arXiv:1403.7222 · Physical Review Letters

Finite-Temperature Spin Dynamics in a Perturbed Quantum Critical Ising Chain with an E8 Symmetry

Jianda Wu, Márton Kormos, Qimiao Si A

Published Dec 9, 2014• 36 citations

Abstract

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.

DOI: 10.1103/physrevlett.113.247201

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arXiv:1403.7222 · Physical Review Letters

Finite-Temperature Spin Dynamics in a Perturbed Quantum Critical Ising Chain with an E8 Symmetry

Jianda Wu, Márton Kormos, Qimiao Si A

Published Dec 9, 2014• 36 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevlett.113.247201

Related papers

Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E 8 Symmetry

R. Coldea, D. M. Tennant, Elisa M. Wheeler +6 more

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.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.

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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.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.

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Julia Steinberg, N. P. Armitage, Fabian H. L. Eßler +1 more

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 CoNb ${}_{2}$ O ${}_{6}$ . 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 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 CoNb${}_{2}$O${}_{6}$. 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.

arXiv:1810.11466Physical review. B./Physical review. BJan 29, 201922 citations

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.

{E}_{8}

Lie algebra. Here we report on high-resolution terahertz spectroscopy of quantum spin dynamics in the ferromagnetic Ising-chain material

{\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}

. At 0.25 K in the magnetically ordered phase we identify characteristics of the first six

{E}_{8}

particles,

{\mathrm{m}}_{1}

{\mathrm{m}}_{6}

, and the two-particle

({\mathrm{m}}_{1}+{\mathrm{m}}_{2})

continuum in an applied transverse magnetic field of

{B}_{c}^{\text{1D}}=4.75 \mathrm{T}

, before the three-dimensional magnetic order is suppressed above ${B}_{c}^{\text{3D}}\ensuremath{\approx}5.3 \mathrm{T}$. The observation of the higher-energy particles

({\mathrm{m}}_{3} \mathrm{to} {\mathrm{m}}_{6})

above the low-energy two-particle continua features quantum many-body effects in the exotic dynamic spectrum.

1/{T}_{1}

of probe nuclei in the QC phase of 1D HAFs is obtained by considering Raman scattering processes which induce nuclear spin flips. The relaxation rate expression, which involves the temperature and the chemical potential, predicts scaling behavior of

1/{T}_{1}

consistent with recent experimental findings for quasi-1D HAF systems. A simple relationship between

1/{T}_{1}

and the deviation of the magnetization from saturation (${M}_{S}\ensuremath{-}M$) is predicted for the QC region.

{}_{2}

{}_{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.