Institute of Advanced Study, Tsinghua, April 19, 017 Discrete scale invariance and Efimov bound states in Weyl systems with coexistence of electron and hole carriers Haiwen Liu ( 刘海文 ) Beijing Normal University 1
Collaborators Theorists: X. C. Xie ICQM, Peking University Robert Joynt University of Wisconsin-Madison Z. Q. Wang Boston College Hua Jiang Soochow University Experimentalists: Jian Wang, Huichao Wang ICQM, Peking University Junfeng Wang, Liang Li Wuhan National High Magnetic Field Center Jiaqiang Yan, David Mandrus Oak Ridge National Laboratory References: arxiv:1704.00995 Supported by
Outline 1. Experimental results and general background Log-periodic magneto-resistance in ultra-quantum ZrTe5 Brief introduction to discrete scale invariance and Efimov physics. Possible Fermionic Efimov bound states Three-body Schrödinger case with resonant scattering Two-body Weyl Hamiltonian with Coulomb attraction Influence of magnetic field and comparison to experiments 3. Conclusions 3
Log-B periodic magneto-resistance oscillation in ZrTe 5 Why? Log-B periodic MR oscillations from 0.T to 55T. 4
Discrete Scale Invariance in ZrTe 5 beyond the quantum limit Why? Discrete Scale Invariance in ultra-quantum in ZrTe 5 with coexistence 5 of heavy electron (low mobility) and light hole (high mobility).
Discrete scale invariance Fibonacci (Spirals) Fractals Efimov spectrum Mandelbrot set D. Sornette, Physics Reports 97 39 (1998). E. Braaten, H.-W. Hammer, Physics Reports 48 59 (006). 6
Bosonic Efimov trimers (resonant scattering) Vitaly Efimov When bosons interact with infinite scattering length a, 3 bosons always form a geometric series of bound states. ee ππ ss 0.7 with ss 0 1.006 Discrete Scale Invariance 7
Resonant scattering (obtained by Feshbach resonance ) 1. Square potential well. Feshbach resonance E. Braaten, H.-W. Hammer, Physics Reports 48 59 (006). 8
9 E. Braaten, H.-W. Hammer, Physics Reports 48 59 (006). Efimov trimer of identical bosons (near resonance) 1. 1/R^ attraction is a requisite.. Energy spectrum also show Discrete Scale Invariance.
Efimov trimer of identical bosons (experimental realization) Rudolf Grimm, Nature 440, 315 (006); PRL 11, 190401 (014); Reinhard Dörner, Science 348, 551 (015); Matthias Weidemüller, PRL 11, 50404 (014); Cheng Chin, PRL 113, 4040 (014). 10
1. Experimental results and general background Log-periodic magneto-resistance in ultra-quantum ZrTe5 Brief introduction to discrete scale invariance and Efimov physics. Possible Fermionic Efimov bound states Three-body Schrödinger case with resonant scattering Two-body Weyl Hamiltonian with Coulomb attraction Influence of magnetic field and comparison to experiments 11
Efimov attraction from Born-Oppenheimer approximation Trimer wave-function ψψ TT RR, rr = FF ee RR φφ h rr, RR Schrödinger equation for hole wave-function ħ MM h rr φφ h rr, RR = εε h φφ h rr, RR Hole wave-function: φφ h rr, RR = exp kk h rr RR rr RR + exp kk h rr + RR rr + RR Decay wave-number: kk h = 1 ħ MM hεε h Bethe-Peierls boundary condition: rr rrφφ h rr 0 1 aa kk h RR ee kk hrr = RR aa Energy of hole: kk h RR = ΩΩ 0.567 Solution of ΩΩ = ee ΩΩ when scattering length aa ± εε h RR ħ ΩΩ MM h RR 1
Efimov attraction from Born-Oppenheimer approximation Schrödinger equation for two electrons (center of mass): ħ MM ee RR + εε h RR FF ee RR = EE TT FF ee RR Wave-function for two electrons (center of mass): FF ee RR = RR 1 ff RR YY LLLL θθ Differential equation for radical part: Efimov attraction: VV RR = dd ddrr + VV RR MM eeee TT ħ ff RR = 0 LL LL + 1 RR MM eeωω MM h RR = ss 0 1 4 RR Definition: (real ss 0 guarantee a solution) ss 0 = MM ee MM h ΩΩ LL LL + 1 1 4 Identical fermions: LL = 1 real ss 0 contraint: MM ee MM h > 13.6 This constraint can be matched in solid state systems. 13
Efimov Trimers from the Born-Oppenheimer approximation Differential equation for radical part (with real positive ss 0 ): dd ddrr ss 0 + 1 4 RR MM eeee TT ħ ff RR = 0 Definitions and differential equation for auxiliary function: EE TT = ħ kk TT ff RR = RR 1 gg RR RR = kk MM TT RR ee gg RR + 1 RR gg RR + 1 + ss 0 RR gg RR = 0 [Landau & Lifshitz QM] E=EE TT Solution: ff RR = RR 1 KK iiss0 kk TT RR E 0 Solution: ff RR RR 1 sin ss 0 ln kk TT RR + αα ff 1 RR AARR 1 sin ss 0 lnkk RR + αα Match the wave-function at boundary. Continuous condition of boundary. RR ff RR ff RR at Discrete Scale Invariance kk TT,nn = ee ππ ss 0 nn nn0 ee αα ss 0kk EE BB,nn = ee ππ ss nn nn 0 ħ kk 0 MM ee RR nn kk TT,nn RR nn+1 RR nn = ee ππ ss 0 14
Influence of magnetic field on Efimov trimers: small B HH = 1 MM h iii 1 ee cc AA 1 + 1 MM ee iii + ee cc AA + 1 MM ee iii 3 + ee cc AA 3 Under small magnetic field, the binding energy of Efimov bound states EE TT is much larger than the Landau level spacing. EE TT,nn = ħ kk TT,nn MM ee ħωω 0 = ħ MM ee ll RR nn = ss 0 kk 1 TT,nn ss 0 ll BB BB Size of Efimov trimers The magnetic field can be treated as perturbation, and the system still possesses approximate hyper-spherical symmetry. In hyper-spherical coordinates: harmonic oscillator potential hyper-spherical operators: rriiii ρρiiii,kk +RR 5 + MM ee ωω BB RR 8ħ rriiii ρρiiii,kk + MM eeωω BB ħ = RR + 15 4RR RR5 + LL zz iiii,kk LL iiii ħ RR sin + θθ kk 1 RR sin θθ kk MM eeee TT ħ Ψ = 0 LL iiii,kk ħ RR cos θθ kk θθ kk 4 sin θθ kk 15
Influence of magnetic field on Efimov trimers: small B The Faddeev decomposition: Ψ = φφ 1 rr 3, ρρ 3,1 + φφ rr 13, ρρ 13, φφ rr 1, ρρ 1,3 The Faddeev equation: Solution ansatz: rr ρρ + MM ee ωω BB RR φφ ii rr, ρρ = φφ 0 8ħ + MM eeωω BB ħ ii (RR,θθ, ρρ) = 1 RR 5/ sin θθ The hyper-angle equation (ll = 1): ll ll+1 θθ + cos θθ Bethe-Peierls boundary condition: 1 zz LL iiii,kk RR 5/ sin θθ nn,ll,mm ff nn,ll ii φφ ii nn,ll θθ; RR = ss ii nn,ll φφ nn,ll rrψ MM eeee TT ħ φφ ii rr, ρρ = 0 ii RR φφ nn,ll θθ; RR rψ rr 0 1 aa θθ; RR YY llll ( ρρ) sin γγ = MM e MM h + MM ee Eigenvalues: 1 ss nn ss nn tan ss nn ππ cos(γγss nn) sin γγ cos(ss nn ππ + sin(γγss nn)/ss nn ) sin ππ γγ cos(ss nn ) = 0 The above secular equation have imaginary solution ss nn = iiss 0 if γγ > 1.199, corresponding to MM 16 ee MM h > 13.6.
Influence of magnetic field: approximate discrete scale invariance The hyper-radial equation: ss 0 +1/4 ddrr dd RR + RR 4ll BB 4 + 1 ll BB Influence of magnetic field ff nn RR = MM EEEE TT ħ ff nn RR Attractive potential condition: ss 0 + 1/4 RR + RR 4ll BB 4 + 1 ll BB < 0 WKB solution of Efimov trimers: RR < RR cc ss 0 ll BB RR nn ss 0 pppppp ħ RR 0 RR 1 RR ss 0 ll BB dddd = ππππħ Approximate discrete scale invariance for RR nn < RR cc : ln RR nn RR nn 1 = ππ ss 0 + RR nn 4ss 0 ll RR nn 1 BBnn 4ss ππ 1, ππ + 1 0 ll BBnn 1 ss 0 ss 0 ss 0 ss 0 ln BB nn+1 BB nn ππ ss 0 1 ss 0, ππ ss 0 + 1 ss 0 17
Problem, comparison and solution from 3-body to -body 1. How to obtain the resonant scattering condition aa ± with large magnetic field? Not clear. Thus, the prerequisite for Efimov trimers is not met.. In the large magnetic field limit, the system become quasi-1d with cylindrical symmetry. No Efimov trimers in 1D. 3. ZrTe5 is identified as Dirac semimetal by ARPES [Nature Physics 1, 550 (016)], what s the influence of Dirac physics? Efimov equation: dd ddrr ss 0 + 1 4 RR MM eeee TT ħ ff RR = 0 Scale Invariant Dirac equation: mm ħvv FF σσ kk ħvv FF σσ kk mm ΨΨ = EE VV RR ΨΨ Scale invariance is obtained, if mm = 0 and VV RR RR 1. Weyl semimetal with Coulomb attraction. 18
Dirac particle with Coulomb attraction Dirac-Kepler problem at large Z, atomic collapse 3D W. Greiner, B. Muller & J. Rafelski, QED of Strong Fields (Springer, 1985). Y. B. Zeldovich and V. S. Popov, Usp. Fiz. Nauk 105, 403 (1971). Subcritical: ZZee 4ππεε 0 ħcc < 1 Ze is central charge. Supercritical: ZZee 4ππεε 0 ħcc > 1 atomic collapse for Z > 137. Atomic collapse in graphene D A. V. Shytov, M. I. Katsnelson, and L. S. Levitov, PRL 99, 36801 (007). A. V. Shytov, M. I. Katsnelson, and L. S. Levitov, PRL 99, 4680 (007). V. M. Pereira, J. Nilsson, and A. H. Castro Neto, PRL 99, 16680 (007). Quasi-bound states in the supercritical regime of graphene. 19
3D Weyl particle + Coulomb attraction + magnetic field 0 ħvv FF σσ kk + ee ħcc σσ AA ħvv FF σσ kk + ee ħcc σσ AA 0 ΨΨ 1 ΨΨ = EE VV(RR) ΨΨ 1 ΨΨ Under small magnetic field, the Coulomb attraction EE TT larger than the Landau level spacing. VV RR nn is much = ZZααħvv FF RR nn EE BB = ħvv FF ll BB RR nn ZZααll BB αα = ee 4ππεε 0 ħvv FF Expanding in spinor spherical harmonic function: Radial equation: dd dddd uu 1 RR uu RR = λλ + RR RR ll BB EE + ZZZZ ħvv FF RR ΨΨ 1 RR ΨΨ RR = YY λλ 1,mm θθ, φφ uu λλ 1 1 RR /RR λλ 1 iiii,mm λλ θθ, φφ uu RR /RR EE + ZZZZ ħvv FF RR λλ RR + RR ll BB uu 1 RR uu RR 0
Approximate discrete scale invariance Radial equation (i=1,): dd ddrr uu ii RR = λλ RR + RR ll BB uu ii RR EE ħvv FF + ZZZZ RR uu ii RR Magnetic field B=0: pp RR ħss 0 RR ss 0 (ZZZZ) λλ WKB solution: RR nnpprr ddr = nnnnħ R 0 RR nn /RR nn 1 = ee ππ/ss 0 EE nn = ħvv FFZZZZ RR nn = ħvv FFZZZZ RR 0 ee nnnn/ss 0 With small B: RR < RR cc ss 0 ll BB pp RR ħ RR (ZZZZ) λλ + RR WKB analysis gives the approximate discrete scale invariance: ll BB Bound state dissolves if RR nn > ss 0 ll BB. ln RR nn RR nn 1 ππ ss 0 1 ss 0, ππ ss 0 + 1 ss 0 ln BB nn+1 BB nn ππ ss 0 1 ss 0, ππ ss 0 + 1 ss 0 1
Weyl particle + Coulomb attraction + magnetic field 0 ħvv FF σσ kk + ee ħcc σσ AA ħvv FF σσ kk + ee ħcc σσ AA 0 ΨΨ 1 ΨΨ = EE VV(RR) ΨΨ 1 ΨΨ RR nn ZZααll BB Under very large magnetic field with, we consider the lowest landau level solution in cylindrical symmetry ρρ, zz, φφ. ΨΨ 1 = 0 uu ρρ, zz ee iiiiφφ ΨΨ = 0 vv ρρ, zz ee iiiiφφ uu ρρ, zz = ρρ mm ee ρρ 4ll BBgg zz vv ρρ, zz = iiρρ mm ee ρρ 4ll BBff zz EE VV zz ħvv FF gg zz + dd ff zz = 0 dddd EE VV zz ħvv FF ff zz dd gg zz = 0 dddd VV zz = 0 ZZZZħvv FF zz ρρ mm+1 ee ρρ ll BB 1 mm + 1 ll BB zz ZZZZħvv FF dddd ρρ + zz mm! mm mm+ ll BB Scale Invariant if ll BB zz with 1D Coulomb attraction.
Weyl particle + Coulomb attraction + magnetic field ddzz + dd VV zz dddd dd EE VV zz dddd + dd EE VV zz ħvv FF gg zz = 0 dd ddzz 3 4 VV zz EE VV zz VV zz EE VV zz + EE VV zz ħvv FF gg zz EE VV zz = 0 For zz ll BB and EE 0, attractive potential: VV zz = ZZZZħvv FF zz dd ddzz + ZZZZ + 1 4 zz zzgg zz = 0 pp zz ħzzzz zz WKB solution: zz nnppzz ddzz = nnnnħ z 0 zz nn /zz nn 1 = ee ππ/zzzz Discrete scale invariance 3
Approximate discrete scale invariance Attractive potential condition: Large magnetic field limit VV zz nn VV zz = ZZZZħvv FF zz 1 ll BB zz = ZZααħvv FF zz nn < EE BB = ħvv FF ll BB ll BB zz nn < 1 ll BB < zz nn ZZZZ WKB solution: zz nnppzz ddz = nnnnħ z 0 Approximate discrete scale invariance: Bound state forms if zz nn > ZZααll BB. pp zz ħzzzz zz 1 ll BB zz ln zz nn = ππ zz nn 1 ZZZZ + ll BB nn zz nn ll BB nn 1 ππ zz nn 1 ZZZZ 4 ZZZZ, ππ ZZZZ + 4 ZZZZ ln BB nn+1 BB nn ππ ZZZZ 8 ZZZZ, ππ ZZZZ + 8 ZZZZ 4
Comparison to the experimental results vv FF 4.5 10 5 mm/ss for the Dirac bands in ZrTe 5 [Nature Physics 1, 550 (016)] Approximate discrete scale invariance: ZZZZ 4.87 and ss 0 = ZZZZ 1 4.77 Theoretical ratio: Dissolution: Formation: ln BB nn+1 BB nn ππ ss 0 1 ss 0, ππ ss 0 + 1 ss 0 BB nn /BB nn+1 (3.03, 4.6) ln BB nn+1 BB nn ππ ZZZZ 8 ZZZZ, ππ ZZZZ + 8 ZZZZ BB nn /BB nn+1 (.59, 5.09) Experimental ratio: BB nn BB nn+1 (.30, 4.00) 5
Experimental results in ZrTe 5 1. Clear demonstration of 5 log-b periodic MR oscillations.. The experimental scaling factor BB nn BB nn+1,30,4.00 is comparable to theoretical estimation. 3. The supercritical Coulomb attraction between hole and electron results in two-body Efimov bound states with discrete scale invariance. 4. Under magnetic field, the formation and dissolution of Efimov bound states give rise to log-b periodic MR oscillations beyond the quantum limit. 6
Conclusion 1. Coexistence of electron and hole carriers in Weyl system with supercritical Coulomb attraction = twobody Efimov quasi-bound states with discrete scale invariance.. The formation and dissolution of two-body Efimov quasi-bound states under magnetic field give rise to novel log-b periodic oscillations of 3D systems beyond the quantum limit, where the Landau level physics is inapplicable. Thank you! 7