Nonperturbative Functional Renormalization Group Approach Research Articles

Flowing bosonization in the nonperturbative functional renormalization-group approach

Bosonization allows one to describe the low-energy physics of one-dimensional quantum fluids within a bosonic effective field theory formulated in terms of two fields: the ``density’’ field \varphiφ and its conjugate partner, the phase \varthetaϑ of the superfluid order parameter. We discuss the implementation of the nonperturbative functional renormalization group in this formalism, considering a Luttinger liquid in a periodic potential as an example. We show that in order for \varthetaϑ and \varphiφ to remain conjugate variables at all energy scales, one must dynamically redefine the field \varthetaϑ along the renormalization-group flow. We derive explicit flow equations using a derivative expansion of the scale-dependent effective action to second order and show that they reproduce the flow equations of the sine-Gordon model (obtained by integrating out the field \varthetaϑ from the outset) derived within the same approximation. Only with the scale-dependent (flowing) reparametrization of the phase field \varthetaϑ do we obtain the standard phenomenology of the Luttinger liquid (when the periodic potential is sufficiently weak so as to avoid the Mott-insulating phase) characterized by two low-energy parameters, the velocity of the sound mode and the renormalized Luttinger parameter.

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.

Bose-glass phase of a one-dimensional disordered Bose fluid: Metastable states, quantum tunneling, and droplets.

We study a one-dimensional disordered Bose fluid using bosonization, the replica method, and a nonperturbative functional renormalization-group approach. We find that the Bose-glass phase is described by a fully attractive strong-disorder fixed point characterized by a singular disorder correlator whose functional dependence assumes a cuspy form that is related to the existence of metastable states. At nonzero momentum scale k, quantum tunneling between the ground state and low-lying metastable states leads to a rounding of the cusp singularity into a quantum boundary layer (QBL). The width of the QBL depends on an effective Luttinger parameter K_{k}∼k^{θ} that vanishes with an exponent θ=z-1 related to the dynamical critical exponent z. The QBL encodes the existence of rare "superfluid" regions, controls the low-energy dynamics, and yields a (dissipative) conductivity vanishing as ω^{2} in the low-frequency limit. These results reveal the glassy properties (pinning, "shocks," or static avalanches) of the Bose-glass phase and can be understood within the "droplet" picture put forward for the description of glassy (classical) systems.

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 .

Glassy properties of the Bose-glass phase of a one-dimensional disordered Bose fluid.

We study a one-dimensional disordered Bose fluid using bosonization, the replica method, and a nonperturbative functional renormalization-group approach. The Bose-glass phase is described by a fully attractive strong-disorder fixed point characterized by a singular disorder correlator whose functional dependence assumes a cuspy form that is related to the existence of metastable states. At nonzero momentum scale, quantum tunneling between these metastable states leads to a rounding of the nonanalyticity in a quantum boundary layer that encodes the existence of rare superfluid regions responsible for the ω^{2} behavior of the (dissipative) conductivity in the low-frequency limit. These results can be understood within the "droplet" picture put forward for the description of glassy (classical) systems.

Nonperturbative Functional Renormalization-Group Approach to the Sine-Gordon Model and the Lukyanov-Zamolodchikov Conjecture.

We study the quantum sine-Gordon model within a nonperturbative functional renormalization-group approach (FRG). This approach is benchmarked by comparing our findings for the soliton and lightest breather (soliton-antisoliton bound state) masses to exact results. We then examine the validity of the Lukyanov-Zamolodchikov conjecture for the expectation value ⟨e^{(i/2)nβφ}⟩ of the exponential fields in the massive phase (n is integer and 2π/β denotes the periodicity of the potential in the sine-Gordon model). We find that the minimum of the relative and absolute disagreements between the FRG results and the conjecture is smaller than 0.01.

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 $N_f =2$.

Superuniversal transport near a (2+1) -dimensional quantum critical point

We compute the zero-temperature conductivity in the two-dimensional quantum $\mathrm{O}(N)$ model using a nonperturbative functional renormalization-group approach. At the quantum critical point we find a universal conductivity $\sigma^*/\sigma_Q$ (with $\sigma_Q=q^2/h$ the quantum of conductance and $q$ the charge) in reasonable quantitative agreement with quantum Monte Carlo simulations and conformal bootstrap results. In the ordered phase the conductivity tensor is defined, when $N\geq 3$, by two independent elements, $\sigma_{\mathrm{A}}(\omega)$ and $\sigma_{\mathrm{B}}(\omega)$, respectively associated to $\mathrm{O}(N)$ rotations which do and do not change the direction of the order parameter. Whereas $\sigma_{\mathrm{A}}(\omega\to 0)$ corresponds to the response of a superfluid (or perfect inductance), the numerical solution of the flow equations shows that $\lim_{\omega\to 0}\sigma_{\mathrm{B}}(\omega)/\sigma_Q=\sigma_{\mathrm{B}}^*/\sigma_Q$ is a superuniversal (i.e. $N$-independent) constant. These numerical results, as well as the known exact value $\sigma_{\mathrm{B}}^*/\sigma_Q=\pi/8$ in the large-$N$ limit, allow us to conjecture that $\sigma_{\mathrm{B}}^*/\sigma_Q=\pi/8$ holds for all values of $N$, a result that can be understood as a consequence of gauge invariance and asymptotic freedom of the Goldstone bosons in the low-energy limit.

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.

Nonperturbative functional renormalization-group approach to transport in the vicinity of a(2+1)-dimensional O(N)-symmetric quantum critical point

Using a nonperturbative functional renormalization-group approach to the two-dimensional quantum $\mathrm{O}(N$) model, we compute the low-frequency limit $\ensuremath{\omega}\ensuremath{\rightarrow}0$ of the zero-temperature conductivity in the vicinity of the quantum critical point. Our results are obtained from a derivative expansion to second order of a scale-dependent effective action in the presence of an external (i.e., nondynamical) non-Abelian gauge field. While in the disordered phase the conductivity tensor $\ensuremath{\sigma}(\ensuremath{\omega})$ is diagonal, in the ordered phase it is defined, when $N\ensuremath{\ge}3$, by two independent elements, ${\ensuremath{\sigma}}_{\mathrm{A}}(\ensuremath{\omega})$ and ${\ensuremath{\sigma}}_{\mathrm{B}}(\ensuremath{\omega})$, respectively associated to $\mathrm{SO}(N$) rotations which do and do not change the direction of the order parameter. For $N=2$, the conductivity in the ordered phase reduces to a single component ${\ensuremath{\sigma}}_{\mathrm{A}}(\ensuremath{\omega})$. We show that ${lim}_{\ensuremath{\omega}\ensuremath{\rightarrow}0}\ensuremath{\sigma}(\ensuremath{\omega},\ensuremath{\delta}){\ensuremath{\sigma}}_{\mathrm{A}}(\ensuremath{\omega},\ensuremath{-}\ensuremath{\delta})/{\ensuremath{\sigma}}_{q}^{2}$ is a universal number, which we compute as a function of $N$ ($\ensuremath{\delta}$ measures the distance to the quantum critical point, $q$ is the charge, and ${\ensuremath{\sigma}}_{q}={q}^{2}/h$ the quantum of conductance). On the other hand we argue that the ratio ${\ensuremath{\sigma}}_{\mathrm{B}}(\ensuremath{\omega}\ensuremath{\rightarrow}0)/{\ensuremath{\sigma}}_{q}$ is universal in the whole ordered phase, independent of $N$ and, when $N\ensuremath{\rightarrow}\ensuremath{\infty}$, equal to the universal conductivity ${\ensuremath{\sigma}}^{*}/{\ensuremath{\sigma}}_{q}$ at the quantum critical point.

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)].

Activated dynamic scaling in the random-field Ising model: A nonperturbative functional renormalization group approach

The random-field Ising model shows extreme critical slowdown that has been described by activated dynamic scaling: the characteristic time for the relaxation to equilibrium diverges exponentially with the correlation length, $\ln \tau\sim \xi^\psi/T$ , with $\psi$ an \textit{a priori} unknown barrier exponent. Through a nonperturbative functional renormalization group, we show that for spatial dimensions $d$ less than a critical value $d_{DR} \simeq 5.1$, also associated with dimensional-reduction breakdown, $\psi=\theta$ with $\theta$ the temperature exponent near the zero-temperature fixed point that controls the critical behavior. For $d>d_{DR}$ on the other hand, $\psi=\theta-2\lambda$ where $\theta=2$ and $\lambda>0$ a new exponent. At the upper critical dimension $d=6$, $\lambda=1$ so that $\psi=0$, and activated scaling gives way to conventional scaling. We give a physical interpretation of the results in terms of collective events in real space, avalanches and droplets. We also propose a way to check the two regimes by computer simulations of long-range 1-$d$ systems.

Spectral functions from the functional renormalization group

We present a viable method to obtain real-time quantities such as spectral functions or transport coefficients at finite temperature and density within a non-perturbative functional renormalization group approach. Our method is based on a thermodynamically consistent truncation of the flow equations for 2-point functions with analytically continued frequency components in the originally Euclidean external momenta. We demonstrate its feasibility by calculating the mesonic spectral functions in the quark–meson model at different temperatures and quark chemical potentials, in particular around the critical endpoint in the phase diagram of the model.

Fluctuations and the axial anomaly with three quark flavors

The role of the axial anomaly in the chiral phase transition at finite temperature and quark chemical potential is investigated within a non-perturbative functional renormalization group approach. The flow equation for the grand potential is solved to leading-order in a derivative expansion of a three flavor quark-meson model truncation. The results are compared with a standard and an extended mean-field analysis, which facilitates the exploration of the influence of bosonic and fermionic fluctuations, respectively, on the phase transition. The influence of U(1)_A-symmetry breaking on the chiral transition, the location of a possible critical endpoint in the phase diagram and the quark mass sensitivity is studied in detail.

Nonperturbative functional renormalization group for random field models and related disordered systems. II. Results for the random fieldO(N)model

We study the critical behavior and phase diagram of the $d$-dimensional random field O(N) model by means of the nonperturbative functional renormalization group approach presented in the preceding paper. We show that the dimensional reduction predictions, obtained from conventional perturbation theory, break down below a critical dimension $d_{DR}(N)$ and we provide a description of criticality, ferromagnetic ordering and quasi-long range order in the whole $(N,d)$ plane. Below $d_{DR}(N)$, our formalism gives access to both the typical behavior of the system, controlled by zero-temperature fixed points with a nonanalytic dimensionless effective action, and to the physics of rare low-energy excitations ("droplets"), described at nonzero temperature by the rounding of the nonanalyticity in a thermal boundary layer.

Nonperturbative functional renormalization group for random field models and related disordered systems. I. Effective average action formalism

We have developed a nonperturbative functional renormalization group approach for random field models and related disordered systems for which, due to the existence of many metastable states, conventional perturbation theory often fails. The approach combines an exact renormalization group equation for the effective average action with a nonperturbative approximation scheme based on a description of the probability distribution of the renormalized disorder through its cumulants. For the random field $O(N)$ model, the minimal truncation within this scheme is shown to reproduce the known perturbative results in the appropriate limits, near the upper and lower critical dimensions and at large number $N$ of components, while providing a unified nonperturbative description of the full $(N,d)$ plane, where $d$ is the spatial dimension.

Fermi-surface renormalization and confinement in two coupled metallic chains

Using a nonperturbative functional renormalization group approach involving both fermionic and bosonic fields we calculate the interaction-induced change of the Fermi surface of spinless fermions moving on two chains connected by weak interchain hopping ${t}_{\ensuremath{\perp}}$. For a model containing interband backward scattering only we show that the distance $\ensuremath{\Delta}$ between the Fermi momenta associated with the bonding and the antibonding band can be strongly reduced, corresponding to a large reduction of the effective interchain hopping ${t}_{\ensuremath{\perp}}^{*}\ensuremath{\propto}\ensuremath{\Delta}$. A self-consistent one-loop approximation neglecting marginal vertex corrections and wave-function renormalizations predicts a confinement transition for sufficiently large interchain backscattering, where the renormalized ${t}_{\ensuremath{\perp}}^{*}$ vanishes. However, a more accurate calculation taking vertex corrections and wave-function renormalizations into account predicts only weak confinement in the sense that $0<\ensuremath{\mid}{t}_{\ensuremath{\perp}}^{*}\ensuremath{\mid}⪡\ensuremath{\mid}{t}_{\ensuremath{\perp}}\ensuremath{\mid}$. Our method can be applied to other strong-coupling problems where the dominant scattering channel is known.

Nonperturbative Functional Renormalization Group for Random-Field Models: The Way Out of Dimensional Reduction

We develop a nonperturbative functional renormalization group approach for the random-field O(N) model that allows us to investigate the ordering transition in any dimension and for any value of N including the Ising case. We show that the failure of dimensional reduction and standard perturbation theory is due to the nonanalytic nature of the zero-temperature fixed point controlling the critical behavior, nonanalyticity, which is associated with the existence of many metastable states. We find that this nonanalyticity leads to critical exponents differing from the dimensional reduction prediction only below a critical dimension dc(N)<6, with dc(N=1)>3.