Biorthogonal Kubo Formalism

• Biorthogonal Kubo Formalism is an extension of equilibrium and linear-response theory that uses left-right eigensystems to define traces, expectation values, and correlation functions.
• It employs a biorthogonal Gibbs functional and a positive metric to establish an analytic KMS condition, ensuring thermodynamic consistency in quasi-Hermitian regimes.
• Applications include non-Hermitian band transport and charge pumping, where response coefficients are determined through biorthogonal matrix elements and Chern numbers.

Searching arXiv for the cited papers and closely related work to ground the article. arXiv_search(query="(Lan et al., 11 Jun 2026)", max_results=5, sort_by="relevance") arXiv_search(query="(Zhang et al., 2024)", max_results=5, sort_by="relevance") arXiv_search(query="non-Hermitian KMS quasi-Hermitian Gibbs detailed balance", max_results=10, sort_by="relevance") In current usage suggested by recent work on non-Hermitian thermal theory and non-Hermitian adiabatic transport, Biorthogonal Kubo Formalism denotes an extension of equilibrium and linear-response theory in which traces, expectation values, correlation functions, and response coefficients are defined with respect to a biorthogonal left-right eigensystem rather than a single orthonormal basis. For diagonalizable non-Hermitian Hamiltonians with real spectra, this program combines a biorthogonal Gibbs functional, a quasi-Hermitian metric formulation, analytic Kubo–Martin–Schwinger (KMS) conditions, and frequency-space detailed-balance relations; in band theory, the same logic appears in biorthogonal charge pumping, where current and pumped charge are expressed as left-right matrix elements and biorthogonal Chern numbers rather than right-right observables (Lan et al., 11 Jun 2026, Zhang et al., 2024).

1. Spectral foundations: biorthogonal eigensystems and quasi-Hermiticity

The basic setting is a non-Hermitian Hamiltonian $H$ with no Jordan blocks, pure point spectrum, real eigenvalues, and a complete biorthogonal system of right and left eigenvectors. A biorthogonal complete structure is

$$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$

with

$$\langle\phi_m|\psi_n\rangle = \delta_{mn},\qquad \sum_n |\psi_n\rangle\langle\phi_n| = \mathbf 1.$$

This pair $\{|\psi_n\rangle,|\phi_n\rangle\}$ replaces an orthonormal basis. It is the basis used to define biorthogonal traces, thermal functionals, and correlation functions. In particular, the biorthogonal trace is

$$\operatorname{Tr}_{\rm bi}[A]=\sum_n \langle \phi_n|A|\psi_n\rangle,$$

and, when the resolution of unity holds, it has completeness and cyclicity properties (Lan et al., 11 Jun 2026).

The quasi-Hermitian sector is singled out by the existence of a positive metric $\eta>0$ such that

$H^\dagger = \eta H \eta^{-1}.$

With the modified inner product

$(\psi,\phi)_\eta := \langle\psi|\eta|\phi\rangle,$

the operator $H$ is self-adjoint on the Hilbert space $\mathcal H_\eta$. Under the boundedness, positivity, and Riesz-basis assumptions used in the KMS analysis, $$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$0 is similar to a Hermitian operator

$$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$1

its spectrum is real, the right eigenvectors form a Riesz basis, and the left and right eigenvectors are related by

$$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$2

This relation is structurally decisive. In a generic biorthogonal problem the left and right eigenvectors are independent data; in the quasi-Hermitian regime they are tied together by a positive metric. That linkage is what allows positivity of thermal expectation values and, ultimately, a genuine KMS structure.

2. Biorthogonal Gibbs functionals and the thermodynamic characterization of quasi-Hermiticity

In finite dimension, assuming only real eigenvalues, a complete biorthogonal system, and a finite partition function, one defines the biorthogonal Gibbs functional

$$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$3

Equivalently,

$$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$4

Formally, this is the thermal ensemble obtained from the spectral decomposition of $$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$5, with Boltzmann weights $$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$6 and biorthogonal matrix elements. It is time-translation invariant and satisfies a KMS-type boundary condition at the level of two-point functions, but it is not automatically positive (Lan et al., 11 Jun 2026).

The central structural result is the finite-dimensional equivalence between three statements: positivity of the biorthogonal Gibbs functional,

$$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$7

existence of a positive metric $$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$8 with $$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$9, and existence of $$\langle\phi_m|\psi_n\rangle = \delta_{mn},\qquad \sum_n |\psi_n\rangle\langle\phi_n| = \mathbf 1.$$0 such that

$$\langle\phi_m|\psi_n\rangle = \delta_{mn},\qquad \sum_n |\psi_n\rangle\langle\phi_n| = \mathbf 1.$$1

This is Theorem 4.4, the “Biorthogonal KMS Structure Theorem.”

The proof gives a thermodynamic characterization of quasi-Hermiticity. Starting from positivity, the Riesz representation theorem for matrices yields a density-matrix representation

$$\langle\phi_m|\psi_n\rangle = \delta_{mn},\qquad \sum_n |\psi_n\rangle\langle\phi_n| = \mathbf 1.$$2

Matching this with the explicit biorthogonal spectral expression reconstructs a positive matrix $$\langle\phi_m|\psi_n\rangle = \delta_{mn},\qquad \sum_n |\psi_n\rangle\langle\phi_n| = \mathbf 1.$$3, and the metric is then built spectrally as

$$\langle\phi_m|\psi_n\rangle = \delta_{mn},\qquad \sum_n |\psi_n\rangle\langle\phi_n| = \mathbf 1.$$4

Because $$\langle\phi_m|\psi_n\rangle = \delta_{mn},\qquad \sum_n |\psi_n\rangle\langle\phi_n| = \mathbf 1.$$5 is a genuine basis, $$\langle\phi_m|\psi_n\rangle = \delta_{mn},\qquad \sum_n |\psi_n\rangle\langle\phi_n| = \mathbf 1.$$6 is Hermitian and positive definite. Using the spectral decompositions

$$\langle\phi_m|\psi_n\rangle = \delta_{mn},\qquad \sum_n |\psi_n\rangle\langle\phi_n| = \mathbf 1.$$7

one obtains

$$\langle\phi_m|\psi_n\rangle = \delta_{mn},\qquad \sum_n |\psi_n\rangle\langle\phi_n| = \mathbf 1.$$8

Conversely, if $$\langle\phi_m|\psi_n\rangle = \delta_{mn},\qquad \sum_n |\psi_n\rangle\langle\phi_n| = \mathbf 1.$$9 with $\{|\psi_n\rangle,|\phi_n\rangle\}$0, then

$\{|\psi_n\rangle,|\phi_n\rangle\}$1

The significance is that positivity of $\{|\psi_n\rangle,|\phi_n\rangle\}$2 is a metric-free certificate of quasi-Hermiticity. No prior choice of $\{|\psi_n\rangle,|\phi_n\rangle\}$3 is assumed; positivity itself forces the existence of a positive metric, constructed directly from the eigenprojectors. In the terminology of the paper, this places the thermodynamic criterion outside the Mostafazadeh–Scholtz framework because the metric is not an input but an output.

3. The $\{|\psi_n\rangle,|\phi_n\rangle\}$4-Gibbs state and the analytic KMS theorem

Once a positive metric is available, the natural equilibrium state is the $\{|\psi_n\rangle,|\phi_n\rangle\}$5-Gibbs state

$\{|\psi_n\rangle,|\phi_n\rangle\}$6

Equivalently, with

$\{|\psi_n\rangle,|\phi_n\rangle\}$7

one has

$\{|\psi_n\rangle,|\phi_n\rangle\}$8

This state is positive and faithful in the $\{|\psi_n\rangle,|\phi_n\rangle\}$9-inner-product sense,

$$\operatorname{Tr}_{\rm bi}[A]=\sum_n \langle \phi_n|A|\psi_n\rangle,$$0

Moreover, the Heisenberg dynamics

$$\operatorname{Tr}_{\rm bi}[A]=\sum_n \langle \phi_n|A|\psi_n\rangle,$$1

is a $$\operatorname{Tr}_{\rm bi}[A]=\sum_n \langle \phi_n|A|\psi_n\rangle,$$2-automorphism for the $$\operatorname{Tr}_{\rm bi}[A]=\sum_n \langle \phi_n|A|\psi_n\rangle,$$3-adjoint: $$\operatorname{Tr}_{\rm bi}[A]=\sum_n \langle \phi_n|A|\psi_n\rangle,$$4

The full KMS statement is formulated through

$$\operatorname{Tr}_{\rm bi}[A]=\sum_n \langle \phi_n|A|\psi_n\rangle,$$5

For bounded $$\operatorname{Tr}_{\rm bi}[A]=\sum_n \langle \phi_n|A|\psi_n\rangle,$$6, the function $$\operatorname{Tr}_{\rm bi}[A]=\sum_n \langle \phi_n|A|\psi_n\rangle,$$7 is analytic in the strip $$\operatorname{Tr}_{\rm bi}[A]=\sum_n \langle \phi_n|A|\psi_n\rangle,$$8, continuous and bounded on its closure, and satisfies the boundary relations

$$\operatorname{Tr}_{\rm bi}[A]=\sum_n \langle \phi_n|A|\psi_n\rangle,$$9

equivalently,

$\eta>0$0

This is proved using a specifically non-Hermitian spectral analysis. A key identity is

$\eta>0$1

whose operator-norm control relies on Bari’s theorem for Riesz bases. The two-point function admits the expansion

$\eta>0$2

and the Hadamard three-line theorem is then used to establish boundedness and uniform convergence on the closed strip (Lan et al., 11 Jun 2026).

A recurrent misconception is that the KMS property should follow trivially by similarity from the Hermitian partner $\eta>0$3. The obstruction is explicit. Writing $\eta>0$4, the transported state

$\eta>0$5

differs from the usual Gibbs state

$\eta>0$6

unless $\eta>0$7. The extra factor $\eta>0$8 in the trace cannot in general be absorbed into the normalization. The KMS theorem for $\eta>0$9 is therefore not a trivial corollary of Hermitian theory.

Under the full quasi-Hermitian assumptions, the biorthogonal and metric-weighted formulations coincide: $H^\dagger = \eta H \eta^{-1}.$0 Thus the metric-free spectral functional and the metric Gibbs state are two representations of the same equilibrium state precisely in the quasi-Hermitian regime.

4. Linear response, correlation functions, and the Kubo structure

The non-Hermitian KMS theory does not derive explicit Kubo formulas for transport coefficients, but it identifies the ingredients required for a linear-response formalism. The equilibrium state is $H^\dagger = \eta H \eta^{-1}.$1 in the spectral setting and, once quasi-Hermiticity holds, equivalently $H^\dagger = \eta H \eta^{-1}.$2. The time evolution is generated by the non-Hermitian Hamiltonian itself,

$H^\dagger = \eta H \eta^{-1}.$3

not by the Hermitian partner $H^\dagger = \eta H \eta^{-1}.$4, unless the observables are simultaneously conjugated by $H^\dagger = \eta H \eta^{-1}.$5. In $H^\dagger = \eta H \eta^{-1}.$6, this dynamics is unitary because $H^\dagger = \eta H \eta^{-1}.$7 is self-adjoint with respect to $H^\dagger = \eta H \eta^{-1}.$8 (Lan et al., 11 Jun 2026).

The KMS analysis supplies well-defined two-point functions

$H^\dagger = \eta H \eta^{-1}.$9

together with the spectral density

$(\psi,\phi)_\eta := \langle\psi|\eta|\phi\rangle,$0

In the frequency domain, the KMS condition yields

$(\psi,\phi)_\eta := \langle\psi|\eta|\phi\rangle,$1

which is the detailed-balance-type relation underlying Kubo relations and fluctuation–dissipation theory.

In Hermitian theory, response kernels are typically written as

$(\psi,\phi)_\eta := \langle\psi|\eta|\phi\rangle,$2

In the quasi-Hermitian setting, the natural analogue is

$(\psi,\phi)_\eta := \langle\psi|\eta|\phi\rangle,$3

where the appropriate notion of adjoint or commutator must be chosen relative to the $(\psi,\phi)_\eta := \langle\psi|\eta|\phi\rangle,$4-inner product, for example by using $(\psi,\phi)_\eta := \langle\psi|\eta|\phi\rangle,$5 and $(\psi,\phi)_\eta := \langle\psi|\eta|\phi\rangle,$6. The paper does not fix a unique $(\psi,\phi)_\eta := \langle\psi|\eta|\phi\rangle,$7-commutator, but it does establish the algebraic prerequisites: $(\psi,\phi)_\eta := \langle\psi|\eta|\phi\rangle,$8 is a $(\psi,\phi)_\eta := \langle\psi|\eta|\phi\rangle,$9-automorphism for $H$0, and $H$1 is $H$2-positive.

The decisive thermodynamic constraint for linear response is the equivalence

$H$3

This suggests that a physically consistent biorthogonal Kubo theory, including positive spectral densities and non-negative dissipative parts where appropriate, is available only when the non-Hermitian Hamiltonian is in fact quasi-Hermitian. A naive insertion of a non-positive biorthogonal Gibbs functional into response formulas does not yield the usual thermodynamic interpretation.

5. Band-theoretic realization: biorthogonal charge pumping as a Kubo-type response

A concrete realization of the formalism appears in non-Hermitian Thouless pumping. For a time-dependent one-dimensional non-Hermitian lattice Hamiltonian $H$4, one introduces instantaneous left and right eigenstates

$H$5

with biorthonormality

$H$6

For a fully filled band under periodic boundary conditions, the biorthogonal average position is

$H$7

and the pumped charge over one cycle is

$H$8

Using the Schrödinger evolution for both $H$9 and $\mathcal H_\eta$0, the instantaneous current becomes

$\mathcal H_\eta$1

with the biorthogonal velocity or current

$\mathcal H_\eta$2

In the adiabatic limit, with the parallel-transport gauge $\mathcal H_\eta$3, one has

$\mathcal H_\eta$4

where

$\mathcal H_\eta$5

For a filled band, the group-velocity term integrates to zero over the Brillouin zone and a full cycle, so

$\mathcal H_\eta$6

with $\mathcal H_\eta$7 the biorthogonal Chern number (Zhang et al., 2024).

This provides an explicit Kubo-type response formula in a non-Hermitian band problem: the observable is the current operator $\mathcal H_\eta$8, the expectation value is biorthogonal, and the integrated response is a geometric coefficient. A related Wannier-center formulation yields

$\mathcal H_\eta$9

and again the shift over one cycle equals the same Chern number.

A second recurrent misconception is that right-right observables should suffice. In general they do not. The naive right-right average position satisfies

$$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$00

which is not a simple current expectation value. Quantization is therefore not generally protected in the right-right formulation. The paper attributes observed right-state quantization in a lossy reciprocal Rice–Mele experiment to a special situation with nearly flat imaginary dispersion in the occupied band, not to a generic topological principle.

When the non-Hermitian skin effect is present under open boundary conditions, the Bloch description fails and the generalized Brillouin zone (GBZ) must replace the ordinary Brillouin zone. The pumped charge is then controlled by the non-Bloch Chern number

$$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$01

and the average biorthogonal displacement satisfies

$$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$02

Thus the formalism has two geometrically distinct regimes: Bloch Chern numbers under periodic boundary conditions and non-Bloch Chern numbers on GBZ-time space under open boundary conditions with the skin effect.

6. Obstructions, scope, and relation to broader equilibrium frameworks

The formalism has a sharply delimited domain of validity. The decisive distinctions are summarized below.

Regime	Spectral or dynamical feature	Consequence for KMS/Kubo structure
Quasi-Hermitian, real spectrum	Diagonalizable; complete biorthogonal basis; $$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$03 exists	Positive equilibrium state and analytic KMS structure are available
Exceptional points	Jordan blocks; polynomial growth in time	Biorthogonal completeness, cyclicity, and bounded-strip arguments fail
Complex spectra	Complex Boltzmann weights; exponential growth/decay factors	Probabilistic interpretation and standard KMS boundedness fail

At exceptional points, the Hamiltonian is non-diagonalizable and the spectral resolution is replaced by Jordan chains. The completeness relation $$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$04 fails in its simple form, biorthogonal collapse can occur through $$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$05, and the time evolution acquires polynomial growth,

$$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$06

Then the pure exponential spectral identity used in the KMS proof breaks down, correlation functions are no longer bounded on the real axis, and the Hadamard three-line argument fails.

For genuinely complex eigenvalues $$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$07, the Boltzmann weights become

$$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$08

so the partition function and thermal functionals are no longer real and positive. In correlation functions, factors of the form

$$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$09

produce exponential growth unless all $$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$10 coincide. Analyticity may persist, but boundedness and the standard probabilistic equilibrium interpretation do not. The construction of a consistent thermal theory in this regime is explicitly identified as an open problem (Lan et al., 11 Jun 2026).

The relation to the Haag–Hugenholtz–Winnink $$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$11-algebraic KMS framework is precise but incomplete. The non-Hermitian analysis establishes the analytic core of KMS theory—strip analyticity, boundedness, and the boundary relation—for bounded operators on a Hilbert space. What remains open is the full $$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$12-algebraic structure: a complete observable algebra including thermodynamic limits, strong or $$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$13-weak continuity of $$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$14, and the Tomita–Takesaki modular objects for $$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$15.

A distinct but related thermodynamic framework appears in open systems through the Fagnola–Umanità quantum detailed balance condition for GKSL generators,

$$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$16

For Davies generators, this condition is characterized through commutation and detailed-balance relations involving the Gibbs state of the Hermitian system Hamiltonian $$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$17, not an effective non-Hermitian Hamiltonian alone. This distinction is conceptually important. A biorthogonal Kubo framework for non-Hermitian closed systems and a QDB-based response theory for open systems both use KMS-type structures, but they operate at different dynamical levels.

Taken together, these developments define the sense in which a biorthogonal Kubo formalism exists. In the minimal spectral setting, one has a formally thermal biorthogonal Gibbs functional. In the quasi-Hermitian setting, positivity promotes it to a genuine equilibrium state and yields a full analytic KMS theorem. In non-Hermitian band transport, the same left-right logic produces quantized pumped charge through biorthogonal and non-Bloch Chern numbers. Beyond that domain—at exceptional points, for complex spectra, or in the full thermodynamic-limit $$H|\psi_n\rangle = E_n|\psi_n\rangle,\qquad H^\dagger|\phi_n\rangle = E_n^*|\phi_n\rangle,$$18-algebraic setting—the formalism encounters explicit obstructions or remains structurally incomplete.