Spectroscopy of spinons in Coulomb quantum spin liquids Quantum spin ice Chris R. Laumann (Boston University) Josephson junction arrays Interacting dipoles Work with: Primary Reference: Siddhardh Morampudi Morampudi, Wilzcek, CRL arXiv:1906.01628 Frank Wilzcek Les Houches School: Topology Something Something September 5, 2019 Collaborators Siddhardh Morampudi Frank Wilczek Summary Emergent photon in the Coulomb spin liquid leads to characteristic signatures in neutron scattering Outline 1. Introduction A. Emergent QED in quantum spin ice B. Spectroscopy 2. Results A. Universal enhancement B. Cerenkov radiation C. Comparison to numerics and experiments 3. Summary New phases beyond broken symmetry paradigm Fractional Quantum Hall Effect Quantum Spin Liquids D.C. Tsui; H.L. Stormer; A.C. Gossard - Phys. Rev. Lett. (1982) Balents - Nature (2010) Savary et al.- Rep. Prog. Phys (2017) Knolle et al. - Ann. Rev. Cond. Mat. (2019) Theoretically describing quantum spin liquids • Lack of local order parameters • Topological ground state degeneracy • Fractionalized excitations Interplay in this talk • Emergent gauge fields How do we get a quantum spin liquid? (Emergent gauge theory) Local constraints + quantum fluctuations + Luck Rare earth pyrochlores Classical spin ice 4f rare-earth Non-magnetic Quantum spin ice Gingras and McClarty - Rep. Prog. Phys. (2014) Rau and Gingras (2019) Pseudo-spins in rare-earth pyrochlores Free ion Pseudo-spins in rare-earth pyrochlores Free ion + Spin-orbit ~ eV Pseudo-spins in rare-earth pyrochlores Free ion + Spin-orbit + Crystal field Single-ion anisotropy ~ eV ~ meV Rau and Gingras (2019) Allowed NN microscopic Hamiltonian Doublet = spin-1/2 like Kramers pair Ising + Heisenberg + Dipolar + Dzyaloshinskii-Moriya Ross et al - Phys. Rev. Lett. 2011 Pyrochlore XXZ as a simple model Local constraints Quantum fluctuations Ring exchange model at low energies Huse et al - Phys. Rev. Lett 2003 Hermele et al - Phys. Rev. B 2004 Emergent gauge symmetry! Small bandwidth Emergent photon? Verified in numerics! Slow photon Banerjee et al - Phys. Rev. Lett. 2008 Shannon et al - Phys. Rev. Lett 2012 Huse et al - Phys. Rev. Lett 2003 Hermele et al - Phys. Rev. B 2004 Usual QED Photon Quantum spin ice Spinon Photon What should we see in neutron scattering? Neutron produces pair of spinons Energy Momentum Fractionalization seen in spectroscopy Multiple excitations = Continuum! ) Fractionalization seen in spectroscopy Spin waves Fractionalized spinons M. Mourigal et al. - Nature Physics (2013) 2 2130 2000 115 90 ⇥ ⇥ ⇥ ⇥ z z /J /J ! ! Density of states at mean field level 2 2130 2000 115 90 6.4 5.5 ⇥ ⇥ 2.3 ⇥ 2.4 ⇥ ⇥ ⇥ ⇥ ⇥ z z /J /J ! ! z z /J /J 6.4 5.5 2.3 2.4 ! ! ⇥ ⇥ ⇥ ⇥ z z /J /J ! ! (1,1,1) 2⇡(0, 2k, 0)/82⇡(k, k, k)/8 2⇡(0, 2k, 0)/82⇡(k, k, k)/8 2⇡(0, 2k, 0)/82⇡(k, k, k)/8 FIG. 2. Left panel: (a,b) Dynamic2 spin⇡(0 structure, 2k, factor0)/S82zz(q, !) Szz (q, !)⇡ obtained(k, k, from k)/ QMC-SAC8 at temperature ⌘ ↵ ↵↵ T1 along high symmetry cubic directions (010) and (111). The photon appears as a gapless branch of excitation with dispersion P starting from Brillouin Zone center. White dots mark the position of spectralHuang peaks. et al Pink- Phys. open Rev. circlesLett. (2018) show the integrated spectral weight at each momentum point with maximal spectral weight rescaled to 1. (a’,b’) Photon spectra calculated from a zz zz + + Gaussian QED model. Right panel: (c,d) Dynamic spin structure factor S −(q, !) S −(q, !) obtained from QMC-SAC FIG. 2. Left panel: (a,b) Dynamic spin structure factor S (q, !) ↵ S↵↵(q, !) obtained from QMC-SAC⌘ ↵ ↵↵ at temperature at T1. The⌘ spectra show a dispersive continuum of two-spinon excitations. (c’,d’) The results from a tight-binding model T1 along high symmetry cubic directions (010) and (111). Thecalculation, photon where appears the spinons are as modeled a gapless as free particles. branch The calculated of excitationspectra areP then broadened with with dispersion a Lorentzian to mimic interactionP e↵ects. The spinon continuum boundaries calculated from the tight-binding model are marked as white dots starting from Brillouin Zone center. White dots mark the positionin both QMC-SAC of spectra spectral (c,d) and peaks. the theoretical Pink spectra (c’,d’). open circles show the integrated spectral weight at each momentum point with maximal spectral weight rescaled to 1. (a’,b’) Photon spectra calculated from a z that preserves all symmetries+ of the system. Viewing S+i tice [25], Gaussian QED model. Right panel: (c,d) Dynamic spin structureas the electric factor fieldS and the−( iceq, rule!) as Gauss’s↵ lawS↵↵ in −(q, !) obtained from QMC-SAC + z z electrostatics [31], the spin liquid ground⌘ state is analo- = J (Si Sj− + h.c.)+JzSi Sj . (1) at T1. The spectra show a dispersive continuum of two-spinon excitations. (c’,d’) The results fromH a− ± tight-binding model gous to the vacuum state of the quantum electrodynamics i,j P hXi calculation, where the spinons are modeled as free particles.(QED) The [25 calculated]. spectra are then broadenedx,y,z with a Lorentzian to Here, Si are the Cartesian components of S =1/2 mimic interaction e↵ects. The spinon continuum boundaries calculatedThree types of topological from the excitations tight-binding can emerge from modelspin operator are on site markedi, and the as summation white is over dots all nearest-neighbor pairs. Jz,J > 0 are spin exchange the QSI ground state [25]: The photon, analogous to the ± in both QMC-SAC spectra (c,d) and the theoretical spectra (c’,d’).electromagnetic wave, is a gapless, wavelike disturbance constants. within the ice manifold. The spinon is a gapped point de- To set the stage, we briefly review the thermody- fect that violates the ice rule within a tetrahedron [Fig. 1 namic phase diagram of the model in Eq. (1), which has been well established by QMC calculations [41–44]. At (a2)]. In the QED language, spinons are sources of the zero temperature, a critical point on the J /Jz axis at electric field; the charge carried by a spinon is taken to ± z J ,c/Jz =0.052(2) [43] separates the XY ferromagnet be Q↵ on the tetrahedron occupied by it. The monopole, ± that preserves all symmetries of the system. Viewing Si tice [25], state (J >J ,c) and the QSI ground state (J <J ,c). also a gapped point defect, is the source of the gauge mag- ± ± ± ± netic field, whose presence is detected by the Aharonov- On the QSI side, with fixed J /Jz, three regimes exist on as the electric field and the ice rule as Gauss’s law in the temperature axis. At high± temperature k T J , Bohm phase of the spinon. (The monopole is also referred+ z z B z to as vison in some literature= [30].) J (S Sthe− system+ h.c. is in)+ the trivialJzS paramagneticS . regime(1) with electrostatics [31], the spin liquid ground state is analo- i entropyj Nk ln 2, N beingi thej number of spins. H − ± S ⇡ B gous to the vacuum state of the quantum electrodynamics The abundant theoretical predictionsi,j [25, 29, 32–35] on When kBT decreases to O(Jz), the system crosses over the QSI topological excitations naturallyhXi call for numer- to a classical spin ice (CSI) regime where it thermally (QED) [25]. ical scrutiny. Yet, theirx,y,z dynamical properties so far have fluctuates within the ice manifold [24]. Since the num- only beenHere, indirectlyS inferred fromare the numerical the analysis Cartesianber of the components spin configurations in the of iceS manifold=1 is/ ex-2 of the ground state ori toy models [36–40]. Here, we di- ponentially large in N, the entropy is still extensive: Nk ln(3/2)/2[45]. As T further decreases, the rectly addressspin the operator dynamics problem on by unbiased site QMCi, andS ⇡ theB summation is over all Three types of topological excitations can emerge from simulation of a QSI model. system approaches the QSI regime through a second crossover with limT 0 = 0. Figure 1(b) shows the nearest-neighbor pairs. J ,J > 0! are spin exchange Model — We study the XXZ model on pyrochlore lat- zentropy and specificS heat C as a function of T for the QSI ground state [25]: The photon, analogous to the ± S electromagnetic wave, is a gapless, wavelike disturbance constants. within the ice manifold. The spinon is a gapped point de- To set the stage, we briefly review the thermody- fect that violates the ice rule within a tetrahedron [Fig. 1 namic phase diagram of the model in Eq. (1), which has been well established by QMC calculations [41–44]. At (a2)]. In the QED language, spinons are sources of the zero temperature, a critical point on the J /Jz axis at electric field; the charge carried by a spinon is taken to ± J ,c/Jz =0.052(2) [43] separates the XY ferromagnet be Q↵ on the tetrahedron occupied by it. The monopole, ± state (J >J ,c) and the QSI ground state (J <J ,c). also a gapped point defect, is the source of the gauge mag- ± ± ± ± netic field, whose presence is detected by the Aharonov- On the QSI side, with fixed J /Jz, three regimes exist on the temperature axis. At high± temperature k T J , Bohm phase of the spinon. (The monopole is also referred B z to as vison in some literature [30].) the system is in the trivial paramagnetic regime with entropy Nk ln 2, N being the number of spins. S ⇡ B The abundant theoretical predictions [25, 29, 32–35] on When kBT decreases to O(Jz), the system crosses over the QSI topological excitations naturally call for numer- to a classical spin ice (CSI) regime where it thermally ical scrutiny.