Non-Hermitian Quantum Materials Overview

• Non-Hermitian quantum materials are systems modeled by non-self-adjoint Hamiltonians that yield complex energy spectra, exceptional points, and biorthogonal state properties.
• They feature unique topological invariants and phase transitions driven by engineered gain/loss and open-system dynamics in crystalline and synthetic platforms.
• Experimental realizations exhibit non-reciprocal edge modes and non-Hermitian skin effects, advancing applications in topological lasers and quantum information devices.

Non-Hermitian quantum materials are crystalline, mesoscopic, or engineered media whose effective quantum Hamiltonians are intrinsically non-Hermitian, typically due to environment-induced decoherence, engineered gain/loss, finite quasiparticle lifetimes, or open-system dynamics. These systems exhibit spectra, band topology, and collective phenomena fundamentally distinct from those in Hermitian quantum matter, including spectral nonorthogonality, exceptional points (EPs), non-Hermitian skin effects (NHSE), complex-valued geometric tensors and response functions, and the robust emergence of new topological and correlated phases. This article provides a comprehensive survey of the theoretical framework, physical realizations, topological invariants, dynamical phenomena, and current research frontiers in non-Hermitian quantum materials.

1. Theoretical Foundations and Band Structure

Non-Hermitian quantum materials are modeled by effective Hamiltonians $H \neq H^\dagger$. Crucially, right and left eigenstates are individually nonorthogonal and form a biorthogonal basis: $$H|\psi^R_n\rangle = E_n|\psi^R_n\rangle, \qquad H^\dagger|\psi^L_n\rangle = E_n^*|\psi^L_n\rangle, \qquad \langle\psi^L_n|\psi^R_m\rangle = \delta_{nm}$$ The lack of Hermiticity gives rise to several distinct phenomena:

• Complex energy spectra: Eigenvalues can be complex-valued, encoding both energy and decay/gain rates.
• Exceptional points: Parameters where two or more eigenvalues and eigenstates coalesce, forming branch points with non-diagonalizable Hamiltonians.
• Spectral topology: Both point gaps (regions where $\det(H-E_0)\neq0$ for some $E_0$) and line gaps (no eigenvalue crosses a reference line in the complex plane) determine distinct topological classes (Fan et al., 2021).

The presence of non-Hermitian terms may result from engineered gain/loss, asymmetric tunneling (non-reciprocity), Lindbladian dissipators, self-energies from many-body interactions, or coupling to reservoirs. Theoretical treatments for both open quantum systems and equilibrium strongly correlated systems can yield equivalent non-Hermitian Hamiltonians under appropriate conditions (Michishita et al., 2020).

2. Topological Phases and Invariants

Non-Hermitian quantum materials can support a rich variety of topological phases absent in Hermitian settings, often classified by generalizations of Berry curvature, Wilson loops, and associated invariants:

• Non-Hermitian Chern numbers and biorthogonal Berry phase: Quantities defined using both right and left eigenstates—e.g., complex-valued Berry curvature and phase (Fan et al., 2021).
• $\mathbb{Z}_2$ and spin Chern invariants: Generalized to non-Hermitian time-reversal-symmetric systems using non-Hermitian Wilson loops and biorthogonal decompositions of the Bloch bundle (Hou et al., 2019).
• Many-body topological invariants: Non-Hermitian many-body Chern numbers have been computed for fractionalized and interacting systems, as in open-system fractional quantum Hall states or dissipative quantum dot chains (Yoshida et al., 2019, Hyart et al., 2021).

Spectral topology is sensitive to both point and line gap closures, with exceptional points signifying topological transitions unique to the non-Hermitian regime.

3. Non-Hermitian Dynamics in Real and Synthetic Materials

Non-Hermitian phenomena manifest both in genuine quantum materials and in synthetic quantum platforms:

• Topological insulators and anomalous Hall systems: Magnetically doped (Bi,Sb)$_2$Te$_3$ heterostructures realize quantum anomalous Hall (QAH) edge states whose conductance matrix is generically non-Hermitian due to non-reciprocal edge transport, leading to experimentally observed NHSE and boundary-driven localization (Yi et al., 1 Oct 2025).
• Proximity-induced non-Hermitian broadening: In Dirac surface states of topological insulators adjacently coupled to metallic ferromagnets, the self-energy naturally introduces non-Hermitian broadening, leading to the breakdown of quantized Hall conductivity even when classified as a Chern insulator (Philip et al., 2018).
• Quantum dots and interacting chains: Engineered dissipation in quantum dot arrays produces many-body non-Hermitian topological edge modes robust to strong interactions, supported by both exact diagonalization and tensor-network calculations (Hyart et al., 2021).
• Altermagnets and multi-orbital systems: Non-Hermitian extensions of spin-orbital Hamiltonians capture complex order parameter symmetry, complex quantum geometric tensor structure, and topological transitions governed by exceptional points and modified Hall conductance (Goswami, 3 Sep 2025).

Table: Experimental Platforms and Non-Hermitian Phenomena

Platform	Non-Hermitian Feature	Reference
Magnetically doped TIs (QAH)	NHSE, non-reciprocal edge G-matrix	(Yi et al., 1 Oct 2025)
TI/ferromagnet interfaces	Loss of Hall quantization, point/line gaps	(Philip et al., 2018)
Quantum dot chains	Many-body NH edge modes, robust interactions	(Hyart et al., 2021)
Cold atoms with controlled dissipation	Dissipative FQH via quantum Zeno effect	(Yoshida et al., 2019)
Altermagnetic and multi-band crystal	NH geometry, anomalous Hall / EP transitions	(Goswami, 3 Sep 2025)

4. Novel Quantum Geometric and Transport Responses

Non-Hermitian quantum materials exhibit unique geometric and transport phenomena:

• Complex quantum geometric tensor (QGT): The QGT acquires complex-valued quantum metric and Berry curvature, directly governing intrinsic, $\tau$-independent nonlinear electrical conductivity and introducing wavepacket-width-dependent responses unique to non-Hermitian systems (Chen et al., 15 Sep 2025).
• Generalized quantum Hall admittance: The Hall response is promoted from a real conductance $\sigma_H$ to a complex admittance $Y_H = \sigma_H + i B_H$, where $\Im\Phi_C$ yields intrinsic quantum capacitance or inductance, with physical consequences tunable by non-Hermitian parameters (Fan et al., 2021).
• Non-Floquet engineering: Periodically driven non-Hermitian systems can be characterized by non-unitary frequency-space Floquet Hamiltonians. Non-Floquet protocols reveal topological Wannier-Stark localization and enable phase-detection not possible in Hermitian Floquet engineering (Wang et al., 2021).

5. Many-Body Correlated and Composite Phases

Interaction effects and many-body physics intertwine non-trivially with non-Hermiticity:

• Non-Hermitian fractional quantum Hall phases: Open-system models with two-body loss exhibit stabilized FQH ground-state multiplets and topological Chern numbers, with degeneracies protected by many-body translation symmetry and gaps induced via the continuous quantum Zeno effect (Yoshida et al., 2019).
• Squeezed polaron states: Non-reciprocal hopping and impurity interactions in open cold-atom chains generate dipole-like "squeezed polarons," which are bulk-localized and fundamentally distinct from conventional skin modes and Hermitian polarons (Qin et al., 2022).
• Composite quantum phases: Non-Hermitian many-body spin chains realize phases where left and right ground states are individually in distinct Hermitian SPT phases, yielding composite symmetry-protected topological order classified by pairs $(\omega_L, \omega_R)$, a phenomenon absent from Hermitian classification (Guo et al., 2023).

6. Exceptional Flat Bands, Emergent Symmetry, and Outlook

Non-Hermitian materials exhibit the following advanced phenomena:

• Exceptional flat bands: The sublattice-mismatch principle for flat-band formation generalizes to the non-Hermitian case, with exceptional points generating long-lived, biorthogonal flat bands whose energies and lifetimes are tunable (Esparza et al., 14 Aug 2025).
• Emergent Lorentz and Yukawa–Lorentz symmetry: In non-Hermitian Dirac materials, RG analysis reveals that Fermi velocity flows to the speed of light, reinstating Lorentz symmetry in the deep IR, even as non-Hermitian effects are manifest at higher energy scales (Murshed et al., 2023). With local interactions, critical points can flow to either non-Hermitian Yukawa-Lorentz fixed points or return to the Hermitian regime depending on operator commutation relations (Juricic et al., 2023).
• Device and platform implications: Non-Hermitian topological materials offer potential for topological lasers, non-reciprocal signal transmission, exceptional-point-based sensors, and quantum information processing architectures exploiting robust edge modes and enhanced susceptibility at EPs (Hou et al., 2019, Yi et al., 1 Oct 2025).

Open questions include the robustness and tunability of interaction-driven non-Hermitian topological order, the interplay between disorder and the skin effect, extension of classification frameworks to higher dimensions and to strongly correlated phases, and experimental realization in artificial and natural materials. Progress in fabricating and probing engineered dissipation, modulation, and reservoir coupling will shape the ongoing development and application of non-Hermitian quantum materials.