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Universal aspects of non-equilibrium quantum many-body systems in low dimension

Benjamin Liégeois

Published Jan 1, 2026• 0 citations
Publisher

Abstract

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 $\lambda$). 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 $\omega=0$. In a similar way, these observations are also reflected in the dramatic decline in long-range spatial coherence with $\lambda$. 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 $z\nu$ which encompasses the adiabatic limit of more complex drives through the gapless point; the effective statistics parameter $\lambda$ 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 $\delta$ of the slow drive or the amplitude $h$ or base frequency $\omega_0$ 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.

DOI
10.3929/ethz-c-000800840

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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)O(2)O(2)-symmetry, SO(2,1)SO(2,1)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=4d=4d=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\lambdaλ). 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\omega=0ω=0. In a similar way, these observations are also reflected in the dramatic decline in long-range spatial coherence with λ\lambdaλ. 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 zνz\nuzν which encompasses the adiabatic limit of more complex drives through the gapless point; the effective statistics parameter λ\lambdaλ of the degrees of freedom; the characteristic Mathieu exponent rrr 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\deltaδ of the slow drive or the amplitude hhh or base frequency ω0\omega_0ω0​ 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.
The theory of statistical mechanics provides a powerful conceptual framework within which the relevant (macroscopic) features of systems at equilibrium can be described. As there is currently no equivalent capable of encompassing the much richer class of non-equilibrium phenomena, research in this direction proceeds mainly on an instance-by-instance basis. The aim of this Thesis is to describe in some detail three such attempts, which involve different dynamical aspects of classical and quantum systems. As summarised below, each of the last three Chapters of this document delves into one of these different topics, while Chapter 2 provides a brief introduction on the study of non-equilibrium dynamics. In Chapter 3 we investigate the purely relaxational dynamics of classical critical ferromagnetic systems in the proximity of surfaces, paying particular attention to the effects that the latter induce on the early stages of the evolution following an abrupt change in the temperature of the sample. When the latter ends close enough to the critical value which separates the paramagnetic from the ferromagnetic phase, it effectively introduces a temporal boundary which can be treated as if it were a surface. Within this picture, we highlight the emergence of novel effects near the effective edge formed by the intersection of the two spatial and temporal boundaries. Our findings are apparently in disagreement with previous predictions which were based on the assumption that the presence of such an edge would not affect the scaling behaviour of observables; in order to explain this discrepancy, we propose an alternative for the original power-counting argument which, at least, correctly predicts the emergence of novel field-theoretical divergences in our one-loop calculations. We show that said singularities are associated with the scaling at the edge. Moreover, by encoding our findings in a boundary renormalisation group framework, we argue that the new predicted behaviour represents a universal feature associated to the short-distance expansion of the order parameter of the transition near the edge; we also calculate explicitly its anomalous dimension at the first-order in a dimensional expansion. As a qualitative feature, this anomalous dimension depends on the type of phase transition occurring at the surface. We exploit this fact in order to provide numerical support to our predictions via Monte Carlo simulations of the dynamical behaviour of a three-dimensional Ising model. The main results reported in Chap. 3 have appeared in Ref. [1]. In Chapter 4 we revisit the Euclidean mapping to imaginary times which has been recently proposed [2, 3] as an alternative for approaching the problem of quantum dynamics following a quench. This is expected to allow one to reformulate the original problem as a static one confined in a film geometry. We show that this interpretation actually holds only if the initial state of the dynamics is pure. Statistical mixtures, instead, intertwine the effects due to the two boundaries, which therefore cannot be regarded as being independent. We emphasize that, although the aforementioned reinterpretation as a confined static problem fails, one is still able, in principle, to write down and solve the corresponding equations. We also discuss in some detail the relation between this approach and the real-time field-theoretical one which makes use of the two-time Keldysh contour. For this purpose, we study the analytical structure of relevant observables — such as correlation functions — in the complex plane of times, identifying a subdivision of this domain into several sectors which depend on the ordering of the imaginary parts of the involved time coordinates. Within each of these subdomains, the analytic continuation to the real axis provides in principle a different result. This feature allows one to reconstruct from the Euclidean formalism all possible non-time-ordered functions, which in particular include all those which can be calculated via the Keldysh two-time formalism. Moreover, we give a prescription on how to retrieve response functions, discussing some simple examples and rationalising some recent numerical data obtained for one of these observables in a one-dimensional quantum Ising chain [4]. We also highlight the emergence of a light-cone effect fairly similar to the one previously found for correlation functions [2], which therefore provides further confirmation to the fact that information travels across the system in the form of the entanglement of quasi-particles produced by the quenching procedure. We have reported part of this analysis in Ref. [5]. Chapter 5 presents part of our recent work on effective relaxation in quantum systems following a quench and on the observed prethermalisation. We analyse the effects caused by the introduction of a long-range integrability-breaking interaction in the early stages of the dynamics of an otherwise integrable quantum spin chain following a quench in the magnetic field. By employing a suitable transformation, we redefine the theory in terms of a fully-connected model of hard-core bosons, which allows us to exploit the (generically) low density of excitations for rendering our model exactly solvable (in a numerical sense, i.e., by numerically diagonalising an exact matrix). We verify that, indeed, as long as the parameters of the quench are not too close to the critical point, the low-density approximation captures the dynamical features of the elementary operators, highlighting the appearance of marked plateaux in their dynamics, which we reinterpret as the emergence of a prethermal regime in the original model. As expected, the latter behaviour is reflected also on extensive observables which can be constructed as appropriate combinations of the mode populations. For these quantities, the typical approach to the quasi-stationary value is algebraic with exponent a ≈ 3, independently of the size of the system, the strength of the interaction and the amplitude of the magnetic field (as long as it is kept far from the critical point). The plateaux mentioned above last until a recurrence time — which can be approximately identified with tR ≈ N/2 for single modes and t′R ≈ N/4 for extensive quantities — after which quantum oscillations due to the finite size of the chain reappear. Our procedure allows us to shed some light over prethermal features without having to considerably limit the size of the system, which we can choose to be quite large, as we discuss in Ref. [6].
In this thesis we study three examples of interacting many-body systems undergoing a non equilibrium time evolution. Firstly we consider the time evolution in an integrable system: the sine-Gordon field theory in the repulsive regime. We will focus on the one point function of the semi-local vertex operator eiβφ(x)/2 on a specific class of initial states. By analytical means we show that the expectation value considered decays exponentially to zero at late times and we determine the decay time. The method employed is based on a form-factor expansion and uses the "Representative Eigenstate Approach" of Ref. [73] (a.k.a. "Quench Action"). In a second example we study the time evolution in models close to "special" integrable points characterised by hidden symmetries generating infinitely many local conservation laws that do not commute with one another, in addition to the infinite commuting family implied by integrability. We observe that both in the case where the perturbation breaks the integrability and when it breaks only the additional symmetries maintaining integrability, the local observables show a crossover behaviour from an initial to a final quasi stationary plateau. We investigate a weak coupling limit, identify a time window in which the effects of the perturbations become significant and solve the time evolution through a mean-field mapping. As an explicit example we study the XYZ spin-1/2 chain with additional perturbations that break integrability. Finally, we study the effects of integrability breaking perturbations on the non-equilibrium evolution of more general many-particle quantum systems, where the unperturbed integrable model is generic. We focus on a class of spinless fermion models with weak interactions. We employ equation of motion techniques that can be viewed as generalisations of quantum Boltzmann equations. We benchmark our method against time dependent density matrix renormalisation group computations and find it to be very accurate as long as interactions are weak. For small integrability breaking, we observe robust prethermalisation plateaux for local observables on all accessible time scales. Increasing the strength of the integrability breaking term induces a "drift" away from the prethermalisation plateaux towards thermal behaviour. We identify a time scale characterising this crossover.
In this thesis we examine universal scaling properties of strongly-correlated systems near and far from equilibrium. We discuss quantum phase transitions at vanishing temperature, multicritical and dynamic critical behavior near thermal equilibrium, and scaling properties of nonequilibrium steady states. We employ nonperturbative methods including the functional renormalization group as well as Monte Carlo simulations. A general outline of the functional renormalization group is given in the introductory chapters. \n \nIn the first part of this thesis, we investigate spinless fermions on the honeycomb lattice interacting via short-range repulsive interactions. Such a system can be seen as a simple model for suspended graphene. The short-range interactions control the ground state properties of the system that may lead to a chiral phase transition from the semimetal to the charge density wave (CDW)/Kekulé ordered state. We determine the universal scaling properties at the chiral transition, and establish the presence of large anomalous dimensions indicating the importance of strong fluctuations. \n \nThe competition of two nonvanishing order parameters and their corresponding multicritical behavior are investigated in the subsequent chapter. We characterize the bicritical and tetracritical behavior in the purely bosonic O(N_1) + O(N_2) symmetric model and comment on possible applications to condensed-matter and high-energy physics. \n \nIn the following chapter we discuss the long-time relaxational behavior at criticality of an order parameter with O(N) symmetry coupled to an additional conserved density. We find an anomalous diffusion phase with new dynamic scaling properties. Using the functional renormalization group we determine the complete dynamic critical behavior of the model in 2 < d < 4 dimensions and compare our results to experiments. \n \nFinally, we investigate the scaling properties of stationary states far from equilibrium. At the example of the one-dimensional Burgers’ equation we develop a novel approach to hydrodynamic turbulence using lattice Monte Carlo methods. We apply these techniques to determine the statistical properties of small-scale fluctuations in this model and identify the anomalous scaling behavior. \n
Topological phases of matter are understood to be characterized by particular configurations of entanglement encoded within the ground states of many-body quantum systems. In this thesis, we discuss how novel phenomena associated with topological phases can arise in systems that are driven out of equilibrium or coupled to their surroundings, which are not described by ground state physics. Firstly, we consider systems undergoing unitary time evolution due to some external driving. We show how the entanglement features of the time-dependent wavefunction can be topologically characterized, and demonstrate that these topological properties are reflected in the dynamics of various quantities usually associated with topological phases in equilibrium. We introduce a new non-equilibrium topological classification scheme which can be used to predict whether or not the topological features of a given system will be preserved under time evolution. In brief, this classification captures the fact that certain symmetries (namely antiunitary symmetries) are inevitably broken once the system is driven out of equilibrium; thus any topological phenomena protected by such symmetries cannot be expected to persist in non-equilibrium scenarios. Secondly, we investigate how topologically non-trivial systems are affected by dissipative effects, i.e.~coupling to an external environment. We demonstrate that system-environment interactions facilitate processes in the system that effectively break any antiunitary symmetries, regardless of the symmetries of the microscopic Hamiltonian. Accordingly, those phases that were shown to be unstable against time-dependent driving are also fragile against coupling to an environment. To illustrate the consequences of this fragility, we consider the effects of dissipation on the coherence properties of topological bound states, as well as the conductance properties of chiral and helical topological edge modes. We find that the decoherence rate of the former and the deviation from quantized conductivity of the latter are (not) exponentially suppressed in the inverse temperature when the phase in question is protected by unitary (antiunitary) symmetries. These results regarding open systems can be connected to the same non-equilibrium classification developed in the context of isolated systems undergoing unitary dynamics. Our findings highlight a distinction between topological phases that are either robust or fragile against non-equilibrium effects and/or system-environment coupling. The ramifications for the use of such systems in quantum technologies are discussed.
Quantum matter emerges from the interplay of fluctuations, topology, and entanglement, which - in equilibrium - governs quantized transport, universal criticality, and topological classification. Non-Hermitian systems, widely explored in platforms ranging from electric circuits to photonics, are intrinsically out-of-equilibrium, and display fundamentally new phenomena, including complex spectra, spectral winding, exceptional topology, and non-unitary dynamics. A central challenge is understanding how the complex single-particle spectrum governs universal many-body behavior. We introduce a symmetry-protected dynamical topological index derived directly from the complex spectrum. Through the lens of algebraic topology, more specifically Morse theory, we identify critical points in the spectrum with topological defects, whose curvature and stability are protected under continuous deformations. This links spectral geometry to many-body observables, unifying non-Hermitian band topology, entanglement, and transport. We demonstrate that non-Hermitian quantum criticality in non-interacting systems is controlled by gain-and-loss-selected non-equilibrium steady states, which dynamically generate an emergent imaginary Fermi surface whose Fermi points host scale-invariant gapless modes with logarithmic entanglement scaling and algebraic correlations. Our work establishes a unified framework for non-Hermitian quantum matter, connecting spectral topology to Morse theory, revealing a topological foundation of non-equilibrium quantum criticality.
We study the dynamics in low dimensional correlated quantum systems. Advances in this thesis are three-fold. First, as a methodical point we extend the real-time formulation of the functional renormalization group approach introduced in [1]. Describing the transient time dependence of a given microscopic model prepared in an initial non-equilibrium state poses a formidable challenge in the presence of interaction. We successfully extend the functional renormalization group approach of [1] to finite temperature as well as extended interacting geometries. Furthermore, for both extensions we test the validity of the perturbatively motivated treatment by explicitly benchmarking to complementary methods and find that for not too large interactions the functional renormalization group proves to be quantitatively accurate. Second, we examine the non-equilibrium dynamics of open as well as closed quantum systems in the presence of correlations. We study a prominent prototype model for coherence in small quantum systems coupled to dissipative environments, namely the spin-boson model. We reveal that prevailing literature does not cover the full complexity of the dynamics encountered. The relaxation dynamics formerly classified as coherent and incoherent should be refined according to the results presented in this thesis: we propose a subdivision of the coherent regime into partially and asymptotically coherent. For the former the coherence signature of the dynamics is restricted only to intermediate times, while in the latter it extends in the asymptotic regime. Interestingly, we explicitly show that elevating temperature might be conducive to coherence, which is opposite to the expected behavior. For those closed quantum systems known to fall into the so called Luttinger Liquid universality class in equilibrium, we provide evidence that the equilibrium concept of universality seemingly carries over to the non-equilibrium steady-state. This is rather counter-intuitive, as universality is usually attributed to the low energy degrees of freedom only. Third, we study the thermoelectric transport properties of quantum dots. Quantum dots are promising candidates for highly efficient thermoelectric devices as they are not bounded by the Wiedemann-Franz law which strongly limits the efficiency of ordinary bulk systems. For this we study a charge fluctuating model and a model which additionally allows for spin fluctuations and phononic degrees of freedom. We find that in certain regimes quantum dots show very high efficiency when operated as power engines. [1] D. M. Kennes, ‘A Renormalization Group Approach to the Time Evolution of Correlated Quantum Dots’ Master’s thesis, RWTH Aachen (2011).