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 in the quantum spin dimer material 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 $O(N)$ 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, which are in excellent agreements with experiment observations. Their $T\ln T$ dependence is predicted to be dominant at very low temperatures, which is to be tested in future experiments. Our work provides confidence that quantum criticality as a theoretical framework, 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.Comment: 9 pages, 7 figure

Some notes on commutators of the fractional maximal function on variable Lebesgue spaces

Let $0<\alpha<n$ and $M_{\alpha}$ be the fractional maximal function. The nonlinear commutator of $M_{\alpha}$ and a locally integrable function $b$ is given by $[b,M_{\alpha}](f)=bM_{\alpha}(f)-M_{\alpha}(bf)$. In this paper, we mainly give some necessary and sufficient conditions for the boundedness of $[b,M_{\alpha}]$ on variable Lebesgue spaces when $b$ belongs to Lipschitz or BMO(\rn) spaces, by which some new characterizations for certain subclasses of Lipschitz and BMO(\rn) spaces are obtained.Comment: 20 page

Crossovers and critical scaling in the one-dimensional transverse-field Ising model

We consider the scaling behavior of thermodynamic quantities in the one-dimensional transverse-field Ising model near its quantum critical point (QCP). Our study has been motivated by the question about the thermodynamical signatures of this paradigmatic quantum critical system and, more generally, by the issue of how quantum criticality accumulates entropy. We find that the crossovers in the phase diagram of temperature and (the non-thermal control parameter) transverse field obey a general scaling ansatz, and so does the critical scaling behavior of the specific heat and magnetic expansion coefficient. Furthermore, the Gr\"{u}neisen ratio diverges in a power-law way when the QCP is accessed as a function of the transverse field at zero temperature, which follows the prediction of quantum critical scaling. However, at the critical field, upon decreasing the temperature, the Gr\"uneisen ratio approaches a constant instead of showing the expected divergence. We are able to understand this unusual result in terms of a peculiar form of the quantum critical scaling function for the free energy; the contribution to the Gr\"uneisen ratio vanishes at the linear order in a suitable Taylor expansion of the scaling function. In spite of this special form of the scaling function, we show that the entropy is still maximized near the QCP, as expected from the general scaling argument. Our results establish the telltale thermodynamic signature of a transverse-field Ising chain, and will thus facilitate the experimental identification of this model quantum-critical system in real materials.Comment: 7 pages, 5 figure

Finite temperature spin dynamics in a perturbed quantum critical Ising chain with an E8E_8 symmetry

A spectrum exhibiting $E_8$ symmetry is expected to arise when a small longitudinal field is introduced in the transverse-field Ising chain at its quantum critical point. Evidence for this spectrum has recently come from neutron scattering measurements in cobalt niobate, a quasi one-dimensional Ising ferromagnet. Unlike its zero-temperature counterpart, the finite-temperature dynamics of the model has not yet been determined. We study the dynamical spin structure factor of the model at low frequencies and nonzero temperatures, using the form factor method. Its frequency dependence is singular, but differs from the diffusion form. The temperature dependence of the nuclear magnetic resonance (NMR) relaxation rate has an activated form, whose prefactor we also determine. We propose NMR experiments as a means to further test the applicability of the $E_8$ description for CoNb$_2$O$_6$.Comment: 5 pages 2 figures - Supplementary Material 11 page