PT-Symmetric Non-Hermitian Superconductor

• PT-symmetric non-Hermitian superconductors are defined by a Bogoliubov–de Gennes Hamiltonian that, despite being non-Hermitian, maintains real quasiparticle spectra through an antiunitary symmetry.
• They exhibit distinctive real-to-complex spectral transitions, exceptional degeneracies, and modified thermodynamics that result in first-order superconducting transitions and unique transport phenomena.
• Practical realizations include lattice models with balanced gain and loss and one-dimensional Kitaev chains, which reveal non-Bloch topologies, unconventional Majorana modes, and novel Andreev physics.

A PT-symmetric non-Hermitian superconductor is a superconducting system whose effective Bogoliubov–de Gennes Hamiltonian is not Hermitian, $H\neq H^\dagger$, yet is constrained by an antiunitary symmetry so that part of its quasiparticle spectrum remains real. In the literature this appears in closely related forms: explicitly PT-symmetric mean-field superconductors with anti-Hermitian pairing, CP-symmetric non-Hermitian BdG systems that are topologically equivalent to PT-symmetric ones under the mapping $H\to iH$, and lattice superconductors with balanced gain and loss. Across these formulations, the defining consequences are real-to-complex spectral transitions, exceptional degeneracies of codimension one, modified thermodynamics, non-Bloch skin phenomena, and unconventional Majorana and Andreev physics (Ghatak et al., 2017, Okugawa et al., 2018, Liu et al., 3 Sep 2025).

1. Symmetry framework and defining Hamiltonians

In the single-band mean-field construction, the superconducting block can be written in Nambu space as

$$\mathbf{h}_{\mathbf{k}}= \begin{pmatrix} \varepsilon_{\mathbf{k}} & \Delta_{\mathbf{k}}\ \tilde{\Delta}_{\mathbf{k}} & -\varepsilon_{\mathbf{k}} \end{pmatrix},$$

with $\varepsilon_{\mathbf{k}}\in\mathbb{R}$. PT symmetry is implemented by $\mathcal P=\sigma_z$, $\mathcal T=\mathcal K$, so that $[\mathcal{PT},\mathbf h_{\mathbf k}]=0$ requires $$\sigma_z\mathbf h_{\mathbf k}^*\sigma_z=\mathbf h_{\mathbf k}$$. Real quasiparticle energies are then possible when the order parameter is either Hermitian, $$\tilde{\Delta}_{\mathbf{k}}=\Delta_{\mathbf{k}}^\dagger$$, or anti-Hermitian, $$\tilde{\Delta}_{\mathbf{k}}=-\Delta_{\mathbf{k}}^\dagger$$; the latter defines the non-Hermitian PT-symmetric superconducting state in that formulation (Ghatak et al., 2017).

In BdG systems, particle-hole symmetry is intrinsic, so many non-Hermitian superconducting realizations are most naturally expressed through CP symmetry rather than explicit PT symmetry. For even-parity pairing with $H\to iH$0, the $H\to iH$1 BdG Hamiltonian block-diagonalizes into

$H\to iH$2

with $H\to iH$3 in a suitable basis and $H\to iH$4. Under the mapping $H\to iH$5, these CP-symmetric BdG systems have the same exceptional-surface topology as PT-symmetric non-Hermitian systems (Okugawa et al., 2018).

A complementary microscopic route introduces non-Hermiticity through balanced gain and loss. On a honeycomb lattice with on-site $H\to iH$6-wave pairing, staggered imaginary onsite potentials $H\to iH$7 and $H\to iH$8 on the two sublattices yield a PT-symmetric non-Hermitian superconductor with $H\to iH$9, where parity exchanges sublattices and time reversal complex conjugates $$\mathbf{h}_{\mathbf{k}}= \begin{pmatrix} \varepsilon_{\mathbf{k}} & \Delta_{\mathbf{k}}\ \tilde{\Delta}_{\mathbf{k}} & -\varepsilon_{\mathbf{k}} \end{pmatrix},$$0 (Liu et al., 3 Sep 2025).

2. Quasiparticle spectrum, PT-unbroken sector, and PT-broken sector

For the anti-Hermitian pairing state, the quasiparticle spectrum is

$$\mathbf{h}_{\mathbf{k}}= \begin{pmatrix} \varepsilon_{\mathbf{k}} & \Delta_{\mathbf{k}}\ \tilde{\Delta}_{\mathbf{k}} & -\varepsilon_{\mathbf{k}} \end{pmatrix},$$1

whereas the Hermitian counterpart has $$\mathbf{h}_{\mathbf{k}}= \begin{pmatrix} \varepsilon_{\mathbf{k}} & \Delta_{\mathbf{k}}\ \tilde{\Delta}_{\mathbf{k}} & -\varepsilon_{\mathbf{k}} \end{pmatrix},$$2. The non-Hermitian state therefore has a PT-unbroken or “paired” region $$\mathbf{h}_{\mathbf{k}}= \begin{pmatrix} \varepsilon_{\mathbf{k}} & \Delta_{\mathbf{k}}\ \tilde{\Delta}_{\mathbf{k}} & -\varepsilon_{\mathbf{k}} \end{pmatrix},$$3 defined by

$$\mathbf{h}_{\mathbf{k}}= \begin{pmatrix} \varepsilon_{\mathbf{k}} & \Delta_{\mathbf{k}}\ \tilde{\Delta}_{\mathbf{k}} & -\varepsilon_{\mathbf{k}} \end{pmatrix},$$4

where the spectrum is real, and a PT-broken or “unpaired” region $$\mathbf{h}_{\mathbf{k}}= \begin{pmatrix} \varepsilon_{\mathbf{k}} & \Delta_{\mathbf{k}}\ \tilde{\Delta}_{\mathbf{k}} & -\varepsilon_{\mathbf{k}} \end{pmatrix},$$5 where the energies are complex and pairing is excluded from the mean-field construction (Ghatak et al., 2017).

This same square-root structure reappears in explicitly one-dimensional PT-symmetric models. In the linearized S–PTS–S junction problem, the central PT-symmetric superconductor is described by

$$\mathbf{h}_{\mathbf{k}}= \begin{pmatrix} \varepsilon_{\mathbf{k}} & \Delta_{\mathbf{k}}\ \tilde{\Delta}_{\mathbf{k}} & -\varepsilon_{\mathbf{k}} \end{pmatrix},$$6

with spectrum

$$\mathbf{h}_{\mathbf{k}}= \begin{pmatrix} \varepsilon_{\mathbf{k}} & \Delta_{\mathbf{k}}\ \tilde{\Delta}_{\mathbf{k}} & -\varepsilon_{\mathbf{k}} \end{pmatrix},$$7

For $$\mathbf{h}_{\mathbf{k}}= \begin{pmatrix} \varepsilon_{\mathbf{k}} & \Delta_{\mathbf{k}}\ \tilde{\Delta}_{\mathbf{k}} & -\varepsilon_{\mathbf{k}} \end{pmatrix},$$8, the energies are real; for $$\mathbf{h}_{\mathbf{k}}= \begin{pmatrix} \varepsilon_{\mathbf{k}} & \Delta_{\mathbf{k}}\ \tilde{\Delta}_{\mathbf{k}} & -\varepsilon_{\mathbf{k}} \end{pmatrix},$$9, they are purely imaginary, producing a PT-broken momentum interval (Kornich et al., 2022).

A continuum $\varepsilon_{\mathbf{k}}\in\mathbb{R}$0-wave PT-symmetric non-Hermitian superconductor has

$\varepsilon_{\mathbf{k}}\in\mathbb{R}$1

Here again, the PT-unbroken regime has real $\varepsilon_{\mathbf{k}}\in\mathbb{R}$2, the PT-broken regime has purely imaginary $\varepsilon_{\mathbf{k}}\in\mathbb{R}$3, and the boundary consists of exceptional points at $\varepsilon_{\mathbf{k}}\in\mathbb{R}$4 (Kornich, 2023).

A direct consequence is that non-Hermitian superconductors can develop Bogoliubov Fermi surfaces or $\varepsilon_{\mathbf{k}}\in\mathbb{R}$5-gaps even when the pairing symmetry itself is fully specified. In the anti-Hermitian single-band theory, zeros of $\varepsilon_{\mathbf{k}}\in\mathbb{R}$6 arise from competition between $\varepsilon_{\mathbf{k}}\in\mathbb{R}$7 and $\varepsilon_{\mathbf{k}}\in\mathbb{R}$8, rather than only from nodes of the gap form factor (Ghatak et al., 2017).

3. Exceptional surfaces, exceptional lines, and non-Hermitian quasiparticles

In a generic $\varepsilon_{\mathbf{k}}\in\mathbb{R}$9 non-Hermitian Hamiltonian,

$\mathcal P=\sigma_z$0

band degeneracy requires two real conditions: $\mathcal P=\sigma_z$1 PT or CP symmetry automatically enforces $\mathcal P=\sigma_z$2, so only one scalar condition remains. The degeneracy locus therefore has codimension one, giving $\mathcal P=\sigma_z$3-dimensional exceptional surfaces in $\mathcal P=\sigma_z$4 spatial dimensions (Okugawa et al., 2018).

In the superconducting CP-symmetric realization,

$\mathcal P=\sigma_z$5

and exceptional degeneracy occurs on

$\mathcal P=\sigma_z$6

In $\mathcal P=\sigma_z$7, this yields exceptional lines in the Brillouin zone; inside the enclosed region, the real part of the quasiparticle gap vanishes and the energies are purely imaginary. The paper characterizes these regions as hosting “drumhead”-like bulk zero-gap quasiparticles bounded by exceptional lines rather than conventional nodal points or nodal lines (Okugawa et al., 2018).

The exceptional structures are topologically protected by $\mathcal P=\sigma_z$8 invariants defined pointwise in momentum space. For PT symmetry one may use

$\mathcal P=\sigma_z$9

while for CP symmetry the invariant is

$\mathcal T=\mathcal K$0

Crossing an exceptional surface flips the corresponding $\mathcal T=\mathcal K$1 index, so the exceptional set is a topological phase boundary in momentum space (Okugawa et al., 2018).

An explicitly solvable PT-symmetric non-Hermitian Kitaev chain with imaginary $\mathcal T=\mathcal K$2-wave pairing exhibits the same structure in one dimension. Its quasiparticle energies are

$\mathcal T=\mathcal K$3

and the PT-unbroken and PT-broken regions are separated by the hyperbola

$\mathcal T=\mathcal K$4

At the exceptional point, the coalescing eigenstate is

$\mathcal T=\mathcal K$5

and the resulting Jordan-block dynamics supports resonant generation of that $\mathcal T=\mathcal K$6-wave Cooper-pair state from the vacuum (Yang et al., 2019).

4. Topological superconductivity, Majorana modes, and non-Bloch structure

A PT-symmetric non-Hermitian topological superconductor is realized by a Kitaev chain with balanced gain at one end and loss at the other,

$\mathcal T=\mathcal K$7

Its Hermitian bulk is that of the Kitaev chain, so the topological region remains $\mathcal T=\mathcal K$8. The non-Hermitian boundary terms generate two distinct edge structures: complex-energy edge modes and nonorthogonal Majorana zero modes. The latter remain pinned at zero by chiral symmetry and produce nonlocal particle transport with currents localized at the edges and absent in the bulk (Kawabata et al., 2018).

The nonorthogonality is not a minor algebraic detail. In the PT-symmetric chain, the edge Majorana operators satisfy modified anticommutation relations rather than the canonical Hermitian Majorana algebra, and the edge currents reach their maximum at the PT transition where nonorthogonality is strongest. This ties PT breaking directly to boundary transport rather than only to spectral complexification (Kawabata et al., 2018).

For one-dimensional non-Hermitian superconductors more generally, non-Bloch band theory replaces Bloch momentum by a complex variable $\mathcal T=\mathcal K$9. Particle-hole symmetry then forces reciprocal generalized Brillouin-zone loops for particles and holes. The quantum phase transition occurs when these loops intersect and the critical GBZ satisfies $[\mathcal{PT},\mathbf h_{\mathbf k}]=0$0; in the models analyzed, $[\mathcal{PT},\mathbf h_{\mathbf k}]=0$1. The corresponding bulk topology is encoded in a $[\mathcal{PT},\mathbf h_{\mathbf k}]=0$2 non-Bloch Majorana Pfaffian invariant, which restores bulk-boundary correspondence for open chains (Cao et al., 2021).

This suggests that PT-symmetric non-Hermitian superconductors in one dimension inherit two intertwined structures when PT symmetry coexists with BdG particle-hole symmetry: a PT-controlled real/complex spectral organization and a PHS-controlled non-Bloch $[\mathcal{PT},\mathbf h_{\mathbf k}]=0$3 topology. The cited non-Bloch framework was developed for PHS-protected non-Hermitian superconductors rather than a single PT implementation, but its reciprocal particle-hole loop structure is directly compatible with PT-symmetric BdG chains (Cao et al., 2021).

5. Thermodynamics, first-order PT transitions, and dissipation-enhanced pairing

Within single-band BCS mean-field theory, Hermitian and PT-symmetric non-Hermitian pairing differ sharply in thermodynamics. The free-energy expansion can be organized as

$[\mathcal{PT},\mathbf h_{\mathbf k}]=0$4

For Hermitian pairing, $[\mathcal{PT},\mathbf h_{\mathbf k}]=0$5; for PT-symmetric non-Hermitian pairing, $[\mathcal{PT},\mathbf h_{\mathbf k}]=0$6. The odd terms therefore change sign in the non-Hermitian case, and the resulting free-energy landscape supports a robust first-order superconducting transition rather than the second-order transition of the Hermitian state. The entropy is discontinuous at the transition, and the Meissner kernel is modified because only the PT-unbroken paired region contributes to the superfluid response (Ghatak et al., 2017).

A microscopic lattice realization of this logic was developed on the honeycomb lattice with on-site attraction and balanced gain/loss. Using right-eigenstate-based non-Hermitian mean-field theory, the quasiparticle energies are

$[\mathcal{PT},\mathbf h_{\mathbf k}]=0$7

and PT symmetry is unbroken when

$[\mathcal{PT},\mathbf h_{\mathbf k}]=0$8

The central result is a first-order phase transition that coincides exactly with the PT-symmetry-breaking line $[\mathcal{PT},\mathbf h_{\mathbf k}]=0$9. In the PT-symmetric phase, moderate non-Hermitian dissipation enhances superconductivity, while in the PT-broken phase stronger dissipation suppresses it. The superconducting spectral gap in the PT-unbroken sector is

$$\sigma_z\mathbf h_{\mathbf k}^*\sigma_z=\mathbf h_{\mathbf k}$$0

and both $$\sigma_z\mathbf h_{\mathbf k}^*\sigma_z=\mathbf h_{\mathbf k}$$1 and the condensed-pair density show discontinuous jumps at the transition (Liu et al., 3 Sep 2025).

The same paper further distinguishes stable and metastable superconducting sectors through the real part of a condensation energy,

$$\sigma_z\mathbf h_{\mathbf k}^*\sigma_z=\mathbf h_{\mathbf k}$$2

using $$\sigma_z\mathbf h_{\mathbf k}^*\sigma_z=\mathbf h_{\mathbf k}$$3 as the stability criterion. This yields PT-symmetric stable and metastable superconducting phases, PT-broken stable and metastable superconducting phases, and a PT-broken normal phase within one phase diagram (Liu et al., 3 Sep 2025).

At the fluctuation level, a different PT-symmetric superconducting regime arises in a biased weak link described by an effective non-Hermitian operator

$$\sigma_z\mathbf h_{\mathbf k}^*\sigma_z=\mathbf h_{\mathbf k}$$4

The odd-parity imaginary potential $$\sigma_z\mathbf h_{\mathbf k}^*\sigma_z=\mathbf h_{\mathbf k}$$5 is PT-symmetric, and the two lowest fluctuation eigenvalues merge at a critical field

$$\sigma_z\mathbf h_{\mathbf k}^*\sigma_z=\mathbf h_{\mathbf k}$$6

beyond which they form a complex-conjugate pair. This PT-symmetry breaking suppresses fluctuation superconductivity and produces a characteristic evolution of the differential resistance in mesoscale superconducting wires (Chtchelkatchev et al., 2010).

6. Junction physics, transport formalisms, and spectroscopic signatures

The S–PTS–S junction displays Andreev physics with no Hermitian analogue. In the one-dimensional setup with a PT-symmetric non-Hermitian superconductor between two conventional superconductors, Andreev bound states exist only for discrete phase pairs,

$$\sigma_z\mathbf h_{\mathbf k}^*\sigma_z=\mathbf h_{\mathbf k}$$7

For $$\sigma_z\mathbf h_{\mathbf k}^*\sigma_z=\mathbf h_{\mathbf k}$$8, the short-junction energies are purely imaginary,

$$\sigma_z\mathbf h_{\mathbf k}^*\sigma_z=\mathbf h_{\mathbf k}$$9

so one state grows and the other decays in time. For $$\tilde{\Delta}_{\mathbf{k}}=\Delta_{\mathbf{k}}^\dagger$$0, the bound-state energy is exactly zero and the junction supports a unidirectional Majorana zero mode propagating through the structure in only one direction, giving a directional quasiparticle supercurrent (Kornich et al., 2022).

Transport through a normal-metal–insulator–PT-symmetric-superconductor junction also depends on the non-Hermitian formalism used to compute observables. In the right-right basis, Andreev reflection enhances subgap current as in generalized BTK theory, and the growth and decay associated with imaginary-energy bands are exactly balanced by a Hermitian source term. In the left-right biorthogonal basis, by contrast, Andreev-reflected particles move in the opposite direction and the corresponding current-voltage characteristics become unphysical. The right-right formalism therefore yields the physically consistent steady-state transport picture for this junction (Kornich, 2023).

Angle-resolved photoelectron fluctuation spectroscopy provides a direct spectroscopic signature of PT-symmetric non-Hermitian superconductivity. For a single-band PT-symmetric non-Hermitian superconductor, the connected ARPFS signal for opposite momenta and opposite spins is

$$\tilde{\Delta}_{\mathbf{k}}=\Delta_{\mathbf{k}}^\dagger$$1

which is negative in the real-spectrum regime. The negative sign is traced to a pairing mechanism in which an attractive interaction for electrons implies a repulsive interaction for holes, leading to negative cross correlations. The same work proposes spatiotemporal modulation of the material as a route to the required odd interaction $$\tilde{\Delta}_{\mathbf{k}}=\Delta_{\mathbf{k}}^\dagger$$2 (Kornich et al., 2021).

Proposed microscopic routes across the literature include antisymmetric interaction potentials, Dzyaloshinskii–Moriya interaction supplemented by a bath or complex potential, superconducting nanowires with spatiotemporal modulation, complex pairing generated by non-equilibrium conditions or engineered dissipation, and balanced gain/loss on lattice substructures or in hybrid platforms (Ghatak et al., 2017, Kornich et al., 2022, Kornich et al., 2021, Liu et al., 3 Sep 2025). Together these works establish PT-symmetric non-Hermitian superconductivity as a regime in which superconducting order, open-system symmetry, and non-Hermitian spectral topology are inseparable rather than perturbatively related.