---
title: Non-Hermitian Quantum Geometric Tensor (NH-QGT)
url: https://www.emergentmind.com/topics/non-hermitian-quantum-geometric-tensor-nh-qgt
type: topic
---

# Non-Hermitian Quantum Geometric Tensor (NH-QGT)

Searching arXiv for the cited NH-QGT literature and related recent work.
The non-Hermitian quantum geometric tensor (NH-QGT) is a biorthogonal extension of the quantum geometric tensor to non-Hermitian Hamiltonians, particularly relevant when right and left eigenstates are distinct. In the formulations used across \(\mathcal{PT}\)-symmetric, pseudo-Hermitian, Bloch-band, and wave-packet settings, the NH-QGT unifies two geometric structures: its real part defines a quantum metric, while its imaginary or antisymmetric part defines a Berry curvature [1811.04638], [2305.17675], [2306.00351]. In quasi-Hermitian or unbroken \(\mathcal{PT}\)-symmetric regimes with real spectra, this construction recovers a Hermitian-like tensor structure once the appropriate biorthogonal inner product or metric operator is introduced [1811.04638], [2509.17043], [2606.15922]. More generally, non-Hermiticity permits pseudo-Riemannian, complex, or degenerate geometries, and singularities of the metric identify exceptional points, spontaneous \(\mathcal{PT}\)-symmetry breaking, topological transitions, localization transitions, mobility edges, and many-body criticality [2305.17675], [2404.15628], [2606.15922].

## 1. Formal definition and biorthogonal setting

For a non-Hermitian Hamiltonian \(H(\lambda)\) depending on real parameters \(\lambda=(\lambda^1,\lambda^2,\dots)\), the right and left eigenstates satisfy
\[
H(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle = E_n(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle,\qquad
\bigl\langle\psi_n^L(\lambda)\bigr|\,H(\lambda)=E_n(\lambda)\,\bigl\langle\psi_n^L(\lambda)\bigr|
\]
with biorthonormality
\[
\bigl\langle \psi_n^L(\lambda)\bigm|\psi_m^R(\lambda)\bigr\rangle=\delta_{nm}.
\]
A standard left-right definition is
\[
Q_{ij}(\lambda)\equiv \bigl\langle \partial_i \psi^L \bigm| \partial_j \psi^R \bigr\rangle
-\bigl\langle \partial_i \psi^L \bigm|\psi^R \bigr\rangle
\bigl\langle \psi^L \bigm|\partial_j \psi^R \bigr\rangle,
\]
with \(\partial_i\equiv \partial/\partial \lambda^i\) [2305.17675], [2306.00351], [2404.15628]. Equivalently, in projector form,
\[
Q^{LR}_{n,ij}\equiv \langle\partial_i \psi_n^L | \Pi_n | \partial_j \psi_n^R\rangle,
\qquad
\Pi_n=1-|\psi_n^R\rangle\langle\psi_n^L|,
\]
which makes explicit that the tensor probes variations orthogonal, in the biorthogonal sense, to the reference band [2306.00351].

In \(\mathcal{PT}\)-symmetric quantum mechanics, the same object can be constructed using a \(\lambda\)-dependent inner product induced by a positive operator \(W(\lambda)\) satisfying
\[
W(\lambda)\,H(\lambda)=H^\dagger(\lambda)\,W(\lambda),
\]
with inner product
\[
\langle\psi|\phi\rangle_\lambda:=\langle\psi|W(\lambda)|\phi\rangle.
\]
If \(|\Psi_n(\lambda)\rangle\) are normalized eigenvectors of \(H(\lambda)\) and \(|\Phi_n(\lambda)\rangle:=W(\lambda)|\Psi_n(\lambda)\rangle\), then \(\langle \Phi_n|\Psi_m\rangle=\delta_{nm}\), and the extended tensor is defined by
\[
Q_{n,\mu\nu}
= \tfrac12 \Big[
\langle\partial_\mu\Phi_n|\partial_\nu\Psi_n\rangle
-\langle\partial_\mu\Phi_n|\Psi_n\rangle\langle\Phi_n|\partial_\nu\Psi_n\rangle
+(\Phi\leftrightarrow\Psi)
\Big].
\]
In that formulation, \(Q_{n,\mu\nu}=Q^*_{n,\nu\mu}\) and is independent of the particular choice of \(W(\lambda)\) [1811.04638].

Recent work in the quasi-Hermitian regime reformulates the same geometry through a Dyson map \(\eta(\lambda)\), where \(H(\lambda)=\eta(\lambda)\,\mathcal H(\lambda)\,\eta(\lambda)^{-1}\) with \(H=H^\dagger\). In that setting the NH-QGT is written as
\[
Q_{\mu\nu}
=\langle\partial_\mu L|\bigl(\mathbb I-|R\rangle\langle L|\bigr)|\partial_\nu R\rangle,
\]
and the Dyson map becomes the central object for a gauge-covariant description of non-Hermitian geometry [2606.15922].

## 2. Metric, curvature, and gauge structure

The NH-QGT is decomposed into metric and curvature components in close analogy with the Hermitian case. Several equivalent conventions appear in the literature. A common one identifies
\[
g_{ij}\equiv \Re\,Q_{ij},\qquad \Omega_{ij}\equiv -2\,\Im\,Q_{ij},
\]
while in other conventions the antisymmetric part in parameter indices is emphasized directly [2305.17675], [2306.00351], [2404.15628]. In the \(\mathcal{PT}\)-symmetric construction,
\[
g_{n,\mu\nu}\equiv \Re\,Q_{n,\mu\nu},\qquad
\Omega_{n,\mu\nu}\equiv \Im\,Q_{n,\mu\nu},
\]
with the connection one-form
\[
A_{n,\mu}= \Im\,\langle\Psi_n|\partial_\mu\Psi_n\rangle_\lambda,
\qquad
\Omega_{n,\mu\nu}=\partial_\mu A_{n,\nu}-\partial_\nu A_{n,\mu}
\]
[1811.04638].

The gauge structure differs from the Hermitian case because left and right eigenstates transform independently but in a constrained biorthogonal manner. In the gauge-covariant framework, the transformation
\[
|R\rangle\to e^{\alpha(\bm\lambda)}|R\rangle,\qquad
\langle L|\to e^{-\alpha(\bm\lambda)}\langle L|
\]
leaves the NH-QGT invariant owing to the projector \(\mathbf1-|R\rangle\langle L|\) [2606.15922]. In the Dyson-map description, the connection
\[
\Gamma_\mu=(\partial_\mu\eta)\eta^{-1}
\]
splits into Hermitian and anti-Hermitian pieces,
\[
\Gamma_\mu=S_\mu+K_\mu,\qquad
S_\mu=\tfrac12(\Gamma_\mu+\Gamma_\mu^\dagger),\quad
K_\mu=\tfrac12(\Gamma_\mu-\Gamma_\mu^\dagger),
\]
identified respectively as stretching and rotation components. This decomposition separates metric deformation from unitary gauge redundancy [2606.15922].

A related point of comparison concerns multiple non-Hermitian generalizations. Two principal variants recur in the literature: the left-right tensor \(Q^{LR}\), built from biorthogonal states, and the right-right tensor \(Q^{RR}\), built from right eigenstates normalized in the usual Hermitian sense [2306.00351], [2412.08141]. The \(Q^{RR}\) tensor remains Hermitian, whereas \(Q^{LR}\) is generally non-Hermitian and can have complex metric and curvature components [2412.08141]. This distinction is operational rather than merely formal, because the two tensors can control different observables in dynamical settings [2306.00351], [2412.08141].

## 3. Relation to fidelity, adiabatic transport, and wave-packet dynamics

The quantum metric component of the NH-QGT is closely tied to infinitesimal state distinguishability. In the \(\mathcal{PT}\)-symmetric formulation, if one regards
\[
\rho_n=|\Psi_n\rangle\langle\Phi_n|
\]
as a pure-state density operator and defines the fidelity
\[
F(\rho_n(\lambda),\rho_n(\lambda+\delta\lambda))
=\sqrt{\langle\Phi_n(\lambda+\delta\lambda)|\Psi_n(\lambda)\rangle
\langle\Phi_n(\lambda)|\Psi_n(\lambda+\delta\lambda)\rangle},
\]
then expansion to second order gives
\[
ds^2=2[1-F]=g_{n,\mu\nu}\,\delta\lambda^\mu\delta\lambda^\nu.
\]
Thus the real part of the tensor is a Bures distance element on parameter space [1811.04638].

A related self-normalized construction shows that, even in non-Hermitian systems, the diagonal metric component can coincide with fidelity susceptibility:
\[
g^{(n)}_{\mu\mu}=\chi_F^{(n)}
=\lim_{d\mu\to0}\frac{-2\ln|\langle\psi_n(\mu)|\psi_n(\mu+d\mu)\rangle|}{(d\mu)^2}
\]
[2404.15628]. This relation underlies the use of the metric as a diagnostic for non-Hermitian criticality.

The curvature component emerges in adiabatic evolution. In \(\mathcal{PT}\)-symmetric quantum mechanics, the time-dependent Schrödinger-like equation is
\[
i\partial_t|\psi(t)\rangle=[\,H(\lambda_t)+iK(\lambda_t)\,]|\psi(t)\rangle,
\]
where
\[
K=-\tfrac12\,W^{-1}\partial_tW
\]
is a Hermitian gauge term guaranteeing unitary evolution with respect to \(\langle\cdot|\cdot\rangle_\lambda\). Under slow variation along a closed loop \(C\), the geometric phase is
\[
\gamma_n=-\oint_C A_n
=-\oint \Im\,\langle\Psi_n|\partial_\mu\Psi_n\rangle_\lambda\,d\lambda^\mu,
\]
and Stokes’ theorem yields \(\gamma_n=-\iint_S\Omega_n\) [1811.04638].

In semiclassical wave-packet dynamics, non-Hermiticity makes the distinction between \(Q^{RR}\) and \(Q^{LR}\) physically consequential. For a narrow Gaussian packet in a two-band model under external force \(\mathbf F\), the center-of-mass equation contains the usual anomalous velocity term controlled by the RR Berry curvature, but also additional terms involving RR and LR connections and QGT corrections [2306.00351], [2412.08141]. One formulation gives
\[
\hbar\,\dot{\mathbf r}_c
=\nabla_k \Re\bigl[E_n(k_c)+F\cdot(A_n^{RR}-A_n^{LR})\bigr]
-F\times \Omega_n^{RR}(k_c),
\]
while an equivalent rewriting highlights \(\Re\,\Omega^{LR}\) instead [2306.00351]. First-order perturbation theory further shows that the RR QGT produces nonadiabatic shifts of anomalous velocity, while the LR QGT enters field-induced corrections to the intraband Berry connection and dynamical phase [2412.08141]. This is one reason later work states that both right-only and biorthogonal QGTs play a significant role in non-Hermitian wave-packet dynamics [2412.08141].

## 4. Geometric types specific to non-Hermitian systems

In Hermitian quantum mechanics the quantum metric is positive semidefinite and thus Riemannian. Non-Hermiticity broadens this structure. In non-Hermitian Su-Schrieffer-Heeger systems, the metric built from both left and right eigenvectors correctly identifies topological phases and topological phase transitions, but the resulting geometry can become pseudo-Riemannian or complex [2305.17675]. Specifically, after diagonalization the diagonal components of \(g_{ij}\) can have indefinite signs, and in some non-Hermitian phases
\[
\det g_{ij}=0,
\]
so that there exists a parameter-space direction \(\delta\lambda^i\) for which
\[
ds^2=g_{ij}\,\delta\lambda^i\,\delta\lambda^j=0
\]
identically. The cited work interprets this as dimensional reduction of the quantum geometry by one [2305.17675].

The same study observes that the topological transition curves \(z=\pm y\pm1\) in a non-Hermitian SSH model appear exactly as the lines along which \(ds^2=0\), and compares these null lines to lightlike paths in general relativity [2305.17675]. This suggests an analogy between non-Hermitian phase boundaries and null structures of pseudo-Riemannian geometry, although the analogy is explicitly mathematical rather than an identification of physical spacetime.

A more systematic operator-level account is developed in the gauge-covariant framework based on the Dyson connection. There the NH-QGT can be decomposed using projected states
\[
|\Phi_\mu\rangle\equiv \mathcal Q\,D_\mu^{(K)}|\Psi_H\rangle,\qquad
|\sigma_\mu\rangle\equiv \mathcal Q\,S_\mu|\Psi_H\rangle,\quad
\mathcal Q=\mathbb I-|\Psi_H\rangle\langle\Psi_H|,
\]
so that
\[
Q_{\mu\nu}
=\langle\Phi_\mu|\Phi_\nu\rangle
-\langle\sigma_\mu|\sigma_\nu\rangle
-i\Bigl[\langle\Phi_\mu|\sigma_\nu\rangle-\langle\sigma_\mu|\Phi_\nu\rangle\Bigr].
\]
In this formulation, the quantum metric is generally indefinite, and the non-Hermitian Berry curvature originates from the non-commutativity of the stretching components \(S_\mu\) at the operator level [2606.15922].

The literature also contains sector-resolved generalizations beyond the standard LR tensor. In the non-Hermitian Zeeman QGT, the tensor is generically non-Hermitian and splits into normal and anomalous sectors, producing an imaginary symmetric metric-like tensor and a real antisymmetric curvature-like tensor with no counterpart in the standard Hermitian QGT [2604.09725]. This is not the same object as the usual NH-QGT of band geometry, but it illustrates how non-Hermitian geometry can support additional symmetry-resolved structures.

## 5. Criticality, exceptional points, and phase transitions

A central application of the NH-QGT is the detection of critical points through singularities of the metric. In \(\mathcal{PT}\)-symmetric systems, when the ground state undergoes a level crossing with an excited state or coalesces at a \(\mathcal{PT}\)-breaking exceptional point, the ground-state metric
\[
g_{0,\mu\nu}
=\Re\sum_{n\neq0}
\frac{
\langle\Psi_0|\partial_\mu H|\Psi_n\rangle
\langle\Phi_n|\partial_\nu H|\Phi_0\rangle
+(\mu\leftrightarrow\nu)
}{
2[\,E_0-E_n\,]^2
}
\]
diverges because \(E_n\to E_0\) in the denominator [1811.04638]. The same work states that the metric diverges both at conventional quantum phase transition points and at spontaneous \(\mathcal{PT}\)-breaking points [1811.04638].

Near exceptional points, several works emphasize distinct singular mechanisms. In the biorthogonal picture, derivatives diverge as left and right eigenvectors coalesce and \(\langle\psi^L|\psi^R\rangle\to0\) [2404.15628]. In the gauge-covariant framework, the metric deformation operator and stretching sector dominate the leading divergence, and for a generic second-order exceptional point one finds
\[
g_{\mu\nu}\sim |\varepsilon_\mu|^{-2},\qquad
\Omega_{\mu\nu}\sim |\varepsilon_\mu|^{-3/2}
\]
with \(\varepsilon_\mu\) denoting displacement from the exceptional point along parameter axis \(\mu\) [2606.15922]. The B-VQE study likewise reports
\[
g^{11}_{\mathrm{bio}}(\lambda)\sim |\lambda-\lambda_{\mathrm{EP}}|^{-2}
\]
as an operational signature used to locate exceptional points [2606.18916].

The scope of NH-QGT diagnostics extends well beyond isolated exceptional-point physics. In non-Hermitian generalized Aubry-André models, the quantum metric exactly identifies localization transitions and mobility edges [2404.15628]. In a non-Hermitian cluster Ising model, peaks of \(g_{\lambda\lambda}\) and \(g_{\Gamma\Gamma}\) coincide with real or imaginary gap closings and with changes in the string order \(O_x\) or staggered magnetization \(m_y\) [2404.15628]. In a non-Hermitian mixed-field Ising model, the real-to-complex transition of the ground-state energy is accompanied by a divergence of \(g_{h_zh_z}\) [2404.15628]. The same study further reports finite-size scaling at single-particle localization transitions,
\[
g_{\mu\mu}(\mu_c;L)\sim L^\kappa,
\]
with measured \(\kappa\approx1.99\) for one non-Hermitian generalized Aubry-André model and \(\kappa\approx2.14\) for another [2404.15628].

These results counter a possible misconception that NH-QGT singularities are limited to topological band touchings or exceptional degeneracies. The cited evidence shows that localization transitions, mobility-edge crossings, and many-body gap-closing transitions can also be encoded in the non-Hermitian metric [2404.15628].

## 6. Representative models and experimental or computational access

Several model systems illustrate the range of NH-QGT phenomena.

In the dimerized XY chain with alternating complex field,
\[
H = \sum_{l=1}^L
[\, (J + (-1)^lJ_s)/2\,](\sigma_l^x\sigma_{l+1}^x + \sigma_l^y\sigma_{l+1}^y)
+ [\,(\Gamma + (-1)^l\Gamma_s)/2\,](\sigma_l^x\sigma_{l+1}^x - \sigma_l^y\sigma_{l+1}^y)
- (h - i(-1)^l\eta)/2\,\sigma_l^z,
\]
one obtains four bands \(\pm\Lambda_\pm(k)\), with unbroken \(\mathcal{PT}\) regime \(|\eta|<\eta_c=\min\{2J,2J_s\}\) [1811.04638]. The ground-state metric per site \(\bar g_{\mu\nu}\) diverges at two circular quantum phase transition loci in the anisotropic case, over a finite critical region in the pseudo-isotropic case, and at the \(\mathcal{PT}\)-breaking threshold \(\eta=\eta_c\) [1811.04638].

In non-Hermitian SSH models, the metric identifies all topological phase transitions only when both left and right eigenvectors are used [2305.17675]. For real non-reciprocal hoppings, one finds four transition lines
\[
z=\pm y\pm1,
\]
and in the non-Hermitian topological regions the metric is pseudo-Riemannian and satisfies \(\det g=0\) [2305.17675]. For complex non-reciprocal hoppings, the transition lines become
\[
|u-v|=1,\qquad |u^*+v^*|=1
\]
with the same pseudo-Riemannian and degenerate features [2305.17675].

In exciton-polariton systems, the generalized QGT components can be reconstructed from experimental observables [2306.00351]. Angle- and polarization-resolved spectroscopy yields the Stokes intensities \(I_H,I_V,I_{D,A},I_{L,R}\), from which one forms the right-eigenstate pseudospin components
\[
S^R_x = (I_H-I_V)/(I_H+I_V),\quad
S^R_y = (I_D-I_A)/(I_D+I_A),\quad
S^R_z = (I_L-I_R)/(I_L+I_R).
\]
Using the relation
\[
S^L_\pm(\kappa)=-S^R_\mp(\kappa)
\]
for a two-band system, one reconstructs the left pseudospin, forms the complex biorthogonal pseudospin
\[
S^{LR}(\kappa)=\langle\psi^L(\kappa)|\sigma|\psi^R(\kappa)\rangle/\langle\psi^L|\psi^R\rangle,
\]
extracts complex Bloch angles \(\theta^{LR},\phi^{LR}\), and then obtains \(g^{LR}_{ij}(\kappa)\) and \(\Omega^{LR}_{ij}(\kappa)\) from two-band formulas [2306.00351]. This provides a direct route to imaging both RR and LR geometries in a photonic platform.

On the algorithmic side, direct measurement protocols have been proposed for pseudo-Hermitian systems with real spectra. One study develops two schemes based on generalized expectation values
\[
\langle A\rangle_{gen}\equiv \langle\psi_1|A|\psi_2\rangle/\langle\psi_1|\psi_2\rangle
\]
between two nonadiabatically evolved states, allowing extraction of the full QGT via either generalized energy fluctuations or generalized forces [2509.17043]. Another study introduces B-VQE, which employs independent variational circuits for left and right eigenstates and reads out the NH-QGT on NISQ hardware through generalized parameter-shift overlaps and Hadamard-test-style circuits [2606.18916]. These approaches are specifically formulated for pseudo-Hermitian or quasi-Hermitian settings with real spectra.

## 7. Transport, topology, and broader implications

The NH-QGT is not solely a diagnostic of state-space geometry; it also appears in transport and response theory. In systems with a spectral line gap, the non-Hermitian QGT governs nonlinear electrical responses [2509.11765]. In the narrow-wavepacket limit, the second-order DC conductivity contains a scattering-time-independent term controlled by the band-renormalized non-Hermitian quantum metric \(G_{\mu\nu}^{LR}\), defining an intrinsic nonlinear conductivity [2509.11765]. For finite wavepacket width \(W\), additional nonlinear terms arise that depend on the imaginary part of the Berry curvature, leading to \(W^2\) and \(W^4\) corrections absent in Hermitian systems [2509.11765]. This suggests that non-Hermitian transport depends on geometric data beyond the Hermitian metric-curvature pair in its usual form.

In plasmonic lattices, a biorthogonal QGT analysis shows that a non-zero local Berry curvature can arise even in a square lattice without magnetic field [2305.13244]. There the effective two-band Hamiltonian contains a real pseudospin-orbit coupling term responsible for the quantum metric and an imaginary non-Hermitian term, \(i\,\Omega_y''(\mathbf k)\sigma_y\), arising from radiative and dissipative loss differences between TE and TM modes [2305.13244]. The work attributes the non-zero Berry curvature exclusively to non-Hermitian effects which break time-reversal symmetry, while the quantum metric originates from pseudospin-orbit coupling [2305.13244].

Several recent works also link NH-QGT structures to topological invariants and their bounds. In pseudo-Hermitian settings with real spectra, the Berry curvature obtained from the NH-QGT yields Chern numbers in the usual way [2509.17043]. In interacting non-Hermitian many-body systems, the B-VQE framework distinguishes state-topological and band-topological signatures and computes a state Chern number from the biorthogonal Berry curvature [2606.18916]. A separate line of work proves geometric bounds on non-Hermitian QGTs and response functions, including a Chern-number bound involving RR and LL metric blocks and connection differences [2512.23708].

A recurring misconception is that a single non-Hermitian QGT universally controls all observables. The available formulations suggest a more differentiated picture. In exciton-polariton and wave-packet dynamics, \(g^{LR}\) is associated with fidelity-susceptibility scaling near phase transitions, while transverse drifts appear to couple to \(\Omega^{RR}\) or to equivalent expressions involving LR curvature plus Berry-connection gradients [2306.00351], [2412.08141]. This suggests that non-Hermitian geometry is intrinsically multi-representational: the physically relevant tensor depends on the observable, the normalization convention, and whether the problem is formulated in biorthogonal, right-normalized, pseudo-Hermitian, or gauge-covariant language.

Across these formulations, the common principle remains stable. The NH-QGT is the object that organizes distance, curvature, adiabatic phase, critical singularity, and, in several settings, measurable dynamical or transport response for non-Hermitian quantum states [1811.04638], [2305.17675], [2306.00351].

Source: https://www.emergentmind.com/topics/non-hermitian-quantum-geometric-tensor-nh-qgt