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 (Znojil, 2012, Zhang et al., 2018).

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 (Znojil, 2012). 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 (Zhang et al., 2018).

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 (Sim et al., 2023). 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 (Pal, 24 Jul 2025).

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 (Hu et al., 2024, Ye et al., 2023).

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 $\Theta$0,

$\Theta$1

with the hidden-Hermiticity condition

$\Theta$2

The admissible $\Theta$3 must be Hermitian, invertible, and positive definite, $\Theta$4, $\Theta$5, so that the amended inner product is a genuine Hilbert-space norm (Znojil, 2012).

This operator metric is generically non-unique. In spectral form,

$\Theta$6

so the metric contains free parameters even for fixed $\Theta$7. For finite-dimensional real tridiagonal Hamiltonians with real nondegenerate spectrum, a recurrent solution of the Dieudonné equation $\Theta$8 generates diagonal, tridiagonal, and higher-band metrics directly from a small set of initial data. For a diagonal ansatz,

$\Theta$9

so once $W(\lambda)$0 is chosen, the rest follow recursively (Znojil, 2012). This construction was applied explicitly to Jacobi-polynomial lattice Hamiltonians, where the metric elements are obtained recursively in closed non-numerical form for arbitrary $W(\lambda)$1 (Znojil, 2012).

Time-dependent non-Hermitian dynamics requires an analogous but explicitly dynamical metric. In that setting, the metric operator $W(\lambda)$2 obeys

$W(\lambda)$3

and the physically relevant norm is $W(\lambda)$4. With $W(\lambda)$5, one obtains a Hermitian image Hamiltonian

$W(\lambda)$6

so probability conservation is restored in the $W(\lambda)$7-weighted norm rather than the naive norm of $W(\lambda)$8 (Sim et al., 2023). A related perturbative construction for scattering Hamiltonians uses $W(\lambda)$9, $\rho(t)$0, and $\rho(t)$1 with $\rho(t)$2, giving explicit metric kernels and equivalent Hermitian Hamiltonians for complex point-interaction models (Mehri-Dehnavi et al., 2010).

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 $\rho(t)$3 (Alvarez et al., 30 Dec 2025).

3. Parameter-space QGTs in $\rho(t)$4-symmetric and pseudo-Hermitian systems

A parameter-space non-Hermitian metric tensor is formulated most cleanly in unbroken $\rho(t)$5-symmetric quantum mechanics. There one assumes a positive definite metric operator $\rho(t)$6 satisfying

$\rho(t)$7

which induces the physical inner product $\rho(t)$8. Right eigenstates $\rho(t)$9 and left states $$\langle \psi,\phi\rangle_\Theta=\langle \psi|\Theta|\phi\rangle,$$0 form a biorthonormal basis, and the extended QGT is defined by

$$\langle \psi,\phi\rangle_\Theta=\langle \psi|\Theta|\phi\rangle,$$1

It decomposes as

$$\langle \psi,\phi\rangle_\Theta=\langle \psi|\Theta|\phi\rangle,$$2

with $$\langle \psi,\phi\rangle_\Theta=\langle \psi|\Theta|\phi\rangle,$$3 real and symmetric and $$\langle \psi,\phi\rangle_\Theta=\langle \psi|\Theta|\phi\rangle,$$4 real and antisymmetric (Zhang et al., 2018).

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

$$\langle \psi,\phi\rangle_\Theta=\langle \psi|\Theta|\phi\rangle,$$5

where $$\langle \psi,\phi\rangle_\Theta=\langle \psi|\Theta|\phi\rangle,$$6. The resulting geometry is not automatically positive semidefinite: the paper states that $$\langle \psi,\phi\rangle_\Theta=\langle \psi|\Theta|\phi\rangle,$$7 may be Riemannian or pseudo-Riemannian, depending on the parameter region, and compares the signatures $$\langle \psi,\phi\rangle_\Theta=\langle \psi|\Theta|\phi\rangle,$$8, $$\langle \psi,\phi\rangle_\Theta=\langle \psi|\Theta|\phi\rangle,$$9, $H^\dagger \Theta=\Theta H$0 to spacelike, lightlike, and timelike intervals (Zhang et al., 2018).

Pseudo-Hermitian band theory gives a closely related but not identical construction. For isolated bands with biorthogonal normalization $H^\dagger \Theta=\Theta H$1, one uses the symmetrized non-Hermitian QGT

$H^\dagger \Theta=\Theta H$2

with $H^\dagger \Theta=\Theta H$3 and $H^\dagger \Theta=\Theta H$4. 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 (Zhu et al., 2021).

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

$H^\dagger \Theta=\Theta H$5

is genuinely non-Hermitian, while $H^\dagger \Theta=\Theta H$6 and $H^\dagger \Theta=\Theta H$7 constructions are “essentially Hermitian.” The same analysis decomposes a non-Hermitian Fubini–Study tensor into four sectors: a real symmetric metric $H^\dagger \Theta=\Theta H$8, an imaginary antisymmetric Berry-curvature sector $H^\dagger \Theta=\Theta H$9, a real antisymmetric “flipped part of the QMT,” and a purely imaginary symmetric “flipped part of the Berry curvature” (Pal, 24 Jul 2025).

A complementary band-theory treatment distinguishes mixed $H$0 QGTs from same-sector $H$1 QGTs. The mixed tensor

$H$2

is generically a non-Hermitian matrix and is not positive semidefinite. By contrast, the same-sector tensors

$H$3

satisfy

$H$4

so $H$5 and $H$6 are positive semidefinite. In that framework, the symmetric same-sector part $H$7 or $H$8 is the physically useful metric entering response bounds, whereas the mixed sector carries the curvature entering non-Hermitian Chern-number bounds (Matraszek et al., 29 Dec 2025).

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

$H$9

with $\rho(t)$0, and

$\rho(t)$1

with

$\rho(t)$2

Here $\rho(t)$3 is real-valued, while $\rho(t)$4 is generally complex-valued; both are gauge invariant under the normalization convention used, and both enter dynamics in different ways (Hu et al., 2024).

By contrast, a criticality-oriented formulation uses only self-normal right eigenstates

$\rho(t)$5

and defines

$\rho(t)$6

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 (Ren et al., 2024).

The SSH literature makes the non-uniqueness operational. For

$\rho(t)$7

one study takes

$\rho(t)$8

as the working metric and finds that only the $\rho(t)$9 metric reproduces the full topological phase diagram of non-Hermitian SSH models; $\Omega$0 and $\Omega$1 each encode only half of the phase boundaries (Ye et al., 2023). A separate and distinct generalization is the Zeeman QGT, where the underlying Hamiltonian remains Hermitian but the tensor

$\Omega$2

is non-Hermitian; it decomposes into a normal metric $\Omega$3, normal curvature $\Omega$4, anomalous metric-like tensor $\Omega$5, and anomalous curvature-like tensor $\Omega$6 (Cui et al., 9 Apr 2026).

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 $\Omega$7, the overlap-based metric obeys

$\Omega$8

so the radial component diverges as $\Omega$9, 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 (Solnyshkov et al., 2020).

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 $H$0 is complex, the RR metric and RR Berry curvature mix through $H$1, and both the real and imaginary parts of the complex LR metric contribute to dynamics (Hu et al., 2024).

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

$H$2

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,

$H$3

In the narrow-wavepacket limit, only $H$4 and $H$5 contribute; for finite wavepacket width, $H$6 and $H$7 enter explicitly through $H$8-dependent terms (Chen et al., 15 Sep 2025).

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

$H$9

and the reported numerical result is that the nonlinear spin conductivity is overwhelmingly dominated by the quantum metric sector (Chen et al., 15 May 2026).

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 $\Theta$00-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 $\Theta$01, $\Theta$02, and $\Theta$03 connect the metric to Abelian, non-Abelian, and tensor Berry curvatures (Zhu et al., 2021).

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, $\Theta$04, 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

$\Theta$05

so the null direction is a zero-excitation direction (Ye et al., 2023).

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

$\Theta$06

so the diagonal quantum metric equals the fidelity susceptibility and peaks or diverges at the relevant critical points (Ren et al., 2024). In unbroken $\Theta$07-symmetric many-body systems, the extended QGT gives a complementary criterion: the ground-state metric

$\Theta$08

becomes singular both at ordinary quantum phase transitions and at spontaneous $\Theta$09-symmetry-breaking points (Zhang et al., 2018).

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,

$\Theta$10

while the metric can also be extracted directly from the generalized-force protocol,

$\Theta$11

The paper demonstrates diagonal and off-diagonal metric measurement in $\Theta$12-deformed pseudo-Hermitian two-band models (Huang et al., 21 Sep 2025). 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 (Cuerda et al., 2023).

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.