We compare the time evolution of entanglement measures after local operator excitation in the critical Ising model with predictions from conformal field theory. For the spin operator and its descendants we find that Renyi entropies of a block of spins increase by a constant that matches the logarithm of the quantum dimension of the conformal family. However, for the energy operator we find a small constant contribution that differs from the conformal field theory answer equal to zero. We argue that the mismatch is caused by the subtleties in the identification between the local operators in conformal field theory and their lattice counterpart. Our results indicate that evolution of entanglement measures in locally excited states not only constraints this identification, but also can be used to extract non-trivial data about the conformal field theory that governs the critical point. We generalize our analysis to the Ising model away from the critical point, states with multiple local excitations, as well as the evolution of the relative entropy after local operator excitation and discuss universal features that emerge from numerics.
The one dimensional (1D) spin-1/2 Heisenberg-Ising model, a prototype quantum many-body system, has been intensively studied for many years. In this review, after a short introduction on some basic concepts of group theory for the octahedral group, a detailed pedagogical framework is laid down to derive the low-energy effective Hamiltonian for the Co-based materials. The 1D spin-1/2 Heisenberg-Ising model is obtained when applying the analysis to quasi-1D antiferromagnetic materials $\rm BaCo_2V_2O_8$ and $\rm SrCo_2V_2O_8$. After the preparation, we review the theoretical progresses of a vari…
We report a high-resolution terahertz spectroscopic study of quantum spin dynamics in the antiferromagnetic Heisenberg-Ising spin-chain compound BaCo$_2$V$_2$O$_8$ as a function of temperature and longitudinal magnetic field. Confined spinon excitations are observed in an antiferromagnetic phase below $T_N\simeq 5.5$ K. In a field-induced gapless phase above $B_c=3.8$ T, we identify many-body string excitations as well as low-energy fractional psinon/antipsinon excitations by comparing to Bethe-Ansatz calculations. In the vicinity of $B_c$, the high-energy string excitations are found to be dy…
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 $E_8$ Lie algebra. Here we report on high-resolution terahertz spectroscopy of quantum spin dynamics in the ferromagnetic Ising-chain material CoNb$_2$O$_6$. At 0.25 K in the magnetically ordered phase we identify characteristics of the first six $E_8$ particles, $\mathbf{m}_1$ to $\mathbf{m}_6$, and the two-particle ($\mathbf{m}_1+\ma…
We consider the time evolution of the gaps of the entanglement spectrum for a block of consecutive sites in finite transverse field Ising chains after sudden quenches of the magnetic field. We provide numerical evidence that, whenever we quench at or across the quantum critical point, the time evolution of the ratios of these gaps allows to obtain universal information. They encode the low-lying gaps of the conformal spectrum of the Ising boundary conformal field theory describing the spatial bipartition within the imaginary time path integral approach to global quenches at the quantum critica…
Starting from the operator algebra of the (1+1)D Ising model on a spatial lattice, this paper explicitly constructs a subalgebra of smooth operators that are natural candidates for continuum fields in the scaling limit. At the critical value of the transverse field, these smooth operators are analytically shown to reproduce the operator product expansions found in the Ising conformal field theory.
We study the dynamics of the quasi-one-dimensional Ising-Heisenberg antiferromagnet BaCo2V2O8 under a transverse magnetic field. Combining inelastic neutron scattering experiments and theoretical analyses by field theories and numerical simulations, we mainly elucidate the structure of the spin excitation spectrum in the high field phase, appearing above the quantum phase transition point mu0Hc ~ 10 T. We find that it is characterized by collective solitonic excitations superimposed on a continuum. These solitons are strongly bound in pairs due to the effective staggered field induced by the n…
We report on spectroscopy study of elementary magnetic excitations in an Ising-like antiferromagnetic chain compound SrCo$_2$V$_2$O$_8$ as a function of temperature and applied transverse magnetic field up to 25 T. An optical as well as an acoustic branch of confined spinons, the elementary excitations at zero field, are identified in the antiferromagnetic phase below the Néel temperature of 5 K and described by a one-dimensional Schrödinger equation. The confinement can be suppressed by an applied transverse field and a quantum disordered phase is induced at 7 T. In this disordered paramagnet…
We study a quantum spin-1/2 chain that is dual to the canonical problem of non-equilibrium Kawasaki dynamics of a classical Ising chain coupled to a thermal bath. The Hamiltonian is obtained for the general disordered case with non-uniform Ising couplings. The quantum spin chain (dubbed Ising-Kawasaki) is stoquastic, and depends on the Ising couplings normalized by the bath's temperature. We give its exact ground states. Proceeding with uniform couplings, we study the one- and two-magnon excitations. Solutions for the latter are derived via a Bethe Ansatz scheme. In the antiferromagnetic regim…
We have used neutron spectroscopy to investigate the spin dynamics of the quantum (S = 1/2) antiferromagnetic Ising chains in RbCoCl3. The structure and magnetic interactions in this material conspire to produce two magnetic phase transitions at low temperatures, presenting an ideal opportunity for thermal control of the chain environment. The high-resolution spectra we measure of two-domain-wall excitations therefore characterize precisely both the continuum response of isolated chains and the "Zeeman-ladder" bound states of chains in three different effective staggered fields in one and the …
For quasi-one dimensional quantum spin systems theory predicts the occurrence of a confinement of spinon excitation due to interchain couplings. Here we investigate the system SrCo2V2O8, a realization of the weakly-coupled Ising-like XXZ antiferromagnetic chains, by terahertz spectroscopy with and without applied magnetic field. At low temperatures a series of excitations is observed, which split in a Zeeman-like fashion in an applied magnetic field. These magnetic excitations are identified as the theoretically predicted spinon-pair excitations. Using a one dimensional Schrödinger equation wi…
The transverse Ising magnet Hamiltonian describing the Ising chain in a transverse magnetic field is the archetypal example of a system that undergoes a transition at a quantum critical point (QCP). The columbite CoNb$_2$O$_6$ is the closest realization of the transverse Ising magnet found to date. At low temperatures, neutron diffraction has observed a set of discrete collective spin modes near the QCP. We ask if there are low-lying spin excitations distinct from these relatively high energy modes. Using the heat capacity, we show that a significant band of gapless spin excitations exists. At…
Emergence of 3D Superconformal Ising Criticality on the Fuzzy Sphere
Westlake Institute of Advanced Study, Westlake University（西湖高等研究院，西湖大学）；Institute for Advanced Study, Kyushu University（九州大学高级研究院）；School of Physics and Astronomy, University of Leeds（利兹大学物理与天文学院）
We investigate the quantum dynamics of the transverse field Ising model on the triangular lattice through large-scale quantum Monte Carlo simulations and stochastic analytic continuation. At weak transverse field, we capture for the first time the excitations related to topological quantum strings, which exhibits continuum features described by XY chain along the strings and those in accord with "Luttinger string liquid" in the perpendicular direction. The continuum features can be well understood from the perspective of topological strings. Furthermore, we identify the contribution of strings…
Using numerical conformal bootstrap technology we perform a non-perturbative study of the Ising CFT and its spectrum from infinitesimal to finite values of $\varepsilon=4-d$. Exploiting the recent navigator bootstrap method in conjunction with the extremal functional method, we test various qualitative and quantitative features of the $\varepsilon$-expansion. We follow the scaling dimensions of numerous operators from the perturbatively controlled regime to finite coupling. We do this for $\mathbb Z_2$-even operators up to spin 12 and for $\mathbb Z_2$-odd operators up to spin 6 and find a goo…
We report thermodynamic and neutron scattering measurements of the triangular-lattice quantum Ising magnet TmMgGaO 4 in longitudinal magnetic fields. Our experiments reveal a quasi-plateau state induced by quantum fluctuations. This state exhibits an unconventional non-monotonic field and temperature dependence of the magnetic order and excitation gap. In the high field regime where the quantum fluctuations are largely suppressed, we observed a disordered state with coherent magnon-like excitations despite the suppression of the spin excitation intensity. Through detailed semi-classical calcul…
Weakly coupled Ising chains provide a condensed-matter realization of confinement. In these systems, kinks and antikinks bind into mesons due to an attractive interaction potential that increases linearly with the distance between the particles. While single mesons have been directly observed in experiments, the role of the multiparticle continuum and bound states of mesons in the excitation spectrum is far less clear. Using time-dependent density matrix renormalization group methods, we study the dynamical structure factors of one- and two-spin operators in a transverse-field two-leg Ising la…
We explore the spin dynamics emerging from the Néel phase of the chain compound antiferromagnet BaCo2V2O8. Our inelastic neutron scattering study reveals unconventional discrete spin excitations, so called Zeeman ladders, understood in terms of spinon confinement, due to the interchain attractive linear potential. These excitations consist in two interlaced series of modes, respectively with transverse and longitudinal polarization. The latter have no classical counterpart and are related to the zero-point fluctuations that weaken the ordered moment in weakly coupled quantum chains. Our analys…
Excitations in the quantum paramagnetic phase of the quasi-one-dimensional Ising magnet CoNb$_2$O$_6$ in a transverse field: Geometric frustration and quantum renormalization effects
准一维Ising铁磁体CoNb $_2$ O $_6$在外加横向磁场的作用下发生了从自发磁序到量子顺磁序的连续量子相变，观察到自旋动力学的显著变化，表现出弱微扰的一维Ising量子临界特性.我们在这里报告广泛的单晶非弹性中子散射测量的磁激发在整个三维（3D）布里渊区的量子顺磁相刚好在临界场以上的有限链间耦合的影响进行表征。在这个阶段，我们观察到，激发有一个尖锐的，分辨率有限的线形在低能量和大部分的色散带宽，自旋翻转准粒子的预期。我们沿着强色散链方向映射的全带宽，并解决明确调制的色散在垂直于链的平面中，在反铁磁等腰三角形晶格中的链间耦合的特性受挫。分散体可以很好地参数化使用线性自旋波模型，包括链间耦合和进一步的邻居交换。沿链方向观察到的色散带宽小于线性自旋波模型使用在零场确定的交换值所预测的，这种效应归因于量子重整化的色散超出自旋波近似在略高于临界场的领域，量子波动仍然显着。
We make precise determinations of the leading scaling dimensions and operator product expansion (OPE) coefficients in the 3d Ising, $O(2)$, and $O(3)$ models from the conformal bootstrap with mixed correlators. We improve on previous studies by scanning over possible relative values of the leading OPE coefficients, which incorporates the physical information that there is only a single operator at a given scaling dimension. The scaling dimensions and OPE coefficients obtained for the 3d Ising model, $(Δ_σ, Δ_ε,λ_{σσε}, λ_{εεε}) = (0.5181489(10), 1.412625(10), 1.0518537(41), 1.532435(19))$, giv…
We compute numerically the dimensions and OPE coefficients of several operators in the 3d Ising CFT, and then try to reverse-engineer the solution to crossing symmetry analytically. Our key tool is a set of new techniques for computing infinite sums of SL(2,R) conformal blocks. Using these techniques, we solve the lightcone bootstrap to all orders in an asymptotic expansion in large spin, and suggest a strategy for going beyond the large spin limit. We carry out the first steps of this strategy for the 3d Ising CFT, deriving analytic approximations for the dimensions and OPE coefficients of se…
We report the neutron scattering results on magnetic orderings and excitations in $\mathrm{Ba}_2\mathrm{CoTeO}_6$ composed of two almost isolated subsystems A and B, which are described as an $S\,{=}\,1/2$ triangular Heisenberg-like antiferromagnet and a frustrated honeycomb Ising-like antiferromagnet, respectively. Stripy ordering of subsystem B was confirmed below $T_{\rm N1}\,{=}\,12.0$ K, whereas sharp streaks were observed along $(1/3, 1/3, L)$ and $(2/3, 2/3, L)$ at 0.3 K (${\ll}\,T_{\rm N2}\,{=}\,3.0$ K). This indicates the two-dimensional nature of ordering in subsystem A. It was found…
It is shown by detailed inelastic neutron scattering experiments that the gapped collective magnetic excitation of the unconventional superconductor CeCoIn$_{5}$, the spin resonance mode, is incommensurate and that the corresponding fluctuations are of Ising nature. The incommensurate peak position of these fluctuations corresponds to the propagation vector of the adjacent field induced static magnetic ordered phase, the so-called Q-phase. Furthermore, the direction of the magnetic moment fluctuations is also the direction of the ordered magnetic moments of the Q-phase. Hence the resonance mod…
We consider the quantum quench dynamics of a Heisenberg-Ising spin ladder which is an archetypal model in which confinement of elementary excitations is triggered by internal interactions rather than an external field. We show that the confinement strongly affects the light cone structure of correlation functions providing signatures of the velocities of the mesons of the model. We also show that the meson masses can be measured from the real time analysis of the evolution of the order parameter.
Combining inelastic neutron scattering and numerical simulations, we study the quasi-one dimensional Ising anisotropic quantum antiferromagnet \bacovo\ in a longitudinal magnetic field. This material shows a quantum phase transition from a Néel ordered phase at zero field to a longitudinal incommensurate spin density wave at a critical magnetic field of 3.8 T. Concomitantly the excitation gap almost closes and a fundamental reconfiguration of the spin dynamics occurs. These experimental results are well described by the universal Tomonaga-Luttinger liquid theory developed for interacting spinl…
Since the seminal ideas of Berezinskii, Kosterlitz and Thouless, topological excitations are at the heart of our understanding of a whole novel class of phase transitions. In most of the cases, those transitions are controlled by a single type of topological objects. There are however some situations, still poorly understood, where two dual topological excitations fight to control the phase diagram and the transition. Finding experimental realization of such cases is thus of considerable interest. We show here that this situation occurs in BaCo$_2$V$_2$O$_8$, a spin-1/2 Ising-like quasi-one di…
Excitations of quantum Ising chain CoNb2O6 in low transverse field: quantitative description of bound states stabilized by off-diagonal exchange and applied field
Inelastic neutron scattering study on the spin-chain compound Sr$_3$NiIrO$_6$ reveals gapped quasi-1D magnetic excitations. The observed one-magnon band between 29.5 and 39 meV consists of magnon modes of the Ni$^{2+}$ ions. The fitting of the spin wave spectrum reveals strongly coupled Ising-like chains along the $c$-axis that are weakly coupled into a frustrated triangular lattice in the $ab$-plane. The magnetic excitations survive up to 200 K well above the magnetic ordering temperature of $T_N \sim 75$ K, also indicating a quasi-1D nature of the magnetic interactions in Sr$_3$NiIrO$_6$. Ou…
We use inelastic neutron scattering to study the temperature and energy dependence of the spin excitation anisotropy in uniaxial-strained electron-doped iron pnictide BaFe$_{1.9}$Ni$_{0.1}$As$_2$ near optimal superconductivity ($T_c=20$ K). Our work has been motivated by the observation of in-plane resistivity anisotropy in the paramagnetic tetragonal phase of electron-underdoped iron pnictides under uniaxial pressure, which has been attributed to a spin-driven Ising-nematic state or orbital ordering. Here we show that the spin excitation anisotropy, a signature of the spin-driven Ising-nemati…
