Finite temperature variational analysis of the tunneling and localization in spin–phonon model

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Physics Letters A 339 (2005) 393–402

www.elsevier.com/locate/pla

Finite temperature variational analysis of the tunneling andlocalization in spin–phonon model

Zoran Ivic a,∗, Dragan Kostic a, Darko Kaporb

a The “Vinca” Institute of Nuclear Sciences, Theoretical Physics Department, 020, 11001 Belgrade, P.O. Box 522, Serbia and Monb Department of Physics, Faculty of Sciences, University of Novi Sad, Trg D. Obradovi´ca 4, Novi Sad, Serbia and Montenegro

Received 4 November 2004; accepted 27 February 2005

Available online 17 March 2005

Communicated by V.M. Agranovich

Abstract

Temperature dependence of the effective tunneling frequency of the spin–phonon model for a few particular types ofhas been studied within the framework of the simple variational method.

Our analysis shows that for the Ohmic type of coupling, temperature drives the system towards delocalization. Ttype of behavior has been noticed for super-Ohmic case, but in the so-called “adiabatic regime”, while in the nonacase we predict localization transition, analogous to that observed for Ohmic coupling, when temperature depenpling constantS(T ) approaches critical valueS(T ) ∼ 1/2. In contrast to these results, for super-Ohmic (r = 3) case, therewas no substantial difference with respect to zero-temperature case. Brief discussion of the validity of the methodgiven. 2005 Published by Elsevier B.V.

PACS:05.30.-d; 03.65.Yz; 71.38.-k

Keywords:Quantum dissipative systems; Spin–boson model; Ohmic dissipation

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entfor

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A simple model representing a particle tunning between the two equivalent minima coupleda collection of the harmonic oscillators, which simulate the influence of the dissipative environm(phonons for example), provides a good basis

* Corresponding author.E-mail address:[email protected](Z. Ivic).

0375-9601/$ – see front matter 2005 Published by Elsevier B.V.doi:10.1016/j.physleta.2005.02.067

the study of the underlying microscopic mechanisdescribing the various phenomena in chemicalphysical [1–9] and even in the biological system[10–12]. Particular interest in this model has oriinated recently due to the development of the ntheoretical techniques and their application inexamination of the modern fundamental concesuch as macroscopic quantum coherence and tuning [1,3].

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mailto:[email protected]

394 Z. Ivic et al. / Physics Letters A 339 (2005) 393–402

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Essential features of the system may be descron the basis of the well-known spin–phonon (or equalently spin–boson) Hamiltonian:

(1)

H = −Jσz − σx

∑q

(λ∗

qbq + λqb†q

) +∑q

hωqb†qbq,

where 2J defines so-called tunneling matrix elemeb

†q andbq are Bose operators creating and annihilat

the phonon quanta of frequencyωq , σi (i = x, y, z)

are Pauli matrices, whileλq denotes the coupling parameter. Each particular application is specified throuthe choice of the explicitq-dependence of theλq andωq or by the particular choice of the spectral densof states whose connection to phonon dispersionand spin–phonon coupling parameter is definedJ (ω) = ∑

q |λq |2δ(ω − ωq). Its particular choice in

the formJ (ω) ∼ ωre−ω/ωB (ωB—cut off frequency),describes a variety of realistic situations in a variocontexts.

The problem of major concern is the examinatof the system dynamics in dependence of (i) the mtual ratio of the values of the basic physical paramedetermining the energy spectrum of the system:ωB—width of the phonon band, tunneling splitting—2J andEB = ∑

q |λq |2/(hωq)—the ground state energy othe system in the absence of tunneling (J = 0), and(ii) the type of coupling and the nature of phonowhich are specified by the explicitq dependence othe ωq andλq or by the particular choice of the exponentr in the expression for the spectral densitystates.

This problem has been extensively studiedmeans of various theoretical tools and it was foundthe system can display regimes with substantiallyferent physical behavior in dependence of the valof coupling constant (S = EB

hωB) and adiabatic parame

ter (B = 2JhωB

). Thus in the weak coupling limit, thmean position of the particle exhibits damped oslations between localized states with transition toexponential decay when the friction constant excethe bare tunneling frequency[6].

On the other hand, in the strong coupling regimparticle substantially influences the phonon substem which in turn affects system dynamics conserably and, in the final instance, the suppressionthe tunneling (localization) may arise[1,3,9,13–16].In that sense, depending on the value of the a

batic parameter, two physically very distinct situatioemerge.

In particular, we found[15] that, irrespectively ofthe type of coupling, the whole parameter spacethe system, comprised by the adiabatic parametercoupling constant ((S,B)-plane in other words), is divided in two distinct regions corresponding to the tdifferent mechanisms of the lattice response (reacfield), which dominate tunneling dynamics and whicin final instance, may lead to a localization. In tfirst one, so-called symmetry preserving region, stem dynamics is determined by the quantum naturthe phonon field and localization is achieved throuthe reduction of the effective tunneling frequency.the second region (symmetry breaking[15]), particlecan highly affect the medium which in turn substatially modifies its dynamics and may cause localition. Transition between these two regions is demined by so-called symmetry breaking boundary. Tis the line in (S,B)-plane, whose explicit form is different for each type of coupling. In the strong coupllimit all these curves converge towards theS � B/4.In this region the tunneling frequency is slightly rduced and tends to its bare value in the strong cpling limit while the system dynamics may be dscribed by a discrete nonlinear Schrödinger equa[17–19]. Analogous, slightly modified, results wefound by Kireev and Mann[20] who used improvedinterpolation variational method taking into accoualso phonon squeezing.

In the present paper we shall examine the sysproperties in the symmetry preserving region. In pticular, we shall restrict ourselves to the examinatof the behavior of the effective tunneling frequenin dependence of system parameters, type of coupand temperature. For that purpose we shall use aperature dependent variational method of Silbeycoworkers[7] utilized in the analysis of the dynamicof the spin–phonon model. In respect to that treatmof the problem which deals with the strict antiadiabalimit (B � 1), here we shall extend such an approatowards the adiabatic limit. In particular, we shall fcus our attention to the examination how the tempature modifies previously[14] observed characteristdependence of the effective tunneling frequency oncoupling constant and adiabatic parameter. Thus,lowing [7], we first perform the modified Lang–Firsounitary transformation[21] of the system Hamiltonian

Z. Ivic et al. / Physics Letters A 339 (2005) 393–402 395

n:

f thele-e—re-

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s-

e

tedinx-of

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andssi-

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er in

H = U†LFHULF, which yields

H = −J

2

(σ+e−2A + σ−e2A

) − V0σx +∑q

hωqb†qbq

(2)+∑q

hωq |fq |2 +∑q

(f ∗

q λq + fqλ∗q

),

where we have introduced the following conventioσ± = σz ± iσy and the operatorA = ∑

q(fqb†q −

f ∗q bq) andV0 = ∑

q [(λq + hωqfq)b†q + h.c.]. ULF =

exp[∑q fqσx(b−q − b†q)], wherefq is variational pa-

rameter whose magnitude determines the degree orenormalization of the effective tunneling matrix ement. It can be found using the standard procedurby minimizing the so-called model free energy corsponding to the effective Hamiltonian:

H0 = −Jeffσz +∑q

hωqb†qbq +

∑q

hωq |fq |2

(3)+∑q

(f ∗

q λq + fqλ∗q

),

which is deduced from Eq.(2) by adding and subtracing the term(Jeff/2)σz. HereJeff = Je−x ≡ J 〈e±2A〉0is the effective tunneling matrix element wherex =2∑

q |fq |2(2νq + 1) denotes so-called dressing pameter, determining the degree of the reduction ofeffective tunneling frequency.〈·〉0 stands for the averaging over the equilibrium phonon ensemble. In sua way transformed Hamiltonian may be written asum of the effective Hamiltonian (H0) and the inter-action one specified as follows:V = −σ+ J

2 (e−2A −〈e−2A〉0) − σ− J

2 (e2A − 〈e2A〉0) − V0σx , which, in thepresent context where we are primarily interestedbehavior of the effective tunneling frequency, is irrevant.

Renormalization of the tunneling matrix elementdetermined by the magnitude of the dressing parater x and in what follows we shall focus our attentioto the analysis of its behavior in dependence of thetem parameters and temperature. For that purposmust find explicitly variational parametersfq by min-imizing the model free energy of the system:

Fo = Fph − kBT ln[2 cosh(βJeff)

](4)+

∑q

[hωq |fq |2 + (

λqf ∗q + λ∗

qfq

)].

At this stage calculations may be highly simplified uing the simple trick introduced in[15]. Thus, insteadof the whole set of variational parametersfq—one foreachq from the entire Brillouin zone, we introducnew variational parametera specifyingfq as: fq =− λq

hωq+a(2νq+1), whereνq = (ehωq/θ − 1)−1 (θ = kT )

is the equilibrium average phonon number. As poinbefore in[15], picking up the variational parameterthis form we did not introduce any additional approimation with respect to the optimization procedurethe previous studies where the same method was[7,14,16]. This is rather an anticipation of the explicform of fq which may be checked easily by demaning the stationarity of the model free energy of tsystem:(∂Fo/∂fq) = 0. In such a way the optimization procedure is simply reduced to minimizing of tmodel free energy of the system(4) overa. Thus de-manding(∂Fo/∂a) = 0 we obtain

a = 2Je−x(a) tanh

(Je−x(a)

θ

),

(5)x(a) =∑q

|λq |2[hωq + a(2νq + 1)]2 ,

while the stability condition(∂2Fo/∂a2) > 0, implies:

(6)1+ a∂x

∂a

[1+ a

2θsinh−2

(Je−x

θ

)]a=a0

> 0,

a0—solution of Eq.(5). Since all the relevant informations are contained in dressing parameter, expknowledge ofa is not necessary, so we proceed wcalculatingx for a few models which are of particular interest for the practical application of the spiphonon model in various contexts. Thus, insertingexplicit form of a into the expression for dressing prameter we obtain the self-consistent equation forlatter which should be solved for each model, i.e.,each particular type of coupling. In general case thequations are rather complicated so thatx cannot beexpressed explicitly through the coupling constantadiabatic parameter, however it occurs that it is poble to find coupling constant as function ofx with B

andT as a parameters. In other words for each fivalue of adiabatic parameter we may find so-caadiabatic curveS = S(x;B,T ) which may be simplyinverted and desired behavior of dressing parametdependence ofS andB may be easily deduced.


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Thus we may proceed in the similar way as in oprevious studies[14–16]: for each particular model foa fixed temperature we plot a set of adiabatic curS(x) for a few chosen values of adiabatic paramespanning the whole range of system parameter sof interest. In order to distinguish the physically meaingful solutions we eliminateB from these curvesusing the stability condition and obtain so-called sbility line which divides(S, x)-plane into two distinc-tive areas where the stable solutions corresponthose points lying below this curve. Let us now dcuss some particular models and we proceed choothe phonon density of states in the formJ (ω) ∼ ωr sothat practically all cases of particular interest mayspecified by suitable choice of the exponentr . Thus,for example, particular choice ofr = 1 correspondsto a so-called Ohmic dissipation and may be asciated with the quantum tunneling of the magneflux in SQUID, particle (electron, proton, hole,. . .) ina 3d crystals piezoelectrically coupled with acousphonons, etc. In such a way the above integralsbe recast into the compact formI ∼ ∫ ωD

0 ωrA(ω)dω,where the explicit form ofA(ω) is determined by thespecific model.

In this way we found self-consistent equationx which cannot be solved explicitly forx as functionof coupling constant, but we may findS = S(x,B,T )

whose inversion gives desired dependence ofx onS,B andT :

(7)S = x

2rJ1(r), εD = hωD

θ.

Stability condition now reads:

1− 2xBe−x tanh(Be−xεD/2

)J2(r)

J1(r)

(8)×[1+ Be−xεD

sinh(Be−xεD)

]� 0.

HereJ1 andJ2 are the following integrals:

J1(r)

(9)

=1∫

0

dωωr coth(ωεD/2)

[ω + Be−x tanh(Be−xεD/2)coth(ωεD/2)]2 ,

J2(r)

(10)

=1∫

0

dωωr coth2(ωεD/2)

[ω + Be−x tanh(Be−xεD/2)coth(ωεD/2)]3 .

The integrals appearing in Eqs.(7), (9), (10) can-not be calculated explicitly analytically in the closform. However, numerical calculations are compatively simple and could be performed easily for eaparticular case. Those results may be visualizedway similar to our previous studies[14–16]by plottingthe set of adiabatic curvesS = S(x) for a few chosenvalues ofεD(T ) andB.

Ohmic dissipation(r = 1)

For Ohmic dissipation (r = 1) our results are plotted in theFigs. 1–3. Here, for the comparison how thtemperature modifies tunneling matrix element wrespect to the zero the temperature case, we first smarize the results of our zero temperature treatmenthe problem[14]. Thus, substitutingεD → ∞ in theexpressions(8)–(11)we easily recover the expressiofor the adiabatic curve and stability line:

S(x) = x

2

1+ Be−x

(1+ Be−x) ln(1+ 1Be−x ) − 1

,

(11)x = √2S

[√2S ln

(1

1− 1√2S

)− 1

].

Our results, for this case, are visualized onFig. 1where we have plotted the set of adiabatic curfor a few representative values of adiabatic parater. Here the dashed curve corresponds to the stabline. Looking at the adiabatic curves as functionalpendence of dressing parameter on coupling conswith B and εD as the parameters (i.e., treating eaadiabatic curve asx = x(S;B,εD)) we observe: thaat T = 0, as long asB < e−1, dressing parameter exhibits continuous growth with the increase of the copling constant. WhenS approaches critical valueSC =1/2, x tends to infinity which corresponds tostrictlocalization due to the vanishing of the effective tuneling frequency:Jeff

h→ 0 for S = 1/2. In the nonadi-

abatic limit we may expand the above analytic relatwhich connects coupling constant, dressing paramand adiabatic parameter in terms of small param


l-

athly-

calo-er,n-tem.

uesuld, inel-

c-

theundsingme-ves

ame-iza-ra-

B � 1, so we found that effective tunneling matrix eement behaves as:Jeff = J (eB)2S/(2S−1). This resultis in very good agreement with the results of the pintegral (PI) and renormalization group (RG) ana

Fig. 1. Adiabatic curves for the Ohmic dissipation(r = 1) at zerotemperature for (a)B = 3; (b)B = 1; (c)B = 1/e; (d) B = 0.1.

sis [1,3,8]. When adiabatic parameter exceeds critivalue (BC = e−1) dressing parameter becomes twvalued function of the coupling constant. Howevonly the lower one of them, satisfying stability codition, characterizes stable eigenstates of the sysAccording to the smallness ofx, which in the highadiabatic region could have considerably small valeven in the strong coupling limit, these states shobe classified as free partially dressed ones.Thusthis case the particle motion exhibits coherent tunning with the renormalized frequency∼ Je−x/h. Obvi-ously in the adiabatic limit (B > e−1) there is no strictlocalizationin the sense of the vanishing of the effetive tunneling frequency sincex always tends to finitevalues.

At finite temperature, as can be seen fromFig. 2 we observe that temperature induces profochanges in the character of dependence of the dresparameter on coupling constant and adiabatic parater. Here we plotted the set of adiabatic lines (curS = S(x,B, εD)) for a few chosen values ofB at twodifferent temperatures (εD = 0.1 and εD = 0.9). Incontrast to the zero temperature case, dressing parter always has finite value so that there is no localtion in the strict sense. Namely, at each finite tempeture and for each value of adiabatic parameter,x is the

me
Fig. 2. Adiabatic curves for Ohmic dissipation(r = 1) for εD = 0.9 (left panel) andεD = 0.1 (right panel). Adiabatic parameter takes the savalues as inFig. 1.


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two-valued function of coupling constant. In the phyically meaningful region of the(S, x)-plane, whichembraces only the lower values of dressing paramfor each value of coupling constant,x continuouslygrows as coupling constant increases in the inte0 � S � SM . Here SM denotes the maximal valuof the coupling constant determined from the contion ∂S/∂x = 0. WhenS exceeds this maximal valupresent method is inapplicable. Consequently thvalues ofS together with the corresponding valuesadiabatic parameter define the region of the paramspace ((S,B)-plane) where the present method holThe lowering of the temperature increases the upbound for the coupling constant and consequentlyregion of the values which dressing parameter mhave is enlarged. Consequences for the particlenamics are manifested through the tunneling slowdown due to the reduction of the effective tunnelimatrix element. This is more expressed at low tempatures wherex could have very high values even fthe very small values of the coupling constant. Quon the contrary, increasing of the temperature suppthe tendency towards the delocalization accordingthe smallness of thex. Therefore we may concludthat, at finite temperatures, although the tunnelingnamics may be significantly slowed down the locization in the strict sense does not occur since dresparameter always tends to finite value, possibly vhigh at low temperatures.

In order to test the validity of our predictions letcompare the present results with those previouslytained by PI and RG approaches. For that purposeshould restrict ourselves to the nonadiabtic high teperature limit in order to calculate the integralsJ1(r)

andJ2(r). In such a way we found the following result:

(12)Jeff = Je− SπkT

2Jeff

[2+ 2Jeff

kTtanh( Jeff

kT)

εD + 2JeffkT

tanh( JeffkT

)

]2S

,

which is precisely the result obtained by Silbey aHarris [12]. In the limit Jeff

kT� 1 effective tunnel-

ing matrix element approaches the well-knownsult [1,7,8]:

Jeff = J

(kT

hωD

)2S

if S � 2J

πkT.

Super-Ohmic dissipation(r = 2)

For this type of coupling our results are visualizin Fig. 3 (T = 0) andFig. 4 for finite temperaturesAt zero temperature we observe continuous localtion transition for all values of adiabatic parameB < BC (BC = 1.25). WhenB exceeds critical valuelocalization occurs through an abrupt transition frofree states (smallx ) to a localized ones (largex). Atfinite temperatures we observe thatx, as function ofcoupling constant, behaves in a manner which is qsimilar to that observed for the Ohmic coupling bfor the zero temperature. Thus, as far asB < BC(T ),it exhibits continuous growth with the increasingthe coupling constant. WhenS approaches some criical value, different for each temperature,x tends toinfinity and localization occurs. WhenB > BC(T )

dressing parameter becomes two-valued functioncoupling constant, however only the lower one of thtwo values ofx characterize the stable eigenstatessystem. Since in this case dressing parameter taknite and considerably low values tunneling frequenis always finite and there is no localization in the strsense.

This similarity with the systems with Ohmic coupling at T = 0 is even more expressed in the hi

Fig. 3. Adiabatic curves for the super-Ohmic dissipation(r = 2)

at T = 0, values of adiabatic parameter are:B = 3 (a); B = 1 (b);
B = 0.35 (c);B = 0.1 (d).


Fig. 4. Adiabatic curves for the super-Ohmic dissipation(r = 2) at finite temperaturesεD = 0.9 (left panel) andεD = 0.1 (right panel). Valuesof adiabatic parameter are:B = 3 (a);B = 1 (b);B = 1/

√e (c); B = 0.1 (d).

e

re:

i-a-y

in-

entat

ed

temperature limit (εD � 1), where we obtained thfollowing analytic relations connectingx with cou-pling constant, adiabatic parameter and temperatu

(13)S(x,T ) = xεD

4

1+ y2

(1+ y2) ln(1+ 1/y2) − 1,

while the stability condition now reads:

(14)S(x,T ) � εD

16

(1+ y2), y = Be−x.

Combining these two relations we may easily elimnatey and derive so-called stability line which seprates those points in(S, x)-plane, where the stabilitcondition is not satisfied:

(15)x = −S(T ) ln

(1− 1

4S(T )

)− 1

4.

From these relations we may, using the proceduretroduced in Ref.[14], find critical value of adiabaticparameterBC = e−1/2. In the nonadiabatic limit (B �BC ) we obtain

(16)Jeff = J(Be1/2) 2S(T )

1−2S(T ) .

Here S(T ) = 4SεD

denotes the temperature dependcoupling constant. This result is very similar to thobtained for the Ohmic dissipation atT = 0. One cansee that localization occurs whenS(T ) approachesS (T ) = 1/2.
C
Fig. 5. Adiabatic curves for the super-Ohmic dissipation(r = 3)

at T = 0, values of adiabatic parameter are:B = 3 (a); B = 1 (b);B = 0.35 (c);B = 0.1 (d).

Super-Ohmic dissipation(r = 3)

Our results, for this type of coupling, are visualizatFig. 5for zero temperature andFig. 6for finite tem-peratures.


Fig. 6. Adiabatic curves for the super-Ohmic dissipation(r = 3) atT = 0, values of adiabatic parameter are:B = 3 (a);B = 1 (b);B = 1/e (c);B = 0.1 (d).

ora-lter

es

d-izedezed

beorra-eex-

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-

o-ng

d

turetheu-wn

n-ra-er-theon, inin

ssi-

ses,asic

It follows that for finite temperatures there is nsubstantial difference in respect with zero tempeture case. Thus, as far asB is less than some criticavalue, different for each temperature, we encounthe continuous increase ofx as a function ofS forfixed B and T (εD). It practically means that whave, for eachB and T gradual transition towardlocalization. WhenB exceeds that critical valueBc

x = x(S) has three solutions, two of them corresponing to stable eigenstates. One of them is characterby a small value ofx, this corresponding to a frestate, while the other one corresponds to a localione. Increasing the temperature (lowering ofεD) fa-vors the tendency towards localization. As it canseen from theFig. 6, at high temperatures, even fa very weak coupling and very high adiabatic pameter,x rapidly tends to infinity highly reducing thregion of parameter space in which free states canist.

In the high temperature region we find the equatfor the adiabatesS = S(x,B,T ) in the relatively sim-ple form:

S � xεD

12

(1+ y2

(1+ y2)2− 5

1+ y2

(17)+ 3y−1 arctany−1)−1

.

Localization arises whenx tends to infinity which happens when adiabate approaches straight lineS = xεD

4 .One may find the condition for the stability of the lcalized states from the inequality (3.9). Thus followithe procedure proposed by Emin[22] we simply sub-stitute S = xεD

4 , in the stability condition (3.9) andisregarding all terms of the order(Be−x)2 we have:

(18)1− 12πS(T )Be−S(T ) � 0.

In the present paper we have studied temperadependence of the effective tunneling frequency ofspin–phonon model for a few particular types of coplings. For that purpose we have used the well-knovariational procedure[7,13,14]. Our primary aim wasto extend that method, which works perfectly in the atiadiabatic limit, towards the adiabatic part of the pameter space of the system including the finite tempatures. The validity of the approach is restricted toso-called “symmetry preserving” case where phonsubsystem manifests its quantum nature maximallycontrast to the so-called “symmetry breaking” casewhich system properties are dominated by the clacal phonon behavior.

Our analysis was focused to three particular cawithin the recent applications commonly knownOhmic (r = 1) and two particular case of super-Ohm


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coupling (r = 2) and (r = 3). For the Ohmic dissipation our results show that the temperature drisystem towards delocalization. More precisely in tcase at finite temperature there is observed the gual decrease of the tunneling frequency but in contto zero-temperature case, there is no strict localizaas manifested through the vanishing of the effectunneling frequency. For the super-Ohmic (r = 2) dis-sipation in the nonadiabatic limit, we observe the stlocalization when the temperature-dependent coupconstant approaches critical valueSc(T ) = 1/2. Inthe adiabatic case, there is no strict localization,rather a gradual slowing down of tunneling frequenHere we observe the remarkable similarity betwethe localization transition in such systems at fintemperatures and the localization in the Ohmic stems atT = 0. This is obvious in the high tempeature limit where the effective tunneling frequenattains practically identical form to the well-knowresult for the Ohmic systems at zero temperatuJeff = J (Be1/2)(2S(T )/(1−2S(T ))).

According to our results, forr = 3 there is nosubstantial difference in the behavior of the systfor zero- and finite-temperatures. Namely, the qutative behavior of the adiabatic curves is identicalboth cases. The only difference is that the temperarise, in the antiadiabatic limit, supports localizatioNamely, at the higher temperatures dressing paramcan have comparably high values even in the wecoupling limit. In other words for the fixed value othe coupling constant higher values of the dressingrameter corresponds to the higher temperaturesthe other hand, in the adiabatic limit (B > Bc(T )—different for each temperature) temperature has a qopposite tendency so that its increasing drives thetem towards the delocalization.

We would like to comment on the applicability othe method: it basic drawback is that at this momein the framework of the present method, we arein a position to conclude whether the above predtions could be accepted as real physical possibilitysimply are the artefact of the, may be oversimplifivariational method which completely relies upon tLF unitary transformation. Limited validity of the LFtransformation in application to a spin–phonon andlated models, is known for a long time and it wrigorously shown by Sonek et al.[23] that the sec-ond unitary transformation of the reflective type

r

needed in order to establish the correct analytichavior of the eigenstates in the opposing paramlimits: vanishing coupling for arbitrary adiabatic parmeter and vanishing adiabatic parameter but arbitcoupling strength. Unfortunately these studies as was the other variational approaches which includetransformation in combination with the squeezed sansatz and other methods[23–27]were devoted to theanalysis of ground state properties (T = 0 K), only andcannot be compared with the present analysis. Tconcerning the applicability of the method, it canestimated only by the comparison with, at present mment the only available, PI and RG finite-temperatresults for Ohmic case, but only for the limited ranof system parameters. Thus, as pointed above, ousults are in accordance for that part of the paramspace. We believe that the extension of the above mtioned more sophisticated methods towards finite tperatures would lead to the results analogous to abpresented ones. A test of that conjecture would becomparison of the results of these approaches withr = 2 characteristic prediction of the strict localizatiat critical coupling value, i.e., the expression(16).

Acknowledgements

This work was supported by the Serbian Minisof Science and Technology under the grants 19611895.

References

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Finite temperature variational analysis of the tunneling and localization in spin–phonon model

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Transcript of Finite temperature variational analysis of the tunneling and localization in spin–phonon model