Holographic Metamagnetism, Quantum Criticality, and Crossover Behavior

TL;DR

This work identifies a holographic quantum critical point in 4D gauge theories at finite density and magnetic field by solving 5D Einstein-Maxwell-Chern-Simons equations with $k=2/\sqrt{3}$. The authors establish a critical magnetic field $\hat{B}_c$ at which the entropy scales as $\hat{s}\sim \hat{T}^{1/3}$, while away from the point the system exhibits Fermi-liquid behavior for $\hat{B}>\hat{B}_c$ and a nonzero ground-state entropy for $\hat{B}<\hat{B}_c$, all governed by a universal scaling form $\hat{s}=\hat{T}^{1/3} f((\hat{B}-\hat{B}_c)/\hat{T}^{2/3})$. The critical theory has $d=1$, $z=3$, and a relevant operator of dimension $\Delta=2$, aligning with Hertz/Millis expectations for metamagnetic quantum criticality, and the results resonate with experimental observations in Sr$_{3}$Ru$_{2}$O$_{7}$. Overall, the paper demonstrates a universal holographic realization of metamagnetic quantum criticality and provides precise scaling data and a framework for connecting gravity duals to condensed-matter metamagnetic phenomena.

Abstract

Using high-precision numerical analysis, we show that 3+1 dimensional gauge theories holographically dual to 4+1 dimensional Einstein-Maxwell-Chern-Simons theory undergo a quantum phase transition in the presence of a finite charge density and magnetic field. The quantum critical theory has dynamical scaling exponent z=3, and is reached by tuning a relevant operator of scaling dimension 2. For magnetic field B above the critical value B_c, the system behaves as a Fermi liquid. As the magnetic field approaches B_c from the high field side, the specific heat coefficient diverges as 1/(B-B_c), and non-Fermi liquid behavior sets in. For B<B_c the entropy density s becomes non-vanishing at zero temperature, and scales according to s \sim \sqrt{B_c - B}. At B=B_c, and for small non-zero temperature T, a new scaling law sets in for which s\sim T^{1/3}. Throughout a small region surrounding the quantum critical point, the ratio s/T^{1/3} is given by a universal scaling function which depends only on the ratio (B-B_c)/T^{2/3}. The quantum phase transition involves non-analytic behavior of the specific heat and magnetization but no change of symmetry. Above the critical field, our numerical results are consistent with those predicted by the Hertz/Millis theory applied to metamagnetic quantum phase transitions, which also describe non-analytic changes in magnetization without change of symmetry. Such transitions have been the subject of much experimental investigation recently, especially in the compound Sr_3 Ru_2 O_7, and we comment on the connections.

Figures (8)

• Figure 1: Schematic phase diagram illustrating the various behaviors of the entropy density versus temperature and magnetic field. The region inside the dotted line is controlled by the quantum critical point at $(\hat{T}=0,\hat{B}=\hat{B}_c)$, and the entropy density can be expressed in terms of a single scaling function $f$ of $(\hat{B} - \hat{B}_c)/T^{2/3}$. We move around inside this region by changing the temperature $\hat{T}$ and the relevant coupling $\hat{B}-\hat{B}_c$. The boundary of the region is defined to be where irrelevant operators become important. The yellow region denotes a regime where temperature is the largest energy scale, corresponding to the argument of the scaling function $f$ being small. Outside the yellow region the low temperature behavior of the entropy density, for fixed $\hat{B}$, is either constant or linear in $\hat{T}$, depending on whether the quantum critical point is approached from below or from above $\hat{B}_c$ as $\hat{T}\rightarrow 0$.
• Figure 2: Plot of entropy versus temperature. On the left we compare $\hat{B}=.53$ to $\hat{B}=0$; this plot is an improved version of Fig. 3 in D'Hoker:2009bc. On the right we exhibit the linear $\hat{s} \sim \hat{T}$ low temperature behavior.
• Figure 3: Plot showing the divergence of $\hat{s}/\hat{T}$ near $\hat{B}_c$ at low temperatures. The straight line through the data points is included to guide the eye.
• Figure 4: Plot showing $\hat{s} \sim \hat{T}^{1/3}$ scaling behavior at $\hat{B}=\hat{B}_c$. The straight line through the data points is included to guide the eye, and has slope $1/3$, consistent with (\ref{['alpha']}).
• Figure 5: The left plot shows the crossover of $\hat{s}$ for low $\hat{T}$. At moderately low temperatures $\hat{s}$ scales as $\hat{T}^{1/3}$ (lower left corner of the plot), while at ultra-low temperatures $\hat{s}$ scales as $\hat{T}$ for $\hat{B} > \hat{B}_c$ (curves a, b, c, d, e, f), and tends to a non-zero constant for $\hat{B} < \hat{B}_c$ (curve g). The right plot shows the crossover for $\hat{s}$ from the moderately low temperature $\hat{T}^{1/3}$ scaling to the high temperature $\hat{T}^3$ behavior. The dots represent numerical data points, while the solid interpolating lines are included to guide the eye.