Quantum oscillations in insulators with neutral Fermi surfaces

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1 Quantum oscillations in insulators with neutral Fermi surfaces ITF-Seminar IFW Institute - Dresden October 4, 2017 Inti Sodemann MPI-PKS Dresden

2 Contents Theory of quantum oscillations of insulators with neutral fermi surfaces. The composite exciton fermi liquid in SmB6. D. Chowdhury T. Senthil Debanjan Chowdhury, Inti Sodemann, T. Senthil, arxiv: (2017) Inti Sodemann, Debanjan Chowdhury, T. Senthil, arxiv: (2017)

3 Insulators in magnetic fields Consider a band insulator at zero T: B B=0i ~ B6=0i Adiabatically connected Adiabaticity implies linear response: E (B) =E 0 + B M = de db 4 M = B +

4 Metals in magnetic fields Consider a metal at zero T: 2D Landau levels: B D(E) ~! c B=0i = B6=0i NOT adiabatically connected! 3D Landau bands: E D(E) E

5 Metals in magnetic fields Energy of 2D metal as function of field: Magnetization of 2D metal as function of field: µ e = 1 2m = N e N = S 2 B cusps = integer filled LLs Amplitude 2D metal 3D metal (T =0,B! 0) 4 M osc n e µ e const 4 M osc L S p B/S B 1/2

6 Beyond band insulators and metals Q: is there a phase of matter that is an insulator but has quantum oscillations? lim (T )=0 j = E T!0

7 Beyond band insulators and metals Q: is there a phase of matter that is an insulator but has quantum oscillations? lim (T )=0 j = E T!0 A: yes! a fractionalized phase of matter: the spinon fermi surface is an electrical insulator displaying quantum oscillations. O. I. Motrunich, PRB (2006) Inti Sodemann, Debanjan Chowdhury, T. Senthil, arxiv: (2017)

8 The spinon fermi surface Fractionalized state Electron Spinon c = f b Chargon (holon) P.W. Anderson Spinfull neutral fermion Spinless charged boson Spinon forms a fermi sea Chargon forms a bosonic Mott insulator Florens & Geoges, PRB (2004)

9 The spinon fermi surface Electron Spinon Chargon (holon) c i = f i b i Physical states satisfy: n c = n f = n b Explicit trial wave-functions can be written as: c(r 1 1,...,r N N )= f (r 1 1,...,r N N ) b(r 1,...,r N ) fermi sea boson Mott insulator

10 The spinon fermi surface Explicit trial wave-functions can be written as: c(r 1 1,...,r N N )= f (r 1 1,...,r N N ) b(r 1,...,r N ) In half-filled band N f = N sites Superfluid hbi 6=0 Mott hbi =0 N b = N sites Bosons can form simple Mott state

11 The spinon fermi surface Explicit trial wave-functions can be written as: c(r 1 1,...,r N N )= f (r 1 1,...,r N N ) b(r 1,...,r N ) In half-filled band N f = N sites Superfluid hbi 6=0 Mott hbi =0 N b = N sites hc i c j i hf i f j ihb i b ji Bosons can form simple Mott state Metal gapped Florens & Geoges, PRB (2004) Senthil, PRB (2008)

12 Electric response of spinon fermi sea Electron Spinon Chargon (holon) U(1) gauge field q i = n fi n bi c i = f i b i Gauge invariant terms L = L f (p a)+l b (p A + a)+ hbi 6=0 hbi =0 Metal Gauge field higgsed Spinon and chargon linearly confined spinon FS Gauge field Landau damped Spinon and chargon de-confined

13 Electric response of spinon fermi sea Mott Electron Spinon Chargon (holon) hbi =0 c i = f i b i L = L f (p a)+l b (p A + a)+ DC insulator lim!!0 (!) =0 lim T!0 (T )=0 Boson is a dielectric b(!) i! 1 4 Constraint j f = j b Charge is not fully frozen

14 Electric response of spinon fermi sea Mott Electron Spinon Chargon (holon) hbi =0 c i = f i b i L = L f (p a)+l b (p A + a)+ DC insulator lim!!0 (!) =0 Ioffe-Larkin rule: = s + b Ioffe & Larkin, PRB (1989). Re[ (!)] =! 2 b 1 T-K Ng & PA Lee, PRL (2007) Re[ s(!)]

15 Magnetism of spinon fermi surface L = L f (p a)+l b (p A + a)+ f B = r A b = r a j b 6=0 j f = j b M f = M b = f (b)+ b (B b)+ f b 2 + b (B b) 2 2 =0 M f = M b equal Equilibrium magnetizations

16 Magnetism of spinon fermi surface Electron Spinon c i = f i b i Chargon (holon) = f (b)+ b (B b)+ f =0 2 + b (B b) 2 2 M f = M b + Equilibrium equal magnetizations b eq = b f + b B Metal spinon FS hbi 6=0 hbi =0 b = 1 b < 1 b eq = B b eq = B

17 Quantum oscillations of spinons = f (b)+ b (B b)+ f b 2 2 f (b) Spinon spectrum is non-perturbative + b (B b) 2 =0 2 b S Two dimensions: f osc(b) osc b 2 2 1X k=1 ( 1) k k 2 kt S 2 b sinh( kt S 2 b ) cos ks b.

18 Quantum oscillations of spinons =0 2 b S 2 2 T/ F = {0.1, 0.5} = f (b)+ b (B b) Two regimes: T! = b m f T! = b m f b eq B several metastable states Oscillation period in 3D: = b f + b

19 Quantum oscillations of spinons Magnetization oscillations of spinons vs metals Spinons 2D 3D period (higher T) S F / S F? / period (low T) S F / 0 S F? / Amplitude Metals B 2 B 2 period Amplitude S F S F? p const B = b f + b Inti Sodemann, Debanjan Chowdhury, T. Senthil, arxiv: (2017)

20 Our proposal Composite exciton Fermi liquid (proposed phase for SmB6 and mixed valence insulators): Fractionalized phase with gapped charge degrees of freedom (insulator) and a neutral fermi surface (fermionic composite excitons) that displays quantum oscillations. A new mechanism within periodic Anderson model is needed because we don t have a half filled band, as in the case of spinon fermi surface.

21 Introduction to SmB6 Simple cubic structure. All action happens in Samarium. Traditional picture of mixed valence insulator: [Xe]4f 6 5d 0 6s 2 Atomic limit (k) d Mixed valence (k) 5d 1 +4f 5 4f 6 F f F

22 SmB6 puzzling behavior Insulating behavior from charge transport: 0 e T 10meV Surface is metallic (proposed to be topological). M. Dzero et al. Ann. Rev. CMP (2016) B. S. Tan et al., Science (2015).

23 SmB 6 magnetic oscillations De Haas-van Alphen effect M(a.u.) visible at B 5T Surface vs bulk picture of dhva effect: G. Li et al. Science 346, 1208 (2014). B. S. Tan et al. Science 349, 287 (2015). J. D. Denlinger et al., arxiv: (2016). B. S. Tan et al., Science (2015).

24 SmB6 puzzles Could be magnetic breakdown? (k) Zhang, Song, Wang, PRL (2016). Knolle and Cooper, PRL (2015). Gap: Cyclotron: 10meV! c 0.2meV B[T ] F Theory oscillations visible at B 50T Experiment oscillations visible at B 5T Other anomalies: Specific heat to temperature ratio has finite intercept: = C T Like in a fermi sea C fermions / T C phonon / T 3

25 Composite exciton Fermi liquid Atomic limit (k) d (k) d electron F f f part-hole conjugation f F hole N d electrons = N f holes f-holes have strong on-site repulsion U ff X U ff!1 i n f i (nf i 1) Hard-core constraint n f i apple 1 Slave bosons: f = b b : spinless boson : neutral spinfull fermion

26 Composite exciton Fermi liquid (k) d electron N d electrons = N b = N F One option: bosons condense hbi 6=0 => Metal ( boring ) Fermi-bose mixture: b : spinless boson : neutral spinfull fermion d : d-electron Slave bosons: f = b

27 Composite exciton Fermi liquid (k) d electron N d electrons = N b = N F One option: bosons condense hbi 6=0 => Metal ( boring ) Fermi-bose mixture: b : spinless boson : neutral spinfull fermion d : d-electron More interesting option: Bosons bind with d electrons b and d attract: U df X i n f i nd i Composite fermionic exciton: Bound state of f-holon" and d electron.

28 Composite exciton Fermi liquid (k) Fermi-bose mixture: b, d b : spinless boson : neutral spinfull fermion charge d : d-electron = b d : fermionic exciton (charge neutral) F charge : ionization energy to un-bind fermionic exciton. charge carrying degrees of freedom gapped: electrical insulator gapless surface of spin-carrying neutral fermions hibridization F semi-metal F

29 Composite exciton Fermi liquid Fractionalized phase with low energy description: Neutral (spinful) fermion: forms fermi surface. Charge 1 (spinless) boson: (gapped). Fermion/boson couple minimally to a gauge field a µ L = X! (p a) 2 i i@ t + µ i a 0 i+ i 2m i + (i@ µ + a µ A µ )b 2 u b 2 g 2 b 4 + Above description implies it is an insulator with a form of de Haas-van Alphen effect. Low energy description is similar to spinon fermi surface although very different in microscopic origin. b

30 Properties of Composite exciton Fermi liquid Fractionalized fermi sea with two pockets ( semi-metal ) Fermionic exciton f-hole spinon Some properties: Essentially linear specific heat: Sub-gap optical conductivity: Upturn might indicate other physics at lower temperature D. Chowdhury, I. Sodemann, T. Senthil, arxiv: (2016). B. S. Tan et al., Science (2015).

31 Properties of Composite exciton Fermi liquid Fractionalized fermi sea with two pockets ( semi-metal ) Some properties: Fermionic exciton f-hole spinon Laurita et al., PRB (2016) Essentially linear specific heat: Sub-gap optical conductivity: Disordered: Clean: Re[ (!)]! 2.33 D. Chowdhury, I. Sodemann, T. Senthil, arxiv: (2016). 1THz

32 Properties of Composite exciton Fermi liquid Linear T heat conductivity: Linear T transverse heat conductivity: Measured heat conductivity: apple i xy =(! c,i i )apple i xx! c,i = e ~ b /m i = e ~ B /m i Y. Xu et al., PRL (2016) Boulanger et al, arxiv: Useful to extend measurements to higher T! D. Chowdhury, I. Sodemann, T. Senthil, arxiv: (2016). B. S. Tan et al., Science (2015).

33 Summary Strong correlations in mixed valence insulators, such as SmB 6, can give rise to state with gapped charge degrees of freedom but with a surface of gapless neutral fermions. Neutral fermion is a superposition of fermionic exciton (bound state of d electron and f holon) with f spinon. Insulators with neutral fermi surfaces coupled minimally to internal gauge field give rise to oscillations reminiscent of de Haas-van Alphen effect. D. Chowdhury, I. Sodemann, T. Senthil, arxiv: (2017). Inti Sodemann, Debanjan Chowdhury, T. Senthil, arxiv: (2017).