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 Spinon Also need or Can be large! Photon Spinons in an effective mass approximation Instantaneous Transverse photon Coulomb Short-range Maxwell term interactions Spinon Transverse photon Coulomb Short-range Spinon-photon Density of states if we neglect interactions Plethora of additional diagrams from interactions Resum dominant ladder diagrams Effective Hamiltonian problem in absence of radiation Free Coulomb Short-range Current-Current interaction from transverse photon Response ∝ relative probability at short distance Short-range interactions dominate? No! Threshold scattering cross-sections are “universal” Universality of Threshold Laws 2 2 p e FGR ~ 2 H = − + VSR(r) 2π∑ |ψf(0)| δ(ϵ − ϵf ) m r f E Quantum Semiclassical r UV a0 V(r) |E| ≪ |V | Universal enhancement from gapless photon Non-perturbative Universal enhancement from gapless photon Emergent Slow photon Rydberg fine structure constant contribution Universal enhancement from gapless photon Near threshold Jump Suppression at discontinuity finite wavevector Universal enhancement from gapless photon Jump Suppression at discontinuity finite wavevector Spinons emit Cerenkov radiation due to slow photon Cerenkov radiation gives a lifetime to the spinons Interplay of enhancement and Cerenkov effects DOS 1 α = 2 ) α =1 q, ω ( S 1 2 3 4 5 ω Interplay ofenhancementandCerenkov effects S(q, ω) 1 2 ω 3 4 α α DOS =1 = 2 1 5 Interplay of enhancement and Cerenkov effects Bound states Consistent with recent numerics and experiments DOS Pyrochlore XXZ Huang et al - Phys. Rev. Lett. (2018) Consistent with recent numerics and experiments DOS QMC Huang et al - Phys. Rev. Lett. (2018) Consistent with recent numerics and experiments Pr2 Hf2 O7 Sibille et al - Nat. Phys. 2018 Summary • Gapless and slow photon significantly modifies scattering cross-section of spinons in Coulomb spin liquids • Universal enhancement near the bottom of band of spinons which can be used to determine the fine-structure constant • Suppression of intensity at finite wavevectors and diffuse cross-section at critical momentum and energy due to Cerenkov radiation • In line with recent numerical and experimental data