Emergent $D_8^{(1)}$ spectrum and topological soliton excitation in CoNb$_2$O$_6$

Abstract

Quantum integrability emerging near a quantum critical point (QCP) is manifested by exotic excitation spectrum that is organized by the associated algebraic structure. A well known example is the emergent $E_8$ integrability near the QCP of a transverse field Ising chain (TFIC), which was long predicted theoretically and initially proposed to be realized in the quasi-one-dimensional (q1D) quantum magnet CoNb$_2$O$_6$. However, later measurements on the spin excitation spectrum of this material revealed a series of satellite peaks that cannot be described by the $E_8$ Lie algebra. Motivated by these experimental progresses, we hereby revisit the spin excitations of CoNb$_2$O$_6$ by combining numerical calculation and analytical analysis. We show that, as effects of strong interchain fluctuations, the spectrum of the system near the 1D QCP is characterized by the $D_{8}^{(1)}$ Lie algebra with robust topological soliton excitation. We further show that the $D_{8}^{(1)}$ spectrum can be realized in a broad class of interacting quantum systems. Our results advance the exploration of integrability and manipulation of topological excitations in quantum critical systems.

Figures (8)

• Figure 1: (a): Sketched phase diagram for CoNb$_2$O$_6$ under a transverse field ($H{/ /}b$), which consists of a 3D AFM phase at low temperatures and a prominent 1D quantum critical regime above the 3D ordering temperature. The (putative) 1D QCP at $H_c^{\rm{1D}}$ is hidden inside the 3D ordered phase close to the 3D QCP. The interchain interaction confines continuous critical excitations into gapped bound states that are characterised by the emergent $\mathcal{D}_8^{(1)}$ Lie algebra. (b): Sketched phase diagram showing crossover between two integrabilities by tuning $J_i/\tilde{h}$ in the minimal model. (c) and (d): Illustration of the crystal structure of CoNb$_2$O$_6$ in the $a$-$b$ plane [in (c)] and in the $b$-$c$ plane [in (d)], showing Co$^{2+}$ ions (purple) inside the edge-sharing O$^{2-}$ octahedra (orange) and translucent coordinate Nb-O octahedra. The Co$^{2+}$ ions construct an isosceles triangular lattice with AFM interchain couplings $J_i$ and $J_i^{\prime}$ in the $a$-$b$ plane, and form a zigzag chain with FM intrachain coupling $J$ alone the $c$ axis.
• Figure 2: (a): Calculated zone-centre spectral functions at $H_c^{\rm{1D}}$ of an Ising ladder with $J=1$ meV, $J_{i}=0.36~J$ and $\tilde{h}=0$; (b): same as (a) but for the minimal model with $\lambda_{i}=0.1~J$ and $\tilde{h}=0$; (c): THz absorption spectrum of CoNb$_2$O$_6$ at $H_{c}=5$ T and $T=0.25$ K, adapted from Ref. zhe2020. (d): Calculated zone-centre spectral functions at $H_c^{\rm{1D}}$ of the single-chain model in Eq. \ref{['Eq2']} with the effective longitudinal field $\tilde{h}=0.034~J$, exhibiting the $E_8$ structure. In each panel, the vertical dashed lines at peak positions correspond to the masses of quasiparticles or bound states of multi-particles in the particular $\mathcal{D}_8^{(1)}$ [in (a)-(c)] or $E_8$ [in (d)] model. Spectra at $k=\pi/2$ are also shown as light-coloured lines in panels (b) and (d) to demonstrate the zone-folding effect. Blue and red arrows refer to peaks associated with detailed microscopic Hamiltonian (see text). Note that many multiparticle modes are located in the shaded regimes, which give rise to multiple peak or plateau like spectrum.
• Figure 3: (a) Field dependence of several peak positions in the measured THz spectra extracted from Refs. zhe2020 and Amelin_2022. We take the lowest-energy peak (i.e. the $m_2$ of the $\mathcal{D}_8^{(1)}$ at QCP) as the energy unit. The low-energy peaks fit to the $\mathcal{D}_8^{(1)}$ algebraic structure best at about $H_{c}^{\rm{1D}} \simeq 5.0$ T. (b) The calculated spectrum of CoNb$_2$O$_6$ at $H_{c}^{\rm{1D}}$ in the entire BZ. The purple dotted lines are analytical predictions of several $\mathcal{D}_8^{(1)}$ quasiparticle dispersions.
• Figure 4: (a) The measured THz spectra of CoNb$_2$O$_6$ at 2.5 K with transverse field ranging from 0 to 12 T in 0.5 T steps, from bottom to top. Data adapted from Ref. Morris2021. (b) The calculated spectra at $k=0$ of the single-chain model with a longitudinal field $\tilde{h}\simeq 0.01 J$ under transverse field ranging from 0 to 6 T in 0.2 T steps, from bottom to top. (c) The calculated spectra at the zone centre of the transverse field Ising ladder with $J_i=0.1J$ under transverse fields ranging from $0.42~J$ to $0.59~J$ in $0.01~J$ steps, from bottom to top.
• Figure 5: (a) Plot of the transverse field evolution of several characteristic energies extracted from peaks of spectral functions at $k=0$. Red points represent the lowest excitation mode of the Ising-type, which becomes critical at $H_c^{\rm{1D}}$. Deep-coloured points are calculated from the single chain model [Eq. (2) in the main text] without an effective longitudinal field $\tilde{h}$, and light-coloured ones are with $\tilde{h}$. Purple points represent modes associated the DW and other non-Ising interactions of the model. Dashed lines are guides to the eye with a factor of 2 difference in their slopes, in accord with the $(1+1)$D Ising universality. (b) The calculated dynamical structure factors of the single chain model at $H_c^{\rm{1D}}$ in the entire BZ. The Ising criticality is characterised by the linear dispersive mode guided by the blue dashed lines.