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Spectroscopy of Spinons in Coulomb Quantum Spin Liquids

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Spectroscopy of Spinons in Coulomb Quantum Spin Liquids MIT-CTP-5122 Spectroscopy of spinons in Coulomb quantum spin liquids Siddhardh C. Morampudi,1 Frank Wilczek,2, 3, 4, 5, 6 and Chris R. Laumann1 1Department of Physics, Boston University, Boston, MA 02215, USA 2Center for Theoretical Physics, MIT, Cambridge MA 02139, USA 3T. D. Lee Institute, Shanghai, China 4Wilczek Quantum Center, Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China 5Department of Physics, Stockholm University, Stockholm Sweden 6Department of Physics and Origins Project, Arizona State University, Tempe AZ 25287 USA We calculate the effect of the emergent photon on threshold production of spinons in U(1) Coulomb spin liquids such as quantum spin ice. The emergent Coulomb interaction modifies the threshold production cross- section dramatically, changing the weak turn-on expected from the density of states to an abrupt onset reflecting the basic coupling parameters. The slow photon typical in existing lattice models and materials suppresses the intensity at finite momentum and allows profuse Cerenkov radiation beyond a critical momentum. These features are broadly consistent with recent numerical and experimental results. Quantum spin liquids are low temperature phases of mag- The most dramatic consequence of the Coulomb interaction netic materials in which quantum fluctuations prevent the between the spinons is a universal non-perturbative enhance- establishment of long-range magnetic order. Theoretically, ment of the threshold cross section for spinon pair production these phases support exotic fractionalized spin excitations at small momentum q. In this regime, the dynamic structure (spinons) and emergent gauge fields [1–4]. One of the most factor in the spin-flip sector observed in neutron scattering ex- promising candidate class of these phases are U(1) Coulomb hibits a step discontinuity, quantum spin liquids such as quantum spin ice - these are ex- 1 q 2 q2 pected to realize an emergent quantum electrodynamics [5– S(q;!) ∼ S0 1 − θ(! − 2∆ − ) (1) 11]. Establishing the exotic phenomena in this context would 4 mc 4m provide a foundation for exploring other conjectured phases rather than the naive square root onset predicted by the density of matter. It will also allow us to explore regimes of quan- of states for spinon pairs [12]. Here, m and ∆ are the effective tum electrodynamics which are theoretically interesting, but mass and gap for the spinons and c is the emergent speed of otherwise inaccessible. At present, the main method to diagnose a spin liquid ex- perimentally is through the the absence of distinct features as- sociated with a local order parameter such as Bragg peaks, and instead the presence of a broad continuum in neutron scatter- ing indicative of a multi-particle continuum. However, broad continua can also arise from other causes and one would like to have more specific signatures which highlight the emergent gauge field. Here we identify and study features in the zero- temperature cross-section for spinon production in Coulomb quantum spin liquids which directly reflect central aspects of the underlying theory, including the existence and the unusual nature of the emergent photon. Threshold behavior— The emergent photon in quantum spin ice arises from coherent ring-exchange processes which lift massive degeneracy within the manifold of spin configu- arXiv:1906.01628v1 [cond-mat.str-el] 4 Jun 2019 rations consistent with classical ice rules. As ring-exchange is FIG. 1. Dynamic structure factor S(q; !) in a Coulomb quantum spin typically a weak process, the photon propagates with a small liquid measuring production of two spinons near threshold. Com- speed c and has a small bandwidth set by the Brillouin zone pared to the naive density of states (inset), the threshold intensity is cutoff. On the other hand, the spinons typically propagate due strongly enhanced for small q due to the emergent Coulomb inter- action over an energy range of order the Rydberg scale α2mc2=4 to direct spin exchange interactions, which can be paramet- above threshold 2∆. Bound Rydberg states (not shown) accumulate rically larger than the ring exchange process. This contrast at the Rydberg scale below the threshold. Breit interactions due to the leads to a strongly non-relativistic theory in which spinons transverse photon reduce the threshold enhancement with increasing readily propagate faster than the emergent speed of light. Tak- q. With larger momentum, the spinons exceed the emergent speed of ing into account both the gapless nature of the photon and its light and emit Cerenkov radiation. This causes the threshold to dis- slow speed leads us to predict distinctive features in the cross appear into a diffuse continuum for q > 2mc and a peak in intensity for ! ∼ ( q − mc)2=m. Both plots use the same scales. section. 2 2 2 2 2 light. The threshold intensity jump, S0 / m e = m cα, part comes from pair production of spinons. Neglecting all is proportional to the strength of the Coulomb interaction and interactions, this is given by the bubble diagram provides a measure of the emergent fine structure constant α. The strong onset in Eqn. (1) is analogous to the Sommerfeld χ(0)(q; i!) = (4) enhancement [13, 14] observed in semiconductor exciton pro- duction [15]. However, here the small speed of light means that transverse photon exchange has non-negligible conse- At zero temperature, this produces the usual 3D density of p quences. Indeed, the transverse interaction is responsible for states, S0(q;!) ∼ m3=2 ! − 2∆ − q2=4m. q 2 The effects of interactions can be taken into account with the suppression of the enhancement with mc at finite mo- mentum q. More dramatically, since spinons emits Cerenkov renormalized propagators G(k; iκ) (double lines) and a vertex radiation when their velocity exceeds the speed of light, there Γ(k1; k2; iκ1; iκ2) (triangle). is a finite lifetime for spinons propagating at high energy and momenta. For momenta q > 2mc, even threshold spinons have a finite Cerenkov lifetime, and then the threshold in the χ(q; i!) = (5) dynamic structure factor becomes entirely diffuse. The universality of the threshold behavior follows from Wigner’s insight, according to which the energy depen- where the vertex Γ is defined through an irreducible 2-particle dence of cross sections just above threshold are governed diagram (hatched square) by long distance interactions between the slowly escaping particles[16, 17]. In our case, these are the Coulomb and Breit interactions expressed in Eq. (10), which can be ana- (6) lyzed semiclassically in the long distance region. The short distance scattering wavefunctions are, of course, sensitive to lattice scale effects, but generically those vary smoothly with We work with a renormalized mass m which takes energy near threshold. Thus, the jump at small q and asso- into account the effect of interactions in G(k; iκ) ≡ ciated low energy spectral weight for spinon production are 1 remarkably direct signatures of the emergent gauge theory. and neglect higher-order cor- iκ − k2=2m − i Im Σ(k; iκ) Computation— A minimal model for the spinon dynam- rections to the dispersion. We discuss lifetime effects in the ics is given in an effective mass approximation by the follow- next section and suppress the self-energy Σ until then. All the ing Lagrangian, momenta have a UV cut-off due to the Brillouin zone. There e 2 are higher order corrections to the photon dispersion on a lat- y j(−ir − σ c A) σj L = σ(i@t − σeφ) σ − tice, but they will not affect the effects described below. 2m Among diagrams resulting from interactions, crossed dia- − j j2 − ∆ σ V ( ) (2) grams in the two-particle irreducible vertex Γ can be neglected (ladder approximation) since they are suppressed by products where σ represent the spinon (σ = +1) and anti-spinon (σ = −1) fields, e is the emergent charge, m is the effective of the Bose occupation factor nB which vanish at temper- mass and ∆ is the spinon gap. The higher-order interaction ature well below the spinon gap 2∆. Thus, the dominant potential V ( ) contains all of the short-range interactions be- diagrams in the perturbative expansion are ladder diagrams tween the spinons including those induced by gapped, weakly with Coulomb interactions (dashed line), transverse photon dispersive visons (magnetic monopoles), which we do not oth- exchange (wavy line) and short-range interactions (line with erwise attempt to model. The emergent scalar φ and vector A star) between the spinon and anti-spinon, potentials are governed by the usual Maxwell Lagrangian (in CGS units), (7) 1 2 1 2 2 LEM = (rφ) + (@tA) − (r × A) (3) 8π c2 Just above threshold, we can neglect the frequency depen- dence of the photon propagator as the spinons separate asymp- with emergent speed of light c. We work in Coulomb gauge totically slowly. In this approximation, the vertex Γ is inde- r · A = 0 throughout. (Note that if the system possesses an pendent of the relative frequency between the two spinons, additional global U(1) spin symmetry, as in the XXZ model and Eq. (5) can be reduced to χ(q; i!) = W (0; q; i!) where for quantum spin ice, then the spinon fields must be doubled to account for the fractional assignment of the global charge. X eik:rΓ(q=2 + k; q=2 − k; i!) W (r; q; i!) ≡ (8) That does not change the results presented here qualitatively.) i! − k2=m − q2=4m − 2∆ The neutron scattering cross-section is proportional to the k dynamic structure factor S(q;!), given by the imaginary part We have introduced W for convenience and temporarily ne- of the dynamic spin susceptibility χ(q;!). The interesting glected radiation effects in the propagators. 3 2 Using the Bethe-Salpeter equation (Eq.
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