%0 Book Section %1 statphys23_0323 %A Santos, E. %A Theumann, A. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K breaking fermionic glass quantum replica representation spin statphys23 symmetry topic-8 %T Quantum M-component spin glass with positive entropy %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=323 %X We consider the quantum version of the M-component spin glass model presented some years ago by De Dominicis and Garel1, that in the limit M$ $ presents a physically acceptable, positive entropy at low temperatures. The original classical model consists of a system of M-components vector spin variables that point to the $ 2^M $ vertices of a M-dimensional hypercube, $S_i^= 1$; $ i= 1...N $;$ \mu= 1...M $ on each of the N sites of a lattice.These spins are coupled through a Sherrington- Kirkpatrick(SK)2 random interaction $J_ij $ with a gaussian distribution of zero mean and variance $<J_ij^2> = J^2/16N $. The free energy is calculated by means of the replica method and two order parameters are considered: one is the overlap among different replicas for arbitrary spin components $q= <S_\mu\alphaS_\mu\beta> $, $\alpha\beta$, and the other is the quadrupolar order parameter $r= <S_\mu\alphaS_\nu\alpha>$, where $\alpha, \beta=1..n $ are replica indices. In ref.1 the definition of $q,r $ are exchanged, but the physics is unchanged. In the limit $Mınfty$ the $ r-q $ vanishes but the replica symmetry breaking parameter $\delta=M(r-q) $ is finite and can be determined from the saddle point equations. A new solution emerges with the $ > 0 $ that has a lower free energy3. The most striking result of this model is a positive entropy at low temperatures that vanishes when $ T=0 $, although the solution continues being unstable. In this communication we consider the quantum version of the model by introducing the coupling to a transverse field $\Gamma $. We use the fermionic representation of spin operators4,5 where they are expressed as bilinear combinations of fermion fields. One problem with this representation is the presence of two spurious states(4S-model) at every site with $ S_z = 0$, that can be supressed by means of an integral constraint that fixes the occupation number $ n_i\uparrow + n_i\downarrow =1 $ (2S-model). We discuss results within the framework of the static ansatz, that describes the singularities of the zero frquency mode and the onset of the phase transition.Like in the classical original model we consider the order parameter $q $ that measures the replicas overlap and the quadrupolar order parameter $ r $ that breaks the rotational symmetry among different spin components. In the limit $ M $, $ r-q 0 $ but $ \delta= M(r-q) $ remains finite, as in the classical model. In this limit the entropy remains positive at low temperatures and it vanishes at $ T=0 $. We show results for the transverse field $ \Gamma= 0.5 J $ .The phase diagram coincides with the one obtained for the quantum Ising model4 with $ M= 1 $, what is expected because at $ T_c $ the quadrupolar overlap vanishes and the critical temperature is independent of the value of $ M $. References:\\ 1)C.De Dominicis and T.Garel, J. de Physique-Lettres,vol.22 p.L-576(1979)\\ 2)D Sherrington and S Kirkpatrick, Phys.Rev.Lett vol.35 p.1792(1975)\\ 3)A.Blandin, M.Gabay and T.Garel,J.Phys.C:Solid St.Phys vol.13 p.403(1980)\\ 4)A.Theumann,A.A. Schmidt, and S.G.Magalhaes, Physica A vol.311 p.498(2002)\\ 5)R.Oppermann and A.Muller Groeling,Nucl.Phys.B vol.401 pag.507(1993)\\ We acknowledge financial support from the Brazilian agencies CNPq and FAPERGS.

@incollection{statphys23_0323, abstract = {We consider the quantum version of the M-component spin glass model presented some years ago by De Dominicis and Garel[1], that in the limit M$ \rightarrow \infty $ presents a physically acceptable, positive entropy at low temperatures. The original classical model consists of a system of M-components vector spin variables that point to the $ 2^M $ vertices of a M-dimensional hypercube, $S_i^\mu = \pm 1$; $ i= 1...N $;$ \mu= 1...M $ on each of the N sites of a lattice.These spins are coupled through a Sherrington- Kirkpatrick(SK)[2] random interaction $J_{ij} $ with a gaussian distribution of zero mean and variance $<J_{ij}^2> = J^2/16N $. The free energy is calculated by means of the replica method and two order parameters are considered: one is the overlap among different replicas for arbitrary spin components $q= <S_{\mu\alpha}S_{\mu\beta}> $, $\alpha\ne \beta$, and the other is the quadrupolar order parameter $r= <S_{\mu\alpha}S_{\nu\alpha}>$, where $\alpha, \beta=1..n $ are replica indices. In ref.[1] the definition of $q,r $ are exchanged, but the physics is unchanged. In the limit $M\rightarrow \infty$ the $ r-q $ vanishes but the replica symmetry breaking parameter $\delta=M(r-q) $ is finite and can be determined from the saddle point equations. A new solution emerges with the $ \delta > 0 $ that has a lower free energy[3]. The most striking result of this model is a positive entropy at low temperatures that vanishes when $ T=0 $, although the solution continues being unstable. In this communication we consider the quantum version of the model by introducing the coupling to a transverse field $\Gamma $. We use the fermionic representation of spin operators[4,5] where they are expressed as bilinear combinations of fermion fields. One problem with this representation is the presence of two spurious states(4S-model) at every site with $ S_z = 0$, that can be supressed by means of an integral constraint that fixes the occupation number $ n_{i\uparrow} + n_{i\downarrow} =1 $ (2S-model). We discuss results within the framework of the static ansatz, that describes the singularities of the zero frquency mode and the onset of the phase transition.Like in the classical original model we consider the order parameter $q $ that measures the replicas overlap and the quadrupolar order parameter $ r $ that breaks the rotational symmetry among different spin components. In the limit $ M \rightarrow \infty $, $ r-q \rightarrow 0 $ but $ \delta= M(r-q) $ remains finite, as in the classical model. In this limit the entropy remains positive at low temperatures and it vanishes at $ T=0 $. We show results for the transverse field $ \Gamma= 0.5 J $ .The phase diagram coincides with the one obtained for the quantum Ising model[4] with $ M= 1 $, what is expected because at $ T_c $ the quadrupolar overlap vanishes and the critical temperature is independent of the value of $ M $. References:\\ 1)C.De Dominicis and T.Garel, J. de Physique-Lettres,vol.22 p.L-576(1979)\\ 2)D Sherrington and S Kirkpatrick, Phys.Rev.Lett vol.35 p.1792(1975)\\ 3)A.Blandin, M.Gabay and T.Garel,J.Phys.C:Solid St.Phys vol.13 p.403(1980)\\ 4)A.Theumann,A.A. Schmidt, and S.G.Magalhaes, Physica A vol.311 p.498(2002)\\ 5)R.Oppermann and A.Muller Groeling,Nucl.Phys.B vol.401 pag.507(1993)\\ We acknowledge financial support from the Brazilian agencies CNPq and FAPERGS.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Santos, E. and Theumann, A.}, biburl = {https://www.bibsonomy.org/bibtex/22322493e824e453dd951826470c77c06/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {bc70594f257359fa5b1d0176f0452bd1}, intrahash = {2322493e824e453dd951826470c77c06}, keywords = {breaking fermionic glass quantum replica representation spin statphys23 symmetry topic-8}, month = {9-13 July}, timestamp = {2007-06-20T10:16:17.000+0200}, title = {Quantum M-component spin glass with positive entropy}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=323}, year = 2007 }