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arXiv:1610.06476 · Physical review. B./Physical review. B

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

Félix Rose, N. Dupuis

Published Jan 17, 2017• 20 citations
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Abstract

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.

DOI
10.1103/physrevb.95.014513

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Using a nonperturbative functional renormalization-group approach to the two-dimensional quantum O(N\mathrm{O}(NO(N) model, we compute the low-frequency limit \ensuremathω\ensuremath→0\ensuremath{\omega}\ensuremath{\rightarrow}0\ensuremathω\ensuremath→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 \ensuremathσ(\ensuremathω)\ensuremath{\sigma}(\ensuremath{\omega})\ensuremathσ(\ensuremathω) is diagonal, in the ordered phase it is defined, when N\ensuremath≥3N\ensuremath{\ge}3N\ensuremath≥3, by two independent elements, \ensuremathσA(\ensuremathω){\ensuremath{\sigma}}_{\mathrm{A}}(\ensuremath{\omega})\ensuremathσA​(\ensuremathω) and \ensuremathσB(\ensuremathω){\ensuremath{\sigma}}_{\mathrm{B}}(\ensuremath{\omega})\ensuremathσB​(\ensuremathω), respectively associated to SO(N\mathrm{SO}(NSO(N) rotations which do and do not change the direction of the order parameter. For N=2N=2N=2, the conductivity in the ordered phase reduces to a single component \ensuremathσA(\ensuremathω){\ensuremath{\sigma}}_{\mathrm{A}}(\ensuremath{\omega})\ensuremathσA​(\ensuremathω). We show that lim\ensuremathω\ensuremath→0\ensuremathσ(\ensuremathω,\ensuremathδ)\ensuremathσA(\ensuremathω,\ensuremath−\ensuremathδ)/\ensuremathσq2{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}lim\ensuremathω\ensuremath→0​\ensuremathσ(\ensuremathω,\ensuremathδ)\ensuremathσA​(\ensuremathω,\ensuremath−\ensuremathδ)/\ensuremathσq2​ is a universal number, which we compute as a function of NNN (\ensuremathδ\ensuremath{\delta}\ensuremathδ measures the distance to the quantum critical point, qqq is the charge, and \ensuremathσq=q2/h{\ensuremath{\sigma}}_{q}={q}^{2}/h\ensuremathσq​=q2/h the quantum of conductance). On the other hand we argue that the ratio \ensuremathσB(\ensuremathω\ensuremath→0)/\ensuremathσq{\ensuremath{\sigma}}_{\mathrm{B}}(\ensuremath{\omega}\ensuremath{\rightarrow}0)/{\ensuremath{\sigma}}_{q}\ensuremathσB​(\ensuremathω\ensuremath→0)/\ensuremathσq​ is universal in the whole ordered phase, independent of NNN and, when N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞, equal to the universal conductivity \ensuremathσ∗/\ensuremathσq{\ensuremath{\sigma}}^{*}/{\ensuremath{\sigma}}_{q}\ensuremathσ∗/\ensuremathσq​ at the quantum critical point.
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We study the ``Higgs'' amplitude mode in the relativistic quantum O(N)O(N)O(N) model in two space dimensions. Using the nonperturbative renormalization group we compute the O(N)O(N)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=2N=2N=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 $N\ensuremath{\ge}3$.
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