Applications of Gaudin models for condensed matter physics

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metodologie fisiche e chimiche, Università di Catania.

1 Applications of Gaudin models for condensed matter physics Luigi Amico MATIS CNR-INFM & Dipartimento di metodologie fisiche e chimiche, Università di Catania. Materials and Technologies for Information and communication Sciences

2 Outline Application 1: Models with pairing force interaction BCS models for superconductivity 6-Vertex models and their quasiclassical descendants Application 2: Spin-boson models

3 Quantum models for pairing force interactions MOTIVATION: Transport through an isolated metallic (Al, Au, Co...) nanograin. Resolution of the level spacing: analysis of (Ralph et al ; Salinas et al. 1999; interaction in disordered metals. Davidovic et al. 1999; Gueron et al ) -V/2 lead Si3 N 4 Vg gate lead (a) Al particle 10 nm size scale +V/2 CL e N ex CR Very small capacitance Number of electrons iscfixed The BCS order g parameter Vg is vanishing. (b) VL VR Fig. 1. (a) Schematic cross section of the ultrasmall SETs studied by RBT in [10], and (b) the corresponding circuit diagram. Question: How to characterize superconductivity in small systems? scales of the SET, such as those set by the bias voltage (V! 1 mv), Evidence for pairing correlations: Parity dependent the temperature (T! 4.2 K) and the bulk superconducting gap for Al results with a finite ( bulk = 0.18 mev), fluctuations in electron number are strongly supspectroscopic gap for even # of electrons (Anderson 1956; Mastellone, Falci, Fazio 1998; pressed. (2) Discrete eigenstates of the Matveev, Glazman Larkin 2000). conduction electron energy spectrum became resolvable their mean level spacing d ranged from 0.02 to 0.3 mev, which Canonical order parameter affects the thermodynamical is in order-of-magnitude agreement with the free-electron estimate of d = 2π 2!2(spin /(mkf Vol) for the single-particle level spacing. d-values are quantity susceptibility, specific heat...;such Di Lorenzo, et al ) much larger than kb T for the lowest temperatures attained (around T! (Review: Ralph and von Delft 2001) 30mK), but on the order of bulk. However, the number of conduction electrons for grains of this size is still rather large (between 104 and 105 ). Since the two scales EC and d differ by at least an order of magnitude, they

4 The models. Since the shape of the grains is random the eigenvalues of the non interacting Hamiltonian are distributed according to the gaussian orthogonal ensemble. Thereby integrating over such an ensemble: Where The terms depend on the specific distribution of the energy levels: Non uniform couplings. Evidence in the transport experiments at large bias voltage. Glazman JLTP 1998; Kurland, Aleiner, Altshuler PRB 2000

5 Exact solution of Huniv Blocking of levels : The diagonalization of the spin part is trivial: Key observation: The BCS Hamiltonian is related to the rational Gaudin model with twisted boundary condition: Richardson Seee also Sklyanin ABA: Amico, Falci Fazio 2000; Zhou, Links, McKenzie, Gould

6 Correlation functions Illustration of the idea for su(2). Sklyanin LMP 1999.

7 SL(2) loop group of G[sl(2)] Gaudin algebra: Loop Lie algebra: Reprentations:

8 Exact BCS correlation functions Amico Osterloh PRL 2001 Generating function: obtained as a BCH decomposition for the SL(2) loop group associated to (Calculated for the Gaudin model by Sklyanin LMP 1999 ) is an auxiliary space

Characterization of superconductivity in the canonical ensemble 0.22 0.25 %# " $ 0.20 0.15 0.05!0.20!# " $ 0.18!0.10 0.00 1/& 0.10 (!%!cr)" 0.20 0.16 0.14 0.12 0.26 0.28 0.30 0.32 0.34 0.36! FIG. 4.

9 Characterization of superconductivity in the canonical ensemble %# " $ !0.20!# " $ 0.18! /& 0.10 (!%!cr)" ! FIG. 4. Data for the pair mixing parameter Ψ, as a function of α = g/δ and Ω = N = even, which show clearly a phase transition (N = 8 ( ), N = 12 (!), N = 16 ("), N = 20 (")). In the inset it is shown the data collapse. (Numerical approach: Mastellone Falci Fazio 1998; Exact: Amico and Osterloh 2001) 10 (Recent: Faribault, Calabrese, Caux, PRB 2008 with Slavnov formula)

10 Fluctuations around Huniv (Amico, Di Lorenzo, Osterloh NPB 2002) Experimental evidence for non uniform coupling: non equilibrium transport. HS and HC are integrable if arbitrary

11 Exact solution of Eigenvalues: Eigenvectors: Bethe equations: (Amico, Di Lorenzo, Osterloh PRL 2001; NPB 2002; see also Dukelski, Essebag, Schuck PRL 2001; Review by Sierra & coathors RMP 2004 )

12 6-Vertex models G: Boundary twist. Exactly solvable St. Petersbourg school ; BOOK: V.E. Korepin, N.M. Bogoliubov, and A.G. Itzergin 1993

13 Integrable boundary conditions (Sklyanin JPA 1989)

14 Disordered 6-Vertex model: Quasiclassical limit and Gaudin models zj: lattice inhomogeneities Hamiltonians can be obtained through the limit η 0 (quasiclassical limit): Hj: Gaudin Hamiltonians: Sklyanin J. Sov. Math. 1989; Babujian et al ; Hikami JPA 1995)

15 Exact solution Di Lorenzo, Amico, Hikami, Osterloh, Giaquinta NPB 2002 Diag. Boundaries: Eigenvalues: Bethe equations:

16 Hamiltonian Di Lorenzo, Amico, Hikami, Osterloh, Giaquinta NPB 2002 where ξ=0: Exact solution of

17 Application 2 Spin-Boson models

18 Structure of the models Rotating terms Counter-rotating terms Traditionally emploied in: (no number cons) Many modes: Dissipative quantum mechanics (Caldeira-Leggett. Ref. U. Weiss ) Quantum optics (single mode: Jaynes/Tavis-Cummings. Ref. Scully, Zubairy) Less traditionally: semiconducting heterostructure(zutic, Fabian, das Sarma 2004) nanocircuits (a lot of work by: G. Falci & coworkers )

19 Example: Spin-orbit coupling in semiconducting heterostructures x Bulk-IA: FM y SC z FM Dresselhaus, 1955 Rashba, 1960 r r r r 1 r2 H= π + gzˆ π S + hs Ω 2m ( In the Landau gauge: Ay=Bx )

20 Rotating Vs Counter-rotating terms Energy shifts due to Rot. or CR terms in perturbation theory: S + a + h.c.: E R = E 0 + g 2 Ω ω CR S + a + + h.c.: E CR = E 0 + h h S + a + + h.c.: E CR 2 2 = E 0 + Ω + ω The counter-rotating terms important if: Ω + ω the corresponding coupling constant h is not small; the frequency of the bosonic fields cannot be adjusted to a resonance. These regimes are going to be the working point for many applications; the dynamics is very complicated and new physics might emerge. It is easy to handle with models with only rotating OR counter rotat. terms. The problem to deal with the terms at the SAME time is unsolved. Ω R ω

21 Integrable spin-boson models Basic idea: limit of infinite representation on a given partition of the lattice: Λ=ΛS+ΛB. I. II. spin su(2) boson H4 H. Yan, Phys. Lett. B 262, 4 (1991) I: Profumo, Ragnisco, Amico 2009

22 State of the art Single bosonic mode interacting with a single spin 2-sites Gaudin model from twisted 6-vertex Jurco 1989; Bogolubov, Bullough, Timonen 1996 Rybin, Kastelewicz, Timonen, Bogolubov 1998; Amico and Hikami sites Gaudin from double row XXX 6-vertex Amico, Frahm, Osterloh, Ribeiro NPB 2007; Amico, Frham, Osterlo, Wirth Many bosonic modes interacting with many spins XXZ Gaudin model from diagonal twist H without CR terms Somma, Viola, Ortiz, Dukelski NPB 2006 XYZ Gaudin with no twist H with CR but with frozen bosonic fields Kundu XXX Gaudin from double row 6-vertex with off-diag K H with CR but ABA is open problem Hibberd, Links 2007; Profumo, Ragnisco, Amico 2009 XXZ Gaudin from double row with diag K H without CR and with finite range interactions Profumo, Ragnisco, Amico 2009

23 Spin-boson from open XXX Amico, Frahm, Osterloh, Ribeiro NPB 2007 Quasi classical expansion: ABA only if ν+=ν-=0 (K± are both triangular):

24 Functional BA of spin-boson model Amico, Frahm, Osterloh, Wirth 2009 Motivation: beyond K± triangular (spin-boson: rotating+counter rotating) FBA: Operator-valued zeros of B(u) Triangular K± corresponds to polynomial Q.

25 Conclusions Both integrable Hamiltonian models with pairing interactions and integrable spin-boson models, interesting for various applications, descends from Gaudin models

The Cooper Problem. Problem : A pair of electrons with an attractive interaction on top of an inert Fermi sea c c FS,

The Cooper Problem. Problem : A pair of electrons with an attractive interaction on top of an inert Fermi sea c c FS, Jorge Duelsy Brief History Cooper pair and BCS Theory (1956-57) Richardson exact solution (1963). Gaudin magnet (1976). Proof of Integrability. CRS (1997). Recovery of the exact solution in applications