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arXiv:2111.15107 · Physical review. B./Physical review. B

Local dynamics and thermal activation in the transverse-field Ising chain

Jiahao Yang, Weishi Yuan, Takashi Imai, Qimiao Si A, Jianda Wu, Márton Kormos

Published Sep 29, 2022• 10 citations
arXivHTMLPDF

Abstract

There has been considerable recent progress in identifying candidate materials for the transverse-field Ising chain (TFIC), a paradigmatic model for quantum criticality. Here, we study the local spin dynamical structure factor of different spin components in the quantum disordered region of the TFIC. We show that the low-frequency local dynamics of the spins in the Ising- and transverse-field directions have strikingly distinctive temperature dependencies. This leads to the thermal-activation gap for the secular term of the nuclear magnetic resonance $1/{T}_{2}^{\ensuremath{'}}$ relaxation rate to be half of that for the $1/{T}_{1}$ relaxation rate. Our findings reveal a surprise in the nonzero-temperature dynamics of the venerable TFIC model and uncover a means to evince the material realization of the TFIC universality.

DOI
10.1103/physrevb.106.125149

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There has been considerable recent progress in identifying candidate materials for the transverse-field Ising chain (TFIC), a paradigmatic model for quantum criticality. Here, we study the local spin dynamical structure factor of different spin components in the quantum disordered region of the TFIC. We show that the low-frequency local dynamics of the spins in the Ising- and transverse-field directions have strikingly distinctive temperature dependencies. This leads to the thermal-activation gap for the secular term of the nuclear magnetic resonance 1/T2\ensuremath′1/{T}_{2}^{\ensuremath{'}}1/T2\ensuremath′​ relaxation rate to be half of that for the 1/T11/{T}_{1}1/T1​ relaxation rate. Our findings reveal a surprise in the nonzero-temperature dynamics of the venerable TFIC model and uncover a means to evince the material realization of the TFIC universality.
The quantum Ising chain in a transverse magnetic field is $t\phantom{\rule{0}{0ex}}h\phantom{\rule{0}{0ex}}e$ paradigm of a quantum phase transition, displaying universal quantum criticality at low temperatures. Insights from this model have informed studies of quantum criticality in numerous modern correlated-electron materials. There have been a number of experimental studies of the quantum Ising chain, most notably in the material CoNb2{}_{2}2​O6{}_{6}6​. It has recently become clear that there was a surprising discrepancy between the observations: the energy gap in the quantum disordered regime as measured by nuclear magnetic resonance was approximately twice that measured by other methods. Here, the authors show via careful analysis of the low-temperature excitations that this is not a discrepancy, rather, it is in fact to be expected from the subtle interactions between the excitations. Although the underlying spin excitations are bosonic, their interactions transmute them into fermions, and this is a key ingredient in understanding the experiments.
The dynamical structure factor of the transverse field Ising model (TFIM) shows universal power-law divergence at its quantum critical point, signatures of which have been arguably observed in inelastic neutron scattering studies of quantum spin chain materials, for example CoNb 2 O 6 . However, it has recently been suggested that its microscopic description is better captured in terms of a twisted Kitaev spin chain (TKSC) with bond-anisotropic couplings. Here, we present exact results for the dynamical structure factor of the TKSC across its quantum critical point, analyzing both the universal low-frequency response and the non-universal high-energy features. In addition, we explore extensions of the model including broken glide symmetry as well as the case of random, and incommensurate magnetic fields. Notably, in the latter case, the fermionic excitations exhibit a localization-delocalization transition, which is manifest in the dynamical response as a distinct signature at finite frequency. We discuss the relevance of these features for the observation of quantum critical response in experiments.
The transverse field Ising chain model is ideally suited for testing the fundamental ideas of quantum phase transitions because its well-known T ¼ 0 ground state can be extrapolated to finite temperatures.Nonetheless, the lack of appropriate model materials hindered the past effort to test the theoretical predictions.Here, we map the evolution of quantum fluctuations in the transverse field Ising chain based on nuclear magnetic resonance measurements of CoNb 2 O 6 , and we demonstrate the finite-temperature effects on quantum criticality for the first time.From the temperature dependence of the 93 Nb longitudinal relaxation rate 1=T 1 , we identify the renormalized classical, quantum critical, and quantum disordered scaling regimes in the temperature (T) vs transverse magnetic field (h ⊥ ) phase diagram.Precisely at the critical field h c ⊥ ¼ 5.25 Æ 0.15 T, we observe a power-law behavior, 1=T 1 ∼ T -3=4 , as predicted by quantum critical scaling.Our parameter-free comparison between the data and theory reveals that quantum fluctuations persist up to as high as T ∼ 0.4J, where the intrachain exchange interaction J is the only energy scale of the problem.
The concept of T = 0 quantum phase transi-tions has emerged as an overarching theme in strongly correlated electron physics [1–9]. The nature of quantum fluctuations near the quantum critical point, however, remains enigmatic [10]. How well does the quantum criticality account for finite temperature properties? How high in tem-perature does the effect of the quantum critical point persist?[10, 11] The dearth of appropriate model materials has not permitted experimental-ists to address these fundamental questions even for the transverse field Ising chain (TFIC), a cel-ebrated textbook example of quantum criticality [6]. Here we use NMR (Nuclear Magnetic Res-onance) to map the evolution of quantum spin fluctuations in CoNb2O6 [12–14], which has re-cently been proposed as an ideal model material for the TFIC [15, 16]. We will demonstrate that the effect of the quantum critical point persists to a remarkably high temperature, T ∼ 0.4J, where the exchange coupling J is the only energy scale in the problem.
A spectrum exhibiting E₈ 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 E₈ description for CoNb₂O₆.
We study the relaxation behaviour of the quantum Ising chain, focusing our attention onto the non-equilibrium dynamics of the transverse magnetization. The initial states, from which the magnetization relaxes, are product states such as those generated by setting in contact several systems, each initially equilibrated at a given temperature. Due to the free fermionic structure of the chain, the dynamics of the transverse magnetization is easily expressed in a compact form. In the completely factorized initial state, corresponding to a situation where all the spins are thermalized independently, we obtain in the scaling limit the Green function associated to the transverse magnetization. The relaxation behaviour is also considered in the system-bath case, where part of the chain called the system is thermalized at a temperature Ts and the remaining part is at a temperature Tb. The magnetization profiles show a scaling behaviour. Moreover, in the extreme case Tb=∞ and Ts=0, it is shown that the magnetization relaxes in quantized steps in the strong transverse field region.