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arXiv:2209.00410 · Physical review. B./Physical review. B

Discrete global symmetries and dynamics of emergent fermions

Fan Yang, Fei Zhou

Published Jun 21, 2023• 0 citations
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Abstract

Global symmetries that define the number of low-energy degrees of freedom have profound consequences on universal properties near topological quantum critical points and in other gapless or nearly gapless states of emergent fermions. We take a ${Z}_{2}$ global symmetry (such as time-reversal) as an example to study its effect on thermodynamic and transport properties. Although the thermal entropy density of ${Z}_{2}$ symmetric systems is simply twice that of their counterparts without any global symmetries or the ${Z}_{1}$ class, the temperature dependence of thermal conductivity $\ensuremath{\kappa}$ is distinctly and drastically different for different symmetries. For systems with dynamic exponent $z=1$, in the ${Z}_{2}$ symmetric class, we have $\ensuremath{\kappa}\ensuremath{\propto}{T}^{\ensuremath{-}(d\ensuremath{-}1)}$ in the quantum critical regime near weakly interacting fixed points, while for systems with no global symmetries (i.e., the ${Z}_{1}$ class), we have $\ensuremath{\kappa}\ensuremath{\propto}{T}^{\ensuremath{-}(d+3)}$, with $d$ being the spatial dimension. Only near strong-coupling fixed points, both cases with or without ${Z}_{2}$ global symmetries follow the same scaling function, $\ensuremath{\kappa}\ensuremath{\propto}{T}^{d\ensuremath{-}1}$. These distinct scalings of thermal conductivity can also appear in gapless surface Majorana states.

DOI
10.1103/physrevb.107.245129

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In standard studies of quantum critical points (QCPs), the dynamic critical exponent zzz is introduced as a fundamental parameter along with global symmetries to identify universality classes. Often, the dynamic critical exponent zzz is set to be one as the most natural choice for quantum field theory representations, which further implies emergence of higher space-time symmetries near QCPs in many condensed matter systems. In this article, we study a family of topological quantum critical points (tQCPs) where the z=1z=1z=1 quantum field theory is prohibited in a fundamental representation by a protecting symmetry, resulting in tQCPs with z=2z=2z=2. We further illustrate that when strong interactions are properly taken into account, the stable weakly interacting gapless tQCPs with z=2z=2z=2 can further make a transition to another family of gapless tQCPs with dynamic critical exponent z=1z=1z=1, without breaking the protecting symmetry. Our studies suggest that dynamic critical exponents, as well as the degrees of freedom in fermion fields, can crucially depend on interactions in topological quantum phase transitions; in tQCPs, to a large extent, they are better thought of as emergent properties.
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