Quantum critical lines in holographic phases with (un)broken symmetry

TL;DR

The paper presents a unified framework for classifying IR scaling behaviors in holographic theories with or without U(1) symmetry, showing that quantum criticality generically appears as scale-invariant fixed points or hyperscaling-violating lines described by the exponents $\theta$, $z$, and $\zeta$. It demonstrates how cohesive and fractionalized phases arise from the IR flux structure, and how the scalar's IR behavior (constant or running) governs whether the geometry is AdS, AdS$_2\times\mathbb{R}^2$, Lifshitz, or hyperscaling-violating. A key methodological contribution is expressing IR couplings with exponential asymptotics and connecting hyperscaling-violating geometries to higher-dimensional, scale-invariant parents through generalized dimensional reduction. The results illuminate the global IR landscape of holographic phases, offer a mechanism for transitions between cohesive and fractionalized states, and provide a basis for exploring connections to strongly correlated systems and semi-local quantum criticality. The work also lays groundwork for multi-scalar extensions and quantum corrections, with potential implications for interpreting experimental quantum criticality in condensed matter systems.

Abstract

All possible scaling IR asymptotics in homogeneous, translation invariant holographic phases preserving or breaking a U(1) symmetry in the IR are classified. Scale invariant geometries where the scalar extremizes its effective potential are distinguished from hyperscaling violating geometries where the scalar runs logarithmically. It is shown that the general critical saddle-point solutions are characterized by three critical exponents ($θ, z, ζ$). Both exact solutions as well as leading behaviors are exhibited. Using them, neutral or charged geometries realizing both fractionalized or cohesive phases are found. The generic global IR picture emerging is that of quantum critical lines, separated by quantum critical points which correspond to the scale invariant solutions with a constant scalar.

Figures (2)

• Figure 1: Depiction of a schematic RG flow at zero temperature with a bifurcation point (see also Donos:2012js). The flow interpolates between AdS$_4$ in the UV and, in the IR, either an unstable, scale invariant ($\theta=0$) critical point, or two hyperscaling violating ($\theta\neq0$) critical lines on each side.
• Figure 2: Plots of the allowed parameter space $(\theta,z)$ for various values of the exponent $\zeta$. The upper left corner is the region where the IR is $r\to+\infty$, the lower right where it is $r\to0$. In red, we depict the region where $\beta_-$ is a real irrelevant deformation; in blue, the region where it is real and relevant; in green, the region where it is complex and relevant. In this case, the geometry \ref{['MassiveSolRunningScalar']} is dynamically unstable.