Nonperturbative Renormalization Group Approach Research Articles

Importance truncation for the in-medium similarity renormalization group

Ab initio nuclear many-body frameworks require extensive computational resources, especially when targeting heavier nuclei. Importance-truncation (IT) techniques allow one to significantly reduce the dimensionality of the problem by neglecting unimportant contributions to the solution of the many-body problem. In this work, we apply IT methods to the nonperturbative in-medium similarity renormalization group (IMSRG) approach and investigate the induced errors for ground-state energies in different mass regimes based on different nuclear Hamiltonians. We study various importance measures, which define the IT selection, and identify two measures that perform best, resulting in only small errors to the full IMSRG(2) calculations even for sizable compression ratios. The neglected contributions are accounted for in a perturbative way and serve as an estimate of the IT-induced error. Overall we find that the IT-IMSRG(2) performs well across all systems considered, while the largest compression ratios for a given error can be achieved when using soft Hamiltonians and for large single-particle bases.

Wrinkling transition in quenched disordered membranes at two loops.

We investigate the flat phase of quenched disordered polymerized membranes by means of a two-loop, weak-coupling computation performed near their upper critical dimension D_{uc}=4, generalizing the one-loop computation of Morse etal. [D. C. Morse etal., Phys. Rev. A 45, R2151 (1992)PLRAAN1050-294710.1103/PhysRevA.45.R2151; D. C. Morse and T. C. Lubensky, Phys. Rev. A 46, 1751 (1992)PLRAAN1050-294710.1103/PhysRevA.46.1751]. Our work confirms the existence of the finite-temperature, finite-disorder wrinkling transition, which has been recently identified by Coquand etal. [O. Coquand etal., Phys. Rev. E 97, 030102(R) (2018)2470-004510.1103/PhysRevE.97.030102] using a nonperturbative renormalization group approach. We also point out ambiguities in the two-loop computation that prevent the exact identification of the properties of the novel fixed point associated with the wrinkling transition, which very likely requires a three-loop order approach.

Hybrid and quark star matter based on a nonperturbative equation of state

With the recent dawn of the multi-messenger astronomy era a new window has opened to explore the constituents of matter and their interactions under extreme conditions. One of the pending challenges of modern physics is to probe the microscopic equation of state (EoS) of cold and dense matter via macroscopic neutron star observations such as their masses and radii. Still unanswered issues concern the detailed composition of matter in the core of neutron stars at high pressure and the possible presence of e.g. hyperons or quarks. By means of a non-perturbative functional renormalization group approach the influence of quantum and density fluctuations on the quark matter EoS in $\beta$-equilibrium is investigated within two- and three-flavor quark-meson model truncations and compared to results obtained with common mean-field approximations where important fluctuations are usually ignored. We find that they strongly impact the quark matter EoS.

Universal behaviors in the wrinkling transition of disordered membranes.

The wrinkling transition experimentally identified by Mutz etal. [Phys. Rev. Lett. 67, 923 (1991)PRLTAO0031-900710.1103/PhysRevLett.67.923] and then thoroughly studied by Chaieb etal. [Phys. Rev. Lett. 96, 078101 (2006)]PRLTAO0031-900710.1103/PhysRevLett.96.078101 in partially polymerized lipid membranes is reconsidered. One shows that the features associated with this transition, notably the various scaling behaviors of the height-height correlation functions that have been observed, are qualitatively and quantitatively well described by a recent nonperturbative renormalization group approach to quenched disordered membranes by Coquand etal. [Phys. Rev E 97, 030102(R) (2018)]2470-004510.1103/PhysRevE.97.030102. As these behaviors are associated with fixed points of renormalization group transformations they are universal and should also be observed in, e.g., defective graphene and graphene-like materials.

Kardar-Parisi-Zhang equation with temporally correlated noise: A nonperturbative renormalization group approach.

We investigate the universal behavior of the Kardar-Parisi-Zhang (KPZ) equation with temporally correlated noise. The presence of time correlations in the microscopic noise breaks the statistical tilt symmetry, or Galilean invariance, of the original KPZ equation with δ-correlated noise (denoted SR-KPZ). Thus, it is not clear whether the KPZ universality class is preserved in this case. Conflicting results exist in the literature, some advocating that it is destroyed even in the limit of infinitesimal temporal correlations, while others find that it persists up to a critical range of such correlations. Using nonperturbative and functional renormalization group techniques, we study the influence of two types of temporal correlators of the noise: a short-range one with a typical timescale τ, and a power-law one with a varying exponent θ. We show that for the short-range noise with any finite τ, the symmetries (the Galilean symmetry, and the time-reversal one in 1+1 dimension) are dynamically restored at large scales, such that the long-distance and long-time properties are governed by the SR-KPZ fixed point. In the presence of a power-law noise, we find that the SR-KPZ fixed point is still stable for θ below a critical value θ_{th}, in accordance with previous renormalization group results, while a long-range fixed point controls the critical scaling for θ>θ_{th}, and we evaluate the θ-dependent critical exponents at this long-range fixed point, in both 1+1 and 2+1 dimensions. While the results in 1+1 dimension can be compared with previous studies, no other prediction was available in 2+1 dimension. We finally report in 1+1 dimension the emergence of anomalous scaling in the long-range phase.

Benchmarking the nonperturbative functional renormalization group approach on the random elastic manifold model in and out of equilibrium

Criticality in the class of disordered systems comprising the random-field Ising model (RFIM) and elastic manifolds in a random environment is controlled by zero-temperature fixed points that must be treated through a functional renormalization group (RG). We apply the nonperturbative functional RG approach that we have previously used to describe the RFIM in and out of equilibrium (Balog et al 2018 Phys. Rev. B 97 094204) to the simpler and by now well-studied case of the random elastic manifold model. We recover the main known properties, critical exponents and scaling functions, of both the pinned phase of the manifold at equilibrium and the depinning threshold in the athermally and quasi-statically driven case for any dimension . This successful benchmarking of our theoretical approach gives strong support to the results that we have previously obtained for the RFIM, in particular concerning the distinct universality classes of the equilibrium and out-of-equilibrium (hysteresis) critical points below a critical dimension .

Stationary, isotropic and homogeneous two-dimensional turbulence: a first non-perturbative renormalization group approach

We study the statistical properties of stationary, isotropic and homogeneous turbulence in two-dimensional (2D) flows, focusing on the direct cascade, that is on large wave-numbers compared to the integral scale, where both energy and enstrophy are provided to the fluid. Our starting point is the 2D Navier–Stokes equation in the presence of a stochastic forcing, or more precisely the associated field theory. We unveil two extended symmetries of the Navier–Stokes action which were not yet identified, one related to time-dependent (or time-gauged) shifts of the response fields and existing in both 2D and 3D, and the other to time-gauged rotations and specific to 2D flows. We derive the corresponding Ward identities, and exploit them within the non-perturbative renormalization group formalism, and the large wave-number expansion scheme developed in Tarpin et al (2018 Phys. Fluids 30 055102). We consider the flow equation for a generalized n-point correlation function, and calculate its leading order term in the large wave-number expansion. At this order, the resulting flow equation can be closed exactly. We solve the fixed point equation for the 2-point function, which yields its explicit time dependence, for both small and large time delays in the stationary turbulent state. On the other hand, at equal times, the leading order term vanishes, so we compute the next-to-leading order term. We find that the flow equations for equal-time n-point correlation functions are not fully constrained by the set of extended symmetries, and discuss the consequences.

Nonperturbative renormalization-group approach preserving the momentum dependence of correlation functions

We present an approximation scheme of the nonperturbative renormalization group that preserves the momentum dependence of correlation functions. This approximation scheme can be seen as a simple improvement of the local potential approximation (LPA) where the derivative terms in the effective action are promoted to arbitrary momentum-dependent functions. As in the LPA, the only field dependence comes from the effective potential, which allows us to solve the renormalization-group equations at a relatively modest numerical cost (as compared, e.g., to the Blaizot--Mend\'ez-Galain--Wschebor approximation scheme). As an application we consider the two-dimensional quantum O($N$) model at zero temperature. We discuss not only the two-point correlation function but also higher-order correlation functions such as the scalar susceptibility (which allows for an investigation of the ``Higgs'' amplitude mode) and the conductivity. In particular, we show how, using Pad\'e approximants to perform the analytic continuation $i{\ensuremath{\omega}}_{n}\ensuremath{\rightarrow}\ensuremath{\omega}+i{0}^{+}$ of imaginary frequency correlation functions $\ensuremath{\chi}(i{\ensuremath{\omega}}_{n})$ computed numerically from the renormalization-group equations, one can obtain spectral functions in the real-frequency domain.

8p-K-4 非摂動くりこみ群によるゲージ理論の解析

8p-K-4 非摂動くりこみ群によるゲージ理論の解析

Glassy phase in quenched disordered crystalline membranes.

We investigate the flat phase of D-dimensional crystalline membranes embedded in a d-dimensional space and submitted to both metric and curvature quenched disorders using a nonperturbative renormalization group approach. We identify a second-order phase transition controlled by a finite-temperature, finite-disorder fixed point unreachable within the leading order of ε=4-D and 1/d expansions. This critical point divides the flow diagram into two basins of attraction: that associated with the finite-temperature fixed point controlling the long-distance behavior of disorder-free membranes and that associated with the zero-temperature, finite-disorder fixed point. Our work thus strongly suggests the existence of a whole low-temperature glassy phase for quenched disordered crystalline membranes and, possibly, for graphene and graphene-like compounds.

Kosterlitz-Thouless signatures in the low-temperature phase of layered three-dimensional systems

We study the quasi-two-dimensional quantum O(2) model, a quantum generalization of the Lawrence-Doniach model, within the nonperturbative renormalization-group approach and propose a generic phase diagram for layered three-dimensional systems with an O(2)-symmetric order parameter. Below the transition temperature we identify a wide region of the phase diagram where the renormalization-group flow is quasi-two-dimensional for length scales smaller than a Josephson length $l_J$, leading to signatures of Kosterlitz-Thouless physics in the temperature dependence of physical observables. In particular the order parameter varies as a power law of the interplane coupling with an exponent which depends on the anomalous dimension (itself related to the stiffness) of the strictly two-dimensional low-temperature Kosterlitz-Thouless phase.

Influence of phonon-assisted tunneling on the linear thermoelectric transport through molecular quantum dots

We investigate the effect of vibrational degrees of freedom on the linear thermoelectric transport through a single-level quantum dot described by the spinless Anderson-Holstein impurity model. To study the effects of strong electron-phonon coupling, we use the nonperturbative numerical renormalization group approach. We also compare our results, at weak to intermediate coupling, with those obtained by employing the functional renormalization group method, finding good agreement in this parameter regime. When applying a gate voltage at finite temperatures, the inelastic scattering processes, induced by phonon-assisted tunneling, result in an interesting interplay between electrical and thermal transport. We explore different parameter regimes and identify situations for which the thermoelectric power as well as the dimensionless figure of merit are significantly enhanced via a Mahan-Sofo type of mechanism. We show, in particular, that this occurs at strong electron-phonon coupling and in the antiadiabatic regime.

Dimensional reduction and its breakdown in the driven random-field O(N) model

The critical behavior of the random field $O(N)$ model driven at a uniform velocity is investigated at zero-temperature. From naive phenomenological arguments, we introduce a dimensional reduction property, which relates the large-scale behavior of the $D$-dimensional driven random field $O(N)$ model to that of the $(D-1)$-dimensional pure $O(N)$ model. This is an analogue of the dimensional reduction property in equilibrium cases, which states that the large-scale behavior of $D$-dimensional random field models is identical to that of $(D-2)$-dimensional pure models. However, the dimensional reduction property breaks down in low enough dimensions due to the presence of multiple meta-stable states. By employing the non-perturbative renormalization group approach, we calculate the critical exponents of the driven random field $O(N)$ model near three-dimensions and determine the range of $N$ in which the dimensional reduction breaks down.

Bose polarons in ultracold atoms in one dimension: beyond the Fröhlich paradigm

Mobile impurity atoms immersed in Bose–Einstein condensates provide a new platform for exploring Bose polarons. Recent experimental advances in the field of ultracold atoms make it possible to realize such systems with highly tunable microscopic parameters and to explore equilibrium and dynamical properties of polarons using a rich toolbox of atomic physics. In this paper we present a detailed theoretical analysis of Bose polarons in one-dimensional systems of ultracold atoms. By combining a non-perturbative renormalization group approach with numerically exact diffusion Monte Carlo calculations we obtain not only detailed numerical results over a broad range of parameters but also qualitative understanding of different regimes of the system. We find that an accurate description of Bose polarons requires the inclusion of two-phonon scattering terms which go beyond the commonly used Fröhlich model. Furthermore we show that when the Bose gas is in the strongly interacting regime, one needs to include interactions between the phonon modes. We use several theoretical approaches to calculate the polaron energy and its effective mass. The former can be measured using radio-frequency spectroscopy and the latter can be studied experimentally using impurity oscillations in a harmonic trapping potential. We compare our theoretical results for the effective mass to the experiments by Catani et al (2012 Phys. Rev. A 85 023623). In the weak-to-intermediate coupling regimes we obtain excellent quantitative agreement between theory and experiment, without any free fitting parameter. We supplement our analysis by full dynamical simulations of polaron oscillations in a shallow trapping potential. We also use our renormalization group approach to analyze the full phase diagram and identify regions that support repulsive and attractive polarons, as well as multi-particle bound states.

Fluctuation-induced continuous transition and quantum criticality in Dirac semimetals

We establish a scenario where fluctuations of new degrees of freedom at a quantum phase transition change the nature of a transition beyond the standard Landau-Ginzburg paradigm. To this end we study the quantum phase transition of gapless Dirac fermions coupled to a $\mathbb{Z}_3$ symmetric order parameter within a Gross-Neveu-Yukawa model in 2+1 dimensions, appropriate for the Kekul\'e transition in honeycomb lattice materials. For this model the standard Landau-Ginzburg approach suggests a first order transition due to the symmetry-allowed cubic terms in the action. At zero temperature, however, quantum fluctuations of the massless Dirac fermions have to be included. We show that they reduce the putative first-order character of the transition and can even render it continuous, depending on the number of Dirac fermions $N_f$. A non-perturbative functional renormalization group approach is employed to investigate the phase transition for a wide range of fermion numbers. For the first time we obtain the critical $N_f$, where the nature of the transition changes. Furthermore, it is shown that for large $N_f$ the change from the first to second order of the transition as a function of dimension occurs exactly in the physical 2+1 dimensions. We compute the critical exponents and predict sizable corrections to scaling for $N_f =2$.

Strong-coupling Bose polarons in a Bose-Einstein condensate

We use a non-perturbative renormalization group approach to develop a unified picture of the Bose polaron problem, where a mobile impurity is strongly interacting with a surrounding Bose-Einstein condensate (BEC). A detailed theoretical analysis of the phase diagram is presented and the polaron-to-molecule transition is discussed. For attractive polarons we argue that a description in terms of an effective Fr\"ohlich Hamiltonian with renormalized parameters is possible. Its strong coupling regime is realized close to a Feshbach resonance, where we predict a sharp increase of the effective mass. Already for weaker interactions, before the polaron mass diverges, we predict a transition to a regime where states exist below the polaron energy and the attractive polaron is no longer the ground state. On the repulsive side of the Feshbach resonance we recover the repulsive polaron, which has a finite lifetime because it can decay into low-lying molecular states. We show for the entire range of couplings that the polaron energy has logarithmic corrections in comparison with predictions by the mean-field approach. We demonstrate that they are a consequence of the polaronic mass renormalization which is due to quantum fluctuations of correlated phonons in the polaron cloud.

In-medium spectral functions of vector- and axial-vector mesons from the functional renormalization group

In this work we present first results on vector and axial-vector meson spectral functions as obtained by applying the non-perturbative functional renormalization group approach to an effective low-energy theory motivated by the gauged linear sigma model. By using a recently proposed analytic continuation method, we study the in-medium behavior of the spectral functions of the $\rho$ and $a_1$ mesons in different regimes of the phase diagram. In particular, we demonstrate explicitly how these spectral functions degenerate at high temperatures as well as at large chemical potentials, as a consequence of the restoration of chiral symmetry. In addition, we also compute the momentum dependence of the $\rho$ and $a_1$ spectral functions and discuss the various time-like and space-like processes that can occur.

Flat phase of quantum polymerized membranes.

We investigate the flat phase of quantum polymerized phantom membranes by means of a nonperturbative renormalization group approach. We first implement this formalism for general quantum polymerized membranes and derive the flow equations that encompass both quantum and thermal fluctuations. We then deduce and analyze the flow equations relevant to study the flat phase and discuss their salient features: quantum to classical crossover and, in each of these regimes, strong to weak coupling crossover. We finally illustrate these features in the context of free-standing graphene physics.

Multilogarithmic velocity renormalization in graphene

We reexamine the effect of long-range Coulomb interactions on the quasiparticle velocity in graphene. Using a nonperturbative functional renormalization group approach with partial bosonization in the forward scattering channel and momentum transfer cutoff scheme, we calculate the quasiparticle velocity, $v (k)$, and the quasiparticle residue, $Z$, with frequency-dependent polarization. One of our most striking results is that $v ( k ) \propto \ln [ C_k (\alpha) / k ]$ where the momentum- and interaction-dependent cutoff scale $C_k (\alpha) $ vanishes logarithmically for $k \rightarrow 0$. Here $k$ is measured with respect to one of the charge neutrality (Dirac) points and $\alpha=2.2$ is the strength of dimensionless bare interaction. Moreover, we also demonstrate that the so-obtained multilogarithmic singularity is reconcilable with the perturbative expansion of $v (k)$ in powers of the bare interaction.

Nonperturbative renormalization group calculation of quasiparticle velocity and dielectric function of graphene

Using a non-perturbative functional renormalization group approach we calculate the renormalized quasi-particle velocity $v (k)$ and the static dielectric function $\epsilon ( k )$ of suspended graphene as functions of an external momentum $k$. Our numerical result for $v (k )$ can be fitted by $v ( k ) / v_F = A + B \ln ( \Lambda_0 / k )$, where $v_F$ is the bare Fermi velocity, $\Lambda_0$ is an ultraviolet cutoff, and $A = 1.37$, $B =0.51$ for the physically relevant value ($e^2/v_F =2.2$) of the coupling constant. In contrast to calculations based on the static random-phase approximation, we find that $\epsilon (k )$ approaches unity for $k \rightarrow 0$. Our result for $v (k )$ agrees very well with a recent measurement by Elias et al. [Nat. Phys. 7, 701 (2011)].