Phase diagram of the dissipative quantum particle in a box

0.2 MB·English

by  J. Sabio

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, Universit¨at 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<by<a1n, etffheectmivoedemlasshsoiunldardeidsusicpeattioveaepnavritriocnle-
k<ωc ment,whichadmitsacompleteanalyticalsolution.20 For
Hint ≡ λq X √k(cid:16)b†k+bk(cid:17) larger couplings we expect, as mentioned, a phase tran-
k<ωc sition as seen in related models. To analyze this region
λ2q2 1 (1) we will concentrate in a small number of sites, ranging
Hct ≡ X between 2 and 6, where both numerical and analytical
k<ωc
calculations are easier to perform.
Heretisthehoppingbetweennearestneighborsites,and
λ the coupling with which the bath couples to the posi-
tion of the particle q (m m )c c , where m
labels the center of the≡chPaimn. Th−e ba0th†mismcharacterized0 III. THE CALCULATIONS.
by a high energy cutoff ω . The last term of the Hamil-
c
tonian is a counter-term introduced in order to preserve A. Preliminary remarks.
thedegeneracybetweentheenergiesofthedifferentsites.
WeassumethatthecouplingleadstoOhmicdissipation,
The existence of a phase transition for a large enough
so that:
strengthcouplingcanbe seensimply bynormalordering
J(ω)=πλ2 kδ(ω k)=2παω , for ω <ω Hamiltonian (3). This operation, which can be regarded
X − | | c asaresummationofaninfiniteseriesoftadpolediagrams
k<ωc fortheinteractingvertex,23givesarenormalizedhopping
(2)
where α λ2/(4π) andJ(ω)is the spectralweightfunc- tren = te−λ22 Pk k1. Performing the summation we can
≡
tion. writefromthisexpressionaflowequationforthehopping
This Hamiltonian can be transformed through a uni- parameter in terms of the cutoff of the bath, giving the
tary operation into one in which all sites are treated on celebrated result:24
the same footing because the transformed bath couples
to the inter-site hopping. In orderto arriveat this form, d ∆(ωc) ∆(ωc)
( )=(α 1) (4)
which can be useful in the study of the phase diagram, dlogω ω − ω
c c c
we use the transformation U =e−λqPk √1k(b†k−bk) on the
Hamiltonian, yielding:13,23 From this equation it follows that a phase transition ex-
istsforα=1fromaregimeinwhicht isfinitetoonein
ren
M which is effectively renormalized to zero, thus suppress-
H= X kb†kbk−t X(c†mcm+1e−λPk √1k(b†k−bk)+h.c.) ingquantumfluctuations. Ingeneral,however,thisisnot
k<ωc m=1 thewholestoryandmoreprecisecalculationsareneeded
(3) in order to get the right critical line, which can have a
Hamiltonians (1) and (3) contain two limiting cases of dependence on the hopping parameter. This is the case,
interest. The case λ = 0 is that of a confined particle as is well known, for the dissipative two level system,
decoupled from the bath. The Hamiltonian is readily wherethetransitioncanbeshowntobeoftheKosterlitz
diagonalized,andthe groundstate correspondsto a par- - Thouless (KT) kind25 when higher corrections to the
ticle delocalized with a density ρm = M2+1sin2(Mπm+1). flow equations are computed.8,9,23 On the contrary, for
Ontheotherhand,thecaset=0correspondstoaparti- the dissipative particle in a periodic potential there are
cle without kinetic energy. FromHamiltonian (3) we see nohigherordercorrectionstothe flowequationsandthe
that the ground state is M-fold degenerate in the sub- transition line in the α t phase diagram can be shown
space of site states adiabatically dressed by a cloud of to be vertical.26 −
bosons, m e−mλPk √1k(b†k−bk) 0 . In general, we will A similar analysis can be tried for the dissipative con-
| i⊗ | i
bemainlyinterestedingenericvaluesoftheparameterst fined particle, but the computation of higher order cor-
andλoftheHamiltonian. Here,thebathcanberegarded rectionsto the transitionresults verytedious, becauseof
as performing repeated measurements of the position of the largenumber of coupledflowequations thatmust be
the particle,localizingit inthe sites basis,as opposedto analyzed. This is due to the lack of symmetries of this
the kinetic term, which tends to delocalize it. As in the model (only parity is preserved),which generates an im-
dissipative two level system and the dissipative particle portantnumberofcounter-termsthatmustbetakeninto
in a periodic potential, the phase diagramis expected to accountinthecriticalregionclosetoα=1. Inparticular
3
we can have higher charges of the form 0.06
α=0.1, t= 0.01
0.4
M 0.05
tl X(c†mcm+le−λlPk √1k(b†k−bk)+h.c.) m00..23
ρ
m=1 0.04 0.1
0
where l > 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 renormal-
ization group, which deal with the whole effective low-
energy Hamiltonian. FIG. 1: Phase diagram predicted by the variational calcula-
tion,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 dis-
sipative 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 2<q>N000...468 22<q>N0.51
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<q 1.5
1.2 2q> 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 sys-
tem,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 val-
ues 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 in- termediate 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. L.B.acknowledgesthe finan-
tent with the expectation that the environment acts as cialsupportoftheJ´anosBolyaiFoundation,theAlexan-
a measurement apparatus on the position of the parti- der von Humboldt Foundation and Hungarian Grants
cle, leading to a more localized distribution. The initial OTKA through projects T048782 and K73361.
1 A. O. Caldeira and A. Leggett, Ann. Phys. 149, 374 7 F. Guinea, Phys. Rev.B 67, 045103 (2003).
(1983). 8 S. Chakravarty,Phys. Rev.Lett. 49, 681 (1982).
2 Y.Makhlin,G.Sch¨on,andA.Shnirman,Rev.Mod.Phys. 9 A. J. Bray and M. A. Moore, Phys. Rev. Lett. 49, 1545
73, 357 (2001). (1982).
3 A.J.Leggett,ChanceandMatter,1986LesHouchesXLVI 10 A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A.
Lectures (North Holland, Amsterdam, 1987). Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59,
4 F. Guinea, R. A. Jalabert, and F. Sols, Phys. Rev. B 70, 1 (1987).
085310 (2004). 11 U.Weiss,Quantum Dissipative Systems (World Scientific,
5 F. Guinea, Phys.Rev.B 71, 045424 (2005). Singapore, 1999).
6 M. A. Cazalilla, F. Sols, and F. Guinea, Phys. Rev. Lett. 12 M. Nielsen and I. L. Chuang, Quantum Computation
97, 076401 (2006). and Quantum Information (Cambridge University Press,
9
2000). (1984).
13 A. Schmid,Phys. Rev.Lett. 51, 1506 (1983). 28 K. Wilson, Rev.Mod. Phys. 47, 773 (1975).
14 M. P. A. Fisher and W. Zwerger, Phys. Rev. B 32, 6190 29 R. Bulla, T. A. Costi, and T. Pruschke, Rev. Mod. Phys.
(1985). 80, 395 (2008).
15 S. Dattagupta and J. Singh, Phys. Rev. Lett. 79, 961 30 R. Bulla, N. H. Tong, and M. Vojta, Phys. Rev.Lett. 91,
(1997). 170601 (2003).
16 C. Callan and D. Freed,Nucl. Phys. B 374, 543 (1992). 31 R.Bulla,H.J.Lee,N.H.Tong,andM.Vojta,Phys.Rev.
17 E. Novais, F. Guinea, and A. H. Castro Neto, Phys. Rev. B 71, 045122 (2005).
Lett. 94, 170401 (2005). 32 T. A. Costi and C. Kieffer, Phys. Rev. Lett. 76, 1683
18 C. L. Kane and M. P. A. Fisher, Phys. Rev. B 46, 15233 (1996).
(1992). 33 T.A.CostiandR.H.McKenzie,Phys.Rev.A68,034301
19 C.Chamon,M.Oshikawa,andI.Affleck,Phys.Rev.Lett. (2003).
91, 206403 (2003). 34 A. H. Castro Neto, E. Novais, L. Borda, G. Zar´and, and
20 V. Hakim and V. Ambegaokar, Phys. Rev. A 32, 423 I. Affleck,Phys.Rev. Lett. 91, 096401 (2003).
(1985). 35 E. Novais, A. H. Castro Neto, L. Borda, I. Affleck, and
21 H. Grabert, P. Schramm, and G. Ingold, Phys. Rep. 168, G. Zar´and, Phys. Rev.B 72, 014417 (2005).
115 (1988). 36 L. Borda, G. Zar´and, and P. Simon, Phys. Rev. B 72,
22 M. Vojta, Phil. Mag. 86, 1807 (2006). 155311 (2005).
23 V. Hakim, A. Muramatsu, and F. Guinea, Phys. Rev. B 37 L. Borda, G. Zar´and, and D. Goldhaber-Gordon, unpub-
30, 464 (1984). lished, arxiv.org:cond-mat/0602019 (2006).
24 D. J. Amit, Y.Y. Goldschmidt, and S. Grinstein, Journal 38 S.Tornow,N.Tong,andR.Bulla,J.Phys: Condens.Mat-
of Physics A Mathematical General 13, 585 (1980). ter 18, 5985 (2006).
25 J. M. Kosterlitz, Phys. Rev.Lett. 37, 1577 (1976). 39 J. S´anchez-Canizares and F. Sols, Physics A 212, 181
26 F.Guinea,V.Hakim,andA.Muramatsu,Phys.Rev.Lett. (1994).
54, 263 (1985).
27 R. Silbey and R. A. Harris, J. Phys. Chem. 80, 2615