Asymptotic freedom in quantum magnets

Abstract

Phase transitions in isotropic quantum antiferromagnets are described by an O(3) nonlinear quantum field theory. In three dimensions, the fundamental property of this theory is logarithmic scaling of the coupling constant. At the quantum critical point the coupling asymptotically vanishes and the quasiparticles become free. This logarithmic decay of the coupling constant has never been observed. In this paper, we derive finite temperature properties of the field theory and use our results to analyze the existing data on the real antiferromagnet TlCuCl3. Including finite temperatures into the theory, we find that agreement between theory and experiment is sufficiently sensitive to unambiguously identify the asymptotic decay of the coupling constant. We also comment on the unique possibility to study Landau pole-physics in quantum magnets.PACS numbers: 64.70. Tg, 75.40.Gb, 75.10.Jm Asymptotic freedom plays a crucial role in Quantum Chromodynamics. The freedom, which is due to nonabelian gauge fields, means a logarithmic decay of the coupling constant at high energies. Ultimately at infinite energies particles do not interact, this is ultraviolet asymptotic freedom [1,2]. In three dimensional (3D) non-gauge quantum field theories as well as in abelian gauge theories, the coupling constant decays logarithmically at low energies [3]. However, usually this decay is terminated because of a low energy cutoff. For example in Quantum Electrodynamics the cutoff is due to the rest energy of the electron. The low energy logarithmic decay can be observed only at a quantum critical point (QCP) where the cutoff energy is zero. It is known to the theory of quantum magnetism that certain antiferromagnetic systems can be driven to a QCP, and that at such a point the characteristic low energy magnetic excitations become non-interacting. At the QCP they experience "infrared asymptotic freedom". Quantum antiferromagnets in the vicinity of a QCP therefore provide a perfect testing ground for such behaviour. The low energy logarithmic decay of the coupling constant at a QCP can, in principle, be observed at zero temperature. The zero temperature case is well understood theoretically, a 3D quantum antiferromagnet at zero temperature is equivalent to a 4D classical antiferromagnet at finite temperature. The 3D QCP corresponds to the Néel transition in the 4D case. Thermodynamic quantities scale as powers of the running coupling constant [4]. Unfortunately, the existing zero temperature experimental data are not sufficient to pin down the logarithmic scaling. On the other hand combined zero and nonzero temperature data on TlCuCl 3 [5][6][7] provide an excellent opportunity to search for fingerprints of asymptotic freedom at the QCP. To perform this search we develop a theory of the QCP which accounts for both quantum and thermal fluctuations. After that we compare the theoretical predictions with experimental data. The comparison unambiguously indicates a logarithmically running cou- pling constant.The phase diagram of the dimerized 3D...

Excitation-Gap Scaling near Quantum Critical Three-Dimensional Antiferromagnets

“…2, that this scaling form does not account for the gap data in the critical region. In fact, the logarithmic decay of the renormalized interaction strength upon approaching the quantum critical point leads to a logarithmic correction to the mean-field scaling behavior,in the vicinity of the quantum critical point [6,10,43,44]. This follows from the quantum-to-classical mapping with a dynamical critical exponent z = 1, and relating ∆ t = ξ −1 τ to the correlation length in the imaginary-time direction.…”

“…in the vicinity of the quantum critical point [6,10,43,44]. This follows from the quantum-to-classical mapping with a dynamical critical exponent z = 1, and relating ∆ t = ξ −1 τ to the correlation length in the imaginary-time direction.…”

“…Also included in Fig. 1 is the field-theory prediction [6,23] for the Higgs mass scaling ∆ H (g) = √ 2∆(g), with ∆(g) = ∆ t (g c +(g c −g)). The extrapolated peak positions closely follow this scaling prediction, indicating that this second feature in S S (ω, Q) indeed signals the amplitude mode of the bicube system.…”

“…Due to a logarithmic decay of the renormalized interaction strength upon approaching the critical point, this field theory exhibits logarithmic corrections to a Gaussian fixed point [7][8][9][10]. This leads to characteristic multiplicative logarithmic scaling corrections to the bare mean-field behavior in various physical quantities that are in principle accessible by several experimental probes, such as in thermodynamic measurements or neutron and light scattering techniques [6,[11][12][13][14], if probed at the relevant energy scales near the quantum critical point.A well characterized example system of this scenario is provided by the dimerized spin-half compound TlCuCl 3 : under the application of hydrostatic pressure, this system features a quantum phase transition from a gapped quantum disordered state into an antiferromagnetically ordered phase [15]. The magnetic excitations across the quantum critical region have been analyzed in detail recently by inelastic neutron scattering [16][17][18].…”

“…In contrast to the situation in lower-dimensional systems, we observe in this three-dimensional coupled-dimer system a distinct signal from the amplitude mode also in the dynamical spin structure factor. The width of the Higgs mode resonance is observed to scale linearly with the Higgs mass near criticality, indicative of this critically well-defined excitation mode of the symmetry broken phase.Quantum critical three-dimensional antiferromagnets provide considerably valuable condensed matter realizations of (infrared) asymptotically free quantum field theories: based on the quantum-to-classical mapping, the critical field theory that describes the underlying quantum critical point is the classical four-dimensional O(3) φ 4 -theory [1][2][3][4][5][6]. Due to a logarithmic decay of the renormalized interaction strength upon approaching the critical point, this field theory exhibits logarithmic corrections to a Gaussian fixed point [7][8][9][10].…”