Fulde−Ferrell−Larkin−Ovchinnikov states in two-dimensional non-Hermitian fermionic superfluidity

Abstract

We investigate a two-dimensional non-Hermitian Fermi system with spin imbalance and dissipative interaction. We employ the mean-field approximation to derive and minimize the thermodynamic potential, and determine the zero-temperature phase diagram upon variation of imbalance and dissipation. Notably, we identify a Fulde−Ferrell−Larkin−Ovchinnikov (FFLO) state characterized by a pairing order parameter with finite center-of-mass momentum, along with the normal, Bardeen−Cooper−Schrieffer (BCS), and metastable BCS states. The FFLO state is favored with increasing spin imbalance but diminishes and eventually disappears with elevated dissipation. Additionally, we present the quasiparticle energy spectrum and momentum distribution, highlighting the spatial asymmetry features that can be used to deterministically identify the FFLO state.

Graphical abstract

Keywords

non-Hermitian systems / superfluids / ultracold gases / Fulde−Ferrell−Larkin−Ovchinnikov (FFLO) states

1 Introduction

In recent years, non-Hermitian physics has emerged as an important extension of quantum mechanics, providing novel insights into open quantum systems and non-equilibrium dynamics [1, 2]. Unlike Hermitian systems, which have strictly real energy spectrum due to conservation laws, non-Hermitian systems incorporate environmental dissipation, resulting in complex eigenvalues [3, 4]. These systems are described by the Lindblad equation rather than the Schrödinger equation to account for gain and loss [5-7]. The distinctive properties of non-Hermitian systems have sparked great interest, leading to extensive theoretical and experimental research on quantum phenomena [8-10], particularly in optical [11-13] and atomic systems [14]. Beyond these platforms, recent efforts have also explored non-Hermitian topological phases and superconductivity [15, 16], while quantum simulators and circuit implementations have demonstrated controllable routes to probe their dissipative and topological properties [17, 18]. However, despite these advances, most studies have focused on single-particle physics, leaving the exploration of many-body non-Hermitian systems relatively underdeveloped. Among the most intriguing phenomena in many-body quantum physics is fermionic superfluidity [19-27]. In the realm of ultracold atomic systems, non-Hermitian Fermi superfluidity with complex-valued interaction represents a rapidly growing field of research, warranting thorough investigation [28-35].

Incorporating non-Hermitian effects into Fermi systems provides a new path to explore exotic pairing mechanisms. Of particular interest is the Fulde−Ferrell−Larkin−Ovchinnikov (FFLO) state [36-38], initially proposed in the context of superconductivity under external magnetic fields, and featured by unmatched Fermi surfaces to accommodate spin imbalance. In contrast to the conventional Bardeen−Cooper−Schrieffer (BCS) pairing, where Cooper pairs possess zero center-of-mass momentum, the FFLO phase is distinguished by finite-momentum pairing and a spatially modulated superconducting order parameter [39, 40]. Although the FFLO state has been extensively studied in a variety of physical systems, the realization and convincible detection of such a state remains challenging due to stringent experimental conditions and stability issues [40, 41]. Theoretical predictions for the FFLO state in ultracold atomic gases have been extensively investigated in different dimensions [42] using either continuum [43-45] or lattice [46-51] models. The experimental observation of the FFLO state is even more challenging. It typically requires sub-micron spatial resolution, ultralow temperatures, and high magnetic fields [52, 53]. Recent advances in local probe techniques, such as scanning tunneling microscopy, quantum spin sensors [54], and cold-atom imaging [55, 56], are helping to overcome these barriers. At the same time, quantum simulation platforms like optical lattices and tweezer arrays offer clean and highly tunable environments to emulate FFLO physics. The study in quasi-one-dimensional arrays of tubes also paves the way to direct observation and characterization of FFLO pairing, with density profiles consistent with partially polarized superfluids and offering evidence of FFLO-like behavior [57]. However, these previous studies have predominantly focused on Hermitian systems.

In this work, we investigate the existence and stability of FFLO phases in a two-dimensional (2D) non-Hermitian Fermi system on a square lattice, focusing on the effect of dissipative interactions on the mechanism and properties of pairing states. Considering a complex-valued interaction and population imbalance, we generalize the extended BCS mean-field theory to non-Hermitian system, and map out the zero-temperature phase diagram by minimizing the mean-field thermodynamic potential. Compared to Ref. [35], which is focused on BCS-type superfluid states formed by pairing two fermions with opposite momenta and zero center-of-mass momentum, this work goes beyond the conventional framework by allowing for pairing states with finite center-of-mass momentum, thereby enabling the emergence of FFLO states in a dissipative environment, existing within a confined parameter space. This allows us to address a distinct set of physical phenomena and stability questions that are beyond the scope [35]. The FFLO range expands with increasing spin imbalance, but shrinks and eventually disappears as two-body loss rates increase, given fixed spin imbalance and interaction strength. Furthermore, we present the quasiparticle energy spectra and momentum distribution of particle number. The spatial asymmetry becomes particularly pronounced compared to the conventional BCS state with zero center-of-mass momentum, and can be used to characterize the FFLO state.

The remainder of this paper is organized as follows. In Section 2, we extend the BCS mean-field theory to non-Hermitian Hamiltonian systems and derive the expression for the mean-field thermodynamic potential and saddle point equations. In Section 3, we analyze the order parameters, center-of-mass momentum, and energies to identify distinct phases, followed by constructing the zero-temperature phase diagram. In Section 4, we present the quasiparticle energy spectra and momentum distribution, highlighting their asymmetric behaviors. Finally, we summarize our findings in Section 5.

2 Hamiltonian and mean-field method

We consider a 2D spin-imbalanced Hubbard model with a complex-valued interaction, which describes particle loss due to inelastic collisions. This non-Hermitian framework is valid within a timescale significantly shorter than the inverse loss rate (\tau \ll {\gamma }^{-1}), during which the quantum jump contributions become negligible [7, 34, 58]. For stronger dissipation, beyond-mean-field corrections become relevant. Possible approaches include Lindblad master-equation formulations and quantum trajectory methods for open many-body systems [6], as well as fluctuation-based extensions of the mean-field framework in non-Hermitian settings [8]. When the characteristic timescale associated with the energy gap \mathrm{\Delta } (\hslash /\mathrm{\Delta }) is also within the dissipation timescale {\gamma }^{-1}, the system can reach equilibrium fast enough such that one can treat the system as a quasi-equilibrium state. Besides, the quasiparticles are interpreted within the framework of the commonly adopted local thermal equilibrium assumption, which is consistent with the standard treatments of BCS and FFLO physics [40, 59, 60]. On this basis, we then employ the effective non-Hermitian Hamiltonian,

(1)\begin{array}{rl}\hat{H}=& \sum _{\mathit{k}\sigma }{\xi }_{\mathit{k},\sigma }{c}_{\mathit{k},\sigma }^{†}{c}_{\mathit{k},\sigma }\\ & -\frac{U}{V}\sum _{\mathit{k}{\mathit{k}}^{\prime }\mathit{q}}{c}_{\mathit{k}+\mathit{q}/2,\uparrow }^{†}{c}_{-\mathit{k}+\mathit{q}/2,\downarrow }^{†}{c}_{-{\mathit{k}}^{\prime }+\mathit{q}/2,\downarrow }{c}_{{\mathit{k}}^{\prime }+\mathit{q}/2,\uparrow }.\end{array}

Here, c_{\mathit{k},\sigma } and c_{\mathit{k},\sigma }^{†} are annihilation and creation operators of fermions. The 2D dispersion relation is described as {\epsilon }_{\mathit{k}}=-2t[\mathrm{c}\mathrm{o}\mathrm{s}(k_{x}d)+\mathrm{c}\mathrm{o}\mathrm{s}(k_{y}d)] with the hopping strength t and lattice constant d, which is shifted by spin-dependent chemical potential {\mu }_{\sigma } as {\xi }_{\mathit{k}\sigma }={\epsilon }_{\mathit{k}}-{\mu }_{\sigma }. For convenience, we define \delta \mu ={\mu }_{\uparrow }-{\mu }_{\downarrow } and \overline{\mu }=({\mu }_{\uparrow }+{\mu }_{\downarrow })/2 as the difference and average of chemical potential of the two spin states. Using this notation, a half filled system corresponds to the case of \overline{\mu }=0. The complex interaction is expressed as U=U_1+\mathrm{i}\frac{\gamma }{2}, which is composed of on-site interaction U_1 and the two-body loss rate \gamma, and V is the quantization volume. The interaction strength U_1 can be tuned continuously by means of magnetic or orbital Feshbach resonances [61-64], while the two-body loss rate \gamma can be independently controlled via photoassociation coupling to short-lived molecular states [65]. This dual controllability allows one to realize a complex-valued interaction parameter U=U_1+\mathrm{i}\frac{\gamma }{2}, with high precision, thereby providing a clean and versatile setting to explore non-Hermitian superfluidity.

For a non-Hermitian system, the left and right BCS wave functions defined respectively as H{\overline{\alpha }}_{\mathit{k}\sigma }|\mathrm{B}\mathrm{C}\mathrm{S}{⟩}_R=E_{\mathit{k}}{\overline{\alpha }}_{\mathit{k}\sigma }|\mathrm{B}\mathrm{C}\mathrm{S}{⟩}_R and H^{†}{\alpha }_{\mathit{k}\sigma }^{†}|\mathrm{B}\mathrm{C}\mathrm{S}{⟩}_L=E_{\mathit{k}}^{\ast }{\alpha }_{\mathit{k}\sigma }^{†}|\mathrm{B}\mathrm{C}\mathrm{S}{⟩}_L, are not typically Hermitian conjugate to each other, but related by the bi-orthogonal condition {}_L⟨{\mathrm{B}\mathrm{C}\mathrm{S}}_m|{\mathrm{B}\mathrm{C}\mathrm{S}}_n{⟩}_R={\delta }_{m,n} [66], where {\overline{\alpha }}_{\mathit{k}\sigma } and {\alpha }_{\mathit{k}\sigma }^{†} create the right and left eigenstates. Therefore, the expectation value of an operator needs to be calculated by the joint left and right wave functions, and the extended BCS theory for such a non-Hermitian system defines a mean-field pairing order parameter with center-of-mass momentum \mathit{q} as [7, 34, 35]

(2)\begin{array}{rl}\mathrm{\Delta }=& -\frac{U}{V}\sum _{\mathit{k}}{}_L⟨c_{-\mathit{k}+\mathit{q}/2,\downarrow }{c}_{\mathit{k}+\mathit{q}/2,\uparrow }{⟩}_R={\mathrm{\Delta }}_0{\mathrm{e}}^{\mathrm{i}\theta },\\ \tilde{\mathrm{\Delta }}=& -\frac{U}{V}\sum _{\mathit{k}}{}_L⟨c_{\mathit{k}+\mathit{q}/2,\uparrow }^{†}{c}_{-\mathit{k}+\mathit{q}/2,\downarrow }^{†}{⟩}_R={\mathrm{\Delta }}_0{\mathrm{e}}^{-\mathrm{i}\theta }.\end{array}

Notably, \mathrm{\Delta } and \tilde{\mathrm{\Delta }} are in general not complex conjugate, since {\mathrm{\Delta }}_0 is already a complex number and \theta is the U(1) phase.

We then obtain the mean-field Hamiltonian as

(3)\begin{array}{rl}{\hat{H}}_{\mathrm{M}\mathrm{F}}=& \frac{V}{U}{\mathrm{\Delta }}_0^2+\sum _{\mathit{k}}({\xi }_{\mathit{k}+\mathit{q}/2,\uparrow }{c}_{\mathit{k}+\mathit{q}/2,\uparrow }^{†}{c}_{\mathit{k}+\mathit{q}/2,\uparrow }\\ & +{\xi }_{-\mathit{k}+\mathit{q}/2,\downarrow }{c}_{-\mathit{k}+\mathit{q}/2,\downarrow }^{†}{c}_{-\mathit{k}+\mathit{q}/2,\downarrow })\\ & +\sum _{\mathit{k}}\left(\mathrm{\Delta }{c}_{\mathit{k}+\mathit{q}/2,\uparrow }^{†}{c}_{-\mathit{k}+\mathit{q}/2,\downarrow }^{†}+\tilde{\mathrm{\Delta }}{c}_{-\mathit{k}+\mathit{q}/2,\downarrow }{c}_{\mathit{k}+\mathit{q}/2,\uparrow }\right).\end{array}

This effective Hamiltonian is now quadratic and can be diagonalized exactly via the standard Bogoliubov transformation as

(4)\begin{array}{rl}{\hat{H}}_{\mathrm{M}\mathrm{F}}=& \sum _{\mathit{k}}(E_{\mathit{k}\mathit{q},+}{\overline{\alpha }}_{\mathit{k}+\mathit{q}/2,\uparrow }{\alpha }_{\mathit{k}+\mathit{q}/2,\uparrow }\\ & -E_{\mathit{k}\mathit{q},-}{\overline{\alpha }}_{-\mathit{k}+\mathit{q}/2,\downarrow }{\alpha }_{-\mathit{k}+\mathit{q}/2,\downarrow })\\ & +\sum _{\mathit{k}}\left({\xi }_{-\mathit{k}+\mathit{q}/2,\downarrow }+E_{\mathit{k}\mathit{q},-}\right)+\frac{V}{U}{\mathrm{\Delta }}_0^2,\end{array}

where

(5)\left(\begin{array}{c}{\alpha }_{\mathit{k}+\mathit{q}/2,\uparrow }\\ {\overline{\alpha }}_{-\mathit{k}+\mathit{q}/2,\downarrow }\end{array}\right)=\left(\begin{array}{cc}{u}_{\mathit{k}\mathit{q}}& -v_{\mathit{k}\mathit{q}}\\ {\overline{v}}_{\mathit{k}\mathit{q}}& u_{\mathit{k}\mathit{q}}\end{array}\right)\left(\begin{array}{c}{c}_{\mathit{k}+\mathit{q}/2,\uparrow }\\ c_{-\mathit{k}+\mathit{q}/2,\downarrow }^{†}\end{array}\right).

Here, {\alpha }_{\mathit{k}\sigma } and {\overline{\alpha }}_{\mathit{k}\sigma } are quasiparticle operators, and the transformation coefficients are

(6)\begin{array}{rl}{u}_{\mathit{k}\mathit{q}}=& \sqrt{\frac{E_{\mathit{k}\mathit{q}}+{\xi }_{\mathit{q},+}}{2E_{\mathit{k}\mathit{q}}}},\\ v_{\mathit{k}\mathit{q}}=& \sqrt{\frac{E_{\mathit{k}\mathit{q}}-{\xi }_{\mathit{q},+}}{2E_{\mathit{k}\mathit{q}}}}\sqrt{\frac{\mathrm{\Delta }}{\tilde{\mathrm{\Delta }}}},\\ {\overline{v}}_{\mathit{k}\mathit{q}}=& \sqrt{\frac{E_{\mathit{k}\mathit{q}}-{\xi }_{\mathit{q},+}}{2E_{\mathit{k}\mathit{q}}}}\sqrt{\frac{\tilde{\mathrm{\Delta }}}{\mathrm{\Delta }}},\end{array}

in which we further define E_{\mathit{k}\mathit{q}}=\sqrt{{\xi }_{\mathit{q},+}^2+{\mathrm{\Delta }}_0^2} and {\xi }_{\mathit{q},+}=({\xi }_{\mathit{k}+\mathit{q}/2,\uparrow }+{\xi }_{-\mathit{k}+\mathit{q}/2,\downarrow })/2, and the eigenenergies of the quasiparticles E_{\mathit{k}\mathit{q},\pm }=({\xi }_{\mathit{k}+\mathit{q}/2,\uparrow }-{\xi }_{-\mathit{k}+\mathit{q}/2,\downarrow })/2\pm E_{\mathit{k}\mathit{q}}.

Although these quasiparticles obey neither Fermi nor Bose statistics as {\alpha }_{\mathit{k}\sigma }^{†}\ne {\overline{\alpha }}_{\mathit{k}\sigma }, we note that the quasiparticle operators satisfy an anticommutation relation \{{\alpha }_{\mathit{k}\sigma },{\overline{\alpha }}_{{\mathit{k}}^{\prime }{\sigma }^{\prime }}\}={\delta }_{{\mathit{k}\mathit{k}}^{\prime }}{\delta }_{\sigma {\sigma }^{\prime }}, resulting in {\overline{\alpha }}_{\mathit{k}\sigma }{\alpha }_{\mathit{k}\sigma } = 0 or 1 [7, 34]. We then obtain the mean-field grand partition function {\mathcal{Z}}_{\mathrm{M}\mathrm{F}} from its relation to the effective action {\mathcal{S}}_{\mathrm{e}\mathrm{f}\mathrm{f}} as

(7)\begin{array}{rl}{\mathcal{Z}}_{\mathrm{M}\mathrm{F}}=& {\mathrm{e}}^{-{\mathcal{S}}_{\mathrm{e}\mathrm{f}\mathrm{f}}(\tilde{\mathrm{\Delta }},\mathrm{\Delta })}\\ =& \prod _{\mathit{k}}[(1+{\mathrm{e}}^{-\beta E_{\mathit{k}\mathit{q},+}})(1+{\mathrm{e}}^{\beta E_{\mathit{k}\mathit{q},-}})\\ & \cdot {\mathrm{e}}^{-\beta ({\xi }_{-\mathit{k}+\mathit{q}/2,\downarrow }+E_{\mathit{k}\mathit{q},-})}]{\mathrm{e}}^{-\beta \frac{V}{U}{\mathrm{\Delta }}_0^2},\end{array}

where \beta =1/(k_{B}T) is the inverse temperature. Using the expression {\mathrm{\Omega }}_{\mathrm{M}\mathrm{F}}=-\frac{1}{\beta }\mathrm{l}\mathrm{n}{\mathcal{Z}}_{\mathrm{M}\mathrm{F}}, we finally reach the thermodynamic potential

(8)\begin{array}{rl}{\mathrm{\Omega }}_{\mathrm{M}\mathrm{F}}=& -\frac{1}{\beta }\sum _{\mathit{k}}\mathrm{l}\mathrm{n}(1+{\mathrm{e}}^{-\beta E_{\mathit{k}\mathit{q},+}})(1+{\mathrm{e}}^{\beta E_{\mathit{k}\mathit{q},-}})\\ & +\sum _{\mathit{k}}({\xi }_{-\mathit{k}+\mathit{q}/2,\downarrow }+E_{\mathit{k}\mathit{q},-})+\frac{V}{U}{\mathrm{\Delta }}_0^2.\end{array}

For a given set of parameters, the ground state of this system can be determined by minimizing the thermodynamic potential or self-consistently solving the saddle-point equations. In the numerical calculation below, we use lattice constant d and hopping parameter t as length and energy units, respectively.

3 FFLO state and zero-temperature phase diagram

We consider a system at half-filling with \overline{\mu }=0 but a non-zero \delta \mu to introduce spin-imbalance. At zero temperature, we obtain the simplified thermodynamic potential as

(9)\begin{array}{rl}{\mathrm{\Omega }}_{\mathrm{M}\mathrm{F}}=& \sum _{\mathit{k}}[E_{\mathit{k}q,+}\mathrm{\Theta }(-E_{\mathit{k}q,+})-E_{\mathit{k}q,-}\mathrm{\Theta }(E_{\mathit{k}q,-})\\ & +{\xi }_{-\mathit{k}+\mathit{q}/2,\downarrow }+E_{\mathit{k}\mathit{q},-}]+\frac{V}{U}{\mathrm{\Delta }}_0^2,\end{array}

in which the Fermi function is replaced by the Heaviside step function. The discussion can be easily generalized to other filling with \overline{\mu }\ne 0, where the energy bands for spin-up and spin-down particles shift upward or downward simultaneously, with corresponding changes of {\xi }_{\uparrow ,\downarrow }. Meanwhile, an extension to a finite temperature must be cautious, although the mean-field thermodynamic potential Eq. (8) is obtained by assuming a finite \beta. This is because at a finite temperature, the system is no longer restricted to the ground state and may involve a significant occupation of excited states. A key question is whether these excited states can still be effectively captured by the non-Hermitian Hamiltonian. Thus, our discussion should be valid at a temperature much lower than the energy gap so that the system is well described by its ground state.

Under a symmetry analysis, the direction of center-of-mass momentum can be confined between the \mathrm{\Gamma }−X direction and \mathrm{\Gamma }−M direction. A detailed numerical analysis shows that the state with \mathit{q}\parallel \mathrm{\Gamma }−X is always energetically favorable. Thus, in the following discussion we set \mathit{q}=(q_x=q,q_y=0), which leads to {\epsilon }_{\mathit{k}+\mathit{q}/\mathbf{2}}=-2t[\cos (k_x+q/2)+\cos (k_y)] and {\epsilon }_{\mathbf{-}\mathit{k}+\mathit{q}/\mathbf{2}}=-2t[\cos (-k_x+q/2)+\cos (-k_y)]. From the stationary condition \mathrm{\partial }{S}_{\mathrm{e}\mathrm{f}\mathrm{f}}/\mathrm{\partial }{\mathrm{\Delta }}_0=0 and \mathrm{\partial }{S}_{\mathrm{e}\mathrm{f}\mathrm{f}}/\mathrm{\partial }q=0, we obtain the saddle point equations for {\mathrm{\Delta }}_0 and q,

(10a)\begin{array}{c}{\displaystyle \frac{V}{U}}={\displaystyle \sum _{\mathit{k}}{\displaystyle \frac{1}{2E_{\mathit{k}q}}}\left[\mathrm{\Theta }(-E_{\mathit{k}q,-})-\mathrm{\Theta }(-E_{\mathit{k}q,+})\right],}\end{array}

(10b)\begin{array}{c}\begin{array}{rl}& \sum _{\mathit{k}}\{[\sin (k_x+q_x/2)+\sin (k_x-q_x/2)]\\ & [\mathrm{\Theta }(-E_{\mathit{k}q,+})+\mathrm{\Theta }(-E_{\mathit{k}q,-})]\\ & +\frac{{\xi }_{\mathit{q},+}}{E_{\mathit{k}q,+}}[\sin (k_x+q_x/2)-\sin (k_x-q_x/2)]\\ & \cdot [\mathrm{\Theta }(-E_{\mathit{k}q,+})-\mathrm{\Theta }(-E_{\mathit{k}q,-})]-2\sin (k_x-q_x/2)\}=0,\end{array}\end{array}

which can be self-consistently solved to obtain all stationary solutions.

In order to determine the full phase diagram including both ground state and possible metastable state, it is necessary to evaluate the thermodynamic potential {\mathrm{\Omega }}_{\mathrm{M}\mathrm{F}} as well. In Fig. 1, we present two contour plots of the real part of thermodynamic potential \mathrm{R}\mathrm{e}({\mathrm{\Omega }}_{\mathrm{M}\mathrm{F}}) as a function of the real part of order parameter Re({\mathrm{\Delta }}_0) and wave vector q for different parameter sets \gamma /t=1,\delta \mu /t=2,U_1/t=4.2 and \gamma /t=1,\delta \mu /t=4,U_1/t=6.6. The imaginary part of the order parameter is obtained via gap equation Eq. (10a). The thermodynamic potential is symmetric at q=\pi due to the periodicity of dispersion relation {\epsilon }_{\pm \mathit{k}+\mathit{q}/\mathbf{2}}. There are two local minima in both contours, including the FFLO state with finite q and {\mathrm{\Delta }}_0 as labeled by the red dot, and a BCS state with q=0 and {\mathrm{\Delta }}_0\ne 0 as depicted by the green square. To further distinguish different phases, we introduce two energy differences from normal state \mathrm{\Delta }{\mathrm{\Omega }}_q={\mathrm{\Omega }}_{\mathrm{M}\mathrm{F}}({\mathrm{\Delta }}_0,q)-{\mathrm{\Omega }}_{\mathrm{M}\mathrm{F}}(0,0) and \mathrm{\Delta }{\mathrm{\Omega }}_0={\mathrm{\Omega }}_{\mathrm{M}\mathrm{F}}({\mathrm{\Delta }}_0,0)-{\mathrm{\Omega }}_{\mathrm{M}\mathrm{F}}(0,0), representing the condensation energy of the FFLO and the BCS states, respectively. While both {\mathrm{\Delta }}_0 and q are finite values and , the ground state of the system stays at FFLO state; the case of {\mathrm{\Delta }}_0\ne 0 but q=0 represents the ground BCS state for , and conversely \mathrm{R}\mathrm{e}(\mathrm{\Delta }{\mathrm{\Omega }}_0)>0 corresponds to the metastable BCS state, where the true ground state is a normal state labeled as N(BCS). When {\mathrm{\Delta }}_0 and q are both zero, the system is the normal state. In both cases shown in Fig. 1, we have and \mathrm{R}\mathrm{e}(\mathrm{\Delta }{\mathrm{\Omega }}_0)>0, indicating that the FFLO state is the ground state and the BCS state is only metastable (FFLO(BCS)).

With this classification criteria, we map out the phase diagrams for two representative chemical potential differences \delta \mu /t=2 and 4 on the U_1−\gamma plane in Fig. 2. For a sufficiently strong attractive interaction, the ground state predominantly stabilizes as a BCS state where the fermions form tightly bound molecules. In the intermediate interaction region, we observe a stable FFLO phase for a sufficiently weak two-body loss, which will transit to a metastable BCS state with increasing \gamma. The mechanism behind this transition is that the FFLO pair becomes unstable with increasing dissipation due to the enhanced inelastic particle collisions, and gives its way to the metastable BCS state with on-site molecular pairing which inhibit two-body losses. By comparing Figs. 2(a) and (b), we notice that the system requires larger interaction strength U_1 to enter the metastable BCS and BCS phases with increasing \delta \mu. Meanwhile, an enhanced chemical potential difference \delta \mu promotes the formation of FFLO pairs, resulting in an expanded FFLO phase. This suggests that with increasing population imbalance, the FFLO state gradually becomes more favorable than the conventional BCS state.

In order to better characterize the phase transitions, we present the order parameter {\mathrm{\Delta }}_0/t and center-of-mass momentum q as functions of interaction strength U_1/t in Fig. 3, with a typical loss rate \gamma /t=1 indicated by the black dotted lines in Fig. 2. In Figs. 3(a) and (b), we show both the real and imaginary parts of the ground state order parameter by solid lines for (a) \delta \mu /t=2 and (b) \delta \mu /t=4. It is clear that the phase transitions between different states are of first order, characterizing by the sudden changes of the order parameter. The metastable BCS solutions are indicated by the dashed lines, which are smoothly connected to the BCS results with increasing interaction strength. Notably, the imaginary part of the BCS order parameter shows negligible response to the change of real part of interaction (U_1). It is worth mentioning that, an imbalanced Fermi gas with fixed total density and magnetization may lower its energy through phase separation, typically resulting in coexisting domains of the normal and BCS paired phases [48, 67-69]. In our analysis, we minimize the thermodynamic potential with respect to the order parameter and center-of-mass momentum, under fixed average chemical potential and chemical potential difference. This procedure avoids inconsistencies between the gap and number equations, which may otherwise occur in homogeneous systems with fixed density and magnetization at first-order transition point.

From Figs. 3(c) and (d), we find that a large chemical potential imbalance would lead to a large center-of-mass momentum for FFLO pairs. Specifically, for \delta \mu /t=4, the center-of-mass momentum is nearly a constant q=\pi /d for the entire FFLO region, indicating a special \pi-FFLO phase. When pairing occurs between fermions occupying different orbital bands, a special \pi-FFLO phase with center-of-mass momentum q=\pi /d can be stabilized in a large parameter regime [49, 70-72]. The emergence of such an exotic state is attributed to the nesting effect induced by the relative inversion of single-particle band structures of the two spin components [48, 73, 74]. As shown in Ref. [49], the \pi-FFLO phase presents near the regime \delta \mu /J=4, where the Fermi surfaces for the two spin species are close in the first Brillouin zone, which is consistent with our results.

Meanwhile, we plot the energy difference between ground state and the normal state \mathrm{\Delta }\mathrm{\Omega }={\mathrm{\Omega }}_{\mathrm{G}}-{\mathrm{\Omega }}_{\mathrm{N}} in Figs. 3(e) and (f). The background colors are used to distinguish different phases, with the inset showing a zoomed-in of the phase transition region. The metastable BCS state energy is given by the dashed lines, which also indicates a smooth energy change for the BCS state.

4 Quasiparticle spectrum and momentum distribution

In this section, we investigate the behavior of quasiparticle spectra E_{\mathit{k}q,\pm } and momentum distribution n_{\sigma }(\mathit{k}) for the FFLO state. For this purpose, we set \gamma /t=1,\delta \mu /t=2 and U_1/t=4.2 as a typical set of parameters in the FFLO regime. The corresponding order parameter and center-of-mass momentum are indicated by the red dot in Fig. 1(a). In Figs. 4(a) and (b), we show the real and imaginary parts of quasiparticle energy spectrum, which characterize the excitation energy and quasiparticle lifetime, respectively [35]. The real part Re(E_{\mathit{k}q,\pm }) shows a direct gap, which is similar to the BCS state. We then focus on E_{\mathit{k}q,+} as an example and show the distribution within the first Brillouin zone in Figs. 4(c) and (d). We observe that the center of Re(E_{\mathit{k}q,+}) is shifted along the x-axis due to the center-of-mass momentum \mathit{q}=(q,0), but the imaginary part is still symmetric. This anisotropy arises from the non-zero chemical potential difference that depends on the spin imbalance, which is a real number and therefore does not affect the imaginary part of the other parameter.

Compared to the BCS state with n_{\mathit{k},\uparrow }=n_{\mathit{k},\downarrow }, the momentum distribution n_{\sigma }(\mathit{k}) in such a spin-imbalanced system needs to be calculated by taking the partial derivative of the thermodynamic potential {\mathrm{\Omega }}_{\mathrm{M}\mathrm{F}} over {\mu }_{\uparrow } and {\mu }_{\downarrow } at zero temperature respectively, leading to

(11)\begin{array}{c}\begin{array}{rl}{n}_{\uparrow }& =u_k^2\mathrm{\Theta }(-E_{\mathit{k}q,+})+v_k^2\mathrm{\Theta }(-E_{\mathit{k}q,-}),\\ n_{\downarrow }& =u_k^2\mathrm{\Theta }(E_{\mathit{k}q,-})+v_k^2\mathrm{\Theta }(E_{\mathit{k}q,+}).\end{array}\end{array}

Here,

(12)\begin{array}{c}\begin{array}{rl}{u}_k^2& =\frac{1}{2}(1+\frac{{\xi }_{\mathit{k}+\mathit{q}/\mathbf{2},\uparrow }+{\xi }_{-\mathit{k}+\mathit{q}/\mathbf{2},\downarrow }}{2\sqrt{({\xi }_{\mathit{k}+\mathit{q}/\mathbf{2},\uparrow }+{\xi }_{-\mathit{k}+\mathit{q}/\mathbf{2},\downarrow }{)}^2}+{\mathrm{\Delta }}_0^2}),\\ v_k^2& =\frac{1}{2}(1-\frac{{\xi }_{\mathit{k}+\mathit{q}/\mathbf{2},\uparrow }+{\xi }_{-\mathit{k}+\mathit{q}/\mathbf{2},\downarrow }}{2\sqrt{({\xi }_{\mathit{k}+\mathit{q}/\mathbf{2},\uparrow }+{\xi }_{-\mathit{k}+\mathit{q}/\mathbf{2},\downarrow }{)}^2}+{\mathrm{\Delta }}_0^2}).\end{array}\end{array}

In Fig. 5, we show the momentum distribution of n_{\uparrow } and n_{\downarrow }, using the same parameters as in Fig. 4. We find that the momentum distribution exhibits the same properties as the energy spectrum, with an asymmetric distribution along the x-axis and a symmetric behavior in y-direction. Moreover, the quasiparticles are uniformly concentrated in the central region, and present an approximate Fermi−Dirac distribution at the edges. Although the real parts of momentum distribution for spin-up and spin-down components in Figs. 5(a) and (c) show obvious differences, their characteristic behaviors are consistent with the above description. The reason why their imaginary parts are the same is that the chemical potential affected by the spin imbalance is a real number and therefore does not change the properties of the imaginary parts, as depicted in Figs. 5(b) and (d).

We emphasize that the complex-valued nature of momentum distribution arises from the non-Hermiticity of Hamiltonian, but particle number should always remain as a real-valued observable, in this case n=\sum _k\mathrm{R}\mathrm{e}(n(k)). In certain non-Hermitian topological systems, the imaginary part is related to the non-Hermitian skin effect, characterizing the directional asymmetry of state accumulation. In momentum space, non-Hermitian dynamics lead to preferential occupation or depletion at specific momentum, manifesting in real space as localization or unidirectional accumulation of particles [15, 16, 75]. Experimentally, the radio-frequency spectroscopy has been widely used to measure both real and imaginary parts of complex energy spectra of a non-Hermitian quantum system for either bosonic or fermionic atoms [76]. In terms of realistic parameter regimes, spin imbalance is often quantified by the polarization P=\frac{N_{\uparrow }-N_{\downarrow }}{N_{\uparrow }+N_{\downarrow }}, with a wide range P=0-0.7 realized in ultracold Fermi gas experiments [77], while recent theoretical and simulated experimental analyses suggest that the effective two-body loss rate \gamma /t can be tuned from 0 to 10 [78]. Nevertheless, the precise parameter regimes in non-Hermitian cold-atom setups still call for further experimental exploration.

5 Conclusion

We investigate the emergence of Fulde−Ferrell−Larkin−Ovchinnikov (FFLO) states in a 2D non-Hermitian fermionic superfluid on a square lattice, focusing on the effect of complex-valued interaction and spin imbalance. By extending Bardeen−Cooper−Schrieffer (BCS) mean-field theory to non-Hermitian Hamiltonian, we derive the mean-field thermodynamic potential and corresponding saddle point equations. To ensure physical relevance and stability, we solve these equations self-consistently while minimizing the energy to classify different phases and map out a zero-temperature phase diagram. Our analysis identifies critical parameters that govern FFLO states with finite center-of-mass momentum. These states are enhanced by increasing spin imbalance but suppressed by dissipation. The results underscore the pronounced dependence of FFLO states on both population imbalance and dissipative effects, providing valuable insights into pairing physics in non-Hermitian systems. Additionally, we present the quasiparticle spectra and momentum distribution for different spin components, revealing spatial asymmetries distinct from those of conventional BCS superfluidity. Compared with Hermitian systems, the non-Hermitian FFLO state exhibits an asymmetric structure not only in the real part of the quasiparticle spectrum and momentum distribution, but also in the imaginary parts of momentum distributions. These imaginary contributions imply finite lifetimes and dissipative processes, which are expected to affect dynamical properties such as transport and superfluid responses. While a detailed analysis of these aspects is beyond the scope of the present work, our results provide clear indications that the non-Hermitian character may lead to qualitatively new dynamical signatures, offering potential directions for future experimental and theoretical studies.

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