We review analyses of open quantum systems. We show how non-Hermiticity arises in an open quantum system with an infinite environment, focusing on the one-body problem. One of the reasons for taking the present approach is that we can solve the problem completely, making it easier to see the structures of problems involving open quantum systems. We show that this results in the discovery of a new complete set, which is one of the main topics of the present article. Another reason for focusing on the one-body problem is that the theory permits the strong coupling between the system and the environment. In the current research landscape, it is valuable to revisit the one-body problem for open quantum systems, which can be solved accurately for arbitrary strengths of the system-environment couplings. A rigorous understanding of the problem structures in the present approach will be helpful when we tackle problems with many-body interactions. First, we consider potential scattering and directly define the resonant state as an eigenstate of the Schrödinger equation under the Siegert outgoing boundary condition. We show that the resonant eigenstate can have a complex energy eigenvalue, even though the Hamiltonian is seemingly Hermitian. Second, we introduce the Feshbach formalism, which eliminates the infinite degrees of freedom of the environment and represents its effect as a complex potential. The resulting effective Hamiltonian is explicitly non-Hermitian. By unifying these two ways of defining resonant states, we obtain a new complete set of bases for the scattering problem that contains all discrete eigenstates, including resonant states. We finally mention the non-Markovian dynamics of open quantum systems. We emphasize the time-reversal symmetry of the dynamics that continuously connects the past and the future. We can capture it using the new complete set that we develop here.
Non-Hermitian (NH) Hamiltonians describe open quantum systems, nonequilibrium dynamics, and dissipative processes. Although a rich range of single-particle NH physics has been uncovered, many-body phenomena in strongly correlated NH systems have been far less well studied. The Kondo effect, an important paradigm for strong correlation physics, has recently been considered in the NH setting. Here, we develop an NH generalization of the numerical renormalization group and use it to solve the NH Kondo model. Our nonperturbative solution applies beyond weak coupling, and we uncover a nontrivial phase diagram. The method is showcased by application to the NH pseudogap Kondo model, which we show supports a completely novel phase with a genuine NH stable fixed point and complex eigenspectrum. Our NH numerical renormalization group code, which can be used in regimes and for models inaccessible to, e.g., perturbative scaling and Bethe ansatz, is provided open source.
For decades, Hermiticity was considered an immutable axiom of quantum mechanics, essential for ensuring real energies and unitary evolution. This perspective has shifted radically, driven by the realization that non-Hermitian Hamiltonians provide a powerful effective description of open quantum systems, granting access to unique phenomena such as Exceptional Points and the Non-Hermitian Skin Effect. In this Perspective, we chart the trajectory of this field, moving from its established foundations in single-particle, linear models to the emerging frontier of interacting many-body systems. We first clarify the physical origins of non-Hermitian dynamics, distinguishing between mean-field approximations, conditional "no-click" evolution, and exact Liouvillian dynamics. We then focus on the rich phenomenology arising from the interplay of non-Hermiticity and interactions. We discuss interaction-induced topological phases, the generalization of skin effects to the many-body Hilbert space, and the distinct signatures of dissipative quantum chaos and complexity. Finally, we highlight collective phenomena in nonlinear regimes, including skin solitons and dissipative phase transitions. We also comment on measurement-induced entanglement transitions and their relation to non-Hermitian spectra and topology. By synthesizing these diverse developments, we provide a roadmap for the future of non-Hermitian physics.
The study of quantum criticality has historically been confined to Hermitian systems, where closed-system approximations neglect environmental interactions. However, a significant portion of quantum phenomena occurs in open systems, necessitating a departure from this conventional framework. This paper explores non-Hermitian quantum criticality in open many-body systems, delving into the unique phases and phase transitions that emerge due to dissipation, gain, and other non-conservative interactions. We establish a comprehensive framework that integrates concepts from biorthogonal quantum mechanics, Liouvillian dynamics, and tensor network methods to characterize critical phenomena, including dynamical quantum phase transitions and the emergence of non-Hermitian quantum many-body scars. Through an extensive literature review, we highlight how the interplay of non-Hermiticity with many-body correlations leads to novel critical behaviors, distinct from their Hermitian counterparts, and in some cases, surprisingly resilient or even enhanced. We discuss the role of exceptional points as hallmarks of non-Hermitian systems, their influence on energy spectra, and their implications for criticality and topological invariants. Furthermore, we examine the methodologies for defining observables and statistical ensembles in these non-Hermitian settings, emphasizing the distinctions and occasional controversies that arise. The findings provide critical insights into the fundamental principles governing quantum criticality in dissipative environments and pave new avenues for engineering and understanding open quantum technologies, from quantum sensing to robust quantum information processing.
In this thesis, we explore various universal aspects of non-equilibrium quantum many-body systems in low dimensions. The study of non-equilibrium settings is rendered possible by the existence of exact solutions in the presence of dynamical symmetries on one hand and through modeling via non-Hermitian Hamiltonians on the other. Low dimensions further offer the chance to study the interplay between the associated phenomenological curiosities, such as topological scaling or generalized statistics, and non-equilibrium dynamics. Emphasis was placed on the effects of symmetries (such as PT-symmetry, continuous
O(2)-symmetry,
SO(2,1) dynamical symmetry), quantum statistics and spatio-temporal dimension, which are shown to largely capture the universality of our results. In particular, we have successively explored the critical properties and renormalization group features of paradigmatic non-Hermitian PT-symmetric field theories within the framework of the functional renormalization group. Particular emphasis was placed on the interplay between PT-symmetry breaking and continuous symmetries as well as topological scaling. In this spirit, this led to our study of the non-Hermitian O(2)-symmetric model and non-Hermitian sine-Gordon model as well as its extensions and the corresponding fermionic duals. Our analysis not only manages to reproduce predictions made by previous studies, it extends them to new methodologies and settings and produces new results. For the O(2)-symmetric problem, our study constitutes as far as we know the first analysis based on FRG and the first exploration of critical behavior altogether. While our results for the RG flows are consistent with previous perturbative studies, it extends them to non-perturbative regimes. In addition, the fixed-point structure of the flow near
d=4 is explored for the first time in the case of Hermitian interactions and features new non-trivial non-Hermitian fixed points, which are located in the PT-broken sector. This is as far as we know the first time that the occurrence of PT-broken fixed points is predicted in a quantum many-body system. %Given their relative stability, the latter may only accessible provided significant fine-tuning of parameters. %this might not be the case in dimensions other than 4 or in the presence of non-Hermitian interactions (which might also trigger the presence of other non-trivial fixed points). Our analysis of the modified sine-Gordon models (embodying the celebrated BKT transition paradigmatic of topological phase transitions) as well as its extensions (including the Pokrovsky-Talapov and massive sine-Gordon models) is consistent with previous studies and extends the latter to non-Hermitian settings and/or to non-perturbative regimes. It predicts novel non-perturbative PT-broken phases that prove robust to the presence of mass terms and finite soliton densities. In the fermionic setting, this phase was further shown to display tachyonic and monstronic propagators and its onset was associated with the interactions between fermions (traditionally corresponding to bosonic solitons) becoming repulsive. In this case, the corresponding extensions respectively model the coupling of fermions to background and dynamical gauge fields. In both cases, the occurrence of a PT-broken phase was shown to robustly occur and gauge fields were even predicted to provide a means of controlling the imaginary mass governing the asymptotic behaviors of propagators in the PT-broken sector. Finally, we have investigated the interplay between dynamical symmetries and generalized statistics in the harmonically confined driven Calogero model. Both periodic and adiabatic drives of the trapping potential were considered, with particular focus on observables such as the ground-state echo amplitudes/fidelity and pair-momentum distribution. The former are significantly suppressed with increasing exclusion statistics (embodied by the interaction strength
λ). Given a natural trap frequency commensurate with a periodic drive, we have predicted sharp suppressions of the likelihood of intermediate echoes (between those imposed by commensurability) in the presence of interactions as well as a substantial drop in the asymptotic ground-state fidelity reached following a slow drive through
ω=0. In a similar way, these observations are also reflected in the dramatic decline in long-range spatial coherence with
λ. This is exemplified, in the case of the slow drive through the gapless point, through a reduction of the central peak of the dynamical pair-momentum distribution with heightened exclusion and interactions. These results are consistent with our interpretation in terms of defects proliferation and hindered annihilation as a direct consequence of non-trivial statistics and interactions. Universality is transversely ubiquitous in our work. Our predictions of stable PT-broken phases and fixed points in non Hermitian many-body systems embody novel universal large-scale behaviors. Moreover, the robustness of these phases was proven under various perturbations, further increasing their universal character. Our additional forecast of critical exponents further captures the nature of these novel universal behaviors. In the case of our study of driven scale-invariant fluids, universality manifests in the sole dependence of our results on universal quantities: the exponent
zν which encompasses the adiabatic limit of more complex drives through the gapless point; the effective statistics parameter
λ of the degrees of freedom; the characteristic Mathieu exponent
r associated with a periodic drive. Conversely it also manifests itself in the independence of our results on irrelevant microscopic details such as the driving rate
δ of the slow drive or the amplitude
h or base frequency
ω0 of the periodic drive. The universality of our results is also embodied by the fact that our core interpretations in terms of defect proliferation are reflected through various observables such as fidelities/echo amplitudes and pair-momentum distributions. All in all, this thesis has produced several novel results regarding universal aspects of non-equilibrium quantum many-body systems in low dimensions, which have turned into two accepted publications in high-impact journals and more in preparation. We hope that this work represents a meaningful step forward in our understanding of collective emergent phenomena in low-dimensional systems driven out of thermal equilibrium.
Critical transition points between symmetry-broken phases are usually characterized as fixed points in the renormalization group (RG) theory. We show that, following the standard Wilsonian procedure that iteratively traces out the large-momentum modes, this well-known fact can easily break down in non-Hermitian systems under open boundary conditions. Based on non-Hermitian Su-Schrieffer-Heeger-type models, we propose a RG scheme based on real-space decimation. We provide concrete examples and an analytic proof to show that the real-space scheme always respects the critical points under RG transformations. We investigate the correlation length of order parameters in the vicinity of critical points. Its divergence and scaling behavior remain intact under the real-space decimation, indicating that our method can be successfully applied to study the critical phenomena in non-Hermitian systems. These results suggest that the real-space RG method will find versatile applications in interacting non-Hermitian quantum systems.