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Spectroscopy of Spinons in Coulomb Quantum Spin Liquids Spectroscopy of Spinons in Coulomb Quantum Spin Liquids The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Morampudi, Siddhardh C., Wilczek, Frank, and Laumann, Chris R. "Spectroscopy of Spinons in Coulomb Quantum Spin Liquids." Physical Review Letters 124, 9 (March 2020): 097204 © 2020 American Physical Society As Published https://dx.doi.org/10.1103/PhysRevLett.124.097204 Publisher American Physical Society Version Final published version Citable link https://hdl.handle.net/1721.1/125409 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW LETTERS 124, 097204 (2020) 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, Massachusetts 02215, USA 2Center for Theoretical Physics, MIT, Cambridge Massachusetts 02139, USA 3T. D. Lee Institute, Shanghai 200240, China 4Wilczek Quantum Center, Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China 5Department of Physics, Stockholm University, Stockholm 114 19, Sweden 6Department of Physics and Origins Project, Arizona State University, Tempe Arizona 25287, USA (Received 11 June 2019; revised manuscript received 6 December 2019; accepted 20 February 2020; published 6 March 2020) We calculate the effect of the emergent photon on threshold production of spinons in Uð1Þ 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. DOI: 10.1103/PhysRevLett.124.097204 Quantum spin liquids are low temperature phases of spinons typically propagate due to direct spin exchange magnetic materials in which quantum fluctuations interactions, which can be parametrically larger than the prevent the establishment of long-range magnetic order. ring exchange process. This contrast leads to a strongly Theoretically, these phases support exotic fractionalized nonrelativistic theory in which spinons readily propagate spin excitations (spinons) and emergent gauge fields [1–4]. faster than the emergent speed of light. Taking into account One of the most promising candidate class of these phases both the gapless nature of the photon and its slow speed are U(1) Coulomb quantum spin liquids such as quantum leads us to predict distinctive features in the cross section. spin ice—these are expected to realize an emergent The most dramatic consequence of the Coulomb inter- quantum electrodynamics [5–11]. Establishing the exotic action between the spinons is a universal nonperturbative phenomena in this context would provide a foundation for enhancement of the threshold cross section for spinon pair exploring other conjectured phases of matter. It also allows production at small momentum q. In this regime, the us to explore regimes of quantum electrodynamics which dynamic structure factor in the spin-flip sector observed are theoretically interesting, but otherwise inaccessible. in neutron scattering exhibits a step discontinuity, At present, the main method to diagnose a spin liquid experimentally is through the absence of distinct features 1 q 2 q2 Sðq; ωÞ ∼ S0 1 − θ ω − 2Δ − ; ð1Þ associated with a local order parameter such as Bragg 4 mc 4m peaks, and instead the presence of a broad continuum in neutron scattering indicative of a multiparticle continuum. rather than the naive square root onset predicted by the However, broad continua can also arise from other causes density of states for spinon pairs [12]. Here, m and Δ are and one would like to have more specific signatures which the effective mass and gap for the spinons and c is the highlight the emergent gauge field. Here we identify and emergent speed of light. The threshold intensity jump, 2 2 2 study features in the zero-temperature cross section for S0 ∝ m e ¼ m cα, is proportional to the strength of the spinon production in Coulomb quantum spin liquids which Coulomb interaction and provides a measure of the directly reflect central aspects of the underlying theory, emergent fine structure constant α. including the existence and the unusual nature of the The strong onset in Eq. (1) is analogous to the emergent photon. Sommerfeld enhancement [14,15] observed in semicon- Threshold behavior.— The emergent photon in quantum ductor exciton production [16]. However, here the small spin ice arises from coherent ring-exchange processes speed of light means that transverse photon exchange has which lift the massive degeneracy within the manifold of non-negligible consequences. Indeed, the transverse inter- spin configurations consistent with classical ice rules. As action is responsible for the suppression of the enhance- ring exchange is typically a weak process, the photon ment with ðq=mcÞ2 at finite momentum q. More propagates with a small speed c and has a small bandwidth dramatically, since spinons emit Cerenkov radiation when set by the Brillouin zone cutoff. On the other hand, the their velocity exceeds the speed of light, there is a finite 0031-9007=20=124(9)=097204(6) 097204-1 © 2020 American Physical Society PHYSICAL REVIEW LETTERS 124, 097204 (2020) lifetime for spinons propagating at high energy and momenta. For momenta q > 2mc, even threshold spinons ð5Þ have a finite Cerenkov lifetime, and then the threshold in the dynamic structure factor becomes entirely diffuse. The universality of the threshold behavior follows from where the vertex Γ is defined through an irreducible two- Wigner’s insight, according to which the energy depend- particle diagram (hatched square) ence of cross sections just above threshold are governed by long distance interactions between the slowly escaping ð6Þ particles [17,18]. In our case, these are the Coulomb and Breit interactions expressed in Eq. (10), which can be analyzed semiclassically in the long distance region. The We work with a renormalized mass m which takes into short distance scattering wave functions are, of course, account the effect of interactions in Gðk;iκÞ ≡ fð1Þ=½iκ − sensitive to lattice scale effects, but generically those vary k2=2m − iℑΣðk;iκÞg and neglect higher-order corrections smoothly with energy near threshold. Thus, the jump at to the dispersion. We discuss lifetime effects in the next small q and associated low energy spectral weight for section and suppress the self-energy Σ until then. All the spinon production are remarkably direct signatures of the momenta have a UV cutoff due to the Brillouin zone. There emergent gauge theory. are higher order corrections to the photon dispersion on a — Computation. A minimal model for the spinon dynam- lattice, but they do not affect the effects described below. ics is given in an effective mass approximation by the Among diagrams resulting from interactions, crossed following Lagrangian, diagrams in the two-particle irreducible vertex Γ can be neglected (ladder approximation) since they are suppressed − ∇ − σ e A ψ 2 † jð i c Þ σj by products of the Bose occupation factor n which vanish L ¼ ψ σði∂ − σeϕÞψ σ − B t 2m at temperature well below the spinon gap 2Δ. Thus, the 2 − Δjψ σj − VðψÞ; ð2Þ dominant diagrams in the perturbative expansion are ladder diagrams with Coulomb interactions (dashed line), trans- where ψ σ represent the spinon (σ ¼þ1) and antispinon verse photon exchange (wavy line) and short-range inter- (σ ¼ −1) fields, e is the emergent charge, m is the effective actions (line with star) between the spinon and antispinon, mass and Δ is the spinon gap. The higher-order interaction ψ potential Vð Þ contains all of the short-range interactions ð7Þ between the spinons including those induced by gapped, weakly dispersive visons (magnetic monopoles), which we do not otherwise attempt to model. The emergent scalar ϕ Just above threshold, we can neglect the frequency and vector A potentials are governed by the usual Maxwell dependence of the photon propagator as the spinons Lagrangian (in CGS units), separate asymptotically slowly. In this approximation, Γ 1 1 the vertex is independent of the relative frequency L ∇ϕ 2 ∂ A 2 − ∇ A 2 between the two spinons, and Eq. (5) can be reduced to EM ¼ 8π ð Þ þ 2 ð t Þ ð × Þ ; ð3Þ c χðq;iωÞ¼Wð0; q;iωÞ where with emergent speed of light c. We work in Coulomb gauge X ik:rΓ q 2 k q 2 − k ω ∇ A 0 r; q ω ≡ e ð = þ ; = ;i Þ · ¼ throughout. Wð ;i Þ ω − 2 − 2 4 − 2Δ : ð8Þ The neutron scattering cross section is proportional to the k i k =m q = m dynamic structure factor Sðq; ωÞ, given by the imaginary We have introduced W for convenience and temporarily part of the dynamic spin susceptibility χðq; ωÞ. The neglected radiation effects in the propagators. interesting part comes from pair production of spinons. Using the Bethe-Salpeter equation [Eq. (6)] for the Neglecting all interactions, this is given by the bubble vertex, we see that Wðr; q;iωÞ is the Green’s function diagram for the Schrödinger equation governing the relative spinon motion 3 ð4Þ ðiω − Hs − 2ΔÞWðr; q;iωÞ¼δ ðrÞ; ð9Þ where At zero temperature, this produces the usual 3D density of 2 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qˆ pˆ e 0 3=2 2 ≡ − r states, S q; ω ∼ m ω − 2Δ − q =4m. Hs þ þ Vð Þ ð Þ 4m m r The effects of interactions can be taken into account with e2 qˆ 2 qˆ:r 2 r r pˆ pˆ renormalized propagators Gðk;iκÞ (double lines) and a ð Þ − ˆ 2 − ð · Þ þ 2 2 2 4 þ 4 2 p 2 : ð10Þ vertex function Γðk1; k2;iκ1;iκ2Þ (triangle) [19] m c r r r 097204-2 PHYSICAL REVIEW LETTERS 124, 097204 (2020) Here, qˆ and pˆ correspond to the center of mass and relative momentum.
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