_i^R(\lambda)\rangle \langle \psi_i^L(\lambda)| = 1,$$9 (Dash et al., 2 Sep 2025). Since the BFFS denominator contains

$$H(\lambda)=\sum_i E_i(\lambda)\,|\psi_i^R(\lambda)\rangle\langle \psi_i^L(\lambda)|.$$0

the quantity becomes large as the effective quasienergy gap collapses, and the paper states that at an EP “the denominator in the expression of biorthogonal fidelity susceptibility approaches zero,” yielding a divergence (Dash et al., 2 Sep 2025).

The static fidelity literature gives two complementary singularity criteria. In the metricized non-Hermitian framework, the generalized susceptibility is introduced as

$$H(\lambda)=\sum_i E_i(\lambda)\,|\psi_i^R(\lambda)\rangle\langle \psi_i^L(\lambda)|.$$1

and the paper states that EPs are found when

$$H(\lambda)=\sum_i E_i(\lambda)\,|\psi_i^R(\lambda)\rangle\langle \psi_i^L(\lambda)|.$$2

(Tzeng et al., 2020). For the $$H(\lambda)=\sum_i E_i(\lambda)\,|\psi_i^R(\lambda)\rangle\langle \psi_i^L(\lambda)|.$$3-symmetric $$H(\lambda)=\sum_i E_i(\lambda)\,|\psi_i^R(\lambda)\rangle\langle \psi_i^L(\lambda)|.$$4 toy model

$$H(\lambda)=\sum_i E_i(\lambda)\,|\psi_i^R(\lambda)\rangle\langle \psi_i^L(\lambda)|.$$5

the explicit result is

$$H(\lambda)=\sum_i E_i(\lambda)\,|\psi_i^R(\lambda)\rangle\langle \psi_i^L(\lambda)|.$$6

which tends to $$H(\lambda)=\sum_i E_i(\lambda)\,|\psi_i^R(\lambda)\rangle\langle \psi_i^L(\lambda)|.$$7 at the EPs $$H(\lambda)=\sum_i E_i(\lambda)\,|\psi_i^R(\lambda)\rangle\langle \psi_i^L(\lambda)|.$$8 (Tzeng et al., 2020).

The $$H(\lambda)=\sum_i E_i(\lambda)\,|\psi_i^R(\lambda)\rangle\langle \psi_i^L(\lambda)|.$$9-symmetric perturbative theory sharpens this statement. For PT-broken states, $G$0 is generally complex, but

$G$1

is interpreted as the PT-symmetrized susceptibility including the PT-partner state (Tu et al., 2022). Approaching an EP from the PT-broken side, the paper proves that

$G$2

under the stated generic conditions (Tu et al., 2022). It also proves that at a second-order EP,

$G$3

for fidelity between PT-unbroken and PT-broken states across the EP (Tu et al., 2022).

This $G$4 feature reappears in the nonreciprocal AAH model. For the first excited state on odd lattices, the biorthogonal fidelity satisfies

$G$5

at the critical point, essentially independent of system size; therefore the corresponding biorthogonal fidelity susceptibility diverges as $G$6 for any finite $G$7, written in the paper’s table notation as

$G$8

(Zeng et al., 2024). This ties the $G$9 fidelity limit directly to singular susceptibility in a concrete non-Hermitian lattice problem.

A plausible implication is that the same mechanism can operate in Floquet problems at second-order Floquet exceptional points, though the directly Floquet paper supports this through momentum-resolved peaks and summed divergences rather than through an explicit $\frac{\partial}{\partial t}G=i(GH-H^\dagger G),$0 theorem (Dash et al., 2 Sep 2025).

5. Scaling laws and diagnostic behavior

The static many-body perturbation theory gives the spectral representation

$\frac{\partial}{\partial t}G=i(GH-H^\dagger G),$1

and uses the finite-size scaling form

$\frac{\partial}{\partial t}G=i(GH-H^\dagger G),$2

for the non-Hermitian transverse-field Ising chain (Sun et al., 2020). In that model, the peak scales linearly,

$\frac{\partial}{\partial t}G=i(GH-H^\dagger G),$3

implying $\frac{\partial}{\partial t}G=i(GH-H^\dagger G),$4, while the second derivative of the ground-state energy yields $\frac{\partial}{\partial t}G=i(GH-H^\dagger G),$5; the transition is concluded to belong to the Ising universality class (Sun et al., 2020).

The overlap-based nonreciprocal AAH implementation uses the finite-size scaling ansatz

$\frac{\partial}{\partial t}G=i(GH-H^\dagger G),$6

and finds a strong state- and parity-dependent structure (Zeng et al., 2024). The summary table is:

States	Even lattice	Odd lattice
$\frac{\partial}{\partial t}G=i(GH-H^\dagger G),$7, $\frac{\partial}{\partial t}G=i(GH-H^\dagger G),$8	$\frac{\partial}{\partial t}G=i(GH-H^\dagger G),$9	$G=\sum_i \ket{L_i}\bra{L_i}$0
$G=\sum_i \ket{L_i}\bra{L_i}$1, $G=\sum_i \ket{L_i}\bra{L_i}$2	$G=\sum_i \ket{L_i}\bra{L_i}$3	$G=\sum_i \ket{L_i}\bra{L_i}$4

For the ground state, both self-normal and biorthogonal susceptibilities obey

$G=\sum_i \ket{L_i}\bra{L_i}$5

with $G=\sum_i \ket{L_i}\bra{L_i}$6 (Zeng et al., 2024). For the first excited state on even lattices, both also scale as $G=\sum_i \ket{L_i}\bra{L_i}$7 with $G=\sum_i \ket{L_i}\bra{L_i}$8 (Zeng et al., 2024). For odd lattices, however, the biorthogonal susceptibility diverges while the self-normal susceptibility follows the new scaling law

$G=\sum_i \ket{L_i}\bra{L_i}$9

(Zeng et al., 2024).

In the direct Floquet setting, the susceptibility is not presented through an $\langle\psi|G|\varphi\rangle$0-scaling analysis but as a momentum-resolved and momentum-summed diagnostic. The paper defines the Brillouin-zone-summed scalar

$\langle\psi|G|\varphi\rangle$1

and reports two key behaviors: the momentum-resolved absolute BFFS shows large non-zero peaks at the momenta where FEPs are located, and the momentum-summed absolute BFFS diverges sharply when the number of FEPs changes with the driving half-period $\langle\psi|G|\varphi\rangle$2 (Dash et al., 2 Sep 2025). Thus the Floquet diagnostic is chiefly organized around momentum-space singularity structure and drive-induced pair creation and annihilation of FEPs, rather than conventional many-body finite-size scaling.

6. Physical scope, applications, and limitations

Within the Floquet FEP study, BFFS is one probe among several. The authors first identify FEPs analytically from the quasienergy conditions and defectiveness of $\langle\psi|G|\varphi\rangle$3, then characterize them topologically by integer winding numbers,

$\langle\psi|G|\varphi\rangle$4

with $\langle\psi|G|\varphi\rangle$5 or $\langle\psi|G|\varphi\rangle$6, and finally use BFFS as a complementary state-sensitive diagnostic of where these singularities occur in momentum space and when they appear or disappear as the period changes (Dash et al., 2 Sep 2025). The same work connects these spectral singularities to zero-energy edge states, Dirac-like dispersion around $\langle\psi|G|\varphi\rangle$7, and a higher-order skin effect with skin modes localized at both edges and corners (Dash et al., 2 Sep 2025). In that sense, BFFS functions as a bulk state-sensitivity probe within a broader Floquet non-Hermitian phase characterization.

The static works delimit the conditions under which biorthogonal fidelity susceptibility is most rigorously controlled. The perturbative many-body theory of (Sun et al., 2020) is explicitly framed for arbitrary interacting non-Hermitian many-body systems with real eigenvalues, in a nondegenerate and diagonalizable regime. The metricized EP analysis of (Tzeng et al., 2020) states that the convenient biorthogonal overlap formula is valid away from EPs; at the EP itself one should in principle solve the metric equation of motion explicitly. The PT-symmetric results of (Tu et al., 2022) rely on PT symmetry for the strongest theorems, including reality in the PT-unbroken sector, the PT-partner interpretation of $\langle\psi|G|\varphi\rangle$8, negative divergence near EPs, and the $\langle\psi|G|\varphi\rangle$9 criterion for second-order EPs.

Transferred to Floquet systems, several caveats remain explicit in the literature. The static AAH analysis notes that in Floquet problems one must decide whether the parameter changes the one-period operator $\mathcal{PT}$0, the micromotion, or an effective Hamiltonian branch; quasienergies are defined modulo $\mathcal{PT}$1; non-unitary Floquet operators can exhibit branch-cut ambiguities and additional spectral winding; exceptional points may occur at quasienergy $\mathcal{PT}$2 or $\mathcal{PT}$3; and finite-size scaling may be altered by drive-induced resonances (Zeng et al., 2024). The many-body static blueprint similarly notes that a finished Floquet susceptibility theory requires derivations specific to quasienergy spectra and non-unitary Floquet evolution, even though the left-right overlap logic carries over directly (Sun et al., 2020).

The resulting consensus is methodologically narrow but robust. Bi-orthogonal Floquet fidelity susceptibility is a momentum-resolved, parameter-derivative-sensitive measure built from left and right Floquet eigenstates of a non-Hermitian effective Floquet Hamiltonian or Floquet operator; it becomes sharply enhanced when quasienergy bands approach an exceptional degeneracy; and it serves as a practical probe of the creation, annihilation, and momentum-space location of zero and $\mathcal{PT}$4 Floquet exceptional points (Dash et al., 2 Sep 2025). The static literature strengthens that interpretation by showing that biorthogonal fidelity, rather than self-normal fidelity, is the quantity aligned with the spectral geometry of non-Hermitian criticality (Sun et al., 2020, Zeng et al., 2024), and by establishing the characteristic exceptional-point signatures $\mathcal{PT}$5 and, at second-order EPs, $\mathcal{PT}$6 in $\mathcal{PT}$7-symmetric settings (Tzeng et al., 2020, Tu et al., 2022).