Right-Eigenstate-Based Mean-Field Theory

• Right-eigenstate-based mean-field theory is a framework that uses the spectral properties of right eigenstates to describe many-body quantum systems, especially in non-Hermitian or far-from-equilibrium environments.
• It extends traditional mean-field methods by incorporating complex eigenvalue spectra to effectively capture loss, gain, and transient dynamics in experimental setups.
• The approach enables both analytic and numerical treatments across various systems, including superfluids, correlated materials, and matrix models of quantum chaos.

Right-eigenstate-based mean-field theory is a framework for describing many-body quantum systems that utilizes the spectral properties of right eigenstates—often in non-Hermitian settings or far-from-equilibrium regimes—where conventional Hermitian approaches can fail to capture essential dynamic, dissipative, or statistical aspects. By formulating self-consistency conditions, observables, and even parent Hamiltonian reconstruction directly in a right-eigenstate basis, this approach enables both analytic and numeric treatments of systems featuring loss, gain, non-equilibrium excitation spectra, or open-system quantum dynamics. It extends traditional mean-field methodologies (such as dynamical mean-field theory, Hartree-Fock, Bardeen–Cooper–Schrieffer, and algebraic mean-field theories) and provides tools for studying strongly correlated matter, non-Hermitian quantum phases, dissipative superfluidity, electronic structure, and matrix-model realizations of statistical chaos.

1. Foundations and Spectral Representation

The central premise is that, rather than always assuming Hermitian evolution and real eigenvalues, one can construct mean-field theory by diagonalizing the effective (possibly non-Hermitian) Hamiltonian in terms of its right (and in general, also left) eigenstates. The Green’s function in this setting adopts a generalized spectral representation,

$$G(\omega) = \sum_n \frac{ |\psi_n^R\rangle \langle\psi_n^L| }{ \omega + \mu - E_n }$$

where $|\psi_n^R\rangle$ and $\langle\psi_n^L|$ are sets of right and left eigenstates of the effective operator $H_\text{eff}(\omega)$, and $E_n$ are generally complex eigenvalues with the imaginary parts encoding lifetimes and dissipation (Vollhardt et al., 2011). Biorthogonality is maintained via $\langle\psi_n^L | \psi_m^R\rangle = \delta_{nm}$.

This spectral reformulation is especially crucial for open-system or non-equilibrium problems: it enables incorporation of decay rates, transient dynamics, and effects arising from external driving or loss. In conventional dynamical mean-field theory, such a representation allows treating local (impurity) problems, hybridization, and bath coupling within a framework that explicitly tracks the full (possibly complex) excitation spectrum.

2. Extended Self-Consistency in Non-Hermitian and Dissipative Systems

The right-eigenstate-based approach modifies mean-field self-consistency cycles. For example, in DMFT, the self-consistency condition,

$$[G(\omega)]^{-1} = \omega + \mu - \Delta(\omega) - \Sigma(\omega),$$

is replaced with one where $G(\omega)$ is constructed in the right-eigenstate spectral basis, and the impurity solver must output the set $\{ E_n, |\psi_n^R\rangle, \langle\psi_n^L| \}$ (Vollhardt et al., 2011). The mapping between impurity and lattice problems becomes,

$$\mathcal{G}^{-1}(\omega) = G^{-1}(\omega) + \Sigma(\omega),$$

and one matches the impurity Green’s function—now formulated in the right-eigenstate basis—to the local lattice Green’s function via integration over the density of states.

This formalism accommodates non-Hermitian effects arising from dissipation, environmental couplings, or engineered loss/gain terms. The complex eigenvalue spectrum provides direct access to lifetimes, transient relaxation, and dissipation rates, especially relevant for nonequilibrium "quench" dynamics, open quantum dots, photonic lattices, and ultracold atom experiments.

3. Application to Fermionic and Bosonic Systems

In bosonic systems, right-eigenstate-based mean-field theory extends the validity of the Hartree or Gross-Pitaevskii equations, including environments with external fields or magnetic couplings (Luhrmann, 2012). Rigorous results demonstrate that marginals and energy converge toward those predicted by mean-field equations at quantifiable rates, even as the single-particle operator is generalized with, for example, magnetic vector potentials. The many-body state remains close to a coherent state generated from the right-eigenstate solution of the effective nonlinear equation.

For fermionic systems, the mean-field limit coexists with the semiclassical regime due to large kinetic energies and antisymmetry. While bosonic coherence is captured by right-eigenstate methods, fermionic systems are described via Slater determinants and quasi-free states, and the reduced one-particle density matrix evolves via the Hartree-Fock equation (with commutator structure in its evolution) (Benedikter et al., 2013). In this context, the right-eigenstate-based philosophy motivates the explicit use of projection operators and Bogoliubov transformations for particle-hole excitations.

In non-Hermitian superfluid models, the right-eigenstate mean-field method defines order parameters using expectation values over a right ground state $|\Psi_0\rangle$, yielding,

$$\Delta = -\frac{U}{N} \sum_{k} \frac{\langle\Psi_0| c_{-k\downarrow} c_{k\uparrow} |\Psi_0 \rangle}{\langle\Psi_0|\Psi_0\rangle}, \qquad \bar{\Delta} = -\frac{U}{N} \sum_{k} \frac{\langle\Psi_0| c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger |\Psi_0 \rangle}{\langle\Psi_0|\Psi_0\rangle}$$

with complex-valued interaction $U$ and nonunitary Bogoliubov transformations, yielding gap equations and condensation energies that remain continuous under moderate dissipation, unlike their biorthogonal counterparts (Liu et al., 5 Oct 2025).

4. Parent Hamiltonian Reconstruction and Experimental Quantum Systems

A striking recent advance is the demonstration that under locality constraints, the full local non-Hermitian Hamiltonian can be reconstructed from just a single right (or left) eigenstate (Xie et al., 28 Aug 2024). The procedure relies on enforcing zero energy variance via the quantum covariance matrix constructed from expectation values in the right eigenstate:

$$M^{(R)}_{ij} = \langle O_i O_j \rangle_R - \langle O_i \rangle_R \langle O_j \rangle_R$$

and the null space condition,

$M^{(R)} \omega = 0$

where $\omega$ are the coefficients of local operators $O_i$ in $H$. This method is robust against measurement errors, requiring only polynomial scaling of measurements with system size. Its practical implication is direct Hamiltonian learning in experimental quantum systems where only right eigenstates (not biorthogonal pairs) are accessible.

This covariance-matrix-driven strategy naturally integrates with right-eigenstate mean-field theory, providing a systematic way to adjust mean-field parameters from experimentally accessible observables and to update effective Hamiltonians within self-consistency cycles.

5. Algebraic and Geometric Extensions

Mean-field approaches based on algebraic structures, notably the SO(2N+1) Lie algebra for fermionic operators, extend the mean-field concept to encompass both paired and unpaired modes (Nishiyama et al., 2018). Diagonalization of generalized Hartree-Bogoliubov mean-field Hamiltonians in these algebraic settings provides explicit amplitudes for unpaired modes, governed by self-consistent field parameters and group-theoretical variables. The Killing potential constructed on the SO(2N)/U(N) coset space is shown to be equivalent to the generalized density matrix, embedding geometric features directly in the mean-field theory and endowing it with a symplectic structure via coadjoint orbits.

The geometric perspective yields additional self-consistency conditions and collective-mode dynamics governed by Hamiltonian equations, supporting the unified treatment of quasi-particle spectra in both even and odd sectors.

6. Statistical and Matrix-Model Realizations

Right-eigenstate-based mean-field theory is also prominent in the statistical physics of quantum chaos and thermalization. In the matrix-model formalism for the eigenstate thermalization hypothesis (ETH), joint probability distributions over energy eigenvalues and operator matrix elements, augmented with non-Gaussian corrections, reproduce the correct higher moments necessary for thermal mean-field theory, crossing symmetry of four-point functions, and nontrivial out-of-time-order correlators (Jafferis et al., 2022). The matrix-integral approach captures the complete set of disk and higher-genus correlators in models such as JT gravity with matter.

The higher statistical moments encoded in ETH matrix models ensure a full mean-field description that remains valid for generalized free fields and semiclassical limits of quantum gravity, uniting concepts from quantum chaos, effective field theory, and spectral theory.

7. Impact, Practical Applications, and Open Directions

Right-eigenstate-based mean-field theory is impactful wherever non-Hermitian dynamics, dissipation, loss/gain processes, or far-from-equilibrium phenomena play a role. It underpins the analysis of:

• Superfluids under two-body loss and impurity-induced destabilization (Liu et al., 5 Oct 2025)
• Correlated materials with bath couplings and nontrivial spectral features (Vollhardt et al., 2011)
• Quantum chemistry via variational geminal-product wavefunctions (Richardson-Gaudin ansatz) (Johnson et al., 2020, Fecteau et al., 2022)
• Learning parent Hamiltonians from quantum measurements in experiments (Xie et al., 28 Aug 2024)
• Statistical modeling of ergodic eigenstate properties and quantum gravity (Jafferis et al., 2022)

By recasting mean-field theory in terms of right-eigenstates and employing spectral, algebraic, and covariance-matrix techniques, the framework is adaptable across condensed matter, chemical physics, atomic systems, and statistical/gravitational theories. Ongoing developments are focused on integrating these perspectives with fluctuation corrections, improved impurity solvers, and geometric quantization on coadjoint orbits, as well as expanding the analytic toolkit for handling complex spectra in open quantum systems.