arXiv subjects

N. Dupuis

Publications and source records attributed to N. Dupuis.

At least 19 recordsLinked to original sources

arXiv2021

The nonperturbative functional renormalization group and its applications

The renormalization group plays an essential role in many areas of physics, both conceptually and as a practical tool to determine the long-distance low-energy properties of many systems on the one hand and on the other hand search for viable ultraviolet completions in fundamental physics. It provides us with a natural framework to study theoretical models where degrees of freedom are correlated over long distances and that may exhibit very distinct behavior on different energy scales. The nonperturbative functional renormalization-group (FRG) approach is a modern implementation of Wilson's RG, which allows one to set up nonperturbative approximation schemes that go beyond the standard perturbative RG approaches. The FRG is based on an exact functional flow equation of a coarse-grained effective action (or Gibbs free energy in the language of statistical mechanics). We review the main approximation schemes that are commonly used to solve this flow equation and discuss applications in equilibrium and out-of-equilibrium statistical physics, quantum many-particle systems, high-energy physics and quantum gravity.

cond-mat.stat-mech↗

arXiv2018

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 $\langle e^{\frac{i}{2}nβφ}\rangle$ 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.

cond-mat.stat-mech↗

arXiv2016

First-order phase transitions in spinor Bose gases and frustrated magnets

We show that phase transitions in spin-one Bose gases and stacked triangular Heisenberg antiferromagnets -- an example of frustrated magnets with competing interactions -- are described by the same Landau-Ginzburg-Wilson Hamiltonian with O(3)$\times$O(2) symmetry. In agreement with previous nonperturbative-renormalization-group studies of the three-dimensional O(3)$\times$O(2) model, we find that the transition from the normal phase to the superfluid ferromagnetic phase in a spin-one Bose gas is weakly first order and shows pseudoscaling behavior. The (nonuniversal) pseudoscaling exponent $ν$ is fully determined by the scattering lengths $a_0$ and $a_2$. We provide estimates of $ν$ in $^{87}$Rb, $^{41}$K and $^7$Li atom gases which can be tested experimentally. We argue that pseudoscaling comes from either a crossover phenomena due to proximity of the O(6) Wilson-Fisher fixed point ($^{87}$Rb and $^{41}$K) or the existence of two unphysical fixed points (with complex coordinates) which slow down the RG flow ($^7$Li). These unphysical fixed points are a remnant of the chiral and antichiral fixed points that exist in the O($N$)$\times$O(2) model when $N$ is larger than $N_c\simeq 5.3$ (the transition being then second order and controlled by the chiral fixed point). Finally, we discuss a O(2)$\times$O(2) lattice model and show that our results, even though we find the transition to be first order, are compatible with Monte Carlo simulations yielding an apparent second-order transition.

cond-mat.quant-gas↗

arXiv2016

Simulating frustrated magnetism with spinor Bose gases

Although there is a broad consensus on the fact that critical behavior in stacked triangular Heisenberg antiferromagnets --an example of frustrated magnets with competing interactions-- is described by a Landau-Ginzburg-Wilson Hamiltonian with O(3)$\times$O(2) symmetry, the nature of the phase transition in three dimensions is still debated. We show that spin-one Bose gases provide us with a simulator of the O(3)$\times$O(2) model. Using a renormalization-group approach, we argue that the transition is weakly first order and shows pseudoscaling behavior, and give estimates of the pseudocritical exponent $ν$ in $^{87}$Rb, $^{41}$K and $^7$Li atom gases which can be tested experimentally.

cond-mat.quant-gas↗

arXiv2016

Critical region of the superfluid transition in the BCS-BEC crossover

We determine the size of the critical region of the superfluid transition in the BCS-BEC crossover of a three-dimensional fermion gas, using a renormalization-group approach to a bosonic theory of pairing fluctuations. For the unitary Fermi gas, we find a sizable critical region $[T_G^-,T_G^+]$, of order $T_c$, around the transition temperature $T_c$ with a pronounced asymmetry: $|T_G^+-T_c|/|T_G^--T_c|\sim8$. The critical region is strongly suppressed on the BCS side of the crossover but remains important on the BEC side.

cond-mat.quant-gas↗

arXiv2014

Reexamination of the nonperturbative renormalization-group approach to the Kosterlitz-Thouless transition

We reexamine the two-dimensional linear O(2) model ($φ^4$ theory) in the framework of the nonperturbative renormalization-group. From the flow equations obtained in the derivative expansion to second order and with optimization of the infrared regulator, we find a transition between a high-temperature (disordered) phase and a low-temperature phase displaying a line of fixed points and algebraic order. We obtain a picture in agreement with the standard theory of the Kosterlitz-Thouless (KT) transition and reproduce the universal features of the transition. In particular, we find the anomalous dimension $η(\Tkt)\simeq 0.24$ and the stiffness jump $ρ_s(\Tkt^-)\simeq 0.64$ at the transition temperature $\Tkt$, in very good agreement with the exact results $η(\Tkt)=1/4$ and $ρ_s(\Tkt^-)=2/π$, as well as an essential singularity of the correlation length in the high-temperature phase as $T\to \Tkt$.

cond-mat.stat-mech↗

arXiv2014

Higgs amplitude mode in the vicinity of a $(2+1)$-dimensional quantum critical point

We study the "Higgs" amplitude mode in the relativistic quantum O($N$) model in two space dimensions. Using the nonperturbative renormalization group we compute the O($N$)-invariant scalar susceptibility in the vicinity of the zero-temperature quantum critical point. In the zero-temperature ordered phase, we find a well defined Higgs resonance for $N=2$ with universal properties in agreement with quantum Monte Carlo simulations. The resonance persists at finite temperature below the Berezinskii-Kosterlitz-Thouless transition temperature. In the zero-temperature disordered phase, we find a maximum in the spectral function which is however not related to a putative Higgs resonance. Furthermore we show that the resonance is strongly suppressed for $N\geq 3$.

cond-mat.quant-gas↗

arXiv2014

Nonperturbative renormalization-group approach to fermion systems in the two-particle-irreducible effective action formalism

We propose a nonperturbative renormalization-group (NPRG) approach to fermion systems in the two-particle-irreducible (2PI) effective action formalism, based on an exact RG equation for the Luttinger-Ward functional. This approach enables us to describe phases with spontaneously broken symmetries while satisfying the Mermin-Wagner theorem. We show that it is possible to choose the Hartree-Fock--RPA theory as initial condition of the RG flow and argue that the 2PI-NPRG is not restricted to the weak-coupling limit. An expansion of the Luttinger-Ward functional about the minimum of the 2PI effective action including only the two-particle 2PI vertex leads to nontrivial RG equations where interactions between fermions and collective excitations naturally emerge.

cond-mat.str-el↗

arXiv2013

Quantum XY criticality in a two-dimensional Bose gas near the Mott transition

We derive the equation of state of a two-dimensional Bose gas in an optical lattice in the framework of the Bose-Hubbard model. We focus on the vicinity of the multicritical points where the quantum phase transition between the Mott insulator and the superfluid phase occurs at fixed density and belongs to the three-dimensional XY model universality class. Using a nonperturbative renormalization-group approach, we compute the pressure $P(μ,T)$ as a function of chemical potential and temperature. Our results compare favorably with a calculation based on the quantum O(2) model -- we find the same universal scaling function -- and allow us to determine the region of the phase diagram in the vicinity of a quantum multicritical point where the equation of state is universal. We also discuss the possible experimental observation of quantum XY criticality in a ultracold gas in an optical lattice.

cond-mat.quant-gas↗

arXiv2013

Thermodynamics in the vicinity of a relativistic quantum critical point in 2+1 dimensions

We study the thermodynamics of the relativistic quantum O($N$) model in two space dimensions. In the vicinity of the zero-temperature quantum critical point (QCP), the pressure can be written in the scaling form $P(T)=P(0)+N(T^3/c^2)\calF_N(Δ/T)$ where $c$ is the velocity of the excitations at the QCP and $Δ$ is a characteristic zero-temperature energy scale. Using both a large-$N$ approach to leading order and the nonperturbative renormalization group, we compute the universal scaling function $\calF_N$. For small values of $N$ ($N\lesssim 10$) we find that $\calF_N(x)$ is nonmonotonous in the quantum critical regime ($|x|\lesssim 1$) with a maximum near $x=0$. The large-$N$ approach -- if properly interpreted -- is a good approximation both in the renormalized classical ($x\lesssim -1$) and quantum disordered ($x\gtrsim 1$) regimes, but fails to describe the nonmonotonous behavior of $\calF_N$ in the quantum critical regime. We discuss the renormalization-group flows in the various regimes near the QCP and make the connection with the quantum nonlinear sigma model in the renormalized classical regime. We compute the Berezinskii-Kosterlitz-Thouless transition temperature in the quantum O(2) model and find that in the vicinity of the QCP the universal ratio $\Tkt/ρ_s(0)$ is very close to $π/2$, implying that the stiffness $ρ_s(\Tkt^-)$ at the transition is only slightly reduced with respect to the zero-temperature stiffness $ρ_s(0)$. Finally, we briefly discuss the experimental determination of the universal function $\calF_2$ from the pressure of a Bose gas in an optical lattice near the superfluid--Mott-insulator transition.

cond-mat.quant-gas↗

arXiv2012

Thermodynamics of a Bose gas near the superfluid--Mott-insulator transition

We study the thermodynamics near the generic (density-driven) superfluid--Mott-insulator transition in the three-dimensional Bose-Hubbard model using the nonperturbative renormalization-group approach. At low energy the physics is controlled by the Gaussian fixed point and becomes universal. Thermodynamic quantities can then be expressed in terms of the universal scaling functions of the dilute Bose gas universality class while the microscopic physics enters only {\it via} two nonuniversal parameters, namely the effective mass $m^*$ and the "scattering length" $a^*$ of the elementary excitations at the quantum critical point between the superfluid and Mott-insulating phase. A notable exception is the condensate density in the superfluid phase which is proportional to the quasi-particle weight $\Zqp$ of the elementary excitations. The universal regime is defined by $m^*a^*{}^2 T\ll 1$ and $m^*a^*{}^2|δμ|\ll 1$, or equivalently $|\bar n-\bar n_c|a^*{}^3\ll 1$, where $δμ=μ-μ_c$ is the chemical potential shift from the quantum critical point $(μ=μ_c,T=0)$ and $\bar n-\bar n_c$ the doping with respect to the commensurate density $\bar n_c$ of the T=0 Mott insulator. We compute $\Zqp$, $m^*$ and $a^*$ and find that they vary strongly with both the ratio $t/U$ between hopping amplitude and on-site repulsion and the value of the (commensurate) density $\bar n_c$. Finally, we discuss the experimental observation of universality and the measurement of $\Zqp$, $m^*$ and $a^*$ in a cold atomic gas in an optical lattice.

cond-mat.quant-gas↗

arXiv2012

Universal thermodynamics of a two-dimensional Bose gas

Using renormalization-group arguments we show that the low-temperature thermodynamics of a three- or two-dimensional dilute Bose gas is fully determined by a universal scaling function $\calF_d(μ/k_BT,\tilde g(T))$ once the mass $m$ and the s-wave scattering length $a_d$ of the bosons are known ($d$ is the space dimension). Here $μ$ and $T$ denote the chemical potential and temperature of the gas, and the temperature-dependent dimensionless interaction constant $\tilde g(T)$ is a function of $ma_d^2k_BT/\hbar^2$. We compute the scaling function $\calF_2$ using a nonperturbative renormalization-group approach and find that both the $μ/k_BT$ and $\tilde g(T)$ dependencies are in very good agreement with recent experimental data obtained for a quasi-two-dimensional Bose gas with or without optical lattice. We also show that the nonperturbative renormalization-group estimate of the Berezinskii-Kosterlitz-Thouless transition temperature compares well with the result obtained from a quantum Monte Carlo simulation of an effective classical field theory.

cond-mat.quant-gas↗

arXiv2011

Quantum criticality of a Bose gas in an optical lattice near the Mott transition

We derive the equation of state of bosons in an optical lattice in the framework of the Bose-Hubbard model. Near the density-driven Mott transition, the expression of the pressure P(μ,T) versus chemical potential and temperature is similar to that of a dilute Bose gas but with renormalized mass m^* and scattering length a^*. m^* is the mass of the elementary excitations at the quantum critical point governing the transition from the superfluid phase to the Mott insulating phase, while a^* is related to their effective interaction at low energy. We use a nonperturbative renormalization-group approach to compute these parameters as a function of the ratio t/U between hopping amplitude and on-site repulsion.

cond-mat.quant-gas↗

arXiv2011

Nonperturbative renormalization-group approach to strongly-correlated lattice bosons

We present a nonperturbative renormalization-group approach to the Bose-Hubbard model. By taking as initial condition of the renormalization-group flow the (local) limit of decoupled sites, we take into account both local and long-distance fluctuations in a nontrivial way. This approach yields a phase diagram in very good quantitative agreement with quantum Monte Carlo simulations, and reproduces the two universality classes of the superfluid--Mott-insulator transition. The critical behavior near the multicritical points, where the transition takes place at constant density, agrees with the original predictions of Fisher {\it et al.} [Phys. Rev. B {\bf 40}, 546 (1989)] based on simple scaling arguments. At a generic transition point, the critical behavior is mean-field like with logarithmic corrections in two dimensions. In the weakly-correlated superfluid phase (far away from the Mott insulating phase), the renormalization-group flow is controlled by the Bogoliubov fixed point down to a characteristic (Ginzburg) momentum scale $k_G$ which is much smaller than the inverse healing length $k_h$. In the vicinity of the multicritical points, when the density is commensurate, we identify a sharp crossover from a weakly- to a strongly-correlated superfluid phase where the condensate density and the superfluid stiffness are strongly suppressed and both $k_G$ and $k_h$ are of the order of the inverse lattice spacing.

cond-mat.quant-gas↗

arXiv2011

Infrared behavior of interacting bosons at zero temperature

We review the infrared behavior of interacting bosons at zero temperature. After a brief discussion of the Bogoliubov approximation and the breakdown of perturbation theory due to infrared divergences, we present two approaches that are free of infrared divergences -- Popov's hydrodynamic theory and the non-perturbative renormalization group -- and allow us to obtain the exact infrared behavior of the correlation functions. We also point out the connection between the infrared behavior in the superfluid phase and the critical behavior at the superfluid--Mott-insulator transition in the Bose-Hubbard model.

cond-mat.quant-gas↗

arXiv2011

Infrared behavior of interacting bosons at zero temperature

We review the infrared behavior of interacting bosons at zero temperature. After a brief discussion of the Bogoliubov approximation and the breakdown of perturbation theory due to infrared divergences, we show how the non-perturbative renormalization group enables to obtain the exact infrared behavior of the correlation functions.

cond-mat.quant-gas↗

arXiv2011

Non-perturbative renormalization-group approach to the Bose-Hubbard model

We present a non-perturbative renormalization-group approach to the Bose-Hubbard model. By taking as initial condition of the RG flow the (local) limit of decoupled sites, we take into account both local and long-distance fluctuations in a nontrivial way. This approach yields a phase diagram in very good quantitative agreement with the quantum Monte Carlo results and reproduces the two universality classes of the superfluid--Mott-insulator transition with a good estimate of the critical exponents. Furthermore, it reveals the crucial role of the "Ginzburg length" as a crossover length between a weakly- and a strongly-correlated superfluid phase.

cond-mat.quant-gas↗

arXiv2011

Infrared behavior in systems with a broken continuous symmetry: classical O(N) model vs interacting bosons

In systems with a spontaneously broken continuous symmetry, the perturbative loop expansion is plagued with infrared divergences due to the coupling between transverse and longitudinal fluctuations. As a result the longitudinal susceptibility diverges and the self-energy becomes singular at low energy. We study the crossover from the high-energy Gaussian regime, where perturbation theory remains valid, to the low-energy Goldstone regime characterized by a diverging longitudinal susceptibility. We consider both the classical linear O($N$) model and interacting bosons at zero temperature, using a variety of techniques: perturbation theory, hydrodynamic approach (i.e., for bosons, Popov's theory), large-$N$ limit and non-perturbative renormalization group. We emphasize the essential role of the Ginzburg momentum scale $p_G$ below which the perturbative approach breaks down. Even though the action of (non-relativistic) bosons includes a first-order time derivative term, we find remarkable similarities in the weak-coupling limit between the classical O($N$) model and interacting bosons at zero temperature.

cond-mat.quant-gas↗