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arXiv:2103.09128 · Physical review. B./Physical review. B

Cascade of singularities in the spin dynamics of a perturbed quantum critical Ising chain

Xiao Wang, Haiyuan Zou, Kristóf Hódsági, Márton Kormos, G. Takács, Jianda Wu

Published Jun 10, 2021• 22 citations
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Abstract

When the quantum critical transverse-field Ising chain is perturbed by a longitudinal field, a quantum integrable model emerges in the scaling limit with massive excitations described by the exceptional ${E}_{8}$ Lie algebra. Using the corresponding analytical form factors of the quantum ${E}_{8}$ integrable model, we systematically study the spin dynamic structure factor of the perturbed quantum critical Ising chain, where particle channels with total energy up to $5{m}_{1}$ (${m}_{1}$ being the mass of the lightest ${E}_{8}$ particle) are exhausted. In addition to the significant single-particle contributions to the dynamic spectrum, each two-particle channel with different masses is found to exhibit an edge singularity at the threshold of the total mass and decays with an inverse square root of energy, which is attributed to the singularity of the two-particle density of states at the threshold. The singularity is absent for particles with equal masses due to a cancellation mechanism involving the structure of the form factors. As a consequence, the dynamic structure factor displays a cascade of bumping peaks in the continuum region with clear singular features which can serve as a solid guidance for the material realization of the quantum ${E}_{8}$ model.

DOI
10.1103/physrevb.103.235117

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When the quantum critical transverse-field Ising chain is perturbed by a longitudinal field, a quantum integrable model emerges in the scaling limit with massive excitations described by the exceptional E8{E}_{8}E8​ Lie algebra. Using the corresponding analytical form factors of the quantum E8{E}_{8}E8​ integrable model, we systematically study the spin dynamic structure factor of the perturbed quantum critical Ising chain, where particle channels with total energy up to 5m15{m}_{1}5m1​ (m1{m}_{1}m1​ being the mass of the lightest E8{E}_{8}E8​ particle) are exhausted. In addition to the significant single-particle contributions to the dynamic spectrum, each two-particle channel with different masses is found to exhibit an edge singularity at the threshold of the total mass and decays with an inverse square root of energy, which is attributed to the singularity of the two-particle density of states at the threshold. The singularity is absent for particles with equal masses due to a cancellation mechanism involving the structure of the form factors. As a consequence, the dynamic structure factor displays a cascade of bumping peaks in the continuum region with clear singular features which can serve as a solid guidance for the material realization of the quantum E8{E}_{8}E8​ model.
Experimental Observation of High-Energy E8E_8E8​ Particles in the Ising Chain Ferromagnet CoNb2_22​O6_66​
Close to the quantum critical point of the transverse-field Ising spin-chain model, an exotic dynamic spectrum was predicted to emerge upon a perturbative longitudinal field. The dynamic spectrum consists of eight particles and is governed by the symmetry of the E8E_8E8​ Lie algebra. Here we report on high-resolution terahertz spectroscopy of quantum spin dynamics in the ferromagnetic Ising-chain material CoNb2_22​O6_66​. At 0.25 K in the magnetically ordered phase we identify characteristics of the first six E8E_8E8​ particles, m1\mathbf{m}_1m1​ to m6\mathbf{m}_6m6​, and the two-particle (m1+m2\mathbf{m}_1+\mathbf{m}_2m1​+m2​) continuum in an applied transverse magnetic field of Bc1D=4.75B_c^{1D}=4.75Bc1D​=4.75 T, before the three-dimensional magnetic order is suppressed above Bc3D≈5.3B_c^{3D}\approx 5.3Bc3D​≈5.3 T. The observation of the higher-energy particles (m3\mathbf{m}_3m3​ to m6\mathbf{m}_6m6​) above the low-energy two-particle continua features quantum many-body effects in the exotic dynamic spectrum.
Near the transverse-field-induced quantum critical point of the Ising chain, an exotic dynamic spectrum consisting of exactly eight particles was predicted, which is uniquely described by an emergent quantum integrable field theory with the symmetry of the E8{E}_{8}E8​ Lie algebra, but rarely explored experimentally. Here we use high-resolution terahertz spectroscopy to resolve quantum spin dynamics of the quasi-one-dimensional Ising antiferromagnet BaCo2V2O8\mathrm{BaC}{\mathrm{o}}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{8}BaCo2​V2​O8​ in an applied transverse field. By comparing to an analytical calculation of the dynamical spin correlations, we identify E8{E}_{8}E8​ particles as well as their two-particle excitations.
Emergent Isingh2{\mathrm{Ising}}_{h}^{2}Isingh2​ integrability is anticipated in a quantum Ising ladder composed of two weakly coupled critical transverse field Ising chains. The system is remarkable for including eight types of massive relativistic particles, whose scattering matrix and mass spectrum are characterized by the D8(1){\mathcal{D}}_{8}^{(1)}D8(1)​ Lie algebra. In this article, by computing the spin dynamical structure factors using analytical form-factor approach, we clearly identify dispersive single-particle excitations of (anti) soliton and breathers as well as their multiparticle continua in the spectra, which is further confirmed by numerical simulations. We show that the selection rule inherent in the parity and topological charge of the theory causes a significant result: charge-parity-odd particles, termed dark particles, cannot be directly excited from the ground state through any local or quasilocal operations. This, in turn, suggests the long lifetime of the lightest dark particle.
Close to the quantum critical point of the transverse-field Ising spin-chain model, an exotic dynamic spectrum was predicted to emerge upon a perturbative longitudinal field. The dynamic spectrum consists of eight particles and is governed by the symmetry of the E8{E}_{8}E8​ Lie algebra. Here we report on high-resolution terahertz spectroscopy of quantum spin dynamics in the ferromagnetic Ising-chain material CoNb2O6{\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}CoNb2​O6​. At 0.25 K in the magnetically ordered phase we identify characteristics of the first six E8{E}_{8}E8​ particles, m1{\mathrm{m}}_{1}m1​ to m6{\mathrm{m}}_{6}m6​, and the two-particle (m1+m2)({\mathrm{m}}_{1}+{\mathrm{m}}_{2})(m1​+m2​) continuum in an applied transverse magnetic field of Bc1D=4.75T{B}_{c}^{\text{1D}}=4.75 \mathrm{T}Bc1D​=4.75T, 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 (m3tom6)({\mathrm{m}}_{3} \mathrm{to} {\mathrm{m}}_{6})(m3​tom6​) above the low-energy two-particle continua features quantum many-body effects in the exotic dynamic spectrum.
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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.