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arXiv:1705.03905 · Physical review. B./Physical review. B

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

Félix Rose, N. Dupuis

Published Sep 11, 2017• 11 citations
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Abstract

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 ${\ensuremath{\sigma}}^{*}/{\ensuremath{\sigma}}_{Q}^{}$ (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 $\mathrm{SO}(N)$ 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.

DOI
10.1103/physrevb.96.100501

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We compute the zero-temperature conductivity in the two-dimensional quantum O(N)\mathrm{O}(N)O(N) model using a nonperturbative functional renormalization-group approach. At the quantum critical point we find a universal conductivity \ensuremathσ∗/\ensuremathσQ{\ensuremath{\sigma}}^{*}/{\ensuremath{\sigma}}_{Q}^{}\ensuremathσ∗/\ensuremathσQ​ (with \ensuremathσQ=q2/h{\ensuremath{\sigma}}_{Q}^{}={q}^{2}/h\ensuremathσQ​=q2/h the quantum of conductance and qqq 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≥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 with SO(N)\mathrm{SO}(N)SO(N) rotations which do and do not change the direction of the order parameter. Whereas \ensuremathσA(\ensuremathω\ensuremath→0){\ensuremath{\sigma}}_{\mathrm{A}}(\ensuremath{\omega}\ensuremath{\rightarrow}0)\ensuremathσA​(\ensuremathω\ensuremath→0) corresponds to the response of a superfluid (or perfect inductance), the numerical solution of the flow equations shows that lim\ensuremathω\ensuremath→0\ensuremathσB(\ensuremathω)/\ensuremathσQ=\ensuremathσB∗/\ensuremathσQ{lim}_{\ensuremath{\omega}\ensuremath{\rightarrow}0}{\ensuremath{\sigma}}_{\mathrm{B}}(\ensuremath{\omega})/{\ensuremath{\sigma}}_{Q}^{}={\ensuremath{\sigma}}_{\mathrm{B}}^{*}/{\ensuremath{\sigma}}_{Q}^{}lim\ensuremathω\ensuremath→0​\ensuremathσB​(\ensuremathω)/\ensuremathσQ​=\ensuremathσB∗​/\ensuremathσQ​ is a superuniversal (i.e., NNN-independent) constant. These numerical results, as well as the known exact value \ensuremathσB∗/\ensuremathσQ=\ensuremathπ/8{\ensuremath{\sigma}}_{\mathrm{B}}^{*}/{\ensuremath{\sigma}}_{Q}^{}=\ensuremath{\pi}/8\ensuremathσB∗​/\ensuremathσQ​=\ensuremathπ/8 in the large-NNN limit, allow us to conjecture that \ensuremathσB∗/\ensuremathσQ=\ensuremathπ/8{\ensuremath{\sigma}}_{\mathrm{B}}^{*}/{\ensuremath{\sigma}}_{Q}^{}=\ensuremath{\pi}/8\ensuremathσB∗​/\ensuremathσQ​=\ensuremathπ/8 holds for all values of NNN, a result that can be understood as a consequence of gauge invariance and asymptotic freedom of the Goldstone bosons in the low-energy limit.
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