---
title: PT-Symmetric Non-Hermitian Superconductor
url: https://www.emergentmind.com/topics/pt-symmetric-non-hermitian-superconductor
type: topic
---

# PT-Symmetric Non-Hermitian Superconductor

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 [1708.09108] [1810.03376] [2509.03072].

## 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 [1708.09108].

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 \(\tilde{\Delta}(\mathbf{k})=-\Delta^*(\mathbf{k})\), the \(4\times4\) BdG Hamiltonian block-diagonalizes into
\[
\mathcal H_\pm(\mathbf k)=\pm \mathrm{Im}[\Delta(\mathbf k)]\,i\tau_x \pm \mathrm{Re}[\Delta(\mathbf k)]\,i\tau_y + \xi_{\mathbf k}\tau_z,
\]
with \(CP=K\) in a suitable basis and \((CP)\mathcal H_\pm(CP)^{-1}=-\mathcal H_\pm\). Under the mapping \(H\to iH\), these CP-symmetric BdG systems have the same exceptional-surface topology as PT-symmetric non-Hermitian systems [1810.03376].

A complementary microscopic route introduces non-Hermiticity through balanced gain and loss. On a honeycomb lattice with on-site \(s\)-wave pairing, staggered imaginary onsite potentials \(+i\gamma\) and \(-i\gamma\) on the two sublattices yield a PT-symmetric non-Hermitian superconductor with \([\mathcal{PT},H]=0\), where parity exchanges sublattices and time reversal complex conjugates \(i\to-i\) [2509.03072].

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

For the anti-Hermitian pairing state, the quasiparticle spectrum is
\[
E_{\mathbf k}^{\mathrm{(NH)}}=\pm\sqrt{\varepsilon_{\mathbf k}^2-|\Delta_{\mathbf k}|^2},
\]
whereas the Hermitian counterpart has \(E_{\mathbf k}^{\mathrm{(H)}}=\pm\sqrt{\varepsilon_{\mathbf k}^2+|\Delta_{\mathbf k}|^2}\). The non-Hermitian state therefore has a PT-unbroken or “paired” region \(\mathfrak R_1\) defined by
\[
|\varepsilon_{\mathbf k}|\ge |\Delta_{\mathbf k}|,
\]
where the spectrum is real, and a PT-broken or “unpaired” region \(\mathfrak R_2\) where the energies are complex and pairing is excluded from the mean-field construction [1708.09108].

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
\[
H_{PT}(k)=v_F k\, \tau_z\otimes\sigma_z+\Delta_{PT}\, i\,\tau_0\otimes\sigma_y,
\]
with spectrum
\[
E(k)=\pm\sqrt{(v_F k)^2-\Delta_{PT}^2}.
\]
For \(|v_Fk|>|\Delta_{PT}|\), the energies are real; for \(|v_Fk|<|\Delta_{PT}|\), they are purely imaginary, producing a PT-broken momentum interval [2205.07785].

A continuum \(p\)-wave PT-symmetric non-Hermitian superconductor has
\[
H_{\rm PTS}(k)=
\begin{pmatrix}
\frac{k^2}{2m}-\mu & \Delta k\\
-\Delta k & -\frac{k^2}{2m}+\mu
\end{pmatrix},
\qquad
E_k=\pm\sqrt{\left(\frac{k^2}{2m}-\mu\right)^2-(\Delta k)^2}.
\]
Here again, the PT-unbroken regime has real \(E_k\), the PT-broken regime has purely imaginary \(E_k\), and the boundary consists of exceptional points at \(E_k=0\) [2302.14802].

A direct consequence is that non-Hermitian superconductors can develop Bogoliubov Fermi surfaces or \(k\)-gaps even when the pairing symmetry itself is fully specified. In the anti-Hermitian single-band theory, zeros of \(E_{\mathbf k}\) arise from competition between \(|\varepsilon_{\mathbf k}|\) and \(|\Delta_{\mathbf k}|\), rather than only from nodes of the gap form factor [1708.09108].

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

In a generic \(2\times2\) non-Hermitian Hamiltonian,
\[
H(\mathbf{k})=(a_0+ia_1)\sigma_0+\mathbf b_0\cdot\boldsymbol{\sigma}+i\mathbf b_1\cdot\boldsymbol{\sigma},
\]
band degeneracy requires two real conditions:
\[
\mathbf b_0\cdot\mathbf b_0-\mathbf b_1\cdot\mathbf b_1=0,
\qquad
\mathbf b_0\cdot\mathbf b_1=0.
\]
PT or CP symmetry automatically enforces \(\mathbf b_0\cdot\mathbf b_1=0\), so only one scalar condition remains. The degeneracy locus therefore has codimension one, giving \((d-1)\)-dimensional exceptional surfaces in \(d\) spatial dimensions [1810.03376].

In the superconducting CP-symmetric realization,
\[
E_{\mathbf k}=\pm\sqrt{\xi_{\mathbf k}^2-|\Delta(\mathbf k)|^2},
\]
and exceptional degeneracy occurs on
\[
\xi_{\mathbf k}^2=|\Delta(\mathbf k)|^2.
\]
In \(d=2\), 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 [1810.03376].

The exceptional structures are topologically protected by \(\mathbb Z_2\) invariants defined pointwise in momentum space. For PT symmetry one may use
\[
s=\mathrm{sgn}\,\det(H_{PT}),
\]
while for CP symmetry the invariant is
\[
s'=\mathrm{sgn}\,\det(iH_{CP}).
\]
Crossing an exceptional surface flips the corresponding \(\mathbb Z_2\) index, so the exceptional set is a topological phase boundary in momentum space [1810.03376].

An explicitly solvable PT-symmetric non-Hermitian Kitaev chain with imaginary \(p\)-wave pairing exhibits the same structure in one dimension. Its quasiparticle energies are
\[
\varepsilon_k=2\sqrt{(\mu-J\cos k)^2-\Delta^2\sin^2 k},
\]
and the PT-unbroken and PT-broken regions are separated by the hyperbola
\[
\mu_c^2-\Delta_c^2=J^2.
\]
At the exceptional point, the coalescing eigenstate is
\[
\left(1+c_{k_c}^\dagger c_{-k_c}^\dagger\right)|0\rangle,
\]
and the resulting Jordan-block dynamics supports resonant generation of that \(p\)-wave Cooper-pair state from the vacuum [1912.07227].

## 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,
\[
\hat H_{\rm PT}=
\sum_{j=1}^{L-1}\left(-J\hat c_j^\dagger \hat c_{j+1}+i\Delta \hat c_j\hat c_{j+1}+{\rm H.c.}\right)
-\mu\sum_{j=1}^L\left(\hat c_j^\dagger \hat c_j-\frac12\right)
-i\gamma(\hat c_1^\dagger \hat c_1-\hat c_L^\dagger \hat c_L).
\]
Its Hermitian bulk is that of the Kitaev chain, so the topological region remains \(|\mu/2J|<1\). 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 [1801.00499].

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 [1801.00499].

For one-dimensional non-Hermitian superconductors more generally, non-Bloch band theory replaces Bloch momentum by a complex variable \(\beta\). 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 \(|\beta_c|=1\); in the models analyzed, \(\beta_c=\pm1\). The corresponding bulk topology is encoded in a \(\mathbb Z_2\) non-Bloch Majorana Pfaffian invariant, which restores bulk-boundary correspondence for open chains [2103.17157].

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 \(\mathbb Z_2\) 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 [2103.17157].

## 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
\[
F_s-F_n=a(\Delta\tilde\Delta)+b(\Delta\tilde\Delta)^2+c(\Delta\tilde\Delta)^3+d(\Delta\tilde\Delta)^4+\cdots.
\]
For Hermitian pairing, \(\Delta\tilde\Delta=|\Delta|^2\); for PT-symmetric non-Hermitian pairing, \(\Delta\tilde\Delta=-|\Delta|^2\). 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 [1708.09108].

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
\[
E_{\alpha k}= \sqrt{\big|\ |w_k|-i(-1)^\alpha \Delta_0\big|^2-\gamma^2},
\]
and PT symmetry is unbroken when
\[
\operatorname{Re}\Delta_0\ge \gamma.
\]
The central result is a first-order phase transition that coincides exactly with the PT-symmetry-breaking line \(\operatorname{Re}\Delta_0=\gamma\). 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
\[
E_{\rm gap}=\sqrt{(\operatorname{Re}\Delta_0)^2-\gamma^2},
\]
and both \(|\Delta_0|\) and the condensed-pair density show discontinuous jumps at the transition [2509.03072].

The same paper further distinguishes stable and metastable superconducting sectors through the real part of a condensation energy,
\[
E_c=\frac{2}{U}|\Delta_0|^2-\frac1N\sum_{\alpha k}E_{\alpha k}
+\frac{2}{N}\sum_k\sqrt{|w_k|^2-\gamma^2},
\]
using \(\operatorname{Re}E_c<0\) 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 [2509.03072].

At the fluctuation level, a different PT-symmetric superconducting regime arises in a biased weak link described by an effective non-Hermitian operator
\[
\mathcal H_{\rm eff}=-D\nabla_x^2-\tau^{-1}-2i\mathcal E x.
\]
The odd-parity imaginary potential \(-2i\mathcal E x\) is PT-symmetric, and the two lowest fluctuation eigenvalues merge at a critical field
\[
\mathcal E_c\approx 49.25\,\frac{E_{\rm Th}}{L},
\]
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 [1008.3590].

## 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,
\[
\{\varphi_1,\varphi_2\}\in\{0,\pi\},\qquad \{\pi/2,3\pi/2\}.
\]
For \(\{0,\pi\}\), the short-junction energies are purely imaginary,
\[
E=\pm i\,\frac{\Delta_{PT}\Delta d}{v_F},
\]
so one state grows and the other decays in time. For \(\{\pi/2,3\pi/2\}\), 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 [2205.07785].

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 [2302.14802].

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
\[
\Delta I_{\mathbf p\uparrow;-\mathbf p\downarrow}
=
-\frac{M_0^4S_0^4}{4}
\frac{\Delta_{\mathbf p}^2}{\varepsilon_p^2-\Delta_{\mathbf p}^2},
\]
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 \(V(\mathbf q)=-V(-\mathbf q)\) [2112.06497].

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 [1708.09108] [2205.07785] [2112.06497] [2509.03072]. 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.

Source: https://www.emergentmind.com/topics/pt-symmetric-non-hermitian-superconductor