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arXiv:1103.0095

Unveiling the quantum critical point of an Ising chain

Dai, Y. F., H. Zhang, Zhou, S. Y., Bingying Pan, Qiu, X., Xiaochen Hong, T. Y. Guan, J. K. Dong, Y. Chen, S. Y. Li

Published Mar 1, 2011• 1 citation
arXivHTMLPDF

Abstract

Quantum phase transitions occur at zero temperature upon variation of some nonthermal control parameters. The Ising chain in a transverse field is probably the most-studied model undergoing such a transition, from ferromagnetic to paramagnetic state1,2. This model can be exactly solved by using a Jordan-Wigner transformation, which transforms the spins into noninteracting spinless fermions1. At the quantum critical point, the magnetic excitations can carry arbitrarily low energy and dominate the low temperature properties. Here we report the unveiling of such quantum critical point in quasi-one-dimensional Ising ferromagnet CoNb2O6 by ultra-low-temperature thermal conductivity measurements. We find that in the paramagnetic state, phonons are scattered by the magnetic excitations above certain temperature Ts, which corresponds to a gap. As predicted by the theoretical model1, this gap linearly goes to zero with decreasing the transverse field, thus determining the quantum critical point of the Ising chain.

DOI
10.48550/arxiv.1103.0095

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Quantum phase transitions occur at zero temperature upon variation of some nonthermal control parameters. The Ising chain in a transverse field is probably the most-studied model undergoing such a transition, from ferromagnetic to paramagnetic state1,2. This model can be exactly solved by using a Jordan-Wigner transformation, which transforms the spins into noninteracting spinless fermions1. At the quantum critical point, the magnetic excitations can carry arbitrarily low energy and dominate the low temperature properties. Here we report the unveiling of such quantum critical point in quasi-one-dimensional Ising ferromagnet CoNb2O6 by ultra-low-temperature thermal conductivity measurements. We find that in the paramagnetic state, phonons are scattered by the magnetic excitations above certain temperature Ts, which corresponds to a gap. As predicted by the theoretical model1, this gap linearly goes to zero with decreasing the transverse field, thus determining the quantum critical point of the Ising chain.
Regarded as the closest realization of the quantum Ising model in a magnetic field, which features stable quasiparticles with an emergent E8 symmetry, CoNb2O6 keeps being a source of inspiration by manifesting intricacies beyond this model. A notable example is the recent finding of an unexpected band of localized gapless fermionic excitations around the field-induced quantum critical point (QCP), whose origin remains unclear. Despite differences in many aspects compared to CoNb2O6, BaCo2V2O8 in a magnetic field also exhibits the E8 physics, while allowing for better scrutiny of the critical excitations associated with its well-separated one- and three-dimensional QCPs. Here we study the low-lying magnetic excitations within the antiferromagnetic order of BaCo2V2O8 by performing ultralow-temperature heat transport measurements. The field-independent thermal conductivity at the lowest temperatures and the suppressed thermal conductivity with an increasing field at higher temperatures point to heat-carrying phonons scattered by magnetic excitations. For the magnetic excitations, importantly, we find no evidence for the emergence, around the one-dimensional QCP, of gapless fermionic excitations similar to those observed in CoNb2O6. Based on the contrasting cases of BaCo2V2O8 and CoNb2O6, we propose that the frustrated alignment of chains and the proximity of two QCPs in CoNb2O6 may put this system in a unique place to host the gapless fermionic excitations.
Quantum phase transitions take place between distinct phases of matter at zero temperature. Near the transition point, exotic quantum symmetries can emerge that govern the excitation spectrum of the system. A symmetry described by the E8 Lie group with a spectrum of eight particles was long predicted to appear near the critical point of an Ising chain. We realize this system experimentally by using strong transverse magnetic fields to tune the quasi-one-dimensional Ising ferromagnet CoNb2O6 (cobalt niobate) through its critical point. Spin excitations are observed to change character from pairs of kinks in the ordered phase to spin-flips in the paramagnetic phase. Just below the critical field, the spin dynamics shows a fine structure with two sharp modes at low energies, in a ratio that approaches the golden mean predicted for the first two meson particles of the E8 spectrum. Our results demonstrate the power of symmetry to describe complex quantum behaviors.
For a system at absolute zero temperature, all thermal fluctuations are frozen out, while quantum fluctuations prevail.These microscopic quantum fluctuations can induce a quantum phase transition (QPT) in the ground state of a many-body system when the relative strength of two competing energy terms is varied across a critical value.This paper presents a detailed and quantitative study of the magnetic properties of Ising two-spin system at zero temperature in the presence of longitudinal, transverse and mixed fields, and also at finite temperatures.At the critical longitudinal magnetic field, the system is found to undergo a QPT to a ferromagnetic state.For this system, an infinitesimal departure from zero transverse field is found to trigger off QPT and also remove the degeneracies that persisted even in the presence of longitudinal field.These quantum fluctuations, leading to QPT are more evident in the mixed field than the separate effect of longitudinal and transverse fields.The observed quantum effects at zero temperature for this system are found to be completely wiped out at finite temperatures due to thermal fluctuations.A temperature increase favours antiferromagnetic alignment of spins, while a decrease in temperature favours ferromagnetic alignment.
We study an infinite-range ferromagnetic Ising model in the presence of a transverse magnetic field, which exhibits a quantum paramagnetic-ferromagnetic phase transition at a critical value of the transverse field. In the thermodynamic limit, the low-temperature properties of this model are dominated by the behavior of a single large classical spin governed by an anisotropic Hamiltonian. Using this property, we study the quench and ac dynamics of the model both numerically and analytically, and develop a correspondence between the classical phase-space dynamics of a single spin and the quantum dynamics of the infinite-range ferromagnetic Ising model. In particular, we compare the behavior of the equal-time order-parameter correlation function both near to and away from the quantum critical point in the presence of a quench or ac transverse field. We explicitly demonstrate that a clear signature of the quantum critical point can be obtained by studying the ac dynamics of the system even in the classical limit. We discuss possible realizations of our model in experimental systems.
In this work we investigate the quantum phase transitions at zero temperature of the one-dimensional transverse Ising model with an extra term containing four-spin interactions. The competition between the energy couplings of the model leads to an interesting zero-temperature phase diagram. We use a modified Lanczos method to determine the ground state and the first excited state energies of the system, with sizes of up to 20 spins. We apply finite size scaling to the energy gap to obtain the boundary region where ferromagnetic to paramagnetic transition takes place. We also find the critical exponent associated with the correlation length. We find a degenerate ⟨3,1⟩⟨3,1⟩⟨3,1⟩ phase region. The first-order transition boundary between this phase and the paramagnetic phase is determined by analyzing the behavior of the transverse spin susceptibility as the system moves from one region to the other.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.