Stable high-temperature paramagnons in a three-dimensional antiferromagnet near quantum criticality: Application to ${\mathrm{TlCuCl}}_3$

The complete set of hallmarks of the three-dimensional antiferromagnet near the quantum critical point has been recently observed in the spin dimer compound ${\mathrm{TlCuCl}}_3$. Nonetheless, the mechanism, responsible for several distinct features of the experimental data, has remained a puzzle, namely: (i) the paramagnons exhibit remarkable robustness to thermal damping and are stable up to high temperatures, where $k_{B}T$ is comparable with the excitation energy; and (ii) the width to mass ratios of the high-temperature paramagnons are, within the error bars, equal to that of the low-temperature amplitude (or Higgs) mode. We propose such a mechanism and identify two principal factors contributing to the scaling between width to mass ratios of the paramagnon and the amplitude mode: (i) the emergence of the thermal mass scale reorganizing the paramagnon decay processes, and (ii) substantial renormalization of the multimagnon interactions by thermal fluctuations. The study is carried out for the general case of a $D=3+1$ quantum antiferromagnet within the framework of the ${\phi }^4$ model using the hybrid Callan-Symanzik + Wilson thermal renormalization group method. Our approach is tested by demonstrating a good quantitative agreement with available experimental data across the phase diagram of ${\mathrm{TlCuCl}}_3$.

Figure 1

(a) A schematic phase diagram of a three-dimensional antiferromagnet near the quantum critical point: Néel – antiferromagnetic phase, QD – quantum disordered, QC – quantum critical, CC – classical critical. The dotted lines mark crossovers. As one moves along the vertical arrow, the magnetic excitations evolve from spin waves and the amplitude mode at low-temperature to the high-temperature paramagnons. (b) The scaling between the width to mass ratio for the high-temperature paramagnons and that for the low-temperature amplitude mode. Squares are experimental data of Ref. [5] and the dashed straight line ${\alpha }_p={\alpha }_H$ is a guide to the eye. The high-temperature data have been collected at $\sim 11.5$ and $13\mathrm{K}$ for $p=1.75$ and $3.6\mathrm{kbar}$, respectively. The low-temperature amplitude mode width to mass ratio has been measured at $\sim 1.8\mathrm{K}$ in both cases.

Figure 2

Leading-order processes yielding broadening of (a) paramagnons above the Néel temperature and (b) amplitude mode below the Néel temperature. Labeling of the lines: P – paramagnon, AM – amplitude mode, TM – transverse mode.

Figure 3

(a) Ratio of the high- to the low-temperature coupling constants as a function of the low-temperature coupling ${\lambda }_{\mathrm{phys}}(T=0)$. (b) Normalized square of the high-temperature paramagnon mass as a function of the high-temperature coupling constant. TRG – thermal renormalization group. HTL – result of the resummation of the hard thermal loops (see the text).

Figure 4

(a) The relation between the stability parameters of the high-temperature paramagnons and the low-temperature amplitude mode. The black solid line is the solution to the thermal renormalization group (TRG) equations and the dashed line ${\alpha }_p={\alpha }_H$ is a guide to the eye. Solid squares are experimental data of Ref. [5]. The red dot-dashed line shows the relation between ${\alpha }_p(T\gg T_N)$ and ${\alpha }_H(T\ll T_N)$ calculated with discarded thermal flow of the coupling constant. The shaded area is defined by the condition ${\lambda }_{\mathrm{phys}}(T=0)/4<1$ that marks the regime of applicability of the renormalized perturbation theory. (b) The ratio of the thermal- to the zero-temperature coupling constant as a function of temperature for the parameters chosen so that ${\alpha }_H(T=0)=0.15$. (c) The calculated normalized width ${\mathrm{FWHM}}_p/T$ of the paramagnons as a function of temperature (solid line) for ${\alpha }_H(T=0)=0.15$ that roughly corresponds to the experimental value for ${\mathrm{TlCuCl}}_3$ at $p=1.75\mathrm{kbar}$. The green points are the data extracted from Ref. [5]. (d) Thermal flow of the parameters for $T=1.5·T_N$ and ${\alpha }_H(T=0)=0.16$. The dashed lines are guides to the eye.

Figure 5

Fit to the experimental data of Ref. [5]. (solid squares). Blue line – hybrid RG. Black dashed line – fit with a square root form $T_N\left(p\right)=b·\sqrt{p-p_c}$.

Figure 6

(a)–(d) Temperature and pressure dependence of masses and widths of the magnetic excitations in ${\mathrm{TlCuCl}}_3$. The curves in the low-temperature regime above $p_c=1.07\mathrm{kbar}$ correspond to the amplitude (Higgs) mode, whereas in the high-temperature and low-pressure range to the paramagnons. (e)–(f) Masses and widths of the magnetic excitations at critical pressure as a function of temperature. In all panels the solid lines are solutions to the hybrid RG equations, while the solid squares are the experimental data of Ref. [5].

Figure 7

A comparison of the damping rates $\gamma \left(T\right)\equiv {\mathrm{FWHM}}_p\left(T\right)/2$ and the thermal coupling constant $\lambda \left(T\right)$ calculated within the formalism of the present paper and the thermal RG variant of Ref. [14] (other scheme). The calculations have been performed for three field components ($N=3$) and two choices of the zero-temperature coupling constant $\lambda (T=0)=1.0$ and $\lambda (T=0)=0.01$. The thin purple solid lines are the fits to the curves close to the critical point, as discussed in the text.

Figure 8

(a) A representative superdaisy diagram accounted for by resummation. (b) The one-loop contribution to the magnon mass operator in the resummed model.