Non-Hermitian Quantum Metric Tensor

• The paper introduces non-Hermitian quantum metric tensors as dual concepts—one as a metric operator defining physical inner products and the other as the metric sector in quantum geometric tensors.
• It distinguishes operator metrics in quasi-Hermitian and pseudo-Hermitian systems from geometric constructions using biorthogonal left–right eigenstates, each with unique dynamical implications.
• The formulations provide insights into critical transitions, wavepacket dynamics, and nonlinear responses, offering practical avenues for experimental probing in non-Hermitian systems.

The expression non-Hermitian quantum metric tensor denotes two distinct structures in the literature. In quasi-Hermitian, pseudo-Hermitian, and (\mathcal{PT})-symmetric quantum mechanics, it can mean a metric operator such as (\Theta), (W(\lambda)), or (\rho(t)) that defines the physical inner product and renders a manifestly non-Hermitian Hamiltonian Hermitian in an amended Hilbert space. In parameter-space quantum geometry, it denotes the metric sector of a non-Hermitian quantum geometric tensor (QGT), usually built from left and right eigenstates and paired with a Berry-curvature sector. These two uses are directly related only at the level that the physical inner product affects geometry; they are not the same object [1201.2263][1811.04638].

1. Terminological scope and conceptual split

A persistent source of ambiguity is that the word metric is used both for an operator acting on Hilbert space and for a tensor on a parameter manifold. In the operator-theoretic usage, the metric is the object entering
[
\langle \psi,\phi\rangle_\Theta=\langle \psi|\Theta|\phi\rangle,
]
with (H^\dagger \Theta=\Theta H), so that a non-Hermitian (H) becomes Hermitian in the amended inner product [1201.2263]. In the geometric usage, the metric is the real or symmetric part of a QGT built from derivatives of states with respect to external parameters, momenta, or control fields [1811.04638].

This distinction is explicit in work on dynamical metric operators. One formulation states that the time-dependent operator (\rho(t)) “is not a metric in the strict sense of a map in a metric space, and it does not correspond to the quantum geometric tensor discussed in Refs. [47,48]”; its role is instead to define the physical inner product for non-Hermitian dynamics [2301.02247]. A broader state-space analysis reaches the same conclusion from another direction: once left and right states are distinct, there is no single automatic non-Hermitian analogue of the Hermitian Fubini–Study tensor, and one must specify the pairing and normalization before speaking of a quantum metric [2507.18486].

The modern literature therefore treats non-Hermitian quantum metric tensor as a family of related but inequivalent constructions. Some are operator metrics tied to quasi-Hermiticity, some are left-right or same-sector QGT metrics, some are right-state metrics used as criticality diagnostics, and some are complex symmetric metric-like tensors that cease to be Riemannian in the Hermitian sense [2412.08141][2305.17675].

2. Metric operators in quasi-Hermitian and pseudo-Hermitian quantum mechanics

In quasi-Hermitian quantum mechanics, the basic structure is an invertible map (\Omega) relating a non-Hermitian representation (H) to a Hermitian operator (h),
[
H=\Omega^{-1}h\,\Omega,
\qquad
\Theta=\Omega^\dagger\Omega,
]
with the hidden-Hermiticity condition
[
H^\dagger \Theta=\Theta H.
]
The admissible (\Theta) must be Hermitian, invertible, and positive definite, (\Theta=\Theta^\dagger), (\Theta>0), so that the amended inner product is a genuine Hilbert-space norm [1201.2263].

This operator metric is generically non-unique. In spectral form,
[
\Theta=\sum_{n=0}^{N-1} |\psi_n\rangle\,\kappa_n\,\langle\langle \psi_n|,
]
so the metric contains free parameters even for fixed (H). For finite-dimensional real tridiagonal Hamiltonians with real nondegenerate spectrum, a recurrent solution of the Dieudonné equation (H^\dagger\Theta=\Theta H) generates diagonal, tridiagonal, and higher-band metrics directly from a small set of initial data. For a diagonal ansatz,
[
\Theta=\operatorname{diag}(\theta_1,\dots,\theta_N),
\qquad
\theta_{n+1}b_{n+1}=\theta_n c_n,
]
so once (\theta_1>0) is chosen, the rest follow recursively [1201.2263]. This construction was applied explicitly to Jacobi-polynomial lattice Hamiltonians, where the metric elements are obtained recursively in closed non-numerical form for arbitrary (N) [1201.2263].

Time-dependent non-Hermitian dynamics requires an analogous but explicitly dynamical metric. In that setting, the metric operator (\rho(t)) obeys
[
i\dot{\rho}(t)=H^\dagger(t)\rho(t)-\rho(t)H(t),
]
and the physically relevant norm is (\langle \psi(t)|\rho(t)|\psi(t)\rangle). With (\rho(t)=\eta^\dagger(t)\eta(t)), one obtains a Hermitian image Hamiltonian
[
h(t)=\eta(t)H(t)\eta^{-1}(t)+i\dot{\eta}(t)\eta^{-1}(t),
]
so probability conservation is restored in the (\rho(t))-weighted norm rather than the naive norm of (|\psi(t)\rangle) [2301.02247]. A related perturbative construction for scattering Hamiltonians uses (H^\dagger=\eta H\eta^{-1}), (\eta=e^{-Q}), and (h=\rho H\rho^{-1}) with (\rho=\sqrt{\eta}), giving explicit metric kernels and equivalent Hermitian Hamiltonians for complex point-interaction models [1002.1221].

Later operator-metric work extends this regime dependence further. In unbroken, broken, and exceptional-point regimes, metrics are constructed separately so that expectation values, variances, and uncertainty relations remain well defined; in the broken and EP regimes the construction passes through a Krein-space decomposition before arriving at a usable positive metric (\mathcal S=\gamma^\dagger\gamma) [2512.24437].

3. Parameter-space QGTs in (\mathcal{PT})-symmetric and pseudo-Hermitian systems

A parameter-space non-Hermitian metric tensor is formulated most cleanly in unbroken (\mathcal{PT})-symmetric quantum mechanics. There one assumes a positive definite metric operator (W(\lambda)) satisfying
[
W(\lambda)H(\lambda)=H^\dagger(\lambda)W(\lambda),
]
which induces the physical inner product (\langle\cdot,\cdot\rangle_\lambda=\langle\cdot|W(\lambda)|\cdot\rangle). Right eigenstates (|\Psi_n(\lambda)\rangle) and left states (|\Phi_n(\lambda)\rangle=W(\lambda)|\Psi_n(\lambda)\rangle) form a biorthonormal basis, and the extended QGT is defined by
[

Q_{n,\mu\nu}

\frac{1}{2}\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
+

\langle \partial_\mu \Psi_n|\partial_\nu \Phi_n\rangle

\langle \partial_\mu \Psi_n|\Phi_n\rangle
\langle \Psi_n|\partial_\nu \Phi_n\rangle
\Big].
]
It decomposes as
[
Q_{n,\mu\nu}=g_{n,\mu\nu}+i\,\Omega_{n,\mu\nu},
\qquad
g_{n,\mu\nu}=\Re Q_{n,\mu\nu},
\qquad
\Omega_{n,\mu\nu}=\Im Q_{n,\mu\nu},
]
with (g_{n,\mu\nu}) real and symmetric and (\Omega_{n,\mu\nu}) real and antisymmetric [1811.04638].

In that framework the metric is obtained from a fidelity-based line element,
[

ds^2:=2\big[1-F(\rho_n(\lambda),\rho_n(\lambda+\delta\lambda))\big]

g_{n,\mu\nu}\,d\lambda^\mu d\lambda^\nu,
]
where (\rho_n(\lambda)=|\Psi_n(\lambda)\rangle\langle\Phi_n(\lambda)|). The resulting geometry is not automatically positive semidefinite: the paper states that (ds^2) may be Riemannian or pseudo-Riemannian, depending on the parameter region, and compares the signatures (ds^2>0), (ds^2=0), (ds^2<0) to spacelike, lightlike, and timelike intervals [1811.04638].

Pseudo-Hermitian band theory gives a closely related but not identical construction. For isolated bands with biorthogonal normalization (\langle u_m^L|u_n^R\rangle=\delta_{mn}), one uses the symmetrized non-Hermitian QGT
[

Q^n_{\mu\nu}

\frac{1}{2}\Big[
\langle \partial_{\mu} u_n^L | (1-P_n) | \partial_{\nu} u_n^R\rangle
+
\langle \partial_{\mu} u_n^R | (1-P_n^\dagger) | \partial_{\nu} u_n^L\rangle
\Big],
]
with (g_{\mu\nu}=\operatorname{Re}(Q_{\mu\nu}^n)) and (F_{\mu\nu}^n=-2\,\operatorname{Im}(Q_{\mu\nu}^n)). In pseudo-Hermitian topological phases, this metric is gauge invariant, real, and capable of distinguishing band geometries that share the same topological invariants as Hermitian counterparts [2106.09648].

4. Non-uniqueness of non-Hermitian metric tensors

The main structural fact is that there is no unique non-Hermitian quantum metric tensor. One formulation classifies the admissible tensors by the choice of pairing. The left-right tensor
[

FS_{ij}^{LR}

\braket{\partial_{i}\Psi^{L}|\partial_{j}\Psi^{R}}

\braket{\partial_{i}\Psi^{L}|\Psi^{R}}
\braket{\Psi^{L}|\partial_{j}\Psi^{R}}
]
is genuinely non-Hermitian, while (LL) and (RR) constructions are “essentially Hermitian.” The same analysis decomposes a non-Hermitian Fubini–Study tensor into four sectors: a real symmetric metric (g_{ij}^{LR}), an imaginary antisymmetric Berry-curvature sector (\omega_{ij}^{LR}), a real antisymmetric “flipped part of the QMT,” and a purely imaginary symmetric “flipped part of the Berry curvature” [2507.18486].

A complementary band-theory treatment distinguishes mixed (LR/RL) QGTs from same-sector (RR/LL) QGTs. The mixed tensor
[

Q^{LR}_{ij}

\frac{
\bra{\partial_{k_i}\psi_n^L}(1-\ket{\psi_n^R}\bra{\psi_n^L})\ket{\partial_{k_j}\psi_n^R}
}{
\bra{\psi_n^L}\ket{\psi_n^L}\,\bra{\psi_n^R}\ket{\psi_n^R}
}
]
is generically a non-Hermitian matrix and is not positive semidefinite. By contrast, the same-sector tensors
[

Q^{\alpha\alpha}_{\mu\nu}

\frac{
\bra{\partial_\mu \psi^\alpha}(1-P^{\alpha\alpha})\ket{\partial_\nu \psi^\alpha}
}{
\braket{\psi^\alpha}{\psi^\alpha}
},
\qquad
\alpha\in{R,L},
]
satisfy
[

v_\mu^* Q^{\alpha\alpha}_{\mu\nu} v_\nu

\frac{|(1-P^{\alpha\alpha})\,\mathbf v\cdot\nabla \ket{\psi^\alpha}|^2}{\braket{\psi^\alpha}{\psi^\alpha}}
\ge 0,
]
so (Q^{RR}) and (Q^{LL}) are positive semidefinite. In that framework, the symmetric same-sector part (G^{RR}) or (G^{LL}) is the physically useful metric entering response bounds, whereas the mixed sector carries the curvature entering non-Hermitian Chern-number bounds [2512.23708].

Wavepacket dynamics yields yet another split between RR and LR QGTs. One paper defines
[

Q_{n,ij}^{RR}

\langle \partial_{k_i} u_n^R | \partial_{k_j} u_n^R\rangle

\langle \partial_{k_i} u_n^R | u_n^R\rangle
\langle u_n^R | \partial_{k_j} u_n^R\rangle,
]
with (g^{RR}{n,ij}=\operatorname{Re}Q^{RR}{n,ij}), and
[

Q_{n,ij}^{LR}

\langle \partial_{k_i} u_n^L | \partial_{k_j} u_n^R\rangle

\langle \partial_{k_i} u_n^L | u_n^R\rangle
\langle u_n^L | \partial_{k_j} u_n^R\rangle,
]
with
[
g_{n,ij}^{LR}=\frac{1}{2}\left(Q_{n,ij}^{LR}+Q_{n,ji}^{LR}\right).
]
Here (g^{RR}) is real-valued, while (g^{LR}) is generally complex-valued; both are gauge invariant under the normalization convention used, and both enter dynamics in different ways [2412.08141].

By contrast, a criticality-oriented formulation uses only self-normal right eigenstates
[
\braket{\psi_n^R(\boldsymbol\lambda)|\psi_n^R(\boldsymbol\lambda)}=1
]
and defines
[

Q^{(n)}_{\mu\nu}

\braket{\partial_{\lambda_\mu}\psi_n|\partial_{\lambda_\nu}\psi_n}

\braket{\partial_{\lambda_\mu}\psi_n|\psi_n}
\braket{\psi_n|\partial_{\lambda_\nu}\psi_n},
\qquad
g_{\mu\nu}^{(n)}=\Re Q_{\mu\nu}.
]
This is formally identical to the Hermitian expression but is not biorthogonal; it is used as a practical metric for localization transitions, mobility edges, and many-body critical points [2404.15628].

The SSH literature makes the non-uniqueness operational. For
[

\chi_{\mu\nu}^{\alpha\beta}

\langle \partial_\mu v_\alpha|\partial_\nu v_\beta\rangle

\langle \partial_\mu v_\alpha|v_\beta\rangle
\langle v_\alpha|\partial_\nu v_\beta\rangle,
]
one study takes
[

g_{\mu\nu}^{\alpha\beta}

\frac{1}{2}\left(\chi_{\mu\nu}^{\alpha\beta}+\chi_{\nu\mu}^{\alpha\beta}\right)
]
as the working metric and finds that only the (LR) metric reproduces the full topological phase diagram of non-Hermitian SSH models; (LL) and (RR) each encode only half of the phase boundaries [2305.17675]. A separate and distinct generalization is the Zeeman QGT, where the underlying Hamiltonian remains Hermitian but the tensor
[
T^{Z,ab}{nm}=r^a{nm}\sigma^b_{mn}
]
is non-Hermitian; it decomposes into a normal metric (g^N), normal curvature (\Omega^N), anomalous metric-like tensor (g^A), and anomalous curvature-like tensor (\Omega^A) [2604.09725].

5. Dynamical and response roles

Non-Hermitian quantum metrics are not only classificatory. Near exceptional points, the metric can dominate dynamics. In a two-dimensional non-Hermitian two-level model with an exceptional point at (q=0), the overlap-based metric obeys
[
g_{qq}\approx \frac{\alpha^2}{16a^2}+\frac{\alpha}{8aq},
\qquad
g_{\varphi\varphi}\approx \frac{\alpha q}{8a},
]
so the radial component diverges as (g_{qq}\sim q^{-1}), जबकि the angular component remains regular. The paper attributes a constant acceleration with fixed direction and a constant non-vanishing velocity with controllable direction to this singular metric behavior, with both effects independent of wavepacket size [2009.06987].

In semiclassical band dynamics, the metric enters through field-induced interband mixing. For two-band non-Hermitian systems, first-order perturbation theory shows that the RR QGT controls the field-induced positional shift, while the LR QGT controls the field-induced correction to the Berry phase. Because the interband gap (\Delta\epsilon=\epsilon_1-\epsilon_0) is complex, the RR metric and RR Berry curvature mix through (\operatorname{Re}[2Q^{RR}\cdot\mathbf F/\Delta\epsilon]), and both the real and imaginary parts of the complex LR metric contribute to dynamics [2412.08141].

Transport theory yields a still more concrete metric. In line-gapped non-Hermitian Bloch bands, the “band-renormalized non-Hermitian quantum metric”
[

G^{LR}_{n,\mu\nu}

\sum_{m\neq n}
\frac{
A^{LR}{nm,\mu}A^{LR}{mn,\nu}
+
A^{LR}{nm,\nu}A^{LR}{mn,\mu}
}{
2(\xi_n-\xi_m)
}
]
is symmetric but generally complex. It appears in the second-order band-energy shift and produces a scattering-time-independent intrinsic term in the second-order nonlinear dc conductivity,
[

\sigma^{\mathrm{intrinsic}}_{\theta\mu\nu}

-e^3\int_{\mathbf{k}}
f_0\,\mathrm{Re}!\left(

2\partial_\theta G^{LR}_{\mu\nu}

\frac{\partial_\nu G^{LR}{\mu\theta}+\partial\mu G^{LR}_{\nu\theta}}{2}
\right).
]
In the narrow-wavepacket limit, only (\mathrm{Re}\,G^{LR}) and (\mathrm{Re}\,\Omega) contribute; for finite wavepacket width, (\mathrm{Im}\,G^{LR}) and (\mathrm{Im}\,\Omega) enter explicitly through (W)-dependent terms [2509.11765].

A closely related Floquet response theory for line-gapped non-Hermitian altermagnets reaches the same structural conclusion for spin transport. There the intrinsic nonlinear spin conductivity decomposes into geometric, magneto, and polar terms, with the geometric term
[

\Gamma^{\text{geom}}_{i\mu\nu}

e^2 s_{nn}^{\alpha}
\mathrm{Re}!\left[

2\partial_i G_{\mu\nu}^{LR}

\frac{\partial_\nu G_{\mu i}^{LR}+\partial_\mu G_{\nu i}^{LR}}{2}
\right],
]
and the reported numerical result is that the nonlinear spin conductivity is overwhelmingly dominated by the quantum metric sector [2605.15541].

6. Topology, criticality, and experimental access

Topological band geometry is one of the main arenas in which non-Hermitian metrics differ from Hermitian ones. In pseudo-Hermitian Chern-insulator, time-reversal-invariant, Weyl-semimetal, and chiral phases built from (q)-deformed matrices, the topological invariants are the same as in Hermitian counterparts, but the band geometries are different. The non-Hermitian quantum metric reveals this directly: in the Weyl case the state manifold is deformed from a sphere to an ellipsoid, and determinant relations such as (|F_{\mu\nu}^n|=2\sqrt{g}), (|\operatorname{tr}\mathbf B_{xy}|=2\sqrt{g}), and (|\mathcal H_{xyz}|=4\sqrt{g}) connect the metric to Abelian, non-Abelian, and tensor Berry curvatures [2106.09648].

In non-Hermitian SSH systems, the left-right metric provides a phase-sensitive geometry that is Riemannian in Hermitian limits, pseudo-Riemannian in real nonreciprocal models, and complex in models with genuinely complex hopping. In the nonreciprocal case, the phase-transition lines are also null curves of the metric, (ds^2=0), and in non-Hermitian topological phases the metric degenerates so that one effective parameter direction becomes dark. Within linear response, the integrated excitation rate satisfies
[

\Gamma^{\mathrm{int}}

\frac{2\pi E^2}{\hbar^2}g^{LR}_{\lambda_i\lambda_i},
]
so the null direction is a zero-excitation direction [2305.17675].

Criticality detection is another major use. Using self-normal right eigenstates, one study identifies localization transitions in a non-Hermitian generalized Aubry–André model, mobility edges in another generalized Aubry–André model, and many-body gap-closing transitions in non-Hermitian cluster and mixed-field Ising models. In that framework
[

g_{\mu\mu}^{(n)}

\lim_{d\mu\to0}\frac{-2\ln F_n}{d\mu^2}

\chi_F^{(n)},
]
so the diagonal quantum metric equals the fidelity susceptibility and peaks or diverges at the relevant critical points [2404.15628]. In unbroken (\mathcal{PT})-symmetric many-body systems, the extended QGT gives a complementary criterion: the ground-state metric
[

g_{0,\mu\nu}

\Re\sum_{n\neq0}
\frac{
\langle \Phi_0|\partial_\mu H|\Psi_n\rangle
\langle \Phi_n|\partial_\nu H|\Psi_0\rangle
+
\langle \Phi_n|\partial_\mu H|\Psi_0\rangle
\langle \Phi_0|\partial_\nu H|\Psi_n\rangle
}{
2(E_0-E_n)^2
}
]
becomes singular both at ordinary quantum phase transitions and at spontaneous (\mathcal{PT})-symmetry-breaking points [1811.04638].

Experimental access has progressed on several fronts. In pseudo-Hermitian systems with real spectra, two direct measurement schemes reconstruct the full left-right QGT from generalized expectation values of either the energy-fluctuation operator or generalized force operators. For the lowest band,
[
Q_{\mu\nu}^0(\boldsymbol\lambda_{\mathrm{tar}})
\approx
\frac{1}{v^2}
\frac{
\langle \psi_\mu' | (H-E_0)^2 | \psi_\nu \rangle
}{
\langle \psi_\mu' | \psi_\nu \rangle
},
]
while the metric can also be extracted directly from the generalized-force protocol,
[

g_{\mu\nu}^0(\boldsymbol\lambda_{\mathrm{tar}})

\frac{1}{2v}\,
\Im!\left[

\frac{\langle \psi_\nu'' | (-\partial_\mu H) | \psi_\nu \rangle}{\langle \psi_\nu'' | \psi_\nu \rangle}

\langle f_\mu(\boldsymbol\lambda_{\mathrm{tar}})\rangle
\right].
]
The paper demonstrates diagonal and off-diagonal metric measurement in (q)-deformed pseudo-Hermitian two-band models [2509.17043]. Experimentally, the QGT, including the quantum metric and a non-Hermitian Berry curvature, has also been observed in a plasmonic lattice of radiatively coupled nanoparticles, where the Berry curvature is reported to arise solely from non-Hermitian effects while the quantum metric originates from a pseudospin-orbit coupling [2305.13174].

Taken together, these results define the subject as a layered rather than singular concept. In one layer, the non-Hermitian metric is an operator that selects the physical Hilbert space. In another, it is the metric sector of a biorthogonal, same-sector, or right-state QGT. In yet another, it is a complex or pseudo-Riemannian tensor that controls wavepacket motion, nonlinear response, topology, criticality, and direct measurement. The common thread is not uniqueness, but the replacement of the standard Hermitian inner-product geometry by a geometry built from non-Hermitian spectral structure.