We study the relativistic O (N) model, a quantum generalization of the φ⁴ theory used in statistical physics to study some phase transitions. This model describes quantum phase transitions such as the Mott insulator-superluid transition in boson gases trapped in optical latices or the paramagnetic-antiferromagnetic transition in magnets. In two space dimensions, these systems exhibit strong correlations near the transition. We study them using the nonperturbative renormalization group, an implementation of Wilson’s renormalization group. We focus on the universal properties in the vicinity of the zero-temperature quantum phase transition. We determine the universal scaling functions which define the thermodynamics and we show that these functions are related to those describing the critical Casimir forces in a three-dimensional system. Then, we study the excitation spectrum in the zero-temperature ordered phase. For N = 2 and 3, we establish the existence of an amplitude mode, also called “Higgs mode” by analogy with the Higgs mechanism in high-energy physics. For N = 1, we show the existence of a bound state at dimensions close to three. Finally, we compute the frequency-dependent conductivity at zero temperature and confirm its universal character, in particular at the transition. We prove that one of the components of the conductivity tensor in the ordered phase is a “superuniversal” quantity depending neither on the distance to the critical point nor on N.
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 $P(T)=P(0)+N({T}^{3}/{c}^{2}){\mathcal{F}}_{N}(\ensuremath{\Delta}/T)$, where
c is the velocity of the excitations at the QCP and $|\ensuremath{\Delta}|$ 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
FN. For small values of
N ($N\ensuremath{\lesssim}10$) we find that
FN(x) is nonmonotonic in the quantum critical regime ($|x|\ensuremath{\lesssim}1$) with a maximum near
x=0. The large-
N approach---if properly interpreted---is a good approximation both in the renormalized classical ($x\ensuremath{\lesssim}\ensuremath{-}1$) and quantum disordered ($x\ensuremath{\gtrsim}1$) regimes, but fails to describe the nonmonotonic behavior of
FN 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 ${T}_{\mathrm{BKT}}/{\ensuremath{\rho}}_{s}(0)$ is very close to $\ensuremath{\pi}/2$, implying that the stiffness ${\ensuremath{\rho}}_{s}({T}_{\mathrm{BKT}}^{\ensuremath{-}})$ at the transition is only slightly reduced with respect to the zero-temperature stiffness ${\ensuremath{\rho}}_{s}(0)$. Finally, we briefly discuss the experimental determination of the universal function
F2 from the pressure of a Bose gas in an optical lattice near the superfluid--Mott-insulator transition.
We study the Higgs excitation in the Goldstone phase of the relativistic O(
N) model in two spatial dimensions at zero temperature. The response functions of the order parameter, and its magnitude squared, become universal functions of frequency in the vicinity of the quantum critical point described by the Wilson-Fisher fixed point, and we compute them to next-to-leading order in
1/N. The Higgs particle has an infrared singular decay to gapless Goldstone excitations, and its response functions are characterized by a pole in the lower half of the complex frequency plane. The pole acquires a nonzero real part only at next-to-leading order in
1/N, demonstrating that the Higgs excitation has an oscillatory component even in the scaling limit. Both the real and imaginary parts of the pole position vanish with the correlation length exponent $\ensuremath{\nu}$ upon approaching the critical point. We present evidence that the spectral density of the O(
N)-invariant amplitude-squared of the order parameter has a peak at a nonzero frequency in the scaling limit. We connect our results to recent experimental studies of the superfluid-insulator quantum phase transition of ultracold bosonic atoms in optical lattices.
We compute the zero-temperature conductivity in the two-dimensional quantum
O(N) model using a nonperturbative functional renormalization-group approach. At the quantum critical point we find a universal conductivity ${\ensuremath{\sigma}}^{*}/{\ensuremath{\sigma}}_{Q}^{}$ (with ${\ensuremath{\sigma}}_{Q}^{}={q}^{2}/h$ 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 $N\ensuremath{\ge}3$, by two independent elements, ${\ensuremath{\sigma}}_{\mathrm{A}}(\ensuremath{\omega})$ and ${\ensuremath{\sigma}}_{\mathrm{B}}(\ensuremath{\omega})$, respectively associated with
SO(N) rotations which do and do not change the direction of the order parameter. Whereas ${\ensuremath{\sigma}}_{\mathrm{A}}(\ensuremath{\omega}\ensuremath{\rightarrow}0)$ corresponds to the response of a superfluid (or perfect inductance), the numerical solution of the flow equations shows that ${lim}_{\ensuremath{\omega}\ensuremath{\rightarrow}0}{\ensuremath{\sigma}}_{\mathrm{B}}(\ensuremath{\omega})/{\ensuremath{\sigma}}_{Q}^{}={\ensuremath{\sigma}}_{\mathrm{B}}^{*}/{\ensuremath{\sigma}}_{Q}^{}$ is a superuniversal (i.e.,
N-independent) constant. These numerical results, as well as the known exact value ${\ensuremath{\sigma}}_{\mathrm{B}}^{*}/{\ensuremath{\sigma}}_{Q}^{}=\ensuremath{\pi}/8$ in the large-
N limit, allow us to conjecture that ${\ensuremath{\sigma}}_{\mathrm{B}}^{*}/{\ensuremath{\sigma}}_{Q}^{}=\ensuremath{\pi}/8$ 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.
We describe the nonzero temperature
(T), low frequency $(\ensuremath{\omega})$ dynamics of the order parameter near quantum critical points in two spatial dimensions
(d), with a special focus on the regime $\ensuremath{\Elzxh}\ensuremath{\omega}\ensuremath{\ll}{k}_{B}T$. For the case of a ``relativistic,''
O(n)-symmetric, bosonic quantum field theory we show that, for small $\ensuremath{\epsilon}=3\ensuremath{-}d$, the dynamics is described by an effective classical model of waves with a quartic interaction. We provide analytical and numerical analyses of the classical wave model directly in
d=2. We describe the crossover from the finite frequency, ``amplitude fluctuation,'' gapped quasiparticle mode in the quantum paramagnet (or Mott insulator), to the zero frequency ``phase''
(n> 2) or ``domain wall''
(n=1) relaxation mode near the ordered state. For static properties, we show how a surprising, duality-like transformation allows an exact treatment of the strong-coupling limit for all n. For
n=2, we compute the universal T dependence of the superfluid density below the Kosterlitz-Thouless temperature, and discuss implications for the high temperature superconductors. For
n=3, our computations of the dynamic structure factor relate to neutron scattering experiments on
La1.85Sr0.15CuO4, and to light scattering experiments on double layer quantum Hall systems. We expect that closely related effective classical wave models will apply also to other quantum critical points in
d=2. Although computations in appendixes do rely upon technical results on the $\ensuremath{\epsilon}$-expansion of quantum critical points obtained in earlier papers, the physical discussion in the body of the paper is self-contained, and can be read without consulting these earlier works.
Relativistic O(
N) field theories are studied near the quantum-critical point in two space dimensions. We compute dynamical correlations by large-scale Monte Carlo simulations and numerical analytic continuation. In the ordered side, the scalar spectral function exhibits a universal peak at the Higgs mass. For
N=3 and 4, we confirm its ${\ensuremath{\omega}}^{3}$ rise at low frequency. On the disordered side, the spectral function exhibits a sharp gap. For
N=2, the dynamical conductivity rises above a threshold at the Higgs mass (density gap), in the superfluid (Mott insulator) phase. For charged bosons (Josephson arrays), the power-law rise above the Higgs mass increases from two to four. Approximate charge-vortex duality is reflected in the ratio of imaginary conductivities on either side of the transition. We determine the critical conductivity to be ${\ensuremath{\sigma}}_{\mathrm{c}}^{*}=0.3(\ifmmode\pm\else\textpm\fi{}0.1)\ifmmode\times\else\texttimes\fi{}4{e}^{2}/h$ and describe a generalization of the worm algorithm to
N>2. We use a singular value decomposition error analysis for the numerical analytic continuation.
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 $N\ensuremath{\ge}3$.