Abstract
Left and righthanded chiral matter is present at every scale ranging from seashells to molecules to elementary particles. In magnetism, chirality may be inherited from the asymmetry of the underlying crystal structure, or it may emerge spontaneously. In particular, there has been a longstanding search for chiral spin states^{1} that emerge spontaneously with the disappearance of antiferromagnetic longrange order. Here we identify a generic system supporting such a behaviour and report on experimental evidence for chirality associated with the quantum dynamics of solitons^{2,3,4,5} in antiferromagnetic spin chains. The soliton chirality observed by polarized neutron scattering is in agreement with theoretical predictions and is a manifestation of a Berry phase^{6}. Our observations provide the first example of the emergence of spin currents and hidden chiral order that accompany the disappearance of antiferromagnetic order, a scheme believed to lie at the heart of the enigmatic normal state of cuprate superconductors.
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Main
The disappearance, rather than the appearance, of magnetic order is common to some of the most remarkable magnetic systems. Even when spin chains^{2,7}, pyrochlores^{8} and holedoped antiferromagnets show no apparent longrange magnetic order, the excitations out of the seemingly featureless ground states consist of a rich palette of exotic quasiparticles involving collective behaviour: spin solitons with semion statistics in spin1/2 chains^{9}, spin clusters in pyrochlores^{10} and exotic charge and spin excitations^{11,12,13,14} in the normal state of hightransitiontemperature (highT_{c}) superconductors. In these systems, the loss of magnetic order is related to the formation of spin singlets, and numerous descriptions build on the influential, albeit controversial, proposal^{15} of using variational wavefunctions based on fluctuating singlets. There has been a longstanding suspicion that fluctuating singlets can give rise to chiral spin correlations^{1}, an idea that has, however, remained elusive owing to the complexity of the above systems.
For the present study we investigated a model system that allows us to understand unambiguously the interplay of singlet dynamics, the destruction of antiferromagnetic order and the emergence of chirality. We considered an antiferromagnetic anisotropic spin1/2 chain, close to the Ising limit. In the classical limit, the ground state is the familiar twofold degenerate Néel state, with all neighbouring spins aligned antiparallel. The lowest excitation is a domain wall^{3,4,5,16} separating the two ground states as highlighted by different colours in Fig. 1a. As this domain wall or soliton can be placed anywhere along the chain, the resulting state is highly degenerate. Quantum fluctuations, due to transverse exchange, lift this degeneracy, the soliton becomes delocalized, destroys antiferromagnetic order and forms a band^{4} with two degenerate minima. Remarkably, the states of lowest energy correspond to propagating singlets that carry opposite (vector) chirality^{17} or spin current,
transverse to the Ising direction. The sum runs over all sites along the chain and S_{i} denotes a spin1/2 operator at site i.
To test such behaviour experimentally, we used spinpolarized neutrons, known for their ability to distinguish right and lefthanded magnetic structures^{18}. Here, however, we faced the challenge that spin1/2 solitons are fractional objects and couple to neutrons only in pairs. As shown in Fig. 1b, a neutron may either create a soliton pair producing a continuum around the energy required to break two bonds, or it scatters an existing thermally excited soliton within the soliton band, producing a lowenergy response bounded by the ‘Villain mode’^{4}. Owing to the continuum character of this neutron response, chirality is hidden, which explains why it has remained undetected so far. An observation of a chiral signal requires lifting the degeneracy between the contributions of left and righthanded solitons.
The system we chose is CsCoBr_{3}, the bestknown realization of an antiferromagnetic Heisenberg–Ising spin1/2 chain. Together with its isomorphous twin CsCoCl_{3}, it is well characterized and has been shown to support quantum dynamics of solitons^{5,19,20,21,22}. In the onedimensional regime, well above the Néel temperature^{5} T_{N}=28.3 K, the interaction between the lowestlying Co^{2+} Kramers doublets along a chain is described by the spin1/2 Heisenberg–Ising (XXZ) hamiltonian,
with the exchange constants^{5} J_{z}=13.8 meV and J_{t}=1.9 meV. The elementary excitations are solitons rather than conventional magnon excitations of a threedimensional ordered magnet. The decay of a magnon into two solitons is illustrated in Fig. 1b. Quantum dynamics is induced by the transverse exchange that is proportional to J_{t}, which causes adjacent spin pairs to flip. To leading order in J_{t}/J_{z}, this term simply propagates the soliton by two lattice constants, giving rise to the onesoliton dispersion^{4,23}
where the first term is the energy of the broken bond and the last term includes the effect of a small magnetic field B transverse to the Ising direction. Here k is the wavevector, μ_{B} is the Bohr magneton and g is the gyromagnetic ratio. The eigenstates of the hamiltonian H are soliton Bloch states k〉, consisting of a coherent superposition of solitons at all lattice sites (see Methods). Remarkably, these Bloch states are also eigenstates of the chirality or spincurrent operator^{17} satisfying C_{x}k〉=sinkk〉. The states R〉,L〉 at k=±π/2 thus have opposite chirality, 〈C_{x}〉=±1, as shown in Fig. 1a. Quantum fluctuations due to transverse exchange thus generate soliton chirality from the highly degenerate manifold of static Ising solitons.
Figure 1c shows the excellent agreement between the soliton description and the experimentally measured neutron response in CsCoBr_{3}. At temperatures much lower than J_{z}, the soliton band ɛ_{k} is unpopulated and the scattering intensity negligible. With increasing temperature, the soliton band becomes thermally populated. Neutrons are scattered off these solitons, thereby inducing inelastic transitions within the soliton band, and the neutron response consists of a continuum, bounded by the intensity maximum at the Villain mode.
Measuring soliton chirality is far more challenging. Whereas experiments traditionally focus on the diagonal part of the susceptibility, here we take advantage of the fact that an incident beam of spin polarized neutrons couples to the antisymmetric part of the susceptibility. Designating I_{+} (I_{−}) as the scattering intensity of neutrons incident with spin parallel (antiparallel) to the applied magnetic field, the signature of soliton chirality becomes a nonvanishing value of the polarization Π≡(I_{+}−I_{−})/(I_{+}+I_{−}) (see Methods). Note that I_{+}+I_{−} is the conventional scattering intensity for unpolarized neutrons, and Π would be identically zero for unpolarized incident neutrons or for nonchiral soliton states. As a consequence of the hidden character of chirality, a net polarization becomes observable only when the πperiodicity of the dispersion ɛ_{k} is lifted. This is achieved by the external field B, which alters the dispersion as shown in the inset of Fig. 2a; the main panel of the figure shows the calculated scattering intensities I_{±} for q=π/2. Even for infinitesimal B, soliton chirality gives rise to a finite value of Π and hence to a finite real part of the antisymmetric susceptibility^{18,24}. The same quantity vanishes at zero field. This property and the identification of chirality and solitons render this effect fundamentally different from studies of systems that show a chiral response either owing to geometric frustration^{25} or owing to a parity breaking (Dzyaloshinski–Moriya) term in the hamiltonian^{24,26}. The latter would add a contribution proportional to sink to the dispersion ɛ_{k} and hence lead to a Π of the same sign for positive and negative energy transfers.
Our neutronscattering experiments on CsCoBr_{3} were performed at the Institute LaueLangevin in Grenoble, using the world’s most intense beam of polarized neutrons. We first studied the emergence of the Villain mode as a function of temperature (see Fig. 1c) and followed the dispersion between q=π/2 and the antiferromagnetic point q=π, in good agreement with existing data^{5}. Using a beam of polarized neutrons, incident with spin parallel or antiparallel to a magnetic field B=3.3 T, which was small compared with the bandwidth gμ_{B}B/2J_{t}≃0.1, we found the maximal polarization Π at q=π/2. As shown in Fig. 2c, the maximal value of Π at q=π/2 is observed at the energy 4.6±0.2 meV, which is close to the estimated energy 2J_{t}+gμ_{B}B≃4.4 meV required to excite a soliton from the chiral band minimum to the band maximum (see the inset of Fig. 2a).
This observation of soliton chirality has been tested in several independent ways. First, as shown in Fig. 2c,d, we compared the measured polarization to the theoretical prediction having parameters completely fixed from a fit to the unpolarized data I_{+}+I_{−} (see Methods and Supplementary Information, Fig. S1). Second, we examined the behaviour of the polarization as a function of the external field. In zero field, the polarization vanishes; in a reversed field it changes sign, as predicted from equation (2). Third, Fig. 2c,d shows a sign change of the polarization for negative energy transfers. Again, this agrees with the theoretical prediction (see equation (2)) and may be understood from the inset in Fig. 2a. The observation of opposite polarization near ±4.6 meV unambiguously demonstrates the existence of two band minima of opposite chirality.
As a fourth independent test, we investigated whether soliton chirality also affects the neutron response due to solitonpair creation. Indeed, our observations in Fig. 3a clearly show a polarization at the highenergy tail of the measured neutron intensity. Even though quantitative predictions are not yet available, this asymmetry is intuitively understood using the inset in Fig. 2a. The soliton pair created with k_{1}=0, k_{2}=π/2 has a definite chirality and an energy that is higher by gμ_{B}B than that of the remaining manifold with the same centre of mass momentum. This completes the evidence for the chiral nature of soliton excitations.
The chiral soliton states can be expressed as singlets moving through a Néel background, , where is a singlet state of adjacent spins. The resonating character of this state is illustrated by bending an oddnumbered Ising chain into a ring: as the soliton is enforced by boundary conditions, the ground state consists of a singlet that ‘resonates’ around the ring, reminiscent of the ‘resonant valence bond’ state proposed for the description of highT_{c} superconductors^{15}. Our observations imply a degeneracy of such states with respect to chirality.
It is illuminating to interpret the observed chirality in terms of Berry’s phase^{6}. Traditionally, the eigenstates of H are classified according to the total S_{z} component in the Ising direction. Hence they are built from nextnearestneighbour superpositions of solitons. However, such states can be shown to lead to a vanishing Π. Our observation of a nonvanishing chirality thus implies that the observed chiral states involve a coherent superposition of solitons at nearestneighbouring sites, and, remarkably, this remains so even in the limit of an infinitesimal magnetic field. This causes a nonvanishing transition amplitude between soliton states at neighbouring sites, which involves the rotation of one spin1/2 as shown in Fig. 4. This process must involve a Berry phase factor of e^{iπ/2} or e^{−iπ/2}, depending on whether the spin is rotated clockwise or anticlockwise around the quantization x axis. Following the conventional view, one would argue that these phases interfere destructively, causing suppression of nearestneighbour transitions. However, our observations show that the soliton evades this impending localization at the expense of acquiring a spingauge flux^{27}, leading to the two chirally distinct excitations L〉, R〉 that differ by a wavevector of π.
This observation has implications that go far beyond the simple system studied here. Berry phases acquired by electrons moving in the effective field of spin textures have been invoked to explain the anomalous Hall effect in pyrochlores^{28}. In the context of highT_{c} superconductors, the hole motion in a locally antiferromagnetic background involves spinflips for transitions between adjacent sites^{29}, resembling the situation shown in Fig. 4. Whereas it has been argued that the corresponding Berry phases lead to destructive interference that suppresses nearestneighbour transitions of holes^{29}, our results indicate that this interference is lifted, and a nearestneighbour transition does occur at the expense of a spingauge flux and the emergence of spin currents.
We have observed the spontaneous emergence of chirality and spin currents in the disordered phase of a quantum antiferromagnet, associated with the motion of solitons. This state of matter could be exploited for the generation of spin currents in spintronic devices and sheds new light on the elusive nature of quasiparticles in strongly correlated electron systems.
Methods
Theory
The scattering intensity of a polarized beam of neutrons is given by^{18}
where the sign indicates the projection of the spin of the incident neutron onto the x axis parallel to the field direction. Here 〈⋯〉=tr{e^{−βH}⋯}/tr{e^{−βH}} denotes the quantum statistical expectation value. The neutrons transfer momentum ħ Q=ħ(k_{i}−k_{f}), where k_{i} and k_{f} are the wavevectors of the incident and final neutrons respectively, and energy ħ ω to the sample. is the Fourier transform of the spin component perpendicular to the momentum transfer. The second chiral term on the righthand side of equation (1) vanishes in general but is nonzero for helical structures and for chiral soliton states. It is this term that is responsible for the difference between the scattering intensities I_{+} and I_{−}. This is reflected by a nonzero polarization Π=(I_{+}−I_{−})/(I_{+}+I_{−}), with I_{+}+I_{−} being the conventional unpolarized scattering intensity. For the Villain mode, the expectation values in equation (1) are evaluated using the exact form of the chiral onesoliton eigenstates k〉 with Hk〉=ɛ_{k}k〉. For periodic boundary conditions, , with the sum running over all sites m, and k=πν/N, ν=−N+1,…,N, where N is the (odd) number of spins in the chain. Here denotes a soliton at site m of (topological) charge ρ=1 and the corresponding antisoliton m,−1〉 of charge ρ=−1 is obtained by reversing all spins. In the spirit of Villain’s approach^{4}, we perform the trace in equation (1) within the onesoliton subspace, tr{⋯}=∑_{k}〈k⋯k〉, and obtain for the chiral part of the cross section,
the result for I_{+}+I_{−} agrees with the expression that has been previously computed^{4,5} using nonchiral states. Here k,q denote reduced momenta in the chain direction, β=1/k_{B}T where k_{B} is the Boltzmann constant and T the temperature, I_{0} is an intensity scale factor that cancels when computing the polarization Π and f is a projection factor that depends on the total neutron momentum transfer. The sum in equation (2) extends over all processes where the soliton is scattered from initial wavevector k to k′≡k+q−π, and measures the average chirality of the solitons involved in the scattering process. This becomes evident in the limit of k′ approaching k, where each summand measures the chirality of the corresponding soliton state, with maximal contributions at k=±π/2 and zero contribution at the band maxima k=0,π. As chirality is hidden, the maximal contributions near k=±π/2 are of opposite sign and cancel in the absence of external fields. A nonvanishing difference I_{+}−I_{−} results only when the π periodicity of the dispersion is lifted because of a magnetic field. Fitting the theoretical value of I_{+}+I_{−} to the unpolarized data as discussed below, quantitative predictions for I_{+}, I_{−} or the polarization can be derived. It is these predictions for Π and I_{±} that are shown in Fig. 2c,d and in Supplementary Information, Fig. S1, respectively.
Experiments
The neutronscattering experiments were performed on the threeaxis spectrometers IN14 and IN20 at the highflux reactor of the Institute LaueLangevin in Grenoble (France). These instruments provide the highest flux of polarized neutrons presently available in their respective energy domains (cold neutrons on IN14 and thermal neutrons on IN20). In short, the wavevector of the incident neutrons, k_{i}, and their spin polarization are selected by Bragg reflection from a singlecrystal Heusler alloy (IN20) or by a pyrolytic graphite (PG 002) monochromator and a supermirror bender (IN14). After passing a spinflipper, the neutrons arrive at the sample with their spins either parallel or antiparallel to the horizontal magnetic field B. Our highquality CsCoBr_{3} single crystal with mosaic spread η<20′ and a volume of 2 cm^{3} was grown using the Bridgman technique. It was placed into a 4 T horizontal cryomagnet (Oxford Instruments) with the crystallographic axes (h0l) in the scattering plane and B perpendicular to the crystallographic c axis. To gain intensity, there was no analysis of the polarization and no extra Soller collimators were used on IN14 and IN20. The scattered neutrons were reflected from an analysing Bragg crystal of pyrolytic graphite thus defining the wavevector k_{f} at either k_{f}=1.5 Å^{−1} (with Be filter) or 2.662 Å^{−1} (PG filter). All data shown in the main figures are produced directly from raw data without performing any background subtraction. All error bars are statistical in nature and derived from the square root of the detector counts. The unpolarized data are fitted with I_{+}+I_{−} as discussed above, convoluted with a lorentzian^{20} to account for finitelifetime effects and convoluted with the instrument resolution function, using Institute LaueLangevin inhouse software. The fitting parameters are background, peak position, intensity and peak width. These parameters based on the unpolarized data are then used to predict the intensities I_{+} (I_{−}) as shown by red (blue) in Supplementary Information, Fig. S1, or Π shown in Fig. 2c. These predicted intensities contain no free parameters.
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Acknowledgements
We acknowledge illuminating discussions with C. Broholm, J. M. D. Coey, M. Enderle, C. Helm, D. Loss, G. Müller, T. M. Rice and M. Sigrist. We thank the Institute LaueLangevin technical staff, in particular A. Brochier, P. Flores and J. L. Ragazzoni, for their excellent support. This work was supported by the Swiss National Science Foundation, the Center for Theoretical Studies (ETHZ), Enterprise Ireland (IC/2005/0043) and the Science Foundation of Ireland under the Research Frontiers Programme (05/RFP/PHY0023).
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Braun, HB., Kulda, J., Roessli, B. et al. Emergence of soliton chirality in a quantum antiferromagnet. Nature Phys 1, 159–163 (2005). https://doi.org/10.1038/nphys152
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DOI: https://doi.org/10.1038/nphys152
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