Quantum phase transitions in an effective Hamiltonian fast a(4)

An effective Hamiltonian describing interaction between generic "fast" and a "slow" systems is obtained in the strong interaction limit. The result is applied for studying the effect of quantum phase transition as a bifurcation of the ground state of the "

thisimpliesappearanceofaseparatrix,whichleadstothediscontinuityontheenergydensityspectruminthethermodynamiclimit.Itisalsoworthnotingthatthereisalossoftherotationalsymmetryinthisprocess:thenewgroundstateisobviouslynotinvariantunderrotationsaroundaxisx,whiletheinitialgroundstateisclearlyinvariantunderx-rotations.

Itiseasytoseethatthelasttwotermsin(12)areoflowerorderintheparameterAδandcanbeneglectedinthe rstapproximationfordescriptionofQPTatξ=1.

B.

Spin-spininteraction

ingsystemwhenthetotalexcitationnumberisnotpre-served.Analyzingthosee ectiveHamiltoniansinthethermodynamiclimitwehaveobservedabifurcationofthegroundstateleadingtothee ectoftheQuantumPhaseTransitions.

Itisinterestingtonotethat,formultidimensionalsys-tems,whenalgebraicallytheXsystemisadirectsumofseveralnon-interactingsubsystems,aninterestinge ectofgenerationofentangledstates(inthenon-preservingexcitationcase)canbeobserved.Really,letussupposethatin(1)X0,±=X0,±1+X0,±2,[Xj,1,Xj,2]=0,j=0,±;thenthecorrespondinge ectiveHamiltonian(uptoa rstnon-trivialorderinδ)takestheformHeff≈ω1(X0,1+X0,2)+ω2Y0+

gδ[(X+,1+X ,1)2+(X+,2+X ,2)2

+2(X+,1+X ,1)(X+,2+X ,2)] yφy(Y0),

wherewecanclearlyseethatthelasttermcontainstheoperatorproduct～X+,1X+,2which,togetherwithquadratictermsinX±,1(2),impliesaspontaneousgen-erationofentangledstatesofX1andX2startingfromtheminimumenergystate.Thiscanbecorroboratedbytheentanglingpowermeasurebyconsideringauniformdistributionoftheinitialfactorizedstates[12].Thuswecansaythat,intheregimestudiedhere,entanglementcanbegeneratedinabipartitesystemthevicinityofaPhaseTransition.

Asasecondexampleletusconsideradipole-dipolelikeinteraction,thatis,

X0=Sz1,X±=S±1,

Y0=Sz2,Y±=S±2.

Thee ectiveHamiltonianfortheslowspinsystem(afterprojectingontotheloweststateofthefastspinsystemwitheigenvalue A2/2)takestheformsimilarto(10),with

222

Heff=ω 1Sz1 2A2gδSx1+2A2ω1δSz1

4322

+16gA2δ3Sx1+24gδA2{Sx,Sz},

(13)

whereω 1=ω1 2A1ω1δ2 20gδ3A21.The rsttwo

termsaredominantforδ 1anddescribetheLipkin-Meshkovmodel,sothatthecriticalpointisreachedatξ=4A2A2 1=1inthethermodynamicallimit.The1gδ/ω

e ectoftherestofthetermsin(13)isnegligibleinthevicinityofξ=1.

IV.

CONCLUSIONS

Acknowledgments

Wededucethee ectiveHamiltonianofagenericslowquantumsysteminteractingwithanotherfastoscillat-

Oneoftheauthors(I.S.)thanksSTINT(SwedishFoundationforInternationalCooperationinResearchandHigherEducation)forsupport.Thisworkwassup-portedbyGrants:CONACyTNo

1080535.

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