Quantum critical dynamics for a prototype class of insulating antiferromagnets

Quantum criticality is a fundamental organizing principle for studying strongly correlated systems. Nevertheless, understanding quantum critical dynamics at nonzero temperatures is a major challenge of condensed-matter physics due to the intricate interplay between quantum and thermal fluctuations. The recent experiments with the quantum spin dimer material ${\mathrm{TlCuCl}}_3$ provide an unprecedented opportunity to test the theories of quantum criticality. We investigate the nonzero-temperature quantum critical spin dynamics by employing an effective $O\left(N\right)$ field theory. The on-shell mass and the damping rate of quantum critical spin excitations as functions of temperature are calculated based on the renormalized coupling strength and are in excellent agreement with experiment observations. Their $TlnT$ dependence is predicted to be dominant at very low temperatures, which will be tested in future experiments. Our work provides confidence that quantum criticality as a theoretical framework, which is being considered in so many different contexts of condensed-matter physics and beyond, is indeed grounded in materials and experiments accurately. It is also expected to motivate further experimental investigations on the applicability of the field theory to related quantum critical systems.

Figure 1

A generic phase diagram with a second-order quantum phase transition, which arises in broad classes of systems [5, 9, 10, 15, 16]. The variables $r$ and $r_c$ denote the tuning parameter and quantum critical point, respectively. Due to different correlation characters resulting from the competition between thermal and quantum fluctuations, the whole diagram has been de facto divided into different regions, which is manifestly illustrated with different colors. The solid curve is used to denote a second-order classical phase transition, while the dot-dashed curves illustrate crossover regions qualitatively. The dashed line on the top is used to denote the cutoff temperature $T^{*}$, beyond which quantum criticality is negligible. The dark blue arrows, flowing right upward from the QCP, show the region we study in the main text.

Figure 2

Coupling strength at different physical energy scales $\mu$ following Eq. (10). Here, the reference coupling $g\left(T\right)=4.146$, and the Landau pole occurs at ${\mu }_L=12.253T$.

Figure 3

Two-loop renormalized mass vs reduced temperature $t$. The blue curve is based on Eq. (7). It is approximated by Eq. (12) and $M\left(T\right)=0.91t$, illustrated by red and green curves, respectively. The black and orange dashed lines mark the minimal experimental temperature and the crossover temperature when the $TlnT$ behavior starts to become dominant, respectively. The inset zooms in the low-temperature region ($t<0.1$), which clearly shows that the mass behavior significantly deviates from the linear-$T$ behavior.

Figure 4

Temperature dependence of the damping rate. Here, our theoretical result $\mathrm{FWHM}=0.16T$ (red line) is shown along with the experimental result of Ref. [16], which includes the values (without error bars) at different temperatures (black dots) and a fit of these values in terms of $\mathrm{FWHM}=0.15T$ (blue line).

Figure 5

A schematic plot of the dynamical structure factor at zero momentum, describing the behavior in different asymptotic regimes: on-shell ($\omega \sim M$) and off-shell at $\omega \ll M$ and $\omega \gg M$. The dashed lines interpolate among these regimes.

Figure 6

(a) One-loop self-energy diagram. (b) Two-point effective interaction in ${\mathcal{L}}_2$. (c) One-loop correction to the four-point vertex. There are two additional crossed channels, which are not shown. (d) UV vertex counterterm for the interaction. (e) Insertion of the finite two-point interaction in ${\mathcal{L}}_2$ into the one-loop self-energy (bubble) diagram. (f) Two-loop bubble diagram. (g) Self-energy contribution from the UV vertex counterterm. (h) Sunset diagram.

Figure 7

A typical daisy diagram with $m$ dressing bubbles for the ${\mathcal{L}}_0$ [Eq. (2) in the main text].