Non-Hermitian Superfluid Model

• The non-Hermitian superfluid model extends the BCS Hamiltonian with complex interactions that introduce dissipative dynamics and unique phase transitions.
• It employs a mean-field decoupling using biorthogonal left and right eigenstates to derive a complex gap equation that diverges from traditional Hermitian theories.
• The model predicts reentrant superfluidity and observable effects like exceptional points, offering experimental insights via ultracold atomic systems and the quantum Zeno effect.

A non-Hermitian superfluid model is a theoretical framework in which the standard Bardeen–Cooper–Schrieffer (BCS) superfluid Hamiltonian is extended to include complex-valued, and thus non-Hermitian, interactions. Such models are physically motivated by ultracold atomic systems and condensed matter environments where dissipative processes—particularly inelastic two-body scattering—induce effective loss terms in the many-body Hamiltonian. The resulting non-Hermitian physics leads to distinctive modifications of the superfluid ground state, the structure of the excitation spectrum, and the nature of phase transitions, in contrast with conventional Hermitian analogues.

1. Non-Hermitian Hamiltonian Construction and Complex Interactions

The starting point of the non-Hermitian BCS model is an effective Hamiltonian for interacting fermions subject to inelastic, typically two-body, losses. Short-time dynamics under two-body loss can be described by the non-Hermitian effective Hamiltonian: $$H_{\mathrm{eff}} = \sum_{\boldsymbol{k},\sigma} \xi_{\boldsymbol{k}}\, c_{\boldsymbol{k}\sigma}^{\dagger}\, c_{\boldsymbol{k}\sigma} - U \sum_{i} c_{i\uparrow}^{\dagger} c_{i\downarrow}^{\dagger} c_{i\downarrow} c_{i\uparrow}$$ where $U = U_1 + i(\gamma/2)$, $U_1 > 0$ denotes the real attractive interaction and $\gamma > 0$ encodes the dissipation rate from inelastic scattering. Lindblad jump operators in the full open system's master equation justify this effective non-Hermitian description after neglecting quantum jumps and postselecting 'null' measurement outcomes.

The essential non-Hermitian feature is the complex-valued pairing interaction, which gives rise to complex-valued mean fields in the BCS channel and fundamentally changes the self-consistency and symmetry of the order parameters.

2. Mean-Field Decoupling and Non-Hermitian Gap Equation

To access superfluid properties, a mean-field treatment is performed using a Hubbard–Stratonovich transformation in the Cooper channel. This introduces auxiliary pairing fields $\Delta$ and $\bar{\Delta}$, which are not constrained to be mutual complex conjugates due to the NH structure. The path-integral formulation for the partition function employs biorthogonal left (${}_L\langle \cdots |$) and right ($| \cdots \rangle_R$) eigenstates: $$Z = \sum_n {}_L\langle E_n | e^{-\beta H_{\mathrm{eff}}} | E_n \rangle_R .$$ The corresponding effective action reads: $$S_{\mathrm{eff}}(\bar{\Delta}, \Delta) = - \sum_{\omega_n, \boldsymbol{k}} \ln(\omega_n^2 + \xi_{\boldsymbol{k}}^2 + \bar{\Delta} \Delta) + \beta N U \bar{\Delta} \Delta,$$ leading, by saddle point, to the NH gap equation (at zero temperature, $\beta \rightarrow \infty$): $$\frac{N}{U} = \sum_{\boldsymbol{k}} \frac{1}{2E_{\boldsymbol{k}}}, \quad E_{\boldsymbol{k}} = \sqrt{\xi_{\boldsymbol{k}}^2 + \bar{\Delta} \Delta}$$ with order parameters defined via

$$\Delta = - \frac{U}{N}\sum_{\boldsymbol{k}} {}_L\langle c_{-\boldsymbol{k}\downarrow} c_{\boldsymbol{k}\uparrow} \rangle_R, \quad \bar{\Delta} = - \frac{U}{N}\sum_{\boldsymbol{k}} {}_L\langle c_{\boldsymbol{k}\uparrow}^{\dagger} c_{-\boldsymbol{k}\downarrow}^{\dagger} \rangle_R.$$

It is crucial that, unlike in Hermitian BCS theory, $\bar{\Delta} \ne \Delta^*$ in general, reflecting the inequivalence of left and right eigenstates and the breakdown of spectral norm conservation.

3. Quasiparticle Structure, Order Parameter Inequivalence, and Exceptional Points

The mean-field Hamiltonian assumes the non-Hermitian Bogoliubov–de Gennes form: $$H_\mathrm{MF} = \sum_{\boldsymbol{k}} \begin{pmatrix} c_{\boldsymbol{k}\uparrow}^\dagger & c_{-\boldsymbol{k}\downarrow} \end{pmatrix} \begin{pmatrix} \xi_{\boldsymbol{k}} & \Delta \ \bar{\Delta} & -\xi_{\boldsymbol{k}} \end{pmatrix} \begin{pmatrix} c_{\boldsymbol{k}\uparrow} \ c_{-\boldsymbol{k}\downarrow}^\dagger \end{pmatrix}.$$ The spectrum consists of

$$E_{\boldsymbol{k}} = \sqrt{\xi_{\boldsymbol{k}}^2 + \bar{\Delta} \Delta}.$$

Exceptional points (EPs) arise in the momentum space when $$\xi_{\boldsymbol{k}} = \pm \operatorname{Im}\Delta_0$$, coinciding with $\operatorname{Re}\Delta_0 = 0$, rendering the mean-field Hamiltonian nondiagonalizable and coalescing the left and right eigenvectors. This signifies a spontaneous breaking of a discrete CP symmetry of the Hamiltonian. The occurrence of EPs controls the unconventional nature of phase transitions in the NH regime.

4. Reentrant Superfluidity and Quantum Zeno Effect

One of the central results is the non-monotonic, "reentrant" phase structure in the dependence of the superfluid order parameter on dissipation γ for weak attractive interactions. As γ increases:

• For small γ, $\operatorname{Re}\Delta_0$ decreases—dissipation suppresses pair coherence.
• At a critical γ, the gap vanishes and exceptional points appear; the system transitions to a non-superfluid phase.
• For even larger γ, the gap reopens and $\operatorname{Re}\Delta_0$ increases—a reentrant superfluid phase emerges.

This phenomenon is explained by the quantum Zeno effect: strong two-body loss localizes fermions, inhibiting tunneling and promoting on-site molecular pairing, thus stabilizing the superfluid state even under strong dissipation. In the strong-coupling limit, the gap approaches

$\Delta_0 \to U/2 \quad (\textrm{large}~\gamma),$

demonstrating dissipation-enhanced superfluidity within the BCS–BEC crossover.

5. NH Order Parameter Structure and Implications

The distinction between left- and right-eigenstate-based order parameters (Δ, $\bar{\Delta}$) introduces new physical consequences:

• The superfluid gap, determining the excitation spectrum, is governed by $$E_{\boldsymbol{k}} = \sqrt{ \xi_{\boldsymbol{k}}^2 + \bar{\Delta}\Delta }$$.
• The system does not conserve probability in the usual sense; the gap equation and observables must be computed using the biorthogonal structure.
• Gauge choices may permit $\bar{\Delta} = \Delta$ if a residual CP symmetry is enforced, but in general, inequivalence persists.

These features imply not only altered excitation lifetimes and spectral broadening but also fundamentally richer phase transition phenomenology (first/second order dichotomy is replaced by the appearance, annihilation, or reconnection of exceptional manifolds).

6. Connections to Open Quantum Systems and Experimental Realization

The non-Hermitian superfluid model offers a microscopically controlled platform to investigate open-system quantum many-body physics:

• The effective NH Hamiltonian arises naturally in ultracold atomic physics via engineered two-body losses (e.g., photoassociation in optical lattices).
• Dissipation-driven phase transitions can be probed in optical lattices with single-site resolution and projective postselection of quantum trajectories (null jump subensemble).
• Theoretical predictions—the reentrant phase diagram, exceptional point-induced transitions, and dissipation-enhanced superfluidity—provide concrete experimental observables.

Moreover, the NH superfluid model establishes a rigorous foundation for exploring non-equilibrium quantum phases, open-system criticality beyond the Hermitian universality classes, and the interplay between dissipation, topology, and many-body order.

7. Salient Mathematical Relations

Key equations defining the NH superfluid model include:

• Complex-valued interaction:

$U = U_1 + i\frac{\gamma}{2}$

• Gap equation (zero temperature):

$$\frac{N}{U} = \sum_{\boldsymbol{k}} \frac{1}{2 E_{\boldsymbol{k}}}, \quad E_{\boldsymbol{k}} = \sqrt{ \xi_{\boldsymbol{k}}^2 + \bar{\Delta}\Delta }$$

• Order parameters:

$$\Delta = -\frac{U}{N}\sum_{\boldsymbol{k}} {}_L\langle c_{-\boldsymbol{k} \downarrow} c_{\boldsymbol{k} \uparrow} \rangle_R, \quad \bar{\Delta} = -\frac{U}{N}\sum_{\boldsymbol{k}} {}_L\langle c_{\boldsymbol{k} \uparrow}^\dagger c_{-\boldsymbol{k}\downarrow}^\dagger \rangle_R$$

• Mean-field matrix:

$$H_{\mathrm{MF}}(\boldsymbol{k}) = \begin{pmatrix} \xi_{\boldsymbol{k}} & \Delta \ \bar{\Delta} & -\xi_{\boldsymbol{k}} \end{pmatrix}$$

• Exceptional point condition:

$$\xi_{\boldsymbol{k}} = \pm \operatorname{Im}\Delta_0 \quad (\operatorname{Re}\Delta_0 = 0)$$

These encapsulate the distinctive NH corrections to BCS theory and form the analytical basis for describing non-Hermitian fermionic superfluidity.

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In summary, the non-Hermitian superfluid model introduces complex-valued interactions into the BCS Hamiltonian to describe dissipative, open-system quantum dynamics of paired fermions. Its analytic structure features biorthogonal order parameters, exceptional points, and reentrant superfluidity, linked via the quantum Zeno effect to the stabilization of pairing under strong loss. The framework is directly relevant to ultracold atom experiments and provides a robust foundation for exploring non-equilibrium phase transitions in open quantum matter (Yamamoto et al., 2019).