Right-Eigenstate Non-Hermitian Mean-Field Theory

• Right-Eigenstate-Based Non-Hermitian Mean-Field Theory is a framework that uses normalized right eigenstates to define observables, order parameters, and variational criteria.
• Key constructions include right-state expectation values, covariance matrices, and companion Hermitian residual minimization to reconstruct effective Hamiltonians.
• Implementations span coherent-state reductions in non-Hermitian Bose–Hubbard dimers and BCS-type theories for dissipative superconductors and superfluids.

Searching arXiv for papers on right-eigenstate-based non-Hermitian theory, biorthogonal observables, and non-Hermitian mean-field/variational methods. Right-Eigenstate-Based Non-Hermitian Mean-Field Theory denotes a family of non-Hermitian many-body approximations in which effective order parameters, variational states, or thermal/statistical ensembles are constructed primarily from normalized right states or right eigenstates, rather than from explicitly biorthogonal left-right pairs. In this literature, the characteristic prescriptions are right-state expectation values $$\langle O\rangle=\frac{\langle\psi|O|\psi\rangle}{\langle\psi|\psi\rangle}$$, right-state covariance matrices, and residual-minimization principles based on Hermitian companions such as $(H^\dagger-\varepsilon^*)(H-\varepsilon)$. The approach appears in coherent-state reductions of non-Hermitian bosonic dynamics, in right-state variational many-body algorithms, and in explicit BCS-type theories for dissipative superconductors and superfluids (Graefe et al., 2010, Guo et al., 2022, Liu et al., 3 Sep 2025, Liu et al., 5 Oct 2025). The same body of work also shows that right-only constructions are not generically complete: Green’s-function response, Berry phases, and some topological or thermal properties depend on left-right biorthogonal structure, and even the standard non-Hermitian Hartree closure can fail in the large-$N$ limit (Wang et al., 31 Mar 2025, Ginzburg et al., 19 Jan 2026, Luo et al., 5 Jun 2026).

1. Definition and conceptual scope

The starting point is the standard non-Hermitian spectral problem

$$H|R_n\rangle=E_n|R_n\rangle,\qquad H^\dagger|L_n\rangle=E_n^*|L_n\rangle.$$

In Hermitian theory, left and right eigenvectors coincide up to Hermitian conjugation, but in non-Hermitian theory they do not generically do so. A right-eigenstate-based formulation therefore makes a deliberate choice: it treats the normalized right state as the primary object for defining observables, order parameters, or variational criteria, even though the full spectral decomposition of $H$ is biorthogonal (Xie et al., 2024, Wang et al., 31 Mar 2025).

Within this broad label, the literature contains several distinct constructions rather than a single canonical formalism. One class uses normalized right-state expectation values as the fundamental observable prescription. Another uses right-state data only to reconstruct or constrain an effective local Hamiltonian. A third uses right-state residual minimization to target right eigenvectors variationally without introducing an independent left variational manifold. Still another uses right-eigenstate anomalous averages to close superconducting or superfluid mean-field equations (Guo et al., 2022, Liu et al., 3 Sep 2025, Liu et al., 5 Oct 2025).

A central ambiguity concerns state selection. Some works define the non-Hermitian “ground state” as the eigenstate with the smallest real part of the energy, while others emphasize that several physically motivated choices may exist, such as a minimal-real-energy state or a long-time steady state selected by maximal imaginary part (Solinas et al., 1 Aug 2025, Luo et al., 5 Jun 2026). This means that a right-eigenstate-based mean-field theory is never specified solely by the choice “use right states”; it also requires a rule for which right state is to be used.

2. Formal constructions built from right states

Three recurrent right-state constructions organize much of the literature.

Construction	Key formula	Role
Right-state expectation value	$$\langle O\rangle=\frac{\langle\psi|O|\psi\rangle}{\langle\psi|\psi\rangle}$$	Defines observables from a normalized right state
Right-state covariance matrix	$$M^R_{ij}=\langle \hat O_i\hat O_j\rangle_R-\langle \hat O_i\rangle_R\langle \hat O_j\rangle_R$$	Reconstructs local parent Hamiltonians from one right eigenstate
Companion Hermitian residual	$$\mathbb G(H,\varepsilon)=(H^\dagger-\varepsilon^*)(H-\varepsilon)$$	Variationally targets right eigenvectors by residual minimization

The covariance-matrix construction shows that a single right eigenstate can determine a local non-Hermitian parent Hamiltonian within a chosen operator manifold

$$\hat H(\boldsymbol{\omega})=\sum_i \omega_i \hat O_i,$$

because the non-Hermitian variance

$$\triangle H^2=\boldsymbol{\omega}^\dagger \mathbf M^R \boldsymbol{\omega}$$

vanishes iff the state is a right eigenstate of the candidate Hamiltonian. The parent Hamiltonian coefficients are then null vectors of $(H^\dagger-\varepsilon^*)(H-\varepsilon)$0, and uniqueness is controlled by the kernel dimension of that matrix (Xie et al., 2024). This establishes that right-state correlators can encode substantial structural information, although only inside a restricted local ansatz space.

The variational construction based on

$(H^\dagger-\varepsilon^*)(H-\varepsilon)$1

goes further. It replaces the failed non-Hermitian Rayleigh–Ritz logic by residual minimization and converts the right-eigenvalue problem into a Hermitian ground-state problem for $(H^\dagger-\varepsilon^*)(H-\varepsilon)$2. If $(H^\dagger-\varepsilon^*)(H-\varepsilon)$3 is an eigenvalue of $(H^\dagger-\varepsilon^*)(H-\varepsilon)$4, the ground state of $(H^\dagger-\varepsilon^*)(H-\varepsilon)$5 is the corresponding right eigenvector (Guo et al., 2022). This supplies a rigorous right-eigenstate-only variational principle, at least for eigenstate targeting. A plausible implication is that a product-state, Gaussian, or Hartree-Fock restriction of the same residual objective could serve as a non-Hermitian mean-field ansatz, but that extrapolation is not developed explicitly in the cited work.

A related variational lesson comes from biorthogonal neural-network treatments of non-Hermitian many-body systems. There, direct minimization of a complex energy expectation is shown to be inadequate, and the basic stable objective is again a residual/variance operator. Those works also show that pseudo-Hermiticity or $(H^\dagger-\varepsilon^*)(H-\varepsilon)$6 symmetry can collapse a full left-right optimization to an effectively single-state scheme through a relation such as $(H^\dagger-\varepsilon^*)(H-\varepsilon)$7 (Solinas et al., 1 Aug 2025). In symmetry-constrained settings, this is one of the clearest routes by which a right-state-based mean-field theory can remain internally consistent without optimizing an independent left state.

3. Dynamical mean-field reductions and coherent-state limits

A foundational early example is the non-Hermitian Bose–Hubbard dimer, where the mean-field limit is derived from generalized $(H^\dagger-\varepsilon^*)(H-\varepsilon)$8 coherent states rather than by ad hoc complexification of the Gross–Pitaevskii equation. The many-body Hamiltonian contains a complex onsite energy modeling effective decay from one mode, and the coherent-state reduction yields a nonlinear two-mode equation in which the interaction depends on the normalized population imbalance

$(H^\dagger-\varepsilon^*)(H-\varepsilon)$9

not on the naive unnormalized occupations (Graefe et al., 2010).

This construction is right-state-based in a precise sense. Expectation values are defined as

$N$0

using the evolving right state and its ordinary Hilbert-space adjoint, without biorthogonal left eigenstates. The resulting Bloch-sphere dynamics obey

$N$1

$N$2

$N$3

with conserved radius $N$4, and the norm evolves as

$N$5

The mean-field flow can be written as a sum of Hamiltonian flow and metric gradient flow,

$N$6

which is one of the canonical geometric signatures of this non-Hermitian mean-field reduction (Graefe et al., 2010).

The same work also identifies a characteristic physical effect of non-Hermitian mean-field dynamics: non-Hermiticity promotes self-trapping while damping self-trapping oscillations. In the unbiased case, the critical interaction becomes

$N$7

instead of the Hermitian threshold $N$8. This makes clear that a right-state mean-field reduction can yield genuinely new nonlinear dynamics, not just a weakly modified Hermitian picture.

4. Superconducting and superfluid implementations

The most explicit realizations of right-eigenstate-based non-Hermitian mean-field theory occur in paired fermionic systems. In a $N$9-symmetric honeycomb superconductor with balanced gain and loss, the order parameters are defined by normalized right-eigenstate anomalous expectation values,

$$H|R_n\rangle=E_n|R_n\rangle,\qquad H^\dagger|L_n\rangle=E_n^*|L_n\rangle.$$0

with the structure

$$H|R_n\rangle=E_n|R_n\rangle,\qquad H^\dagger|L_n\rangle=E_n^*|L_n\rangle.$$1

The self-consistent gap equation is

$$H|R_n\rangle=E_n|R_n\rangle,\qquad H^\dagger|L_n\rangle=E_n^*|L_n\rangle.$$2

and the quasiparticle spectrum is

$$H|R_n\rangle=E_n|R_n\rangle,\qquad H^\dagger|L_n\rangle=E_n^*|L_n\rangle.$$3

Within this framework, the superconducting transition between $$H|R_n\rangle=E_n|R_n\rangle,\qquad H^\dagger|L_n\rangle=E_n^*|L_n\rangle.$$4-symmetric and $$H|R_n\rangle=E_n|R_n\rangle,\qquad H^\dagger|L_n\rangle=E_n^*|L_n\rangle.$$5-broken regimes is first order and coincides exactly with the $$H|R_n\rangle=E_n|R_n\rangle,\qquad H^\dagger|L_n\rangle=E_n^*|L_n\rangle.$$6-breaking point. The same theory predicts that moderate non-Hermitian dissipation enhances superconductivity in the $$H|R_n\rangle=E_n|R_n\rangle,\qquad H^\dagger|L_n\rangle=E_n^*|L_n\rangle.$$7-symmetric phase but suppresses it strongly after $$H|R_n\rangle=E_n|R_n\rangle,\qquad H^\dagger|L_n\rangle=E_n^*|L_n\rangle.$$8 breaking (Liu et al., 3 Sep 2025).

A closely related implementation treats a lattice Fermi superfluid with two-body loss, encoded as a complex interaction

$$H|R_n\rangle=E_n|R_n\rangle,\qquad H^\dagger|L_n\rangle=E_n^*|L_n\rangle.$$9

The right-eigenstate mean-field theory defines

$H$0

with $H$1, $H$2, and $H$3. The resulting gap equation contains explicit norm and modulus factors,

$H$4

which are absent in the biorthogonal counterpart (Liu et al., 5 Oct 2025).

The comparison between the two prescriptions is one of the main results. In the superfluid model with two-body loss, the right-eigenstate framework yields smooth and continuous nontrivial solutions, whereas the biorthogonal mean-field theory can show missing-solution intervals and abrupt discontinuities under moderate dissipation. The right-eigenstate solution also has lower condensation energy in the reported calculations, and the same framework predicts strong fragility against backscattering, with sufficiently strong backscattering destroying the superfluid state (Liu et al., 5 Oct 2025).

Taken together, these superconducting and superfluid works establish that right-eigenstate-based mean-field theory is not only a conceptual proposal but an operative self-consistent scheme for interacting fermionic phases. They also make clear that the decisive choice is not merely spectral; it is the decision to define pairing amplitudes and condensed densities from normalized right-state observables.

5. Relation to biorthogonality, response theory, and statistical mechanics

The principal objection to a purely right-eigenstate-based theory is that generic non-Hermitian response is biorthogonal. In Green’s-function language,

$H$5

so the response amplitude

$H$6

depends on the rank-one operator $H$7, not on $H$8. The same work shows experimentally that columns of the Green’s-function matrix encode right eigenvectors, while rows encode left eigenvectors, and that the non-Hermitian Berry phase

$H$9

is intrinsically biorthogonal (Wang et al., 31 Mar 2025). This sharply limits any claim that right eigenvectors alone furnish a complete observable theory.

Exactly solvable non-Hermitian XY chains reinforce the same point from the opposite direction. There, expectation values computed from normalized right states and those computed biorthogonally lead to different magnetizations, correlation functions, critical exponents, and even different phase boundaries. The authors of that work argue in favor of the standard right-state prescription for genuine non-Hermitian open-system realizations, but they also show that the physical content depends on which right eigenstate is selected as the analog of the ground state. Minimal-real-energy and steady-state selections can yield distinct phase diagrams (Luo et al., 5 Jun 2026). Right-eigenstate-based mean-field theory therefore resolves one ambiguity only by introducing another: one must specify the preparation principle that selects the right state.

Open-system derivations of thermalization give a dynamical foundation to this distinction. A GKSL analysis of systems coupled to thermal baths derives two inequivalent stationary ensembles: the Boltzmann biorthogonal statistics

$$\langle O\rangle=\frac{\langle\psi|O|\psi\rangle}{\langle\psi|\psi\rangle}$$0

and the Boltzmann right-eigenstate statistics

$$\langle O\rangle=\frac{\langle\psi|O|\psi\rangle}{\langle\psi|\psi\rangle}$$1

Under suitable assumptions, both can arise from bath-induced thermalization, but they are not equivalent and lead to different observables and entropy functionals (Mao et al., 2024). This is highly relevant for non-Hermitian mean-field thermodynamics: a right-eigenstate free-energy-like construction is a distinct physical framework, not a rewriting of biorthogonal statistical mechanics.

6. Limitations, controversies, and broader frameworks

The strongest limitation is that right-state mean-field logic can fail even when it looks formally natural. An exactly solvable bosonic model with purely anti-Hermitian two-body interaction

$$\langle O\rangle=\frac{\langle\psi|O|\psi\rangle}{\langle\psi|\psi\rangle}$$2

shows that the large-$$\langle O\rangle=\frac{\langle\psi|O|\psi\rangle}{\langle\psi|\psi\rangle}$$3 normalized one-particle marginal does not generically follow the standard non-Hermitian Hartree equation. For generic initial conditions the limiting one-particle state remains pure but obeys a different nonlinear law, while for the balanced initial state $$\langle O\rangle=\frac{\langle\psi|O|\psi\rangle}{\langle\psi|\psi\rangle}$$4 the large-$$\langle O\rangle=\frac{\langle\psi|O|\psi\rangle}{\langle\psi|\psi\rangle}$$5 one-particle marginal becomes mixed after the critical time $$\langle O\rangle=\frac{\langle\psi|O|\psi\rangle}{\langle\psi|\psi\rangle}$$6 (Ginzburg et al., 19 Jan 2026). This is a direct refutation of the notion that a single normalized right-state order parameter always captures the mean-field limit of a non-Hermitian bosonic many-body system.

Other frameworks point to the same conclusion from different directions. Real-space DMFT for the non-Hermitian Hubbard model with asymmetric hopping does not formulate self-consistency directly on many-body right eigenvectors, but it shows that once the non-Hermitian skin effect localizes right eigenstates at boundaries, a usable correlated mean-field theory must be real-space, site dependent, and Green’s-function based. Its key diagnostics are end-to-end propagators rather than local densities of states, and the noninteracting spectral representation remains explicitly biorthogonal,

$$\langle O\rangle=\frac{\langle\psi|O|\psi\rangle}{\langle\psi|\psi\rangle}$$7

This suggests that any right-eigenstate-centered mean-field theory for systems with strong nonreciprocity must still reproduce biorthogonal propagation physics at the response level (Rangi et al., 25 Jul 2025).

At an even more collective level, field theory of non-Hermitian disordered systems proceeds not from right eigenstates alone but from Hermitization and replica sigma models. The basic objects are $$\langle O\rangle=\frac{\langle\psi|O|\psi\rangle}{\langle\psi|\psi\rangle}$$8, $$\langle O\rangle=\frac{\langle\psi|O|\psi\rangle}{\langle\psi|\psi\rangle}$$9, and Hermitized operators such as

$$M^R_{ij}=\langle \hat O_i\hat O_j\rangle_R-\langle \hat O_i\rangle_R\langle \hat O_j\rangle_R$$0

That framework identifies saddle manifolds, target spaces, and topological terms across the full 38-fold symmetry classification, but it naturally treats $$M^R_{ij}=\langle \hat O_i\hat O_j\rangle_R-\langle \hat O_i\rangle_R\langle \hat O_j\rangle_R$$1 and $$M^R_{ij}=\langle \hat O_i\hat O_j\rangle_R-\langle \hat O_i\rangle_R\langle \hat O_j\rangle_R$$2 together rather than privileging right eigenvectors alone (Chen et al., 2024). A plausible implication is that a genuinely general right-eigenstate-based mean-field theory would need either an embedded biorthogonal sector or a Hermitized reformulation to recover the correct universality structure.

The resulting picture is therefore sharply delimited. Right-eigenstate-based non-Hermitian mean-field theory is well defined and operational in several important senses: it can reconstruct effective generators from a single right eigenstate, target right eigenvectors variationally, derive coherent-state nonlinear dynamics, and formulate experimentally motivated BCS theories of dissipative paired phases (Xie et al., 2024, Guo et al., 2022, Graefe et al., 2010, Liu et al., 3 Sep 2025, Liu et al., 5 Oct 2025). But it is not a universally sufficient replacement for biorthogonal theory. The most accurate general statement is conditional: right-eigenstate-based mean-field theory is reliable when the target problem is intrinsically right-state defined—such as variational right-eigenvector targeting, no-jump/postselected observables, or symmetry-constrained reductions—but it becomes incomplete whenever response kernels, topology, thermal ensembles, or large-$$M^R_{ij}=\langle \hat O_i\hat O_j\rangle_R-\langle \hat O_i\rangle_R\langle \hat O_j\rangle_R$$3 dynamics depend irreducibly on left-right structure (Wang et al., 31 Mar 2025, Ginzburg et al., 19 Jan 2026, Luo et al., 5 Jun 2026).