Neutral Order Parameters in Metallic Criticality in d=2+1 from a Hairy Electron Star

TL;DR

The paper develops a holographic description of neutral order-parameter condensation in a $2+1$-D metal at finite density by embedding a neutral scalar into the electron star background, whose IR is a Lifshitz fixed point with finite $z$. Condensation induces neutral hair and a zero-temperature quantum phase transition of BKT type, with backreaction driving a change in the dynamical exponent $z$ across the transition. At finite temperature, the transition becomes second order with mean-field exponents, and backreaction continues to support the condensed phase while modifying IR data. The framework suggests holographic routes to model antiferromagnetic and quadrupole-nematic phases and to study competing neutral order parameters, with implications for metallic quantum criticality and Fermi-surface physics.

Abstract

We use holography to study the spontaneous condensation of a neutral order parameter in a (2+1)-dimensional field theory at zero-temperature and finite density, dual to the electron star background of Hartnoll and Tavanfar. An appealing feature of this field theory is the emergence of an IR Lifshitz fixed-point with a finite dynamical critical exponent $z$, which is due to the strong interaction between critical bosonic degrees of freedom and a finite density of fermions (metallic quantum criticality). We show that under some circumstances the electron star background develops a neutral scalar hair whose holographic interpretation is that the boundary field theory undergoes a quantum phase transition, with a Berezinski-Kosterlitz-Thouless character, to a phase with a neutral order parameter. Including the backreaction of the bulk neutral scalar on the background, we argue that the two phases across the quantum critical point have different $z$, a novelty that exists in certain quantum phase transitions in condensed matter systems. We also analyze the system at finite temperature and find that the phase transition becomes, as expected, second-order. Embedding the neutral scalar into a higher form, a variety of interesting phases could potentially be realized for the boundary field theory. Examples which are of particular interest to condensed matter physics include an antiferromagnetic phase where a vector condenses and break the spin symmetry, a quadrupole nematic phase which involves the condensation of a symmetric traceless tensor breaking rotational symmetry, or different phases of a system with competing order parameters.

Figures (10)

• Figure 1: A cartoon of the phase transition in our hologrphic setup as a function of the mass square, $m^2$, of the bulk neutral scalar field $\phi$. Once $m^2$ violates the BF bound, $m_c^2$, of the far-interior Lifshitz region of the electron star background, the boundary theory goes into a phase where the dual operator $\Phi$ spontaneously condenses. The backreaction of the scalar on the geometry changes the value of the dynamical exponent $z$. The condensation of $\Phi$ is controlled by the IR characteristics of the background which remains $d=2+1$ Lifshitz across the transition.
• Figure 2: Plots of $B$ versus $A$ for the neutral scalar field $\phi$ with a mass square $m^2L^2=-2.20$ on an electron star background with $m_{\rm f} =0.36$ and (a) $z=2$ and (b) $z=3$. The BF bounds in the far-interior of the star (with $m_{\rm f}=0.36$) for $z=2$ and $z=3$ are $m_c^2L^2\simeq -2.12$ and $-2.02$, respectively. Since $B$ is non-zero at $A=0$, the plots show the spontaneous condensation of the dual operator $\Phi$ in the respective boundary theories.
• Figure 3: Plots of $m_c^2L^2=-(z+2)^2/4\mathfrak{g}$ as function of $z$ for $m_{\rm f}=0.36$ (red) and $m_{\rm f}=0.7$ (blue). Here $L$ is the curvature radius of the asymptotic AdS$_4$ region. As $z$ increases the curves asymptote to the dashed line, which represents the scalar BF bound in ${\rm AdS}_2$.
• Figure 4: $B$ versus $m^2L^2$ for the neutral scalar field $\phi$ on an electron star background with $m_{\rm f} =0.36$ and $z=2$. Due to numerical limitations we could not access the region of mass square very close to the BF bound, $m_c^2L^2\simeq -2.12$, of the far-interior of the star. The inset shows a close up of the behavior of $B$ for larger values of $m^2L^2$.
• Figure 5: Plot of $\Delta\Omega/\mu^3$ versus $m^2L^2$. For the background, we set $z=2$ and $m_{\rm f}=0.36$. Numerical difficulties prevented us from obtaining more data points for the values of $m^2L^2$ very close to the critical value $m_c^2L^2\simeq -2.12$.