Thermal Hall Effect from Neutral Currents in Quantum Magnets

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1 Princeton Summer Sch, Aug 2016 Thermal Hall Effect from Neutral Currents in Quantum Magnets Hirschberger Krizan Cava NPO Max Hirschberger, Jason Krizan, R. J. Cava, NPO Princeton University Robin Chisnell, Young Lee and Dan Nocera, MIT 1. Berry curvature and finite Hall effect in charge-neutral currents 2. The frustrated pyrochlore magnet Tb 2 Ti 2 O 7 3. Kagome ferromagnet Cu(1,3 -- bdc) Supported by ARO and NSF

2 Pairwise cancellation Local moments on Kagome lattice Predictions: Chiral spin texture leads to K xy in sparse lattices (bonds between asymmetric plaquettes) --- Kagome lattice and pyrochlores Heuristic derivation of K xy in spin liquid (with fermionic spinons) Neglected magnetization current

3 Experimental checks Lu 2 V 2 O 7 Onose et al. observed a weak K xy in insulating pyrochlore ferromagnet Lu 2 V 2 O 7 Katsura, Nagaosa and Lee missed magnetization current term (Matsumoto, Murakami)

4 Luttinger s gravitational method to calculate K ij Luttinger, Phys. Rev. (1964) cool J E Problem in calculating K ij in linear response (Kubo) theory (- T is a statistical force, not dynamical) Luttinger (1964) considered photons in a grav. potential Ψ (r). Blue-shifted photons transport an energy current J E. At equilibrium, J E = -J Q g Instead of J Q, we calculate J E by including a (fictitious) grav. Potential Ψ (r) to H in Kubo approach. The added term (h(r) is energy density) is dr h(r) Ψ (r) warm J Q Photons in a gravitational potential H = H 0 + V V = -ee.r + ½(Hr + rh). Ψ/c 2 In zero B, does not add anything new

5 Gravitational-field method to calculate K xy in strong B Luttinger, Phys. Rev. (1964) Obraztsov, Sov. Phys. Solid State (1965) Luttinger s grav. approach is essential in B field Thermal Hall current, Nernst response are entangled with orbital magnetization current Smrcka, Streda J. Phys. C (1977), (1983) Oji, Streda, PRB (1985) Cooper, Halperin, Ruzin, PRB (1997) Bergman, Oganesyan, PRL (2010) Matsumoto, Murakami PRL, PRB (2011) Revival in analyzing thermal response of 2D gas in QHE experiments To incorporate correctly Berry curvature effects

6 Berry Curvature Ω ε (k) Consequence of ignoring inter-band transitions by constraining dynamics to lowest band. Karplus Luttinger (1954) discovered the term A(k) in theory of AHE x = R + A(k) Wannier coord. Adams-Blount coordinate (1960) A(k) = (u k, i u k ) (Berry vector potential) R A(k) x [x i, x j ] = iε ijk Ω k Ω(k) = A(k) (Berry curvature) v(k) = k ε + ( r V) Ω(k) anomalous velocity v A Berry curvature Ω(k) acts like a magnetic field in k-space (vanishes if both time reversal and inversion symm hold) Ω(k) is a key concept in TopoIog. Insuls., Weyl and Dirac semimetals, valley physics, etc

7 K xy from Berry Curvature (semiclassical) Matsumoto, Murakami, PRL, PRB (2011) Semiclass. Eqns of motion with Berry curvature rr = kk εε kk kk ΩΩ Berry curvature ΩΩ(kk) = ii uu uu kk = UU(rr) Wall potential U(r) (sketch) produces anomalous velocity -kk ΩΩ and edge particle and energy currents. Ω(k) U(r) Wall potentl y Particle current along x is difference between 2 edges x JJ xx = yy 1 VV dddd kk εε kk ρρ(εε, TT(yy)) ΩΩ(kk) ρ = Bose-Einstein distrib. x y T Energy current along x JJ EE,xx = TT (1 TT ) 1 VV dddd εε(εε μμ) εε kk kk ΩΩ zz cc 2 ρρ = dddd εε kk ββ(εε μμ) 2 With J Q = J E μ J, we obtain κκ xxxx = 2kk BB 2 TT ħvv cc 2 ρρ nn IIII nn,kk uu uu kk xx kk yy

8 K xy from Berry Curvature (linear response) Matsumoto, Murakami, PRL, PRB (2011) Add grav. potential Ψ. to linear response MM obtain L xy as sum of S xy and M xy Kubo term L xy = S xy + M xy Magnetization current MM = ee 2ħ IIII kk 2μμ εε k + HH 0 kk SS 12 = ee 2ħ IIII kk εε k + HH 0 kk kk kk M agrees with Berry-phase calc. of orbital magnetization (Xiao, Shi, Niu PRL (2005)) LL 12 = 2μμμμ 2ħ IIII kk kk kk The sum is a simpler expression than S or M κκ xxxx = 2kk BB 2 TT ħvv cc 2 ρρ nn IIII nn,kk uu uu kk xx kk yy Agrees with semiclassical calculation

9 Materials Experiments on a frustrated quantum pyrochlore magnet (spin liquid candidate) And an ordered Kagome magnet 1) Pyrochlore Tb 2 Ti 2 O 7 High-temp suscep yields Curie-Weiss MFT temp of 19 K but fails to order down to 50 mk. Ground state Quantum spin ice, may harbor spin liquid Investig. for 20 years (mostly neutron), but still an enigma 2) Kagome magnet Cu(1,3 bdc) Kagome planes of Cu separated by organic molecule Orders magnetically at 1.8 K

10 Science (2015) B v B v Charged excitation Wave packet of spin exc. In ferromagnet Chiral antiferromagnet

11 Pyrochlores, spin-ice systems Two-in, two-out config. Spin ice ice Ramirez, et al. Nature 1999 Classical Spin ice Dy 2 Ti 2 O 7, Ho 2 Ti 2 O 7 Castelnovo, Moessner, Sondhi Annu. Rev. Cond. Mat Quantum spin ice (no trace of 2-in/2- out) Tb 2 Ti 2 O 7, Yb 2 Ti 2 O 7

12 I) Thermal conductivity vs temp (B=0) in Tb 2 Ti 2 O 7 Tb2Ti2O7 is a very poor thermal conductor below 5 K. Displays unusually large magnetocalorific effect when B is swept

13 Magneto-thermal conductance T< 20 K Large contribution to thermal current from spin excitations Dominant below ~3 K Very large field effect Metamagnetic transition at H s = 2 T leads to step-like increase in K xx

14 Hall effect in a neutral current? Experimental checks Checks Hall signal reverses when gradient (- T) is reversed in same B Hall signal scales linearly with gradient strength Hall signal is 1000 x larger than in nonmag analog Y 2 Ti 2 O 7

15 Thermal Hall conductivity 10 < T < 140 K

16 Thermal Hall conductivity 1 < T < 15 K Field profile of Kxy/T attains a maximum near 15 K. Decreases rapidly below 3 K Except in weak H.

17 Hall angle Below 3 K, tan θ H is H-linear (with constant slope) until H > H p Defines low-t state with largest Hall response. The state is readily destroyed when H exceeds H p (T)

18 Phase diagram of large Hall state in the H-T plane Extent of large-hall response state in T-H plane (shaded). Hall response is strongly suppressed in field-induced metamagnetic state (H>H s )

19 The Quantum Spin Ice Hermele, Fisher, Balents (PRB 2004) Higgs phases (spinons condensed) Coulomb ferromagnet (spinons deconfined) T = 0 Quantum spin liquid (spinons deconfined) Fractional excitations in CFM and QSL states E field monopoles B field monopoles Faux photon

20 Part II. Kagome Magnet B v B v

21 Rare example of Kagome magnet Cu(1,3-bdc), benzenedicarboxylate or bdc Becomes antiferromagnetic (type I) below 1.8 K

22 K vs. Temp in Kagome Magnet (H=0) Max Hirschberger, Robin Chisnell, Young Lee and NPO

23 K vs. Temp in Kagome Magnet

24 Rich behavior of Kxx vs H at low temp (T< 4 K)

25 Behavior of Kxx vs H at low temp (T< 4 K)

26 Thermal Hall conductivity in Kagome magnet

27 Behavior of Kxy across Tc = 1.8 K K xy reverses sign as a function of T, and as a function of H. Rules out skew scattering of phonons

28 Berry curvature and Chern number of adjacent magnon bands Lee, Han and Lee, PRB 2015

29 Computed Kxy of Kagome Magnet Lee, Han and Lee, PRB (2015) Calculation captures main qualitative features, some quantitative discrepancies

30 K xy and near-minkowski gravity, neutral modes Kxy may probe neutral fractional excitations 1) FQHE, Kane and Fisher (1996), Read and Green (2001) 2) Neutral Topological Insulators, Ryu, Moore, Ludwig (2012); Stone (2013) Nomura, Ryu, Furusaki, Nagaosa (2012) Near-Minkowski Ψ > E g gravitoelectic field gµν = ηµν + hµν x B g = -4π GJ m /c 2 gravitomagnetic field Lorentz force F = m(e g + 2 v x B g ) Frame dragging K xy = (π 2 /6) k B 2 T /2h gyroscope B g Bg E g J m Ryu, Moore, Ludwig Nomura, Ryu, Furusaki, Nagaosa M. Stone M

31 Summary Observation of a large Hall effect from neutral spin excitations I) In frustrated magnet Tb 2 Ti 2 O 7 Excitations are not magnons Are they spinons? Other fractional excitations? II) III) In Kagome ferromagnet Cu(1-3, bdc) Hall signal observed both above and below Tc Unexpected sign reversal at Tc Consequence of Berry curvature in different magnon bands Other spin liquid and exotic quantum magnets? Related pyrochlore Yb 2 Ti 2 O 7 yes Herbert smithite ongoing ET salts dmit salts no Han Purple salt SrCu(BO 3 ), Haldane Shastry systems Classical spin ice Dy 2 Ti 2 O 7

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