Spin-Ice State of the Quantum Heisenberg Antiferromagnet on Pyrochlore Lattice Yuan Huang1,2,KunChen1,2, Youjin Deng1,2,NikolayV.Prokof’ev1,3,andBorisV.Svistunov1,3

1 University of Massachusetts 2 University of Science and Technology of China 3 National Research Center “Kurchatov Institute, Moscow”

Abstract

We study the low-temperature physics of the spin-1/2 Heisenberg antiferromagnet on a pyrochlore lattice and find a “fingerprint” evidence for the thermal spin-ice state in this frustrated quantum magnet. The identification of the spin-ice state is done through a remarkably accurate microscopic correspondence for static structure factor between the quantum Heisenberg and classical Heisenberg/Ising models, and the characteristic bow-tie pattern with pinch points. The dynamic structure factor is consistent with di↵usive spinon dynamics at pinch points.

Frustrated Magnets Spin Ice: Static Structure Factor

Frustration Pinch points are direct consequences of ’2-in/2-out’ ice rule. (classical) massive degenerate ground state • Around the pinch points, structure factor has strong correlations/fluctuations • pseudo-dipolar singularities: novel phases, including spin liquids 2 • q common in many real materials S(Qpinch + q) 2 ? 2 • / q + q typical example in 3D: pyrochlore lattice k ? • Model AF quantum Heisenberg model on pyrochlore lattice Spin Ice: Quantum-to-Classical Correspondence

H = J Si Sj (J>0) · Static spin-spin correlation function: X no controlled method to study the low temperature physics so far (r) d⌧(r0, ri; ⌧) ⌘ Z0 Compare the normalized static correlator f(r)=(r) in Diagrammatic Monte Carlo (0) quantum Heisenberg model (QH) • classical Heisenberg model (CH) Spin Fermionization • J i⇡T Ising model H = (fi†↵↵fi) (fj†fj) (ni 1) • 4 · 2 i X↵ X The QCC holds for: complex chemical potential term: exact cancelation of all unphysical contributions in • only static correlations, but not for grand-canonical statistics • equal-time correlations interacting flat-band fermionic Hamiltonian • all distances starting from the nearest- Feynman Diagrams & DiagMC • neighbor sites Full Green’s function: expansion of Feynman diagrams in bare Green’s function and coupling constant not only high-T limit but also intermedi- W • Dyson equations: J/4 ate temperature regime (down to J/6) ... (0) (0) G G↵ = G↵ + G↵ ⌃↵G↵ *SimilarQCCbetweenquantumandclas- ˆ ˆ ˆ ˆ ˆ ˆ sical Heisenberg model was also found on U = J J ⇧ U · · triangular and square lattice [?] Susceptibility: G (r,⌧)= S(ri, 0) S(rj,⌧) ˆ ... h · i G Dyson-like equation ˆ = ⇧ˆ +⇧ˆ Jˆ ˆ · ·

Series Convergence Spin Ice: Dynamics

Uniform susceptibility u Dynamic structure factor S(Q,!) From Matsubara frequency functions to real frequency u = d⌧(r,⌧) (numerical analytic continuation) : r Z0 ! X 1 1 (1 e )! High temperature expansion diverges at T

Structure factor in the momentum–Matsubara-frequency domain 1 i[Q (rj ri)+!n⌧] S(Q, i!n)= d⌧(ri, rj; ⌧)e · References & Acknowledgments V 0 Xi,j Z S(Q, i!n =0)in the ([hh0][00l]) plane (left: T =2J;right:T = J/Appro6) ved Variations [1] Y. Huang, K. Chen, Y. Deng, N. V. Prokof’ev, and B. V. Svistunov, Phys. Rev. Lett. 116,177203(2016). [2] S. A. Kulagin, N. V. Prokof’ev, O. A. Starykh, B. V. Svistunov, and C. N. Varney, Phys. Rev. Lett. 110, 070601 (2013); Phys. Rev. B 87,024407(2013). high-T(2J): [3] N. V. Prokof’ev and B. V. Svistunov, Phys. Rev. B 84,073102(2011). • checkerboard pattern

low-T(J/6): • bow-tie pattern with pinch points

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