Phase diagram of the dissipative quantum particle in a box J. Sabio1, L. Borda2,3, F. Guinea1, and F. Sols.4 1Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, 28049 Madrid. Spain 2Research Group “Theory of Condensed Matter” of the Hungarian Academy of Sciences, TU Budapest, Budapest, H-1521, Hungary 3 Physikalisches Institut, Universität Bonn, Nussallee 12, D-53115 Bonn, Germany and 4Departamento de F´ısica de Materiales. Universidad Complutense de Madrid. 28040 Madrid. Spain. (Dated: January 16, 2009) 9 Weanalyzethephasediagramofaquantumparticleconfinedtoafinitechain,subjecttoadissi- 0 pativeenvironment described by an Ohmic spectral function. Analytical and numerical techniques 0 areemployedtoexploreboththeperturbativeandnon-perturbativeregimeofthemodel. Forsmall 2 dissipation the coupling to the environment leads to a narrowing of the density distribution, and n to a displacement towards the center of the array of accessible sites. For large values of the dissi- a pation, we find a phase transition to a doubly degenerate phase which reflects the formation of an J inhomogeneous effective potential within thearray. 6 1 PACSnumbers: 03.65.Yz;74.78.Na;85.25.-j ] l l I. INTRODUCTION. sition, for a critical value of the coupling to the bath, a to a phase where the particle is localized. This kind of h transition, which belongs to the general class of bound- - Theproblemofaquantumparticleinteractingwithan s aryquantumphasetransitions,22 hasbeenstudiedinthe e environmentdeservesspecialattentionsinceithasimpli- literaturewithmanydifferentapproachesincluding path m cations in fundamental areas as quantum measurement integral, renormalization group and variational Ansatz. theory, quantum dissipation and quantum computation, . t a among others. A quantum particle interacting with an In the present work, we study the phase diagram of a m environmentconsistingofacontinuumofdegreesoffree- particle confined to a finite tight-binding chain coupled dom, the Caldeira-Leggett model,1 is actually the sim- - to an Ohmic dissipative environmentthrough its coordi- d plest model which can be used to study the destruction nate variable. This is the simplest intermediate instance n of quantum coherenceand the emergence of classicalbe- between two of the limiting cases described above, the o havior in the framework of quantum mechanics. c dissipative two level system and a particle in an infinite [ ItisgenerallybelievedthattheCaldeira-Leggettmodel array. It represents a quantum particle interacting with captures the essential features of the behavior of more an ohmic reservoir whose motion is restricted to a fi- 4 complicatedopenquantumsystems. Therefore,itsstudy nite region. As discussed in detail below, the inclusion v 5 asatoymodelcanbejustifiedevenifitsnotconnectedto ofthehardwallboundaryconditionsintroducesinhomo- 2 any particular experimental realization. However, some geneitiesinthedensitydistributionoftheparticle,andit 3 extensions of this model are known to be relevant in yields a non-trivialphase diagram, with a quantum crit- 1 real systems: for instance when analyzing dephasing in ical point which can be characterized in detail by using 1. a qubit2 or in a dissipative Josephson junction.3 An- numerical techniques. 1 otherapplicationrecentlypointedoutisthestudy ofthe 7 decoherence induced in mesoscopic systems by external The paper is organized as follows: First we describe 0 gates.4,5,6 ThiscanbeseenasaconsequenceofCaldeira- the model, and briefly review the main results of related v: Leggett model reproducing the long time dynamics of models obtained in the past. Then, we discuss the main i particles interacting with Ohmic environments.7 techniques employed to analyze the ground state of the X system. The calculated phase diagram is discussed, as Three variations of this model have been particularly ar wellstudied: (i)The dissipativetwolevelsystem,8,9,10,11 wellasthewayvariousobservablequantitiesareaffected bythedissipation. Thelastpartofthepapersummarizes whichisprobablythemostanalyzedmodelinthecontext the main results of the work. of quantum computation, being the archetype of a qubit in the presence of a bath.12 (ii) A particle moving in a periodicpotential,13,14 inamagneticfield,15 andinboth a periodic potential and a magnetic field.16,17 The case ofadissipativeparticleinaperiodicpotentialisrelevant to the study of defects in Luttinger liquids,18 while the case with both a magnetic field and a periodic potential II. THE MODEL. applies to a junction between more than two Luttinger liquids.19 Finally, (iii) the dissipative free particle20,21 of interest in the study of quantum Brownian motion. In As we have noted above,the model we address is that the first two cases the system undergoes a phase tran- ofa particle coupled to a dissipative bath whichcanhop 2 between M sites. The Hamiltonian reads: reflect the two opposing tendencies through a quantum phase transition. H ≡ Hkin+Hbath+Hint+Hct In this work we will analyze the regime t≪ωc, where M weexpectthatthelow-energypropertieswilldependonly Hkin ≡ −t X c†mcm+1+h.c. onthedimensionlessparameterst/ωc andα. Noticethat m=1 in the continuum limit, M >> 1, where the couplings Hbath ≡ X kb†kbk sdaetsicsrfiybeαd 1, generating next-to-nearest neighbor hop- 1 2 3 4 5 t0.03 m α=1.2, t= 0.01 pings and beyond in the low energy theory. However, 0.25 by normal ordering this term can be shown to be irrel- 0.02 0.2 evant close to the transition. Also, a renormalization ρm 0.15 M of the potential can be expected, v c c , with Pm=i m †m m 0.01 0.1 v =v in order to preserve parity symmetry. And fi- 1 2 3 4 5 m m m nally, a−renormalizationof λ whichwe do not show here. 0 0 0.5 1 1.5 2 The complexity of the problem justifies the application α of numerical techniques such as the numerical renormalization group, which deal with the whole effective low- energy Hamiltonian. FIG. 1: Phase diagram predicted by the variational calculation,forachainof5sites. Themodelshowsaphasetransition at the critical coupling αc = 1, where the parity symmetry B. Variational Ansatz. is broken. For α < 1 there is a delocalized phase, with tren finite and an effective quadratic potential dependent on the couplingstrength,thatisresponsiblefortheincreasinglocal- Somedeeperinsightintothephysicsoftheproblemcan ization of the particle at the central sites. For α > 1 there be obtained by using a generalization of the variational is a localized phase, with tren = 0, and a M-fold degenerate Ansatz proposed by Silbey and Harris:27 ground state is predicted. Notice that, in the variational ap- proach, a similar phase diagram is predicted for a chain of 6 |Gi=eqPk fkk(b†k−bk)|0i⊗Xcm|mi (5) sites. m For the application to our problem we have enlargedthe as t is nonzero,but a certain degree of localizationat ren original set of variational parameters, fk, in order to in- the centerarisesfromthe existenceofthe potentialterm cWelnueidtrhegoytuhoteflotohsnse-osaifbtgeoevanemeprparloliitptuyo,dsteehdseoglfraottuthenerdawsrateavtteeakfiuesn:ncrteioanl.,Tcmhe. bagp.epIgrno=athc−hee2lshitmqtr2heiinetGαchtqrrP2ietiniGcma≪lcmva+ωl1cuc,em,th.thisAefslaotcchtaoelriczcoaautnipolnbineogfsshtthorewennpgattrho- E =2t c c + q2 g (6) ticle in the central site becomes sharper, as can be seen G renX m+1 m h iG inFig. 5 for a chainof 5sites. In this figure the position m mean square of the particle is plotted as a function of wherewehavedefinedtren ≡te−21Pk fkk22 andg ≡Pk(λ+ tishecocmoupplelitneglystlorecnaglitzhe.dAint tthhee ccreintticear,l paosinatctohnesepqauretnicclee fk )2. The minimum conditionimposes the followingset of the renormalization to zero of the hopping parameter √k of equations: and the appearance of the confining potential. Forα>1theparticlegetslocalized,witht =0,and t (c +c )+(g(m m )2 E )c =0 (7) ren ren m+1 m 1 0 G m an M-fold degenerate ground state, in which the parity − − − t fk c c q2 ( λ + fk)=0 (8) symmetry is broken, is predicted. This is, actually, the renk2 X m+1 m−h iG √k k result of solving exactly the Hamiltonian (3) in the lim- m iting case t=0, as discussed in Section I. The transition Thefirstsetofequationscanbeseenasthoseofaparticle linepredictedbythisapproachisvertical,nothavingany in a chain with renormalized hopping t and moving dependenceonthehoppingparameter. Aphasediagram ren in a symmetric parabolic potential v = g(m m )2. containing this features is shown in Fig. 1. m 0 − Thesecondequationisaself-consistentconditionforthe The variational calculation shows that despite the parameters fk, once we have determined the lowest EG Caldeira - Leggett model includes a counter-term to en- and the corresponding set of cm from (7). sure homogeneous dissipation, the boundary conditions are responsible of some non homogeneous effects such as thecreationofaneffectiveparabolicpotentialwhichcon- Results finestheparticleatthecenter. However,aswewillseein the next section, the numerical results suggest that the Thesolutionoftheseequationsgivesaphasetransition situation close to the critical point is more complicated, forthecriticalcouplingα =1,asexpectedfromthepre- withnewinhomogeneoustermsplayinganimportantrole c ceding section. For α < 1 the particle is not localized, in the solution. 4 C. Numerical Renormalization Group. a In order to include higher correlation effects it is nec- (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) essary to go beyond the variational solution. A power- (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) ... ful method to study quantum impurity problems is Wil- a a a a 0 1 2 3 son’snumericalrenormalizationgroup(NRG).28,29 Orig- inally conceived to deal with fermionic environments, it has been recently adapted to handle bosonic baths.30,31 Bosonicenvironmentscanalsobeanalyzedusingthewell t t t t t known correspondence between bosons and fermions in b one dimension,32,33,34,35 used e.g. in the study of dissipative gates.36,37 However, the fermionic model covers onlyapartofthebosonicone,asthedissipationstrength is limited to the range 0 < α < 1. Exactly because of this fact, we use the bosonic version of the problem. Whenworkingwithbosonicdegreesoffreedom,weuse a ... thesocalledstar-NRG31method,schematicallyshownin 0 Fig. 2. As opposed to the conventional “chain” NRG, this version allows us to deal with the counter-term in the Hamiltonian. This issue arises due to the fact that the counter-term, which only involve particle operators, is of order of the cutoff. In the chain NRG, the Hamil- FIG.2: Sketchofthestar-NRGHamiltonianusedinthework. tonian is transformed in such a way that all the particle a) Every bosonic site an couples to the particle. In each it- dependence is included in the first iteration of the algo- eration a new site is added and the resulting Hamiltonian rithm. The inclusion of the counter-term here requires is diagonalized, giving the energy spectrum. b) The particle Hamiltonian,coupledtothefirstbosonicsite. Noticethatthe muchgreaterprecisionin the calculations,asopposedto structure of the couplings is the same for the rest of bosons. the star-NRG, where is included iteratively. This is no Seethe text for more details on thecalculations. longer the case in absence of the counter-term,as shown in Ref.38. There the authors used the chain NRG to study dissipative exciton transfer. The star-NRGmethod is based on the introduction of the basisofeigenstatesofparity,cm,p = √12(cm+pc−m). a new basis of bosonic states for which the couplings to For M odd the central site is always guaranteed to be the particle decrease exponentially as Λ n, where Λ is a a parity eigenstate. The states used to diagonalize the − scaling factor between 2 and 3, and n labels the bosonic NRG-Hamiltonianatzeroiterationarethen m,p;n ;P , b sites. Then, the Hamiltonian is diagonalized iteratively, with total parity P = p( 1)nb being a go|od quantumi − adding a new bosonic site in every step. number. The totalHamiltonianthen splits into two sep- The basis is truncated with both an upper cutoff in arated sectors of well defined total parity, reducing by energies,which is progressivelyreduced in eachiteration twothesizeofthematricestobediagonalized. Thesame with the scaling factor Λ, and an upper cutoff in the can be shown to be true at iteration N +1, where the occupation number of the bosonic sites. The first one totalparitystates r,p;n ;P areconstructedadding b N+1 | i is chosen to have a number of kept states in each iter- a new bosonic site to the eigenstates at iterationN with ation of N = 100 120, while for the second we let parity p. The matrix elements of the NRG-Hamiltonian s − N = 30 40 bosons per site. We check that the proce- verify, in this basis: b − duregiveswellconvergedresults,regardlessofthechosen truncation parameters. We exploitthe paritysymmetry ofthe Hamiltonianin hr′,p′;n′b;P′|HN+1|r,p;nb;Pi∝δP,P′ the code,bothforreducingthe sizeofthe matricestobe diagonalizedand for minimizing numericalerrors. Let Π The output of the NRG procedure are the flows of the be the parity operator,under which the operators of the lowest lying energy states as the cutoff is reduced itera- Hamiltonian transform as: tively. At some point the flows are expected to converge to stable (low energy) fixed points. The effective Hamil- Πc†mΠ† =c m tonian can be reconstructed analyzing the evolution of − those flows, as well as the evolution of other observables Πb†kΠ† =−b†k ofthe system. Herewewillusethe evolutionoftheaver- ΠqΠ = q (9) † agedpositionofthe particle, q , andits meansquared − N deviation q2 , evaluated inh tihe ground state. Those N Bosonic states have a well defined parity, Π(b†k)nb|0i = flows are ehnoiugh to characterize the different phases of (−1)nb(b†k)nb|0i. The particle states must be rotated to the system. 5 3.5 2.5 1.5 1 2.53 2N000...468 22N0.51 0.2 0 0 5 10 15 20 25 0 0 5 1015202530 1.5 N ΝΛΕΝ 2 N ΝΛΕΝ 1.5 1 1 0.01 0.5 0.0075 0.005 0.5 (a) (a) 0.0025 10 15 20 25 00 5 10 15 20 25 30 35 40 00 10 20 30 40 N N 1.6 2.5 2.5 1.4 >N 2 N 1 22 0.8 1 < 0.6 0 5 10 15 20 25 1 0.4 1.5 N Ν 0 10 20 ΕΝ ΝΕ 0.8 N ΝΛ Λ 1 0.6 0.02 0.015 0.4 0.01 0.5 0.004 0.005 0.002 0.2 (b) 00 5 10 15 20 0 (b) 00 5 10 15 20 0 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 N N FIG. 4: Flows of the NRG for M = 6 sites. a) Flows in FIG. 3: Representative flows of the Numerical Renormal- the first region of the localized phase (α = 1.2, t = 0.01), ization Group transformations carried out in this work, for where theparticle is confined at thecentral sites. b) Flow in M =5sites. Thehorizontalaxisistheiterationnumber,and the second region of the localized phase (α = 1.4, t = 0.01). the graphs are scaled energy levels. a) Flow towards a non Here the particle is confined in the next to the center sites, degenerate ground state (α=0.8, t=0.01) The inset shows suggesting the formation of a double well effective potential the flow of the mean squared position of the particle, which inthearray. Inbothcasesthelowerinsetsshowindetailthe becomes localized around the center. b) Flow towards a de- flow of the lowest energy levels. Notice that, as happened in generate ground state (α = 1.2, t = 0.01). The lower inset thechainoffivesites,twoenergyscalesapparentlydelimitthe givesdetailsofthewayinwhichthelowest energylevelsflow onsetofanunstablefixedpoint,correspondingtoaneffective tothefixedpoint. Thetopinsetshowsthelocalization ofthe chain of M =4 sites. This is actually well conformed by the particle beyond the center, as indicated by its mean squared valueof themean squared position of the particle (inset) position. Notice that, in this regime, two energy scales are playingapartintheflow,delimitatinganintermediateregime which is dominated byan unstable fixedpoint. tive two level system. In both cases there is a phase transition in which t = 0, but for M = 3 the parity ren Results symmetry is notbroken,because the particle is localized at the center. For M = 4 the phase transition is that of the dissipative two level system, the edge sites being Inthis paper weanalyze chainswith a numberof sites decoupled in energy from the central ones. This is all in ranging between 2 and 6. For two sites the NRG repro- contrastwiththe variationalsolution,whereatransition duces the phase diagramof the dissipative two level system,aswasshownbyBullaetal31. Forlargerchainsand to an M degenerate state is predicted. small dissipation, the results are in quantitative agree- Of more interest are the cases of M = 5,6. Here a ment with the variational solution (see below), predict- new behavior is observed, which should be representa- ing adelocalizedphasewith renormalizedhoppinganda tive of the one expected for larger chains. The energy renormalized potential which tends to localized the par- flows for small and large dissipation are shown in Fig. 3 ticle at the center as the coupling strength is enlarged. and Fig. 4. Again, weak dissipation induces some local- As far as the phase transition is concerned, the case of ization at the center of the array of the particle density, M = 3,4 does not deviate too much from the dissipa- as can be seen in the inset of the figures, where q2 N h i 6 Type M GSdeg hq2iNRG ρˆ 1 VNaRrGia tsioolnuatli osnolution DLoeclo.c. 55 12 hq2i1NRG ρˆρˆ==12|t(r|e2ni,h2α|i+htr|e4ni,h4α||) Deloc. 6 1 hq2iNRG ρˆ=|tren,αihtren,α| Loc. I 6 2 0.25 ρˆ= 1(|3ih3|+|4ih4|) 2 >G Loc. II 6 2 2.25 ρˆ= 12(|2ih2|+|5ih5|) 2q < 0.5 TABLE I: Stable fixed points of the NRG for chains of M = 5,6 sites. The fixed points are characterized by the ground statedegeneracy,GS andthepositionmeansquaredvalue, deg hq2iNRG,whosevaluescanbeobtainedwiththeNRG(inthe localizedphasethereisasinglevalueforeveryα,whileinthe 0 delocalizedonethevaluedependsonthecouplingstrength,as 0 0.2 0.4 0.6 0.8 1 α showninFig. 5forachainof5sites). Thoseareusedtopro- pose an ansatz for the ground state density matrix, ρˆ, which fits it correctly. The states |ii represent a particle sitting at FIG.5: Mean-squaredpositionoftheparticleasafunctionof site i. |tren,αi is the ground state of a free tight-binding the coupling strength in the delocalized phase of the model, chainwithhoppingtren andaparabolicpotential,dependent for M = 5, as predicted by the variational calculation and on the strength coupling α. The latter is the output of the theNRG.Bothapproachesshowanincreasinglocalization of variational calculation for the delocalized phase, which fits theparticleatthecenterofthechainasthecouplingstrength quitewell thenumerical data. getslarger. Thiseffectarisesduetotherenormalizationofthe hopping parameter and the emergence of an effective confin- ingpotential. Inthevariationalcalculationthelocalization is providedby NRG is not enoughto fully characterizethe stronger,suggestingthathigherordercorrectionstotheeffec- nature of this phase transition. tive potential are playing an important role in the numerical This is not the only limitation of the numerical calculation. method. It also does not allow us to study large values of the dissipation, α 1, as the occupancies of the ≫ bosonicstatesbecomealsohigh. Thequestionofwhether is computed. As mentioned above, in this regime the other phase transitions can be ruled out for high values results agree qualitatively with the variational solution, of α remains open, and deserves a separate study with as shown in Fig. 5, where the mean squared position of different techniques. the particle is calculated in both approximations for a chainof5sites(asimilarplotcanbeobtainedfor6sites, -2 themaindifferencebeingthatthemeansquaredposition t = 0.01 tends to a finite value for α 1, having two states in t = 0.02 → -4 the center instead of one). The differences between the localized phase predicted -6 by the variational calculation and the NRG are even * T sharperforthoselongerchains. Aboveacriticalstrength g o L coupling αc of order one, we find a doubly degenerate -8 state, for odd and even number of sites, in which the parity symmetry is broken. For M = 5 the particle lo- -10 calizesnexttothecenter,whileforM =6itdoesinitially in the centralsites, and for largerdissipationin the next tothecenterones. Thisresultfollowsfromanalyzingthe -120 5 10 15 degeneracy of the ground state, extracted from the en- 1/(α-α) c ergy flows, as well as the evolution of the mean squared positionoperator. Theconvergedvaluesofthelattercan ∗ FIG.6: PlotofthedependenceofthecrossoverscaleT onthe be used to make an ansatz of the sort of ground state distance to the critical coupling strength. Here T∗ ∝ Λ−N∗, density matrix to which the flow converges. with N∗ chosen as the iteration for which the first excited InTableI,thezooofstablefixedpointsofthemodelis level verifies ΛN∗EN∗,1 = 0.03. The figure shows the results shown,forchainsofM =5,6sites. Inthe caseofM =6 fortwodifferenthoppingparameters. Inbothcasesthereisa ∗ there is an extra fixed point in the localized regime, cor- goodagreementwithanexponentialdecayofT asafunction ∗ respondingtoasituationinwhichtheparticlefindsmore ofthedistancefromthecriticalcoupling,logT ∝1/(αc−α) favorably to get localized in the sites next to the edges thannexttothecentralsites. Thissecondtransitionalso From the energy flows some extra information can be occurs for a critical value of the coupling strength, but extracted. In the delocalizedphase,a single energyscale thereisneitherasymmetrybreakingnorachangeinthe seems to be playing a part in the evolution from high degeneracy of the ground state. Thus, the information energy to low energy behavior. Actually, the flow in this 7 phaseissimilartothatinthedissipativetwolevelsystem, 0.06 andinthesamewaywecandefineacrossoverscaleT∗ α=0.1, t= 0.01 Λ−N∗ from the iteration N∗ at which the flow change∝s 0.05 00..45 from its initial behavior to the low energy regime. As 0.3 m shown in Fig. 6, T∗ tends to zero exponentially as the 0.04ρ 0.2 0.1 coupling strength approachesthe criticalvalue, logT ∗ ∝ 0 1/(α α). Hence,ourresultssuggestthatthetransition 1 2 3 4 5 isconct−inuous,beingconsistentwiththeexistenceofaKT t0.03 m α=1.2, t= 0.01 transition.30 0.5 0.02 0.4 The flows in the localized phase show a different be- m0.3 ρ havior. Here, two different energy scales appear in the 0.2 0.1 course of the flow, revealing an unstable fixed point in 0.01 0 1 2 3 4 5 an intermediate regime. Those scales are defined now m from the iterations N1∗ and N2∗ at which the energy lev- 00 0.5 1 1.5 2 elsdecouplefromthelowenergysector,givingrisetotwo α crossover temperatures Ti ∝ Λ−Ni∗, i = 1,2. From the 0.06 α=1.2, t= 0.01 mean squared position flows it can be deduced that the upper energy scale marks the decoupling of the sites lo- 0.05 0.5 0.4 catedatthe edges,asthe valuesofthis operatorarewell m0.3 ρ 0.2 fitted to the expected ones in free tight-binding chains 0.04 0.1 with finite hoppings but sites in the edges supressed (ef- 0 1 2 3 4 5 6 fectivelyreducingthechaintoonewithtwolesssites). In m this way, the intermediate fixed point would correspond t0.03 α=0.1, t= 0.01 α=1.4, t=0.01 to an effective cluster of three sites in the M = 5 case, 0.5 0.02 0.5 0.4 and four sites in the M = 6 one, with a renormalized 0.4 m0.3 hopping parameter tren. The lower energy scale corre- ρm00..23 ρ 00..21 sponds to the onset of the phase transition, for here the 0.01 0.1 0 0 1 2 3 4 5 6 parity symmetry is broken and only two sites remain in 1 2 3 4 5 6 m m the low energy regime. 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 α FIG.7: Phasediagram ofthemodel, eq.(1), for M =5(top) IV. PHASE DIAGRAM andM =6(bottom)sites, deducedfrom theNRGflow. The continuouslinesarelinearfittingstothenumericaldata. The A phase diagram of the model, including all the fea- insetsshowrepresentativedensitydistributionsoftheparticle tures discussed above, is presented in Fig. 7. As in the in each phase. In both cases there is a phase transition to widely studied dissipative two level system, there is a a localized phase in which the parity symmetry is broken. phase transition betweena delocalizedregime to a local- The even case shows also a second transition in thelocalized phase,wheretheparticleisconfinedbeyondthecentralsites, izedone. Inthedelocalizedphasetheeffectofthebathis resembling the kindof localization observed in the odd case. that of reducing the effective hopping and of generating a renormalized potential which makes the density of the particlehighercloseto the center. Inthe localizedphase the parity symmetry is broken and in both cases, odd at m = m0 ± q−g61g0 in the localized one, explaining and even, the particle localizes in one of two degenerate not only the doubly degenerate state, but also that in sites. This transition is continuous, and the numerical the case M = 6 there is a second transition to a phase results are consistent with a transition of the KT type, in which the particle localizes in the next to the center as in similar dissipative systems. sites. From the point of view of renormalization theory, The variational calculation shows how a parabolic ef- such an ansatz is reasonable as higher corrections to the fective potential emerges from the coupling to the bath, potential should be more irrelevant. being responsible of some localization of the particle at In the region of the phase diagram which we have an- the center. However, the numerical results suggest the alyzed (α<2), we have not found any further crossover existence of a more complicated renormalized potential to a region where the particle is confined to the edges. which would explain both the almost complete localiza- This could be explained by the role played by the in- tion of the particle as α 1 from below, and the intermediate unstable fixed point in the localized phase. → homogeneous degenerate ground state in the localized By studying the case of three and four sites, the only phase. Asimpleguesswhichworkswellqualitativelyisa way to get a phase transition in which the particle lo- effective potential in the form V = (g /2)(m m )2+ calizes at the edges is by starting with slightly lower site m 0 0 (g /4!)(m m )4. If g (α α), this An−satz has energies here as compared to the center. Hence, the de- 1 0 0 c − ∝ − a minimum at m = m in the delocalized phase, and coupling of the edges would be necessary to give rise to 0 8 such a renormalization of the on-site energies, favoring degeneracy between the sites of the array is broken by the phase transition to a more stable regime. the combination of the abrupt boundary conditions and For larger chains we expect a similar picture to work, the dissipation. As a result, an effective local potential and new features should not appear as far as the con- is induced, with a minimum at the center of the array, tinuous limit is not reached. The phase diagram should leadingtotheconfinementoftheparticlenearthecenter. show a transition to a localized phase for a critical value For values of the dimensionless dissipation strength αc also around the unity. Close to this phase transition, α > 1, we find a transition to a situation with a doubly the renormalized potential gets quartic corrections and de∼generate ground state, in which the parity symmetry the particle becomes localized in the resulting double- is broken. In this case, the combined effect of boundary well profile. As the minimum of this effective potential conditions and dissipation leads to the formation of an dependsonthedissipationstrength,thereshouldbesev- effective double wellpotentialirrespectiveofthe number eralcrossoverstoregionsinwhichtheparticleislocalized of sites in the array. at points increasingly farther from the center. However, Theformationofaninhomogeneouseffectivepotential the particle is never localized at the edges, since their from a spatially homogeneous coupling to a dissipative decoupling seems to be crucial to the realization of this bath should be a generic feature in similar models. A inhomogenous transition. Thus, in the localized phase, more remarkable result is that, in some cases, this effec- two or more energy scales are expected to play a role, tive potential is non-monotonic, so that simple confine- depending on the number of energy levels that are de- ment geometries can lead to complicated patterns in the coupled from the low-energysector until the stable fixed localized regime for high values of the dissipation. An point is reached. In the continuous model, which corre- open question is to what extent these results may de- sponds to the case of an infinite number of sites in the pend on the particular choice of particle-bath coupling. array, we expect only a single phase transition, at αc It has been argued in Ref. 39 that the introduction of around the unity, to a phase where the particle is local- a counter-term is insufficient to introduce full transla- ized in a double well potential profile whose minimum tionalinvariancein some dynamic contexts,suchas that depends on the coupling strength. of a suddenly introduced coupling. An extension of the presentequilibriumstudy to models oftruly translation- ally invariantdissipation39 could shed some light on this V. CONCLUSIONS. issue. We haveanalyzedthe simplestextensionofthe widely studied dissipative two level system, which interpolates VI. ACKNOWLEDGEMENTS. betweenthismodelandthealsoextensivelyanalyzeddis- sipative quantum particle in a periodic potential. The main results are obtained with a numerical renormaliza- We acknowledge A. J. Leggett for valuable discus- tion group technique specifically adapted to bosonic co- sions. This work was supported by MEC (Spain) ordinates. Forsmallandintermediatevaluesofthedissi- through grants FIS2005-05478-C02-01, FIS2004-05120 pationparameter,α,ourresultsprovideawellcontrolled and FIS2007-65723,the Comunidad de Madrid, through approximation to the ground state of the system. theprogramCITECNOMIK,CM2006-S-0505-ESP-0337, Whendissipationisweak,wefindthatthedensitydis- and EU Marie Curie RTN Programme no. MRTN-CT- tributionofthe particlebecomes narrower,andlocalized 2003-504574. J.S.wantstoacknowledgetheI3PProgram around the center of the array. This result is consis- fromthe CSICforfunding. 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Phase diagram of the dissipative quantum particle in a box

Similar Phase diagram of the dissipative quantum particle in a box