It is known that the U ( 2 ) Wess-Zumino-Witten model is dual to the free fermion theory in two dimensions via non-Abelian bosonization. While it is decomposed into the S U ( 2 ) Wess-Zumino-Witten model and a free compact boson, the former is believed to be equivalent to the O ( 3 ) nonlinear sigma model with the theta term at θ = π . In this work, we reexamine this duality through the lens of nonperturbative renormalization group (RG) flow. We analyze the RG flow structure of the O ( 3 ) nonlinear sigma model with the theta term in two dimensions using the functional renormalization group. Our results reveal a nontrivial fixed point with a nonzero value of the topological coupling. The scaling dimensions (critical exponents) at this fixed point suggest the realization of a duality between the O ( 3 ) nonlinear sigma model with the theta term and the free fermion theory, indicating that these models belong to the same universality class.
The two-dimensional
O(N) nonlinear sigma model (NLSM) is asymptotically free for
N>2: it exhibits neither a nontrivial fixed point nor spontaneous symmetry-breaking. Here we show that a nontrivial fixed point generically does exist in the complex coupling plane and is described by a complex conformal field theory (CCFT). This CCFT fixed point is generic in the sense that it has a single relevant singlet operator, and is thus expected to arise in any non-Hermitian model with
O(N) symmetry upon tuning a single complex parameter. We confirm this prediction numerically by locating the CCFT at
N=3 in two non-Hermitian spin-1 antiferromagnetic Heisenberg chains, and in a non-Hermitian spin-
1/2 ladder, finding good agreement between the complex central charge and scaling dimensions and those obtained by analytic continuation of real fixed points from
N≤2. We further construct a realistic Lindbladian for a spin-1 chain whose no-click dynamics are governed by the non-Hermitian Hamiltonian realizing the CCFT. Since the CCFT vacuum is the eigenstate with the smallest decay rate, the system naturally relaxes under dissipative dynamics toward a CFT state, thus providing a route to preparing long-range entangled states through engineered dissipation.
Motivated by the prospect of quantum simulation of quantum field theories, we formulate the
O(N) nonlinear sigma model as a ``qubit'' model with an (
N+1)-dimensional local Hilbert space at each lattice site. Using an efficient worm algorithm in the worldline formulation, we demonstrate that the model has a second-order critical point in (
2+1) dimensions, where the continuum physics of the nontrivial
O(N) Wilson-Fisher fixed point is reproduced. We compute the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$ for the
O(N) qubit models up to
N=8, and find excellent agreement with known results in literature from various analytic and numerical techniques for the
O(N) Wilson-Fisher universality class. Our models are suited for studying
O(N) nonlinear sigma models on quantum computers up to
N=8 in
d=2, 3 spatial dimensions.
Abstract Chapter 6 describes how the perturbative renormalization group (RG) of quantum field theory has made it possible to derive universal properties of continuous macroscopic phase transitions. The RG initially emerged as a consequence of the necessity to cancel infinities that appear in the perturbative expansion (the renormalization procedure) and the possibility of defining the parameters of the renormalized theory at different momentum scales. Although the field theory RG is now understood to be an asymptotic form, it has made it possible to confirm the Wilson–Fisher fixed point and led to an understanding of universality for a large class of critical phenomena. In the framework of dimensional continuation, zeros of RG beta functions, which correspond to Wilson–Fisher’s fixed points, have made it possible to recover Wilson–Fisher’s epsilon expansion, proving scaling relations and calculating critical exponents. Series summation methods have then been used to generate precise values of exponents.
Motivated by the realization of
SU(N) antiferromagnetism with the multirow representations in cold-atom physics, we studied its low-energy nonlinear sigma model defined on the Grassmann manifold $\text{U}(N)/\text{U}(m)\text{U}(N\ensuremath{-}m)$ using the complex projective presentation, which is a direct generalization of the widely studied ${\mathrm{CP}}^{N\ensuremath{-}1}$ model (corresponding to
m=1). Using the large-
N expansion up to the first order of
1/N with fixed
m, and in space dimension
2<d<4, we obtain the critical exponents, including in particular the gauge-independent exponents that are directly relevant for experimental measurements. These exponents are found to depend only on
m/N, indicating that a larger
m effectively reduces
N and thus causes stronger fluctuations about the saddle point.
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.