1 / 19

Integrable spin boson models

Integrable spin boson models. Luigi Amico MATIS – INFM & DMFCI Università di Catania. S uperconductivity M esoscopics T heory group. Collaboration with: K. Hikami (Tokyo) A. Osterloh H. Frahm . Ma terials and T echnologies

galiena
Download Presentation

Integrable spin boson models

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Integrable spin boson models Luigi AmicoMATIS – INFM & DMFCI Università di Catania Superconductivity Mesoscopics Theory group Collaboration with: K. Hikami (Tokyo) A. Osterloh H. Frahm Materials and Technologies for Information and communication Sciences (Hannover)

  2. OUTLINE • The models & their physical origins. • Rotating wave approx.: integrable models of the Tavis-Cummings type. • Integrable models beyond the rotating wave approximation. • Conclusions.

  3. Spin-orbit coupling in semiconducting heterostructures x Bulk-IA: Dresselhaus, 1955 FM SC FM z Rashba, 1960 y In the Landau gauge: Ay=Bx Zutic, Fabian, das Sarma 2004;Shliemann, Egues, Loss 2003

  4. Superconducting nanocircuits Chiorescu et al. 2004 Two SQUID’s The two states are given from the clockwise-anticlockwise currents of the secondary. (Nanocircuits for quantum computation: Maklhlin, Schoen, Shnirman 2001; Murali et. al. 2002; Paternostro et al. 2003). Amico, Hikami 2005

  5. Structure of the models “Rotating terms” “Counter-rotating terms” (no number cons) Traditionally emploied in: Dissipative quantum mechanics (Caldeira-Leggett. Ref. U. Weiss ) Quantum optics (single mode: Jaynes/Tavis-Cummings. Ref. Scully, Zubairy) Less traditionally: semiconducting heterostructure nanocircuits (a lot of work by: G. Falci & coworkers 1993-2005) (Zutic, Fabian, das Sarma 2004)

  6. Rotating Vs Counter-rotating terms Energy shifts due to Rot. or CR terms in perturbation theory: • the corresponding coupling constant h is not small; • the frequency of the bosonic fields cannot be adjusted to a “resonance ”. CR R W w The counter-rotating terms important if: 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. These regimes are going to be the working point for many applications; the dynamics is very complicated and “new” physics might emerge.

  7. Simple example: Tavis-Cummings Tavis-Cummings is solved exactly (T-C 1969; Hepp-Lieb 1973). Constants of the motion: Tavis-Cummings with Counter-Rotating terms: Is not solvable. How to insert CR terms to keep the exact solvability?

  8. Integrability: QIS method Existence of a pair of matrices R(l), T(l) satisfying the Yang-Baxter eq. Transfer matrix. t(l) is taken as generating functional for the Hamiltonian. And for the integrals of the motion. Ex.: H=d/dl[log t(l)] l0. St. Petersbourg group 1980; Korepin et al. book 1993

  9. Tavis-Cummings model from the XXX R-matrix R-matrix Monodromy matrix Lax matrices Comment: the tr0 In the auxiliary space. Bogolubov, Bullough, Timonen 1996

  10. Beyond the RWA.I: Boundary Twist to the XXX Tavis Cummings Previous literature: KB=KS=K diagonal: various type of non-linearities (like ) Rybin, Kastelewicz, Timonen, Bogolubov 1998 In the present case K can be general 2X2 C-number matrices and . “Quantum boundary” Important remark: Fixed in such a way the final model is “interesting”. The genereting function for the integrals is the first order coefficient of t(h). K(h) non-diagonal: no “number” conserv.

  11. Tavis-Cummings type + counter rotatingterms Two different bondary twist for the bosonic and spin degrees of freedom: Constants of the motion: Restrictions on the parameters: u and v have the same sign; l, D, x,z free.

  12. The problem with the twisted XXX chains Because of the relation between the coupling constants the CR terms can be rotated away: • Possible application for nanocircuit: an hidden working point is revealed where the interaction is “effectively weak”: General property: Any non singular twist for XXX chain can be put in a diagonal form unitarily (Amico, Hikami 2003; Ribeiro, Martins,2004). The optimal working point is reached by tuning the capacitance to

  13. Exact solution The idea is to obtain the bosonic problem starting from a suitable “auxiliary” spin problem. We exploit that the bosonic algebra can be obtained via a singular limit (contraction) of su(2): (Dyson-Maleev) Then: The “impurity” is With: The auxiliary monodromy matrix represents 2 sites with 2 different representations:

  14. The solution of the auxiliary problem The transfer matrix fulfills the Baxter eq. Where Q(l) are (2j+2s+2)x(2j+2s+2) matrices satisfying and The eigenvalue of ta is The Bethe eqs. are

  15. The solution of the spin-boson problem The bosonic limit: 1) e infty; 2) h 0 in the energy & the BE. Energy: Bethe eqs:

  16. More general coupling constants: XXX with open boundaries. Idea: non diagonal boundary: Hxxx+ a S+ +bS- +c Sz. Sklyanin 1989; De Vega, Ruiz 1993; Goshal, Zamolodchikov 1994. For spin chain, Algebraic BA by:Melo, Ribeiro, Martins 2004 . The eigenvalue of t(l) is obtained by contractions.

  17. Beyond the XXX models: spin-boson from the XYZ R-matrix a,b,c,d parametrized in terms of theta functions: R-matrix Baxter 1972 spin S: Sklyanin, Takebe 1996; Takebe 1992. Lax matrices Sklyanin 1989; De Vega, Ruiz 1993; Inami, Konno 1994.

  18. The XYZ spin-phase model Work by: Felder Varchenko 1996; Gould, Zhang, Zhao 2002; Fan, Hou, Shi 1997

  19. Conclusions • XXX: Counter rotating terms can be included by applying general boundary twist (restriction on the coefficients). • XYZ: spin boson with CR terms can be obtained with a diagonal boundary

More Related