Non-Bloch Integrated Quantum Metric

• Non-Bloch integrated quantum metric is a quantum-geometric quantity defined via the generalized Brillouin zone and biorthogonal states in non-Hermitian systems.
• It achieves exact equivalence between momentum-space GBZ formulations and real-space integrated approaches, ensuring consistent quantification of phase transitions.
• The framework extends conventional Bloch theory by linking gauge-invariant metrics with localized non-Bloch Wannier functions and Bott metrics in diverse systems.

The non-Bloch integrated quantum metric is a quantum-geometric quantity designed for settings in which conventional Bloch-band formulas are inadequate. In non-Hermitian systems under open boundary conditions, it is defined on the generalized Brillouin zone (GBZ) from a biorthogonal, left–right quantum metric tensor, and it is exactly equivalent to a real-space integrated quantum metric constructed from open-boundary projectors (Sun et al., 19 May 2026). A complementary real-space development in Hermitian but non-periodic systems defines the Bott metric from the amplitude of the Bott plaquette operator and shows that, in the thermodynamic limit, it converges to the trace of the integrated quantum metric even in disordered, quasicrystalline, or amorphous settings (Chatterjee et al., 6 Apr 2026). Together, these constructions extend quantum geometry beyond translationally invariant Bloch theory.

1. Generalized-Brillouin-zone formulation

In the non-Hermitian formulation, non-Bloch states are defined on the GBZ by a complex deformation of momentum,

$\beta = |\beta(k)| e^{ik}, \qquad k \in [0,2\pi),$

such that the open-boundary eigenvalue problem is solved by

$$\tilde H(\beta)\,|u^R_{m,\beta}\rangle = E_{m,\beta}\,|u^R_{m,\beta}\rangle, \qquad \langle u^L_{m,\beta}|\,\tilde H(\beta) = E_{m,\beta}\,\langle u^L_{m,\beta}|,$$

with biorthonormality

$$\langle u^L_{m,\beta}|u^R_{n,\beta}\rangle = \delta_{mn}.$$

The left–right quantum-metric tensor is defined by

$$g^{LR}_{\alpha\beta}(\beta) = \mathrm{Re}\,\chi^{LR}_{\alpha\beta}(\beta),$$

with

$$\chi^{LR}_{\alpha\beta}(\beta) = \langle \partial_\alpha u^L_{m,\beta}| \bigl[I-|u^R_{m,\beta}\rangle\langle u^L_{m,\beta}|\bigr] |\partial_\beta u^R_{m,\beta}\rangle.$$

Under the $GL(1,\mathbb{C})$ gauge

$$|u^R\rangle \to z(\beta)|u^R\rangle, \qquad \langle u^L| \to z(\beta)^{-1}\langle u^L|,$$

$\chi^{LR}_{\alpha\beta}$ and hence $g^{LR}_{\alpha\beta}$ are invariant (Sun et al., 19 May 2026).

The integrated non-Bloch quantum metric is the trace of $g$ over the GBZ, normalized by the GBZ “volume.” In one dimension,

$$\tilde H(\beta)\,|u^R_{m,\beta}\rangle = E_{m,\beta}\,|u^R_{m,\beta}\rangle, \qquad \langle u^L_{m,\beta}|\,\tilde H(\beta) = E_{m,\beta}\,\langle u^L_{m,\beta}|,$$0

and equivalently

$$\tilde H(\beta)\,|u^R_{m,\beta}\rangle = E_{m,\beta}\,|u^R_{m,\beta}\rangle, \qquad \langle u^L_{m,\beta}|\,\tilde H(\beta) = E_{m,\beta}\,\langle u^L_{m,\beta}|,$$1

This formulation replaces the ordinary Brillouin zone by the GBZ appropriate to open-boundary spectra and thereby encodes the quantum geometry of non-Hermitian bands in a representation compatible with the non-Hermitian skin effect.

2. Exact equivalence to the real-space integrated quantum metric

The real-space integrated quantum metric for an open-boundary energy sector is defined as

$$\tilde H(\beta)\,|u^R_{m,\beta}\rangle = E_{m,\beta}\,|u^R_{m,\beta}\rangle, \qquad \langle u^L_{m,\beta}|\,\tilde H(\beta) = E_{m,\beta}\,\langle u^L_{m,\beta}|,$$2

where $$\tilde H(\beta)\,|u^R_{m,\beta}\rangle = E_{m,\beta}\,|u^R_{m,\beta}\rangle, \qquad \langle u^L_{m,\beta}|\,\tilde H(\beta) = E_{m,\beta}\,\langle u^L_{m,\beta}|,$$3 is the total number of sites, $$\tilde H(\beta)\,|u^R_{m,\beta}\rangle = E_{m,\beta}\,|u^R_{m,\beta}\rangle, \qquad \langle u^L_{m,\beta}|\,\tilde H(\beta) = E_{m,\beta}\,\langle u^L_{m,\beta}|,$$4 is the position operator, and

$$\tilde H(\beta)\,|u^R_{m,\beta}\rangle = E_{m,\beta}\,|u^R_{m,\beta}\rangle, \qquad \langle u^L_{m,\beta}|\,\tilde H(\beta) = E_{m,\beta}\,\langle u^L_{m,\beta}|,$$5

is the projector built from right and left real-space eigenvectors satisfying

$$\tilde H(\beta)\,|u^R_{m,\beta}\rangle = E_{m,\beta}\,|u^R_{m,\beta}\rangle, \qquad \langle u^L_{m,\beta}|\,\tilde H(\beta) = E_{m,\beta}\,\langle u^L_{m,\beta}|,$$6

The same projector has the non-Bloch representation

$$\tilde H(\beta)\,|u^R_{m,\beta}\rangle = E_{m,\beta}\,|u^R_{m,\beta}\rangle, \qquad \langle u^L_{m,\beta}|\,\tilde H(\beta) = E_{m,\beta}\,\langle u^L_{m,\beta}|,$$7

with

$$\tilde H(\beta)\,|u^R_{m,\beta}\rangle = E_{m,\beta}\,|u^R_{m,\beta}\rangle, \qquad \langle u^L_{m,\beta}|\,\tilde H(\beta) = E_{m,\beta}\,\langle u^L_{m,\beta}|,$$8

obeying $$\tilde H(\beta)\,|u^R_{m,\beta}\rangle = E_{m,\beta}\,|u^R_{m,\beta}\rangle, \qquad \langle u^L_{m,\beta}|\,\tilde H(\beta) = E_{m,\beta}\,\langle u^L_{m,\beta}|,$$9 and completeness $$\langle u^L_{m,\beta}|u^R_{n,\beta}\rangle = \delta_{mn}.$$0 (Sun et al., 19 May 2026).

Substituting this projector into $$\langle u^L_{m,\beta}|u^R_{n,\beta}\rangle = \delta_{mn}.$$1 and evaluating the commutators via $$\langle u^L_{m,\beta}|u^R_{n,\beta}\rangle = \delta_{mn}.$$2 yields

$$\langle u^L_{m,\beta}|u^R_{n,\beta}\rangle = \delta_{mn}.$$3

Thus,

$$\langle u^L_{m,\beta}|u^R_{n,\beta}\rangle = \delta_{mn}.$$4

This exact identity is the central structural result: the open-boundary quantum geometry can be computed either from real-space projectors or from non-Bloch states on the GBZ, without discrepancy.

3. Relation to localized non-Bloch Wannier functions

A set of localized non-Bloch Wannier functions is obtained by a GBZ Fourier transform,

$$\langle u^L_{m,\beta}|u^R_{n,\beta}\rangle = \delta_{mn}.$$5

$$\langle u^L_{m,\beta}|u^R_{n,\beta}\rangle = \delta_{mn}.$$6

with biorthogonality

$$\langle u^L_{m,\beta}|u^R_{n,\beta}\rangle = \delta_{mn}.$$7

and translational covariance. Their spread functional is

$$\langle u^L_{m,\beta}|u^R_{n,\beta}\rangle = \delta_{mn}.$$8

with Wannier center

$$\langle u^L_{m,\beta}|u^R_{n,\beta}\rangle = \delta_{mn}.$$9

The spread decomposes into a gauge-invariant part $$g^{LR}_{\alpha\beta}(\beta) = \mathrm{Re}\,\chi^{LR}_{\alpha\beta}(\beta),$$0 and a gauge-dependent part $$g^{LR}_{\alpha\beta}(\beta) = \mathrm{Re}\,\chi^{LR}_{\alpha\beta}(\beta),$$1 exactly as in Hermitian theory (Sun et al., 19 May 2026).

The gauge-invariant contribution is

$$g^{LR}_{\alpha\beta}(\beta) = \mathrm{Re}\,\chi^{LR}_{\alpha\beta}(\beta),$$2

Accordingly, the non-Bloch integrated quantum metric is the gauge-invariant lower bound on the spread of localized non-Bloch Wannier functions. In the non-Hermitian SSH example, the real-space profiles of $$g^{LR}_{\alpha\beta}(\beta) = \mathrm{Re}\,\chi^{LR}_{\alpha\beta}(\beta),$$3 display asymmetric exponential decay associated with the non-Hermitian skin effect, while the distribution

$$g^{LR}_{\alpha\beta}(\beta) = \mathrm{Re}\,\chi^{LR}_{\alpha\beta}(\beta),$$4

can be complex, with its real part controlling the real part of $$g^{LR}_{\alpha\beta}(\beta) = \mathrm{Re}\,\chi^{LR}_{\alpha\beta}(\beta),$$5.

4. Real-space plaquette formulation and the Bott metric

A distinct but closely related real-space formulation begins from a two-dimensional single-particle Hamiltonian $$g^{LR}_{\alpha\beta}(\beta) = \mathrm{Re}\,\chi^{LR}_{\alpha\beta}(\beta),$$6 on a torus of linear size $$g^{LR}_{\alpha\beta}(\beta) = \mathrm{Re}\,\chi^{LR}_{\alpha\beta}(\beta),$$7, with area $$g^{LR}_{\alpha\beta}(\beta) = \mathrm{Re}\,\chi^{LR}_{\alpha\beta}(\beta),$$8, Fermi projector

$$g^{LR}_{\alpha\beta}(\beta) = \mathrm{Re}\,\chi^{LR}_{\alpha\beta}(\beta),$$9

and complement $$\chi^{LR}_{\alpha\beta}(\beta) = \langle \partial_\alpha u^L_{m,\beta}| \bigl[I-|u^R_{m,\beta}\rangle\langle u^L_{m,\beta}|\bigr] |\partial_\beta u^R_{m,\beta}\rangle.$$0. Using the position operators $$\chi^{LR}_{\alpha\beta}(\beta) = \langle \partial_\alpha u^L_{m,\beta}| \bigl[I-|u^R_{m,\beta}\rangle\langle u^L_{m,\beta}|\bigr] |\partial_\beta u^R_{m,\beta}\rangle.$$1, one defines the $$\chi^{LR}_{\alpha\beta}(\beta) = \langle \partial_\alpha u^L_{m,\beta}| \bigl[I-|u^R_{m,\beta}\rangle\langle u^L_{m,\beta}|\bigr] |\partial_\beta u^R_{m,\beta}\rangle.$$2 twist operators

$$\chi^{LR}_{\alpha\beta}(\beta) = \langle \partial_\alpha u^L_{m,\beta}| \bigl[I-|u^R_{m,\beta}\rangle\langle u^L_{m,\beta}|\bigr] |\partial_\beta u^R_{m,\beta}\rangle.$$3

the projected twists

$$\chi^{LR}_{\alpha\beta}(\beta) = \langle \partial_\alpha u^L_{m,\beta}| \bigl[I-|u^R_{m,\beta}\rangle\langle u^L_{m,\beta}|\bigr] |\partial_\beta u^R_{m,\beta}\rangle.$$4

and their extensions

$$\chi^{LR}_{\alpha\beta}(\beta) = \langle \partial_\alpha u^L_{m,\beta}| \bigl[I-|u^R_{m,\beta}\rangle\langle u^L_{m,\beta}|\bigr] |\partial_\beta u^R_{m,\beta}\rangle.$$5

The plaquette operator is

$$\chi^{LR}_{\alpha\beta}(\beta) = \langle \partial_\alpha u^L_{m,\beta}| \bigl[I-|u^R_{m,\beta}\rangle\langle u^L_{m,\beta}|\bigr] |\partial_\beta u^R_{m,\beta}\rangle.$$6

with $$\chi^{LR}_{\alpha\beta}(\beta) = \langle \partial_\alpha u^L_{m,\beta}| \bigl[I-|u^R_{m,\beta}\rangle\langle u^L_{m,\beta}|\bigr] |\partial_\beta u^R_{m,\beta}\rangle.$$7 and

$$\chi^{LR}_{\alpha\beta}(\beta) = \langle \partial_\alpha u^L_{m,\beta}| \bigl[I-|u^R_{m,\beta}\rangle\langle u^L_{m,\beta}|\bigr] |\partial_\beta u^R_{m,\beta}\rangle.$$8

The Bott index is

$$\chi^{LR}_{\alpha\beta}(\beta) = \langle \partial_\alpha u^L_{m,\beta}| \bigl[I-|u^R_{m,\beta}\rangle\langle u^L_{m,\beta}|\bigr] |\partial_\beta u^R_{m,\beta}\rangle.$$9

while the Bott metric retains the amplitude information,

$GL(1,\mathbb{C})$0

where $GL(1,\mathbb{C})$1 are the eigenvalues of $GL(1,\mathbb{C})$2 (Chatterjee et al., 6 Apr 2026).

Because $GL(1,\mathbb{C})$3, one has $GL(1,\mathbb{C})$4, and $GL(1,\mathbb{C})$5 measures the total “volume contraction” of $GL(1,\mathbb{C})$6 under one plaquette. A small-twist expansion with $GL(1,\mathbb{C})$7 gives, in the thermodynamic limit,

$GL(1,\mathbb{C})$8

where the real-space quantum-metric tensor is

$GL(1,\mathbb{C})$9

No crystal momentum or Brillouin zone is required in the definition of $$|u^R\rangle \to z(\beta)|u^R\rangle, \qquad \langle u^L| \to z(\beta)^{-1}\langle u^L|,$$0 itself; only the Fermi projector and real-space position operators enter. This real-space plaquette construction therefore provides a route to integrated quantum geometry in disordered, quasicrystalline, or amorphous systems, provided the Fermi projector remains local.

5. Representative models and critical behavior

In the one-dimensional non-Hermitian SSH chain with intracell hoppings $$|u^R\rangle \to z(\beta)|u^R\rangle, \qquad \langle u^L| \to z(\beta)^{-1}\langle u^L|,$$1, intercell hopping $$|u^R\rangle \to z(\beta)|u^R\rangle, \qquad \langle u^L| \to z(\beta)^{-1}\langle u^L|,$$2, and optionally long-range hoppings $$|u^R\rangle \to z(\beta)|u^R\rangle, \qquad \langle u^L| \to z(\beta)^{-1}\langle u^L|,$$3, the open-boundary spectrum exhibits a topological regime with zero-mode end states and a trivial regime as $$|u^R\rangle \to z(\beta)|u^R\rangle, \qquad \langle u^L| \to z(\beta)^{-1}\langle u^L|,$$4 is tuned. Numerically, $$|u^R\rangle \to z(\beta)|u^R\rangle, \qquad \langle u^L| \to z(\beta)^{-1}\langle u^L|,$$5 from real-space eigenvectors, $$|u^R\rangle \to z(\beta)|u^R\rangle, \qquad \langle u^L| \to z(\beta)^{-1}\langle u^L|,$$6 from $$|u^R\rangle \to z(\beta)|u^R\rangle, \qquad \langle u^L| \to z(\beta)^{-1}\langle u^L|,$$7 on the GBZ, and the gauge-independent spread $$|u^R\rangle \to z(\beta)|u^R\rangle, \qquad \langle u^L| \to z(\beta)^{-1}\langle u^L|,$$8 of projected-position Wannier functions agree perfectly; they are large in the topological phase, small in the trivial phase, and diverge near the gap-closing at the transition (Sun et al., 19 May 2026). The GBZ may be a simple circle for $$|u^R\rangle \to z(\beta)|u^R\rangle, \qquad \langle u^L| \to z(\beta)^{-1}\langle u^L|,$$9 and $\chi^{LR}_{\alpha\beta}$0, or a more complicated loop when long-range hoppings are present.

In two-dimensional Hermitian examples, the Bott metric reproduces the behavior of the integrated quantum metric in both periodic and non-periodic settings. In the clean Qi–Wu–Zhang Chern insulator, $\chi^{LR}_{\alpha\beta}$1 and the momentum-space integrated quantum metric $\chi^{LR}_{\alpha\beta}$2 both show sharp cusps at the topological transitions $\chi^{LR}_{\alpha\beta}$3 and otherwise track each other almost perfectly even for modest $\chi^{LR}_{\alpha\beta}$4. In the disordered Qi–Wu–Zhang model with random mass disorder, the disorder-averaged Bott metric $\chi^{LR}_{\alpha\beta}$5 and real-space $\chi^{LR}_{\alpha\beta}$6 exhibit matching ridges at the phase boundaries, while the Bott index $\chi^{LR}_{\alpha\beta}$7 remains quantized in the mobility-gap regime. In the amorphous Chern insulator of Agarwala–Shenoy type, $\chi^{LR}_{\alpha\beta}$8 shows a broad topological plateau, but $\chi^{LR}_{\alpha\beta}$9 varies strongly, peaks sharply at each transition, and reveals an asymmetry in localization properties not visible from $g^{LR}_{\alpha\beta}$0 alone (Chatterjee et al., 6 Apr 2026).

These examples establish a common pattern: integrated quantum-metric diagnostics become large near gap closings and phase boundaries, and they capture localization information that is complementary to topological winding data.

6. Scope, conditions, and conceptual distinctions

The non-Bloch integrated quantum metric is defined for non-Hermitian systems under open boundary conditions, where the GBZ resolves the open-boundary eigenvalue problem and accommodates the non-Hermitian skin effect. The Bott-metric construction applies to two-dimensional real-space systems lacking translational symmetry, provided one can define boundary twists and the Fermi projector is local. In the Bott-metric setting, a spectral or mobility gap is required so that $g^{LR}_{\alpha\beta}$1 remains exponentially local; if this fails, as at a localization transition or in a metal, the small-$g^{LR}_{\alpha\beta}$2 expansion breaks down and $g^{LR}_{\alpha\beta}$3 can diverge. Periodic boundary conditions, or equivalent torus twists, are required so that $g^{LR}_{\alpha\beta}$4 and $g^{LR}_{\alpha\beta}$5 are well defined and $g^{LR}_{\alpha\beta}$6 can be taken small; convergence to $g^{LR}_{\alpha\beta}$7 improves as $g^{LR}_{\alpha\beta}$8, and in typical lattice models $g^{LR}_{\alpha\beta}$9–$g$0 already gives excellent agreement (Chatterjee et al., 6 Apr 2026).

A common source of ambiguity is the term “non-Bloch” itself. In the non-Hermitian literature summarized above, it denotes the GBZ-based description appropriate to open boundaries and skin modes. In the real-space Bott-metric literature, “non-Bloch” refers to disordered or aperiodic settings in which no use of crystal momentum or Brillouin-zone structure is made. This suggests that the term names a broader departure from conventional Bloch band theory rather than a single formalism. In both usages, the central point is the same: integrated quantum geometry can be defined without relying on ordinary translational symmetry, and it retains direct information about localization, gap closings, and phase structure (Sun et al., 19 May 2026).