Non-Hermitian Hamiltonians are relevant to describe the features of a broad class of physical phenomena, ranging from photonics and atomic and molecular systems to nuclear physics and mesoscopic electronic systems. An important question relies on the understanding of the influence of a curved background on the static and dynamical properties of non-Hermitian systems. In this work, we study the interplay of geometry and non-Hermitian dynamics by unveiling the existence of curvature-dependent non-Hermitian phase transitions. We investigate a prototypical model of Dirac fermions on a sphere with an imaginary mass term. This exactly solvable model admits an infinite set of curvature-dependent pseudo-Landau levels. We characterize these phases by computing an order parameter given by the pseudomagnetization and, independently, the non-Hermitian fidelity susceptibility. Finally, we probe the non-Hermitian phase transitions by computing the (generalized) Loschmidt echo and the dynamical fidelity after a quantum quench of the imaginary mass and find singularities corresponding to exceptional radii of the sphere.
Non-Hermitian Hamiltonians are relevant to describe the features of a broad class of physical phenomena, ranging from photonics and atomic and molecular systems to nuclear physics and mesoscopic electronic systems. An important question relies on the understanding of the influence of a curved background on the static and dynamical properties of non-Hermitian systems. In this work, we study the interplay of geometry and non-Hermitian dynamics by unveiling the existence of curvature-dependent non-Hermitian phase transitions. We investigate a prototypical model of Dirac fermions on a sphere with an imaginary mass term. This exactly solvable model admits an infinite set of curvature-dependent pseudo-Landau levels. We characterize these phases by computing an order parameter given by the pseudomagnetization and, independently, the non-Hermitian fidelity susceptibility. Finally, we probe the non-Hermitian phase transitions by computing the (generalized) Loschmidt echo and the dynamical fidelity after a quantum quench of the imaginary mass and find singularities corresponding to exceptional radii of the sphere.
The influence of curved background on the properties of Condensed Matter systems is a significant question for a broad class of physical applications, catching the curiosity because of their interesting features in a rich interface between many research areas. In this thesis, we introduce an analytically solvable model of Dirac fermions with imaginary mass on the sphere. We show the existence of an in nite sequence of exceptional points (EP), which depend on the radius (curvature) of the sphere. We employ quench dynamics to characterize curvaturedependent Non-Hermitian phase transitions. We demonstrate that the existence of singular points of the Loschmidt echo and the fidelity are an unambiguous signature of geometric EPs that distinguish between different phases of the model. Also, we use numerical techniques to solve the Bose-Hubbard hamiltonian that simulates the hyperbolic geometry. In this system, we attempt to verify whether we can observe a bound-states comprising a pair of atoms in the presence of repulsive interactions in a hyperbolic lattice. This system is a playground to simulate a space with negative curvature, that is difficult to simulate in a euclidean space without distortions.
We establish a unified framework for dynamical quantum phase transitions (DQPTs) in non-Hermitian systems that encompasses both biorthogonal and self-norm nonbiorthogonal formulations for pure and mixed states under quantum quench protocols. Our framework provides explicit expressions for the Loschmidt amplitude, Loschmidt echo, and rate function, revealing a universal geometric signature of DQPTs in the two-band model: orthogonality of two related vectors in two-dimensional real space. Strikingly, we demonstrate that nonbiorthogonal quenches from non-Hermitian to Hermitian Hamiltonians under chiral symmetry exhibit emergent topological characteristics of DQPTs, unveiling their fundamental features beyond conventional Hermitian regimes. This work establishes fundamental geometric and topological principles governing quantum criticality in open systems, with implications for quantum sensing and many-body physics in dissipative environments.
The geometric properties of quantum states are fully encoded by the quantum geometric tensor. The real and imaginary parts of the quantum geometric tensor are the quantum metric and Berry curvature, which characterize the distance and phase difference between two nearby quantum states in Hilbert space, respectively. For conventional Hermitian quantum systems, the quantum metric corresponds to the fidelity susceptibility and has already been used to specify quantum phase transitions from the geometric perspective. In this paper, we extend this wisdom to the non-Hermitian systems for revealing non-Hermitian critical points. To be concrete, by employing numerical exact diagonalization and analytical methods, we calculate the quantum metric and corresponding order parameters in various non-Hermitian models, which include two non-Hermitian generalized Aubry-Andr\'e models and non-Hermitian cluster and mixed-field Ising models. We demonstrate that the quantum metric of eigenstates in these non-Hermitian models exactly identifies the localization transitions, mobility edges, and many-body quantum phase transitions with gap closings, respectively. We further show that this strategy is robust against the finite-size effect and different boundary conditions.
We develop a comprehensive, gauge-covariant geometric framework for non-Hermitian quantum systems in the quasi-Hermitian regime, that is, the region of parameter space where the non-Hermitian Hamiltonian admits a real spectrum and a positive-definite metric operator. We build this framework by elevating the Dyson map to a central geometric object. This map is the transformation that converts a non-Hermitian Hamiltonian into an equivalent Hermitian one. From it we construct the Dyson connection and decompose it into Hermitian and anti-Hermitian parts, identified respectively as {\it stretching } and {\it rotation } components. This decomposition cleanly separates the genuine physical metric deformations from the unitary gauge redundancies. Working with manifestly gauge-covariant states, we then derive the complex non-Hermitian Berry phase and the quantum geometric tensor (QGT), and show that the non-Hermitian geometric curvature originates from the non-commutativity of the stretching components at the operator level. We further analyse the geometric singularities near an exceptional point (EP) and uncover a distinct hierarchy of divergences. For a general two-level non-Hermitian model, the quantum metric tensor (QMT) exhibits a leading-order divergence ∼∣εμ∣−2, while the Berry curvature shows a weaker, subleading divergence ∼∣εμ∣−3/2, with εμ denoting the parameter displacement from the EP along an individual parameter axis μ. Finally, we examine physical realizations of this model, including the non-Hermitian Su--Schrieffer--Heeger (SSH) and Hatano--Nelson (HN) models, where exact analytical results confirm the predicted critical scaling laws and illustrate the metric-deformation-driven non-Hermitian geometries.
Composite topological phases with intriguing topology like M\"obius strips emerge in sublattice symmetric non-Hermitian systems due to spontaneous breaking of time-reversal symmetry at some parameter regime. While these phases have been characterized by nonadiabatic complex geometric phases of multiple participating complex bands, the physical properties of these phases remain largely unknown. We explore the dynamical response of these phases by studying Loschmidt echo from an initial state of the Hermitian Su-Schrieffer-Heeger (SSH) model, which is evolved by a non-Hermitian SSH Hamiltonian after a sudden quench in parameters. Topology-changing quenches display nonanalytical temporal behavior of return rates (logarithm of the Loschmidt echo) for the non-Hermitian SSH Hamiltonian in the trivial, M\"obius, and topological phase. Moreover, the dynamical topological order parameter appears only at one side of the Brillouin zone for the M\"obius phase case in contrast to both sides of the Brillouin zone for quench by the trivial and topological phase of the non-Hermitian SSH model. The last feature is a dynamical signature of different symmetry constraints on the real and imaginary parts of the complex bands in the M\"obius phase.
Understanding the interplay between quantum coherence and non-Hermitian features would enable the devising of quantum technologies based on dissipative systems. In turn, quantum coherence can be characterized in terms of the language of multiple quantum coherences (MQCs) originally developed in solid-state nuclear magnetic resonance (NMR), nowadays applied to the detection of quantum chaos and to the study of criticality in many-body quantum systems. Here, we show the usefulness of MQCs for probing equilibrium phase transitions in non-Hermitian systems. To do so, we investigate the connection of quantum coherences and critical points for several paradigmatic non-Hermitian Hamiltonians. For a non-Hermitian two-level system, MQCs witness the parity-symmetry-breaking phase transition from the unbroken to the broken phase. Furthermore, for the non-Hermitian transverse field Ising model, MQCs capture the Yang-Lee phase transition in which the ground state energy acquires a nonzero imaginary component. For the disordered Hatano-Nelson (HN) model with periodic boundary conditions, MQCs testify the emergence of mobility edges in the spectrum of this model. In addition, MQCs signal the topological phase transition exhibited by the complex energy spectra of the disorder-free HN model. Finally, we comment on experimentally probing phase transitions in NMR systems, realizing non-Hermitian Hamiltonians. Our results have applications to non-Hermitian quantum sensing, quantum thermodynamics, and in the study of the non-Hermitian skin effect.