1. Persistent spin current in mesoscopic spin ring Ming-Che Chang Dept of Physics Taiwan Normal Univ Jing-Nuo Wu (NCTU) Min-Fong Yang (Tunghai U.)

2. A brief history persistent current in a metal ring (Hund, Ann. Phys. 1934) related papers on superconducting ring Byers and Yang, PRL 1961 (flux quantization) Bloch, PRL 1968 (AC Josephson effect) persistent current in a metal ring Imry, J. Phys. 1982 diffusive regime (Buttiker, Imry, and Landauer, Phys. Lett. 1983) inelastic scattering (Landauer and Buttiker, PRL 1985) the effect of lead and reservoir (Buttiker, PRB 1985 … etc) the effect of e-e interaction (Ambegaokar and Eckern, PRL 1990) experimental observations (Levy et al, PRL 1990; Chandrasekhar et al, PRL 1991) electron spin and spin current textured magnetic field (Loss, Goldbart, and Balatsky, PRL 1990) spin-orbit coupling (Meir et al, PRL 1989; Aronov et al, PRL 1993 … etc) FM ring (Schutz, Kollar, and Kopietz, PRL 2003) AFM ring (Schutz, Kollar, and Kopietz, PRB 2003) this work: ferrimagnetic ring spin charge

3. Basics of a superconducting ring (Byers and Yang, PRL 1961) the energy levels (and hence the free energy F) are an even periodic function with period  0 =h/2e SC equilibrium state is given by the min of F away from a min, body current the flux inside the SC ring has to be quantized this in turn implies the Meissner effect   E 00

4. Phase coherence in a mesoscopic ring (Buttiker, Imry, and Landauer, Phys. Lett. 1983) Webb et al, PRL 1985

5. Persistent charge current in a normal metal ring I /0/0 1/2-1/2 Smoothed by elastic scattering… etc Persistent current Similar to a periodic system with a large lattice constant R L=2  R Phase coherence length ……

6. Diamagnetic response of an isolated gold ring (Chandrasekhar et al, PRL 1991)

7. Persistent current, Drude weight, and the Meissner fraction for insulators, D=0 for a clean metal (bulk), D≠0 (even at finite T) D≠0 does not imply superconducting behavior (Meissner effect) Refs: W. Kohn, PRB 1964, M. Himmerich’s thesis, Mainz 2004 Meissner fraction Same as the Drude weight when T=0 for a clean metal, ρ s =0 at finite T (no Meissner effect) for mesoscopic normal metal, ρ s ≠ 0 even at finite T (can show Meissner effect) Drude weight (or charge stiffness)

8. Metal ring in a textured B field (Loss et al, PRL 1990, PRB 1992) R After circling once, an electron acquires an AB phase 2πΦ/Φ 0 (from the magnetic flux) a Berry phase ± (1/2)Ω(C) (from the “texture”) B C Ω(C) Electron energy:

9. Persistent charge and spin current (Loss et al, PRL 1990, PRB 1992)

10. Ferromagnetic Heisenberg ring in a non-uniform B field (Schütz, Kollar, and Kopietz, PRL 2003) Large spin limit, using Holstein-Primakoff bosons:

11. Transverse part mimi m i+1 Longitudinal part to order S, Choose the triads such that Then, (rule of “connection”)

12. Anholonomy angle of parallel-transported e 1 = solid angle traced out by m Ω m Local triad and parallel-transported triad Gauge-invariant expression mimi m i+1

13. Hamiltonian for spin wave (NN only, J i.i+1 ≡ － J) Choose a gauge such that Ω spreads out evenly Persistent spin current Magnetization current Schütz, Kollar, and Kopietz, PRL 2003 I m vanishes if T=0 (no zero-point fluctuation!) I m vanishes if N>>1 ka ε(k)

14. Experimental detection (from Kollar’s poster) measure voltage difference ΔV at a distance L above and below the ring magnetic field temperature Estimate: L=100 nm N=100 J=100 K T=50 K B=0.1 T → ΔV=0.2 nV

15. Antiferromagnetic Heisenberg chain (S=1/2) From Broholm’s Cargese lecture Free fermions for XY-model Free fermion excitation Particle-hole excitation Jordan-Wigner transformation

16. Bethe ansatz calculation (Karbach et al, cond-mat/9809162; 0008018) Generation of 2 spinons by a spin flip Dender et al, PRB 1996

17. White and Huse, PRB 1993 AFM spin chain with S=1 (Haldane spin chain) From Zheludev’s poster Low-lying excitation

18. n = number of spins per primitive unit cell S = the spin quantum number m = the magnetization per spin n(S-m) = Oshikawa, Yamanaka, and Affleck (1997) and Oshikawa (2000) gaps in non-magnetized spin chains?  Uniform spin ½ chain 1. ½ = ½ no gap  Alternating spin ½ chain 2. ½ = 1 perhaps  (2n+1) leg spin ½ ladder (2n+1). ½ = n+ ½ no gap  2n leg spin ½ ladder 2n. ½ = nperhaps  Uniform spin 1 chain 1. 1 = 1 perhaps Gapped phases in isotropic spin systems (from Broholm’s talk) LSM theorem: Integer: gap possible Non-Integer: gap impossible

19. Antiferromagnetic Heisenberg ring in a textured B field (Schütz, Kollar, and Kopietz, PRB 2004) half-integer-spin AFM ring has infrared divergence (low energy excitation is spinon, not spin wave) consider only integer-spin AFM ring. need to add staggered field to stabilize the “classical” configuration (modified SW) for a field not too strong Large spin limit v

20. Antiferromagnetic Heisenberg chain (S=1/2 case) With twisted boundary magnetic field Zhuo et al, cond-mat/0501693

21. (S. Yamamoto, PRB 2004) Ferrimagnetic Heisenberg chain, two separate branches of spin wave: Gapless FM excitation well described by linear spin wave analysis Modified spin wave qualitatively good for the gapful excitation

22. Ferrimagnetic Heisenberg ring in a textured B field (Wu, Chang, and Yang, PRB 2005) no infrared divergence, therefore no need to introduce the self-consistent staggered field consider large spin limit, NN coupling only Using HP bosons, plus Bogolioubov transf., one has where

23. Persistent spin current At T=0, the spin current remains non-zero Effective Haldane gap

24. FM limit AFM limit Clear crossover between 2 regions System size, correlation length, and spin current (T=0) no magnon current Magnon current due to zero-point fluctuation

25. Magnetization current assisted by temperature Assisted by quantum fluctuation (similar to AFM spin ring) At low T, thermal energy < field-induced energy gap (activation behavior) At higher T, I max (T) is proportional to T (similar to FM spin ring)

26. Issues on the spin current Charge is conserved, and charge current density operator J is defined through the continuity eq. The form of J is not changed for Hamiltonians with interactions. Spin current is defined in a similar way (if spin is conserved), Even in the Heisenberg model, J s is not unique when there is a non-uniform B field. (Schütz, Kollar, and Kopietz, E.Phys.J. B 2004). Also, spin current operator can be complicated when there are 3- spin interactions (P. Lou, W.C. Wu, and M.C. Chang, Phys. Rev. B 2004). Beware of background (equilibrium) spin current. There is no real transport of magnetization. Spin is not always conserved. Will have more serious problems in spin-orbital coupled systems (such as Rashba system). However,

27. Other open issues: spin ring with smaller spins spin ring with anisotropic coupling diffusive transport leads and reservoir itinerant electrons (Kondo lattice model.. etc) connection with experiments methods of measurement any use for such a ring?

28. Thank You !