Quantum Criticality

TL;DR

Quantum Criticality reviews the theoretical framework for quantum phase transitions in correlated electron systems and their finite-temperature signatures. It contrasts magnetic insulators (e.g., Ising-chain and dimer antiferromagnets) with metallic transitions, highlighting how a tuning parameter $g$ drives a continuous transition at $g_c$ and produces a quantum-critical ground state with diverging length scales. The metallic case introduces strong coupling between fermions and critical modes, yielding non-Fermi-liquid and strange-metal behavior, Fermi-surface reconstruction, and broad experimental evidence across cuprates, pnictides, and related materials. The article also discusses competing orders, such as unconventional superconductivity and nematicity, stressing the need for a comprehensive theory of how multiple orders interact near quantum criticality, including the role of disorder.

Abstract

This is a review of the basic theoretical ideas of quantum criticality, and of their connection to numerous experiments on correlated electron compounds. A shortened, modified, and edited version appeared in Physics Today. This arxiv version has additional citations to the literature.

Figures (7)

• Figure 1: A zig-zag chain of Co$^{2+}$ ions, surrounded by oxygen octahedra in CoNb$_{2}$O$_6$. Coldea et al.coldea showed that the spins on Co$^{2+}$ undergo a quantum phase transition in a magnetic field applied transverse to the chain direction.
• Figure 2: Top: The unit cell of the insulating antiferromagnet TlCuCl$_3$. Bottom: A simple realization of the dimer antiferromagnet on the square lattice. The full red lines represent an exchange interaction $J>0$, while the dashed green lines have exchange $J/g$ with $g>1$. The ellipses represent a singlet valence bond of spins $(|\uparrow \downarrow \rangle - | \downarrow \uparrow \rangle )/\sqrt{2}$.
• Figure 3: Observation of the quantum phase transition in TlCuCl$_3$ in neutron scattering experiments by Ruegg et al.ruegg. At low pressure, the ground state is the $g>g_c$ quantum paramagnet. Its excitations are 'triplons' created by breaking nearest-neighbor singlet bonds (see Fig. \ref{['fig:qc']}), and these are represented by the blue symbols; their energy vanishes as the critical point (QCP) is approached. At high pressure we have the Néel state, which has antiferromagnetic order at low temperatures. Here, there are two types of excitations: the 'spin waves' are slow deformations of the Néel order (see Fig. \ref{['fig:qc']}) which have vanishing energy in the limit of long wavelengths, and these are represented by the white circles. The other excitation is an analog of the Higgs boson: it is an oscillation in the magnitude of the local Néel order, and its energy also vanishes upon approaching the critical point (red symbols).
• Figure 4: Non-zero temperature ($T$) phase diagram for the model in Fig. \ref{['fig:dimer']}. For $g<g_c$, the low energy excitations are slow deformations of the Néel order, the spin waves. These have strong non-linear couplings which destroy the Néel order at all $T>0$, and the long time dynamics have a classical description in a theory of interacting spin waves. For $g>g_c$, the excitations are the triplet $S=1$ excited states of each dimer. These become mobile and form a dilute gas of 'triplons' whose dynamics can be described by the classical Boltzmann equation. Quantum criticality appears in the intermediate orange region, where there is no effective classical theory at the scale of the characteristic spin equilibration time; instead we have the strongly coupled dynamics of the non-trivial critical excitations which have neither a particle or wave interpretation.
• Figure 5: Non-zero temperature properties of the Ising quantum spin chain which models CoNb$_2$O$_6$ shown in Fig. \ref{['fig:conbo']}. Shown are theoretical computations from the exactly solvable spin chain with nearest-neighbor exchange. The color plot indicates the value of the $(4\hbar c /\pi k_B) (d \xi^{-1} /dT)$, where $\xi$ is the spin correlation length and $c$ is the velocity of spin excitations; this dimensionless number has a $T$ dependence similar to that of the $T$ derivative of $\tau_{\rm eq}^{-1}$ of non-integrable strongly-interacting quantum critical points. Also indicated are typical spin configurations in the two low temperature regimes. For $g<g_c$, we have the ferromagnetic configurations of Eq. (\ref{['ferro']}) separated by domain walls, while for $g>g_c$ we have the paramagnetic state of Eq. (\ref{['para']}) with its 'reversed spin' excitations; here $|\rightarrow \rangle = (|\uparrow \rangle + |\downarrow \rangle)/\sqrt{2}$ and $|\leftarrow \rangle = (|\uparrow \rangle - |\downarrow \rangle)/\sqrt{2}$.