Spectroscopy of Spinons in Coulomb Quantum Liquids

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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

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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 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

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Here, qˆ and pˆ correspond to the center of mass and relative momentum. The terms suppressed by c are Breit terms governing the leading velocity dependent interactions between moving charges. Using a spectral representation for W, we find X 2 q ω 2π ψ 0 2δ ω−2Δ− q −ϵ Sð ; Þ¼ j jðr ¼ Þj 4 j ; ð11Þ j m where ψ j are the relative eigenstates of Hs. Physically, Eq. (11) shows that the naive density of states is enhanced by the probability that the two spinons be found together in the eigenfunction. FIG. 1. Dynamic structure factor Sðq; ωÞ in a Coulomb quan- The preceding derivation of the effective two-body tum spin liquid measuring production of two spinons near Schrödinger equation holds for energy close to threshold threshold. Compared to the naive density of states (inset), the and small spinon momenta. In particular, the Breit term, threshold intensity is strongly enhanced for small q due to the corresponding to instantaneous current-current interactions emergent Coulomb interaction over an energy range of order the between the spinons, is only valid when the spinons are Rydberg scale α2mc2=4 above threshold 2Δ. Bound Rydberg moving sufficiently slowly and thus should not be used states (not shown) accumulate at the Rydberg scale below the directly in the short distance region [21,22]. Fortunately, threshold. Breit interactions due to the transverse photon reduce the energy dependence of the probability at the origin the threshold enhancement with increasing q. With larger 2 momentum, the spinons exceed the emergent speed of light jψ ð0Þj is governed by the long range part of the inter- j and emit Cerenkov radiation. This causes the threshold to actions near the threshold. The short distance wave func- disappear into a diffuse continuum for q>2mc and a peak in tion, whatever it is, is rigid due to the large energy costs intensity for ω ∼ ½ðq=2Þ − mc2=m. Both plots span the same associated with amplitude shifts in that region [17,18].We range and use the same color scale. can thus neglect both VðrÞ and the exact form of the Breit Hamiltonian for small r in estimating the energy depend- 2 pattern of bound states is seen in studying defects in ence of jψ ð0Þj . j quantum dipolar spin ice from a mapping to a Coulomb At large separation, the effective potential between the problem on the Bethe lattice [23]. We have not shown spinons decays as 1=r with a q-dependent effective charge pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bound states in Fig. 1 since their lifetime is affected by → 1 − 2 4 2 2 e e ðq = m c Þ restricted to the zero angular temperature, disorder, and other complicating effects. momentum channel. A semiclassical analysis of the mixed Cerenkov radiation.—Since the system is not Lorentz r p terms involving products of and (which are neither pure invariant, the spinons can exceed the effective speed of potential nor kinetic) shows that they only change the light. The resulting Cerenkov radiation carries momentum ψ 0 2 probability j jð Þj by an energy-independent constant and energy away from the spinon and leads formally to a (see the Supplemental Material [13]). finite lifetime (imaginary part of the self-energy) for Using the renormalized effective charge from the Breit k>mc. The leading order contribution to the spinon interactions and the solution of the two particle Coulomb lifetime comes from problem [15], we find pffiffiffiffiffiffiffiffiffi m3=2 2πR q2 ð13Þ Sðq; ωÞ ∼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffi θ ω − 2Δ − ; ð12Þ 2πR 4m 1 − expð− 2 Þ ω−2Δ−q 4m The seagull diagram, although of the same order in α, does 1 2 2 2 2 2 2 where R ¼ 4 mc α ½1 − ðq =4m c Þ is the effective not contribute. There are no self-energy corrections to the Rydberg constant. Here, α ¼ e2=c is the fine-structure photon for temperatures below the magnetic monopole gap. constant. Neglecting the (non-universal) essential singu- After analytic continuation, the imaginary part of larity cutting off the enhancement for energies above R Eq. (13) is given by recovers Eq. (1). 2 2 Z Z The two particle problem also has an infinite set of θ kΛ ℑΣ k ω ∼ e k c θ 3 θ solutions (excitons) at energies ω ¼ 2Δ − ð ; Þ d dKKsin ð Þ n mc 0 0 ðR=n2Þ below the two particle continuum. The q-dependent ðk − KÞ2 charge renormalization implies that the states bend into the × δ ω − − cK ; ð14Þ two-particle continuum as q approaches 2mc. A similar 2m

097204-3 PHYSICAL REVIEW LETTERS 124, 097204 (2020) where θ is the angle between k and the photon momentum K, and θc is the critical angle below which no are radiated (i.e., the Cerenkov cone). The finite bandwidth kΛ can cut off the amount of radiation produced if kΛ < 2ðk − mcÞ. Assuming no such cutoff, Z 2 2 θ e k c ℑΣðk; ωÞ ∼ dθsin3ðθÞ mc 0 k cosðθÞ − mc × 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½k cosðθÞ − mc2 þ 2mω − k2 ð15Þ

If we evaluate the self-energy on shell, cosðθ Þ¼mc=k c ω and we get FIG. 2. Sðq; Þ (arbitrary units) at varying momentum q (Δ ¼ 0.5, m ¼ c ¼ 1). The naive density of states (dashed blue) k2 e2k2 mc 1 mc 3 2 shows a square-root onset. Interaction with the gauge field ℑΣ k; ω ¼ ∼ − − ð16Þ enhances the square root into a jump discontinuity with magni- 2m mc k 3 k 3 tude proportional to the emergent fine-structure constant α ¼ e2=c. The size of the jump decreases with q due to the when k ≥ mc. Close to the threshold for radiation, the 2 2 Breit interaction from transverse photon exchange (second inverse lifetime goes as e ðk − mcÞ =mc, whereas for panel), until the threshold is washed out entirely by the produc- 2 2 k ≫ mc, it goes as 2e k =3mc. tion of Cerenkov radiation at all energies (third panel). The peaks At threshold, the spinons have no relative momentum, in the grayed region arise from spectral transfer from the high and both start emitting Cerenkov radiation at a critical energy regime, where the spinons are broadened significantly by external momentum transfer q ¼ 2mc which gives each of radiation toward the low energy region where no radiation is them momentum of mc and velocity c. Since the relative produced. The exact shape of the peaks in the grayed regime is momentum is zero, the vertex equation can be again solved likely an artifact of the approximations used, which are most accurate close to threshold. including the self-energy effects. The resulting effect is that at momenta beyond 2mc, the threshold becomes increas- near the top of the band the threshold is diffuse for all ingly diffuse (Fig. 1). momentum transfers. The combined effect of the suppres- The spinons also emit Cerenkov radiation at any external sion and Cerenkov radiation at threshold implies that the momentum once they have enough energy above threshold, 2 spectral intensity at the top is heavily suppressed compared i.e., when ω > ω ðqÞ¼½ðq=2Þ − mc =m. The vertex c to the density of states. equation cannot be solved exactly in this regime, but the Application to spin ice.—The results apply to any U(1) dominant effects are captured by considering collinear Coulomb with gapped spinons, such as trajectories of the spinons in the evaluation of the self- have been discussed in models of hard-core bosons on the energies. Additionally, retardation effects from the trans- pyrochlore [5], interacting dipoles [6], quantum rotors [25], verse photon are important away from threshold and and quantum link models [26]. Perhaps the most promising contribute to a cutoff of the Sommerfeld enhancement at experimental application is to quantum spin ice materials relativistic speeds. The qualitative effect is a peak in – ω ∼ ω [11,27 41]. The ideal realization is an XXZ model on the intensity when cðqÞ due to a spectral weight transfer – pyrochlore lattice [9,42 45] with parameters Jzz;J such (Fig. 2). J ≪ J — that zz, and the generic Hamiltonian describing spin Top threshold. Near the top of the band, the effective ice materials contains perturbations to this. The spinons mass is negative. The dominant vertex corrections at the top live on a diamond lattice with the bare mass given by threshold can be taken into account by noticing that the ∼ ℏ2 4 2 Δ ∼ 2 − 12 m = Ja0 and a gap Jzz= J. The photon problem can be mapped to a positive mass spinon and 3 2 has a bandwidth set by the ring-exchange g ≡ 12J =J antispinon interacting with a repulsive Coulomb interac- zz and hence a speed of light c ∼ ξga0=ℏ where ξ is tion. This suppresses the intensity at the top threshold and an Oð1Þ constant and a0 is the lattice constant. There gives [24] would be a significant enhancement in intensity seen in sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi neutron scattering over an energy scale R ¼ α2mc2= pffiffiffiffiffiffiffiffiffi 2π 2 2 5 4 3=2 R 4 ∼ 9ξ α Sðq; ωÞ ∼ m 2πR exp − 2 : ð17Þ ð J=JzzÞ. Additionally, the threshold becomes 2Δ − q − ω 2 ∼ 4m incoherent due to Cerenkov effects at qc ¼ mc 3ξℏ 2 2 ð J=a0JzzÞ. A more accurate estimate would use the However, kinematically the negative mass allows for renormalized parameters determined from data as detailed Cerenkov radiation right at threshold. This means that in the discussion.

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Recent numerical and experimental works have made Energy under Grant Contract No. DE-SC0012567, progress in determining the spinon dynamic structure factor by the European Research Council under Grant in quantum spin ice. Quantum Monte Carlo data from [45] No. 742104, and by the Swedish Research Council under clearly shows a sudden onset in intensity at the bottom Contract No. 335-2014-7424. C. R. L. acknowledges threshold and a drop in intensity at threshold at large wave support from the NSF through Grant No. PHY-1752727. vectors away from the bottom, indicative of possible Cerenkov effects. There have been claims of spinon excitations in neutron scattering data on Pr2 Hf2O7,a candidate quantum spin ice [46,47]. If true, then the large [1] P. W. Anderson, Science 235, 1196 (1987). intensity at the expected threshold is consistent with our [2] L. Balents, Nature (London) 464, 199 (2010). predictions, but more precise data are needed to confirm [3] L. Savary and L. Balents, Rep. Prog. 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Cardona, Fundamentals of Semiconductors: find emergent gauge structure in “mechanical” models has Physics and Materials Properties (Springer, 2010). a glorious history, as it led Maxwell to his equations. Today, [17] E. P. Wigner, Phys. Rev. 73, 1002 (1948). given our increased abilities to engineer interactions and to [18] S. C. Morampudi, A. M. Turner, F. Pollmann, and F. sculpt metamaterials (such as spin-ice physics at room Wilczek, Phys. Rev. Lett. 118, 227201 (2017). temperature [49]), it is newly relevant to practice, even if it [19] See Ref. [20] for diagrammatic conventions. does not lead to new models of microphysics. Emergent [20] H. Bruus, K. Flensberg, and O. U. Press, in Many-Body electrodynamics can give us access to parameter regimes Quantum Theory in : An Intro- duction, Oxford Graduate Texts (OUP Oxford, 2004). and phenomena that are not easily accessible otherwise, [21] G. Breit, Phys. 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