Quantum oscillations in insulators with neutral Fermi surfaces

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

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:1706.00418 (2017) Inti Sodemann, Debanjan Chowdhury, T. Senthil, arxiv:1708.06354 (2017)

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 + B2 2 + 4 M = de db 4 M = B +

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

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

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

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:1708.06354 (2017)

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)

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

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

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)

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

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

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). 4 2 1 Re[ s(!)]

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 = @ f @b + @ b 2 + b (B b) 2 2 + @b =0 M f = M b equal Equilibrium magnetizations

Magnetism of spinon fermi surface Electron Spinon c i = f i b i Chargon (holon) = f (b)+ b (B b)+ f b 2 @ @b = @ f @b + @ b @b =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

Quantum oscillations of spinons = f (b)+ b (B b)+ f b 2 2 f (b) Spinon spectrum is non-perturbative + b (B b) 2 2 + @ @b = @ f @b + @ b @b =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.

Quantum oscillations of spinons f (b) @ @b =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

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:1708.06354 (2017)

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.

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

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).

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:1601.07408 (2016). B. S. Tan et al., Science (2015).

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

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

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

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.

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

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

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:1706.00418 (2016). B. S. Tan et al., Science (2015).

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:1706.00418 (2016). 1THz

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:1709.10456 Useful to extend measurements to higher T! D. Chowdhury, I. Sodemann, T. Senthil, arxiv:1706.00418 (2016). B. S. Tan et al., Science (2015).

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:1706.00418 (2017). Inti Sodemann, Debanjan Chowdhury, T. Senthil, arxiv:1708.06354 (2017).