$\mathcal{PT}$-like phase transitions from square roots of supersymmetric Hamiltonians

Abstract

We introduce a general framework for realizing $\mathcal{PT}$-like phase transitions in non-Hermitian systems without imposing explicit parity--time ($\mathcal{PT}$) symmetry. The approach is based on constructing a Hamiltonian as the square root of a supersymmetric partner energy-shifted by a constant. This formulation naturally leads to bipartite dynamics with balanced gain and loss and can incorporate non-reciprocal couplings. The resulting systems exhibit entirely real spectra over a finite parameter range precisely when the corresponding passive Hamiltonian lacks a zero mode. As the non-Hermitian parameter representing gain and loss increases, the spectrum undergoes controlled real-to-complex transitions at second-order exceptional points. We demonstrate the versatility of this framework through several examples, including well-known models such as the Hatano--Nelson (HN) and complex Su--Schrieffer--Heeger (cSSH) lattices. Extending the formalism to $q$-commuting matrices further enables the systematic realization of higher-order exceptional points in systems with unidirectional couplings. Overall, this work uncovers new links between non-Hermitian physics and supersymmetry, offering a practical route to engineer photonic arrays with tunable spectral properties beyond what is achievable with explicit $\mathcal{PT}$-symmetry.

Figures (4)

• Figure 1: Schematic illustration of our construction of non-Hermitian bipartite tight-binding models. Starting with a supersymmetric (SUSY) Hamiltonian, $H_{\mathrm{SUSY}}$, with real spectrum, we take a matrix square root after a uniform downward shift by $\gamma^2$, where $\gamma \in \mathbb{R}$. The result is a tight-binding model with hopping on a bipartite graph and gain and loss applied to vertices in the color classes $+1$ and $-1$, respectively. Our procedure does not require geometric symmetry or explicit $\mathcal{PT}$-symmetry, thereby introducing a more general condition under which real-to-complex spectral transitions occur in non-Hermitian systems.
• Figure 2: Depiction of four bipartite graphs endowed with non-Hermitian Hamiltonians. Balanced gain, $i \gamma$, and loss, $-i \gamma$, potentials are applied to vertices colored red and blue, respectively. These Hamiltonians admit two distinct nonzero coupling parameters, $t_+$ (solid) and $t_-$ (dashed). These examples showcase how the Hamiltonians in our class may or may not exhibit: non-uniform couplings, as in panels $(a)$ and $(d)$, non-reciprocal couplings, as in panels $(b$–$c)$, or long-range couplings, as in panel $(d)$. The vertices may be elements of spaces with any number of dimensions, highlighted by the graphs in $(c$–$d)$ whose embeddings require more than one dimension. Although panels $(a)$ and $(d)$ depict $\mathcal{PT}$-symmetric dynamics, the Hamiltonians of panels $(b$–$c)$ do not possess a geometric antiunitary symmetry, since no color-reversing weighted graph automorphism exists.
• Figure 3: Spectrum of the Hamiltonians with graph structure depicted in \ref{['fig:bipartite-graphs-JB']}, characterized by the real parameter $\gamma$. The couplings are taken to be $(t_+, t_-) = (1,1/2)$. In all cases, the spectrum exhibits $\mathcal{PT}$-like phase transitions where pairs of real eigenvalues bifurcate into the complex plane in complex-conjugate pairs. As discussed in the main text, exceptional points exist at locations determined by the spectrum of the corresponding passive system, $\gamma = 0$. The threshold of this transition is nonzero if and only if the passive system has no zero modes. One zero mode exists in the passive limit of panel $(b)$; this is because the difference in its number of gain (red) and loss (blue) sites is one, resulting in imbalanced gain and loss and a single purely imaginary eigenvalue without a complex-conjugate counterpart.
• Figure 4: This directed variant of a triangular grid graph is an example of an oriented graph with chromatic number 3. Adapted to this graph is a pair of $q$-commuting matrices: one matrix, $x$, implements nonreciprocal couplings on the plaquettes; the other, $y$ implements an on-site complex potential taking values from the cube roots of unity.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• proof
• proof : Proof of \ref{['thm:Quasi-Hermitian']}
• proof
• Lemma 5
• proof
• proof : Proof of \ref{['thm:multiplicity']}