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Register a new userNonperturbative renormalization group treatment of amplitude fluctuations for $|varphi|^4$ topological phase transitions
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The study of the Berezinskii-Kosterlitz-Thouless transition in two-dimensional $|varphi|^4$ models can be performed in several representations, and the amplitude-phase (AP) Madelung parametrization is a natural way to study the contribution of density fluctuations to nonuniversal quantities. We introduce a functional renormalization group scheme in AP representation where amplitude fluctuations are integrated first to yield an effective sine-Gordon model with renormalized superfluid stiffness. By a mapping between the lattice XY and continuum $|varphi|^4$ models, our method applies to both on equal footing. Our approach correctly reproduces the existence of a line of fixed points and of universal thermodynamics and it allows to estimate universal and nonuniversal quantities of the two models, finding good agreement with available Monte Carlo results. The presented approach is flexible enough to treat parameter ranges of experimental relevance.
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Understanding the robustness of topological phases of matter in the presence of interactions poses a difficult challenge in modern condensed matter, showing interesting connections to high energy physics. In this work, we leverage these connections to present a complete analysis of the continuum long-wavelength description of a generic class of correlated topological insulators: Wilson-Hubbard topological matter. We show that a Wilsonian renormalization group (RG) approach, combined with the so-called topological Hamiltonian, provide a quantitative route to understand interaction-induced topological phase transitions that occur in Wilson-Hubbard matter. We benchmark two-loop RG predictions for a quasi-1D Wilson-Hubbard model by means of exhaustive numerical simulations based on matrix product states (MPS). The agreement of the RG predictions with MPS simulations motivates the extension of the RG calculations to higher-dimensional topological insulators.
Using the nonperturbative renormalization group, we study the existence of bound states in the symmetry-broken phase of the scalar $phi^4$ theory in all dimensions between two and four and as a function of the temperature. The accurate description of the momentum dependence of the two-point function, required to get the spectrum of the theory, is provided by means of the Blaizot--Mendez-Galain--Wschebor approximation scheme. We confirm the existence of a bound state in dimension three, with a mass within 1% of previous Monte-Carlo and numerical diagonalization values.
We show that four-dimensional systems may exhibit a topological phase transition analogous to the well-known Berezinskii-Kosterlitz-Thouless vortex unbinding transition in two-dimensional systems. The realisation of an engineered quantum system, where the predicted phase transition shall occur, is also presented. We study a suitable generalization of the sine-Gordon model in four dimensions and the renormalization group flow equation of its couplings, showing that the critical value of the frequency is the square of the corresponding value in $2D$. The value of the anomalous dimension at the critical point is determined ($eta=1/32$) and a conjecture for the universal jump of the superfluid stiffness ($4/pi^2$) presented.
It is commonly believed that a massive real scalar field $phi$ only mediates short-range interactions on the scale of its Compton wavelength via the Yukawa potential. However, in the nonperturbative regime of nonlinear self coupling, $phi$ can also mediate larger scale interactions. Moreover, the classical potential, namely, the static configuration of $phi$ in the presence of an external source, is not always unique for given boundary conditions. In this paper, a complete set of finite-energy potentials (FEPs) induced by a Gaussian source is identified in one, two, and three spatial dimensions when the nonlinearity is of the Mexican-hat type, which is often prescribed to induce spontaneous symmetry breaking. In the size-strength parameter space of the source, phase boundaries are mapped out, across which the number of FEPs differ. Additionally, softer phase transitions are delineated according to whether the potential exhibits a light horizon at which $phi$ vanishes. The light horizon is of physical significance when $phi$ couples with other particles. For example, when $phi$ is the Higgs field, all elementary particles become massless at the light horizon. It is remarkable that white dwarfs and neutron stars are potentially in a phase where light horizons exist, whose outer radii are a few times the star sizes. Moreover, suppose elementary fermions of mass greater than $sim10^3$ GeV exist, then they may also be surrounded by light horizons with radii comparable to the Higgs Compton wavelength. Finally, nonperturbative states may also be realized in condensed matter systems, wherein phase transitions are controllable using localized sources.
Renormalization group calculations are used to give exact solutions for rigidity percolation on hierarchical lattices. Algebraic scaling transformations for a simple example in two dimensions produce a transition of second order, with an unstable critical point and associated scaling laws. Values are provided for the order parameter exponent $beta = 0.0775$ associated with the spanning rigid cluster and also for $d u = 3.533$ which is associated with an anomalous lattice dimension $d$ and the divergence in the correlation length near the transition. In addition we argue that the number of floppy modes $F$ plays the role of a free energy and hence find the exponent $alpha$ and establish hyperscaling. The exact analytical procedures demonstrated on the chosen example readily generalize to wider classes of hierarchical lattice.
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