Dunsiger and co-workers have criticised our paper on magnetic monopole dynamics spin ice [S. T. Bramwell et al., Nature, 461 956 (2009)]. We consider their criticisms and results but show that these do not warrant any revision of our earlier conclusions. In contrast, we point out that their conclusion of a `new' type of spin dynamics in spin ice is inconsistent with established facts.
Spin ice magnetic monopoles are fractionalized emergent excitations in a class of frustrated magnets called spin ices. The classical spin ice model has an extensive number of ground state spin configurations, whereas magnetic monopoles can be thought of as the endpoints of string operators applied to these ground states. Introducing quantum fluctuations into the model induces monopoles with dynamics, which would normally lift the degeneracy of the two-monopole energy level at the linear order of the perturbation. Contrary to this expectation, we find that quantum fluctuations in the form of a …
We present a proposal for applying nanoscale magnetometry to the search for magnetic monopoles in the spin ice materials holmium and dysprosium titanate. Employing Monte Carlo simulations of the dipolar spin ice model, we find that when cooled to below $1.5\,$K these materials exhibit a sufficiently low monopole density to enable the direct observation of magnetic fields from individual monopoles. At these temperatures we demonstrate that noise spectroscopy can capture the intrinsic fluctuations associated with monopole dynamics, allowing one to isolate the qualitative effects associated with …
Thermal Conductivity and Specific Heat of the Spin-Ice Compound Dy$_2$Ti$_2$O$_7$: Experimental Evidence for Monopole Heat Transport
自旋冰化合物Dy $_2$ Ti $_2$ O $_7$中的基元激发可以描述为在由磁性Dy离子形成的烧绿石晶格内独立传播的磁单极子。我们研究了B的热导率κ（B）与磁场的关系||[001]并观察到源自磁激发的磁热传输的明显证据。磁贡献κ mag与磁场有很强的相关性，并与磁化强度M（B）有关。由κ mag与磁比热之比得到的扩散系数在低于1K时大大增强，表明自旋冰态的自旋激发具有高的迁移率.
Magnetic monopoles are hypothesised elementary particles connected by Dirac strings that behave like infinitely thin solenoids. Despite decades of searches, free magnetic monopoles and their Dirac strings have eluded experimental detection, although there is substantial evidence for deconfined magnetic monopole quasiparticles in spin ice materials. Here we report the detection of a hierarchy of unequally-spaced magnetic excitations \emph{via} high resolution inelastic neutron spectroscopic measurements on the quantum spin ice candidate Pr$_{2}$Sn$_{2}$O$_{7}$. These excitations are well-descri…
We report on the non-equilibrium monopole dynamics in the classical spin ice Dy$_2$Ti$_2$O$_7$ detected by means of high-resolution magnetostriction measurements. Significant lattice changes occur at the transition from the kagome-ice to the saturated-ice phase, visible in the longitudinal and transverse magnetostriction. A hysteresis opening at temperatures below 0.6 K suggests a first-order transition between the kagome and saturated state. Extremely slow lattice relaxations, triggered by changes of the magnetic field, were observed. These lattice-relaxation effects result from non-equilibri…
Magnetic monopoles are hypothetical particles that may exist as quantized sources and sinks of the magnetic field. In materials, they may appear in an emergent quantum electrodynamics described by a U(1) lattice gauge theory. Particularly, quantum spin ice hosts monopoles as bosonic spinons coupled to emergent gauge fields in a U(1) quantum spin liquid, namely, a deconfined Coulomb phase. When monopoles are condensed to form a long-range order, monopoles and gauge fields are screened and confined. Here we show, however, that monopole supercurrent flows across a junction of two ferromagnets tha…
Excitations in a spin ice behave as magnetic monopoles, and their population and mobility control the dynamics of a spin ice at low temperature. CdEr$_2$Se$_4$ is reported to have the Pauling entropy characteristic of a spin ice, but its dynamics are three-orders of magnitude faster than the canonical spin ice Dy$_2$Ti$_2$O$_7$. In this letter we use diffuse neutron scattering to show that both CdEr$_2$Se$_4$ and CdEr$_2$S$_4$ support a dipolar spin ice state -- the host phase for a Coulomb gas of emergent magnetic monopoles. These Coulomb gases have similar parameters to that in Dy$_2$Ti$_2$O…
Critical speeding-up near the monopole liquid-gas transition in magnetoelectric spin-ice
在所谓的自旋冰化合物中，相互竞争的相互作用稳定了一个具有有限零点熵的受抑基态，有趣的是，还出现了磁共振激发。这些单极子的性质是最近研究的焦点，特别强调它们的量子动力学。据预测，每个电介质也具有电偶极矩，这允许通过介电函数ε（ν）来研究它们的动力学。在这里，我们报告的宽带光谱测量的ε（ν）在Dy 2 Ti 2 O 7下降到200 mK的温度与特定的重点上的关键端点存在的磁场沿晶体[111]方向。当磁单极子的密度以临界方式变化时，类似于在液-气转变中，在介电响应中揭示出明确的临界特征。令人惊讶的是，介电弛豫时间τ表现出临界加速，随着温度向临界温度降低，介电弛豫时间τ显著增加1/τ。除了展示了紧急的双激发的磁电特性，我们的结果揭示了独特的临界动力学附近的双凝聚转变。
A Measure of Monopole Inertia in the Quantum Spin Ice Yb$_2$Ti$_2$O$_7$
现代固体物理学的一个重要和持续的主题是在材料（例如准粒子）中实现奇异激发，这些材料在自由空间的实际物理真空中没有类似物（或尚未被观察到）。尽管它们不是基本粒子，但此类准粒子确实构成了它们所处的“真空”的激发态的最基本描述。在这方面的激发态的自旋冰，占主导地位的伊辛相互作用的磁性烧绿石氧化物的磁性纹理，提出了建模为有效的磁荷单极。最近的非弹性中子散射实验已经确定烧绿石材料Yb $_2$ Ti $_2$ O $_7$（YbTO）作为量子自旋冰，其中除了伊辛相互作用之外，还有大量的横向项可以诱导量子动力学和-原则上-相干的自旋运动。在这里，我们报告了相结合的时域太赫兹光谱（THEMS）和微波腔的研究YbTO探测其复杂的动态磁化率。我们发现磁化率的形式与自旋运动相一致，并且可以定义和测量磁自旋电导率。使用这些技术的独特的相敏能力，我们观察到的磁响应的反应部分的符号变化。在惯性运动的一般模型中，这只能通过引入惯性效应（例如，质量相关项）到运动方程中来实现。类似于传统的电荷系统，测量电导率的光谱权重使我们能够推导出磁致伸缩质量的值，我们发现该值约为1800电子质量。我们的结果建立了磁单极子的量子自旋冰作为真正的相干传播准粒子的这个系统。
A non-Ohmic current that grows exponentially with the square root of applied electric field is well known from thermionic field emission (the Schottky effect), electrolytes (the second Wien effect) and semiconductors (the Poole-Frenkel effect). It is a universal signature of the attractive Coulomb force between positive and negative electrical charges, which is revealed as the charges are driven in opposite directions by the force of an applied electric field. Here we apply thermal quenches to spin ice to prepare metastable populations of bound pairs of positive and negative emergent magnetic …
Pyrochlore iridates and pyrochlore ices are two families of materials where novel quantum phenomena are intertwined with strong spin-orbit coupling, substantial electron correlation and geometrical frustration. Motivated by the puzzling experiments on two pyrochlore systems Pr2Ir2O7 and Yb2Ti2O7, we study the proximate Ising orders and the quantum phase transition out of quantum spin ice U(1) quantum spin liquid (QSL). We apply the electromagnetic duality of the compact quantum electrodynamics to analyze the "magnetic monopoles" condensation for U(1) QSL. The monopole condensation transition r…
Frustration in magnetic systems is fertile ground for complex behaviour, including unconventional ground states with emergent symmetries, topological properties, and exotic excitations. A canonical example is the emergence of magnetic-charge-carrying quasiparticles in spin-ice compounds. Despite extensive work, a reliable experimental indicator of the density of these magnetic monopoles in spin-ice systems is yet to be found. Here, using measurements on single crystals of Ho$_{2}$Ir$_{2}$O$_{7}$ in combination with dipolar Monte Carlo simulations, we show that the magnetoresistance is highly s…
The single-ion anisotropy and magnetic interactions in spin-ice systems give rise to unusual non-collinear spin textures, such as Pauling states and magnetic monopoles. The effective spin correlation strength ($J_{eff}$) determines the relative energies of the different spin-ice states. With this work, we display the capability of capacitive torque magnetometry in characterizing the magneto-chemical potential associated with monopole formation. We build a magnetic phase diagram of Ho$_2$Ti$_2$O$_7$, and show that the magneto-chemical potential depends on the spin-sublattice ($α$ or $β$), i.e.,…
Anisotropic Monopole Heat Transport in the Spin-Ice Compound Dy$_2$Ti$_2$O$_7$
本文报道了自旋冰材料Dy $_2$ Ti $_2$ O $_7$的热导率$κ$的研究。从κ $ and by additional measurements on the phononic reference compounds Y$ _2 PDMATHABMQ_2 PDMATHABMR_7 PDMATHABMS_2 PDMATHABMR_7 $, we are able to separate the phononic and the magnetic contributions to the total heat transport, i.e. $ κ_{ph} $ and $ κ_{mag} $, respectively, which both depend on the magnetic field. The field dependent $ κ_{ph} $ arises from lattice distortions due to magnetic-field induced torques on the non-collinear magnetic moments of the Dy ions. For $ κ_{mag} $, we observe a highly anisotropic magnetic-field dependence, which co…
While there is compelling evidence for the existence of magnetic monopoles in spin ice, the direct observation of a point-like source of magnetic field in these systems remains an open challenge. One promising approach is electron holography, which combines atomic-scale resolution with extreme sensitivity to magnetic vector potentials, through the Aharonov-Bohm effect. Here we explore what holography can teach us about magnetic monopoles in spin ice, through experiments on artificial spin ice, and numerical simulations of pyrochlore spin ice. In the case of artificial spin ice, we show that ho…
Quantum spin ice, modeled for magnetic rare-earth pyrochlores, has attracted great interest for hosting a U(1) quantum spin liquid, which involves spin-ice monopoles as gapped deconfined spinons, as well as gapless excitations analogous to photons. However, the global phase diagram under a [111] magnetic field remains open. Here we uncover by means of unbiased quantum Monte-Carlo simulations that a supersolid of monopoles, showing both a superfluidity and a partial ionization, intervenes the kagomé spin ice and a fully ionized monopole insulator, in contrast to classical spin ice where a direc…
The spin ice materials Ho$_{2}$Ti$_{2}$O$_{7}$ and Dy$_{2}$Ti$_{2}$O$_{7}$ are by now perhaps the best-studied classical frustrated magnets. A crucial step towards the understanding of their low temperature behaviour -- both regarding their unusual dynamical properties and the possibility of observing their quantum coherent time evolution -- is a quantitative understanding of the spin-flip processes which underpin the hopping of magnetic monopoles. We attack this problem in the framework of a quantum treatment of a single-ion subject to the crystal, exchange and dipolar fields from neighbourin…
Extensive work on single molecule magnets has identified a fundamental mode of relaxation arising from the nuclear-spin assisted quantum tunnelling of nearly independent and quasi-classical magnetic dipoles. Here we show that nuclear-spin assisted quantum tunnelling can also control the dynamics of purely emergent excitations: magnetic monopoles in spin ice. Our low temperature experiments were conducted on canonical spin ice materials with a broad range of nuclear spin values. By measuring the magnetic relaxation, or monopole current, we demonstrate strong evidence that dynamical coupling wit…
Dirac Strings and Magnetic Monopoles in Spin Ice Dy2Ti2O7
虽然磁场的来源--磁单极子--到目前为止被证明是难以捉摸的基本粒子，但最近在凝聚态物理中已经提出了几种类似单极子的新兴准粒子的方案。一个特别简单的命题涉及到在高度受挫的烧绿石晶格上旋转冰。自旋冰态被认为可以用类似螺线管的偶极子网络来描述--经典的、可观测的狄拉克弦。在这些管子结束的地方，产生的缺陷看起来像一个磁性磁铁。我们证明，通过漫中子散射，存在这样的字符串在自旋冰Dy 2 Ti 2 O 7。这是通过施加一个破磁磁场来实现的，我们可以用它来操纵弦的密度和方向。反过来，热容量由磁单极子通过磁库仑相互作用相互作用的气体描述。
We present the full phase diagram of the dumbbell model of spin ice as a function of temperature, chemical potential and staggered chemical potential which breaks the translational lattice symmetry in favour of charge crystal ordering. We observe a double winged structure with five possible phases, monopole fluid (spin ice), fragmented single monopole crystal phases and double monopole crystal, the zinc blend structure. Our model provides a skeleton for liquid-liquid phase transitions and for the winged structures observed for itinerant magnets under pressure and external field. We relate our …
The complexity embedded in condensed matter fertilizes the discovery of new states of matter, enriched by ingredients like frustration. Illustrating examples in magnetic systems are Kitaev spin liquids, skyrmions phases, or spin ices. These unconventional ground states support exotic excitations, for example the magnetic charges in spin ices, also called monopoles. Beyond their discovery, an important challenge is to be able to control and manipulate them. Here, we propose a new mechanism to inject monopoles in a spin ice through a staggered magnetic field. We show theoretically, and demonstra…
We study the three-dimensional Kasteleyn transition in both nearest neighbours and dipolar spin ice models using an algorithm that conserves the number of excitations. We first limit the interactions range to nearest neighbours to test the method in the presence of a field applied along $[100]$, and then focus on the dipolar spin ice model. The effect of dipolar interactions, which is known to be greatly self screened at zero field, is particularly strong near full polarization. It shifts the Kasteleyn transition to lower temperatures, which decreases $\approx 0.4 K$ for the parameters corresp…
A magnetic monopole in spin ice is a novel quasiparticle excitation in condensed matter physics, and we found that the ac frequency dependent magnetic susceptibility $χ(ω)$ in the two-dimensional (2D) spin ice (so-called kagomé ice) of Dy$_2$Ti$_2$O$_7$ shows a single scaling form. This behavior can be understood in terms of the dynamical scaling law for 2D Coulomb gas (CG) systems [Phys. Rev. B 90, 144428 (2014)], characterized by the charge correlation length $ξ(\propto1/\sqrt{ω_1})$, where $ω_{1}$ is a characteristic frequency proportional to the peak position of the imaginary part of $χ(ω)…
Classical symmetry enriched topological orders and distinct monopole charges for dipole-octupole spin ices
International Center for Quantum Materials, Peking University（北京大学量子材料中心）；Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交通大学李政道研究所）；Collaborative Innovation Center of Quantum Matter（量子物质协同创新中心）；The University of Hong Kong, Shenzhen Institute of Research and Innovation（香港大学深圳研究院）；Beijing Key Laboratory of Quantum Devices, Peking University（北京大学量子器件北京市重点实验室）
We study the U(1) quantum spin liquid on the pyrochlore spin ice systems. For the non-Kramers doublets such as Pr$^{3+}$ and Tb$^{3+}$, we point out that the inelastic neutron scattering not only detects the low-energy gauge photon, but also contains the continuum of the "magnetic monopole" excitations. Unlike the spinons, these "magnetic monopoles" are purely of quantum origin and have no classical analogue. We further point out that the "magnetic monopole" experiences a background dual "$π$" flux due to the spin-1/2 nature of the local moment when the "monopole" hops on the dual diamond latt…
We consider quantum spin ice in a temperature regime in which its response is dominated by the coherent motion of a dilute gas of monopoles. The hopping amplitude of a monopole is sensitive to the configuration of its surrounding spins, taken to be quasi-static on the relevant timescales. This leads to well-known blocked directions in the monopole motion; we find that these are sufficient to reduce the coherent propagation of monopoles to quantum diffusion. This result is robust against disorder, as a direct consequence of the ground-state degeneracy, which disrupts the quantum interference pr…
We study the quantum dynamics of fractional excitations in quantum spin ice. We focus on the density of states in the two-monopole sector, $ρ(ω)$, as this can be connected to the wavevector-integrated dynamical structure factor accessible in neutron scattering experiments. We find that $ρ(ω)$ exhibits a strikingly characteristic singular and asymmetric structure which provides a useful fingerprint for comparison to experiment. $ρ(ω)$ obtained from exact diagonalisation of a finite cluster agrees well with that from the analytical solution of a hopping problem on a Husimi cactus representing co…
Exploring possible magnetic monopoles-induced magneto-electricity in spin ices
University of Geneva（日内瓦大学）；Institut Néel, CNRS and Université Grenoble Alpes（内勒研究所，法国国家科学研究中心与格勒诺布尔阿尔卑斯大学）；ICMMO, Université Paris-Saclay, CNRS（物质化学与分子和超分子研究所，巴黎萨克雷大学，法国国家科学研究中心）
We investigate thermal transport in square ice, a two-dimensional analogue of spin ice, exploring the role played by emergent magnetic monopoles in transporting energy. Using kinetic Monte Carlo simulations based on energy preserving extensions of single-spin-flip dynamics, we explicitly compute the (longitudinal) thermal conductivity, $κ$, over a broad range of temperatures. We use two methods to determine $κ$: a measurement of the energy current between thermal baths at the boundaries, and the Green-Kubo formula, yielding quantitatively consistent values for the thermal conductivity. We inte…
Low-temperature muon spin rotation studies of the monopole charges and currents in Y doped Ho2Ti2O7
在Ho 2 Ti 2 O 7自旋冰的基态，磁矩的无序遵循相同的规则，在水冰的质子无序。激发采用磁单极子的形式，其通过磁库仑相互作用相互作用。本文用μ子自旋转动研究了Ho_（2-x）Y_xTi_2 O_7（x=0，0.1，1，1.6和2）单晶的低温磁性。在非常低的温度下，在x=0和0.1的样品中观察到μ子旋进λ（B）的弛豫速率与线性场的依赖关系，在以前的一些实验中，Dy 2 Ti 2 O 7自旋冰与μ子电流有关。从磁场穿透到银样品板由于晶体的磁化的信号被观察到所有的样品含有Ho，使我们能够研究不寻常的磁性动力学的Y掺杂自旋冰。
