Renormalization and Quantum Scaling of Frenkel-Kontorova Models

0.24 MB·English

by  Nuno R. Catarino

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n Renormalization and Quantum Scaling of
a
J
Frenkel–Kontorova Models
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Nuno R. Catarino∗†& Robert S. MacKay‡,
]
h Mathematics Institute, University of Warwick
c
e Gibbet Hill Road, Coventry CV4 7AL, U.K.
m
- 2nd February 2008
t
a
t
s
.
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a Abstract
m
Wegeneralisetheclassical Transition byBreakingofAnalyticity
- fortheclassofFrenkel–KontorovamodelsstudiedbyAubryandoth-
d
n ers to non-zero Planck’s constant and temperature. This analysis is
o basedonthestudyofarenormalizationoperatorforthecaseofirra-
c tional mean spacing using Feynman’s functional integral approach.
[ Weshowhowexistingclassicalresultsextendtothequantumregime.

In particular we extendMacKay’s renormalization approach for the
1
classical statistical mechanics to deduce scaling of low frequency ef-
v
1 fects and quantum effects. Our approach extends the phenomenon
2 ofhierarchicalmeltingstudiedbyVallet,SchillingandAubrytothe
2 quantum regime.
1 Keywords: Transition by breaking of analyticity; renormaliza-
0 tion; quantum scaling; specific heat.
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0
/ 1 Introduction
t
a
m TheFrenkel–Kontorovamodel(FK)isaone-dimensionallatticemodel
- exhibiting incommensurate structures. It is a system of elastically
d coupledparticlesinanexternalperiodicpotential(adiscreteversion
n
of thesine–Gordon model) with Lagrangian
o
c x˙2
v: L(x,x˙)= 2n −v(xn,xn+1) , (1)
i nX∈Z(cid:26) (cid:27)
X ∗Present address: School of Mathematical and Computer Sciences, Scott Russell
Building,Heriot-WattUniversity,EdinburghEH144AS,U.K.
r
a †e-mail: [email protected]
‡e-mail: [email protected]
1
where
v x,x′ = 1 x′ x 2+P x′ x + u cos(2πx), (2)
2 − − 4π2
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
with constants P (“pressure” or P as tension) and u (amplitude
−
of the onsite potential). We call successive pairs (x ,x ) bonds
n n+1
and v(x ,x ) the (potential) energy of the bond. Notice that in
n n+1
(1)and(2)theparticles’mass,theelasticcouplingstrengthandthe
period of the onsite potential are all scaled to one. The FK model
is a particular case of the broader class of (Generalised) Frenkel–
Kontorova models(GFK)with Lagrangian still givenby(1) butthe
bond action v is a generic 2 function satisfying
C
v x+1,x′+1 =v x,x′
∂(cid:0)2v x,x′ 6(cid:1) C (cid:0)<0. (cid:1)
∂x∂x′ −
(cid:0) (cid:1)
In the space of parameters there are two important limits. In
the integrable limit, the bond energy, v, depends only on (x′ x).
−
For the FK model (2) the integrable limit is attained at u = 0 and
its minimum energy configurations (i.e. x RZ such that M <
N,V (x) N−1 v(x ,x ) is minimu∈m for all variati∀ons of
M,N ≡ n=M n n+1
x withfixedx andx )arearraysofequallyspacedparticleswith
n M N
P
mean spacing
x x
ρ:= lim N − M
−M,N→∞ N M
−
simply P. Intheanti-integrablelimit[4]theonsitetermdominates,
−
which corresponds to u in (2). All particles are then in the
→ ∞
minima of the potential and the mean spacing is the closest integer
to P, or any valuein between thetwo if non-unique.
−
Aninterestingsetofcodimension-2criticalpointsoccurbetween
thesetworegimes,oftencalledTransitionby Breaking ofAnalyticity
(TBA):inthespaceofparameters(u,P),foreachirrational ρthere
isacurveP =P (u)ofmeanspacingρcontainingacriticalvalueu
ρ c
(there may be more than one u depending on the potential). The
c
regime for u less than u is called subcritical (or sliding phase) and
c
above u supercritical (or pinned phase) [5]. FortheFK model with
c
fixed mean spacing ρ = γ−1, where γ = 1+√5 /2 is the golden
mean, the TBA is at thecritical value u 0.971635406 [15]. This
c ≃(cid:0) (cid:1)
is the case most often studied in the literature, since it is presumed
to be thehighest valueof u at which a TBA occurs.
For the case of mean spacing ρ = γ−1, relabelling the bonds v
appropriately [19] as τ and υ results in a Fibonacci sequence of τs
and υs where each υ-bond is always surrounded by τ-bonds. The
TBA point can be viewed as a fixed point of the renormalization
operator that minimises the energy of the sum of successive υ and
τ-bonds with suitably chosen space and energy scalings, α, R
J ∈
2
respectivelydependingonthepair(υ,τ)(actuallyitisalsonecessary
tosubtract aconstant andaquadraticcoboundarybutwewill sup-
press reference tothese inessential terms) [19]. The renormalization
operator has a nontrivial fixed point1 (υ¯,τ¯) with α 1.4148360
≃ −
and 4.3991439. This fixed point corresponds to the critical
u alJong≃the curve in (u,P) for ρ = γ−1, the transition point be-
c
tween the subcritical and supercritical regimes. It has two unsta-
ble directions: one along the curve of constant mean spacing, with
eigenvalue δ 1.6279500 and the other transverse to this curve in
≃
the (u,P) plane (which we’ll call the P direction), with eigenvalue
η= /γ 2.6817384 [19, 20].
−J ≃−
Whereas the classical ground states of Frenkel–Kontorova mod-
els have been extensively studied since the beginning of the 1980’s
[1, 2, 3, 5, 22] (see also the review in chapter 1 of [11], and the
book [10]for severalaspects of theFK model), theextension tothe
quantumregimeoftheclassical Transition byBreakingofAnalytic-
ity is still not fully understood. Most of the previous studies stem
from thework of Borgonovi et al. [8,9], where theauthors doa nu-
mericalstudyofFKmodelinthesupercriticalregion u>u forthe
c
case of mean spacing γ−1. They introduce a ‘quantum hull func-
tion’ for the expected positions x¯ as the extension of Aubry’s hull
n
function (see [11], section 1.2) as well as the ‘quantum g-function’
(whichreducestosin(2πx¯ )intheclassicallimit)andverifythatby
n
increasingℏthequantumhullfunctionbecomesasmoothversionof
theclassical hullfunctionandthatg tendstoasawtooth-likemap.
n
Similar results have been later obtained using various numerical or
a combination of analytical and numerical methods[6, 7, 17, 16].
In a recent numerical study Zhirov et al. [26] claim to observe
a ‘quantum phase transition’ in the FK model at a critical value of
Planck’sconstantbetween‘slidingphonongas’anda‘pinnedinstan-
ton glass’. In fact we expect that due to KAM-type of arguments,
for sufficiently irrational mean spacing thephononenergy band will
survivefor asmall perturbation oftheintegrable limit u=0, where
theinteractionsbetweenphononsofdifferentwavenumberaresmall.
At the anti-integrable limit, on the other hand, when the dominant
interaction is the onsite periodic potential, the quantum spectrum
consists of N degenerate sets of Bloch bands (N being the total
number of particles) whose width is due to tunneling or instanton
effects between distinct minima of the onsite potential. As u is de-
creasedfromtheanti-integrablelimit,thedegeneracybetweenBloch
bands corresponding to distinct particles should be lifted, widening
the Bloch bands until they merge into a unique phonon band for
some non-zerovalue u (ℏ) (possibly not a uniquecurve).
c
In addition to the groundstates, it is physically significant to
1This has now been proved by Koch [18] by reformulation as a renormalization on
continuous–timeHamiltoniansystemsandrigorouscomputer–assisted bounds.
3
study the effect of the TBA on the low temperature statistical me-
chanics of FK models. This was done for the classical case in [21].
The quantum statistical mechanics of FK models was considered in
a series of papers by Giachetti and Tognetti, e.g. [13, 14], but we
arenotawareofanyworkontheeffectsoftheTBAonthequantum
statistical mechanics.
Thegoalofthisarticleistostudythetransitionabovebyextend-
ingtheminimumenergyrenormalizationapproachin[19]tonon-zero
Planck’sconstantandtemperature,whichisdoneinsection2. This
isdoneinawaysimilartotheclassical non-zerotemperatureexten-
sion performed in [21]butrequiresfirst an extensionof theclassical
case from ground states to time-periodic solutions. The strategy is
to construct a renormalization operator, , which reduces to the
groundstateoperator, ,whenℏgoestoRzero,i.e. = .
Then(u,P,...,ℏ=0)iRscagns invariantsubspacefor wRh|ℏic=h0incRlucdgess
R
thecriticalfixedpointofrenormalization correspondingtotheTBA
by construction.
In the main section 2 we start by defining a decimation proce-
dure for pairs of “bond actions” by doing the trace overinterme-
⊕
diate particles corresponding to a partial trace over the kernel and
the renormalization operator by composing the decimation with
R
the appropriate scalings. The quantum eigenvalue κ is also intro-
duced. Next, in subsection 2.1, we show that the classical ground
staterenormalization, ,isobtainedasthelimit ofzeroPlanck’s
cgs
R
constant and frequency of quantum renormalization. In subsection
2.2 the value of the quantum eigenvalue is determined by analysing
the linearised or phonon problem. Finally in section 2.4 the results
obtained are ‘Wick rotated’ to obtain the scaling laws for quantum
thermal quantities, and an extension of the phenomenon of hierar-
chical melting studied by Vallet, Schilling and Aubry[24, 23, 25] to
the quantumregime is proposed.
2 Renormalization and scaling
Webeginbyintroducingtherenormalizationprocedurefortheaction
of time-periodic functions of prescribed period. Consider the class
of models with the following formal sum for theaction
(x;T)= s(x ,x ;T).
n n+1
S
n∈Z
X
for time–periodic functions x of period T, where thebond action s
n
is given by
T x˙2
s x,x′;T = v x,x′ dt, (3)
2 −
Z0 (cid:26) (cid:27)
(cid:0) (cid:1) (cid:0) (cid:1)
4
andvisaGFKbondpotentialenergy. Thequantumrenormalization
will concern thetrace of thekernel which is
K′(T,ℏ)= ′xeℏiS(x;T),
D
Z
wherethenotation ′xmeansthattheintegrationistobeperformed
D
over the space of periodic paths x with period T and includes the
integration over the endpoints dx(0) (i.e. x(T) = x(0) and ′x =
D
dx(0) x). Rescaling time t t/T and defining Ω = 2π/T the
D →
bond action can berewritten as
2π 2π 1 Ω2
s x,x′; = x˙2 v x,x′ dt, (4)
Ω Ω 8π2 −
(cid:18) (cid:19) Z0 (cid:26) (cid:27)
(cid:0) (cid:1)
andnowsactsonthespaceofperiod-onefunctionsx,x′:R/Z R
→
or loops.
Nowletx RZ beaclassicalgroundstateandcallabondaction
∈
s a τ or an υ as for the renormalization for classical ground states
in [19]. For the case of mean spacing γ−1 the sequence of bond
actions then forms an infinite Fibonacci sequence of τ and υ types
of bonds where each υ is always surrounded by τs. At this point
one wants to eliminate all particles z from sequences of the form
υ(x,z;Ω)+τ(z,x′;Ω) (for simplicity we use Ω as an argument of
action bonds instead of 2π/Ω from now on). In order to do this
define the following decimation operator acting on pairs of bond
⊕
actions (υ,τ) as
(υ τ) x,x′;Ω = iℏln ′zeℏi[υ(x,z;Ω)+τ(z,x′;Ω)]. (5)
⊕ − D
Z
(cid:0) (cid:1)
The decimated functional υ τ still has theform of an action bond
inthesensethatitactsonpa⊕irsofloops(x,x′). Therenormalization
operator is defined as the composition of a decimation and scalings
as
τ(x,x′;Ω) (υ τ)(x/α,x′/α;Ω/ε)
R υ(x,x′;Ω) = Jε ⊕τ(x/α,x′/α;Ω/ε) , (6)
(cid:20) (cid:21) (cid:20) (cid:21)
which includes a scaling of frequencies, ε, still undetermined at this
point. The global scaling of bond actions is /ε instead of just
J J
because the renormalization now acts on actions instead of energies
asintheclassicalgroundstatecase. Byperformingthecomposition
of (6) and (5) explicitly one can see that the natural scale factor
κ= /εforPlanck’sconstantarisesand canbeseenasactingon
the eJxtendedspace of (τ,υ,ℏ) for ℏ 0 aRs
≥
τ˜(x,x′;Ω)= iκℏ ′zeκiℏJε{υ(x/α,z;Ω/ε)+τ(z,x′/α;Ω/ε)}
R: υ˜(x,x′;Ω)=−Jετ(xR/αD,x′/α;Ω/ε) .
 ℏ˜=κℏ
(7)

5
We interpret κ as the eigenvalue of in the direction of Planck’s
R
constant.
2.1 Semiclassical approximation and ground
state limit
Asafirststep toconnect thequantumrenormalization (6) withthe
groundstaterenormalization of[19],onecanlookatthedecimation
operator (5) in the stationary phase approximation for small ℏ:
(υ τ) x,x′;Ω = iℏln ′zδ z z x,x′;Ω
cl.
⊕ − D − ×
Z
(cid:0) (cid:1) e(cid:0)ℏi{υ(x,z;Ω(cid:0))+τ(z,x′(cid:1);Ω(cid:1))}. (8)
×
Here z (x,x′;Ω) is the classical path of period one satisfying the
cl.
Euler–Lagrange equations
δ[υ(x,z;Ω)+τ(z,x′;Ω)]
=0. (9)
δz
(cid:12)z=zcl.(x,x′;Ω)
(cid:12)
(cid:12)
In the classical limit one is thus left w(cid:12)ith a dynamical problem (9)
and thedecimated action (8) can berewritten as
sta υ(x,z;Ω)+τ z,x′;Ω , (10)
z∈loops
(cid:2) (cid:0) (cid:1)(cid:3)
the sum of bond actions evaluated at z = z , given by (9), which
cl.
stationarisesthesumυ+τ overthespaceofallloops z:R/Z R .
{ → }
2.1.1 Ground state limit
The ground state decimation can now be taken as the limit Ω 0
→
of(8)or(10). Toseethis,noticethatinthislimit onlytheclassical
ground states contributeto the kerneland
′zeℏi{υ(x,z)+τ(z,x′)} e−2ℏπΩi{v(υ)(x,z)+v(τ)(z,x′)},
D ≃
Z
where (x,z,x′) is a segment of a classical ground state and z(x,x′)
minimises the sum υ+τ. Taking the logarithm and multiplying by
iℏ, this results in the decimation (5) being
−
2π
(υ τ) x,x′;0 = lim min v(υ)(x,z)+v(τ) z,x′ (11)
⊕ Ω→0−Ω z∈R
(cid:0) (cid:1) h (cid:0) (cid:1)i
(the factor 2π/Ω is kept in the above expression simply to con-
trol the divergence of decimation). Apart from the limit factor
6
limΩ→0 2π/Ω this is in fact the classical ground state decimation
−
in [19] and so therenormalization (6) becomes
τ(x,x′;0) lim −2ΩπJ v(υ)⊕cgsv(τ) (x/α,x′/α)
R(cid:20) υ(x,x′;0) (cid:21) ≃ Ω→0" −(cid:16)2ΩπJv(τ)(x/α(cid:17),x′/α) #
2π v(τ)(x,x′)
= lim , (12)
Ω→0−ΩRcgs v(υ)(x,x′)
(cid:20) (cid:21)
where and arethedecimation andrenormalization opera-
cgs cgs
tors for⊕the classRical ground states, and ε R is still undetermined
∈
atthispoint. Therenormalization hasthereforeafixedpoint(the
TBA fixed point), with ground staRtes of mean spacing ρ = γ−1, in
the Ω=0 subspace consisting of
τ¯(x,x′;0) 2π v¯(τ)(x,x′)
= lim ,
υ¯(x,x′;0) Ω→0− Ω v¯(υ)(x,x′)
(cid:20) (cid:21) (cid:20) (cid:21)
where(v¯τ,v¯υ)isthefixedpointofthegroundstaterenormalization
operator , because applying expression (12) at the TBA fixed
cgs
R
point results in
τ¯(x,x′;0) 2π v¯(τ)(x,x′) τ¯(x,x′;0)
= lim = .
R υ¯(x,x′;0) Ω→0−Ω v¯(υ)(x,x′) υ¯(x,x′;0)
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
2.2 Renormalization for the phonon spectrum
Close to the ground state, for small Ω2, the relevant contributions
are approximately given by the normal modes or phonons. For a
GFK model at irrational mean spacing ground state, ρ, the phonon
spectrum includes zero in the subcritical regime, but the minimum
frequency or phonon gap is positive above the critical point u [5].
c
Thephononcontributiontothetraceofthekernelisgivenbydoing
a quadratic approximation which is (with x =x +ξ )
n n n
K′ x,2π e−2ℏπΩi nv(n)(xn,xn+1) (13)
Ω ≃ ×
(cid:18) (cid:19) P
′ξeℏπΩi n 01 4Ωπ22m(n)(ξ˙n,ξ˙n+1)−u(n)(ξn,ξn+1) dt,
× D P R (cid:26) (cid:27)
Z
where the kinetic term was generalised to a symmetric quadratic
form in the velocities
m(n) ξ˙,ξ˙′ =m(n)ξ˙2+2m(n)ξ˙ξ˙′+m(n) ξ˙′2
11 12 22
(cid:16) (cid:17)
(as will become clear later on the renormalization operator intro-
ducescouplingbetweenthevelocitiesofneighbouringparticles),and
u is also a symmetric quadratic form
u(n) ξ,ξ′ =u(n)ξ2 2u(n)ξξ′+u(n)ξ′2.
22 − 12 11
(cid:0) (cid:1)
7
The first term in (13) is simply the classical ground state and
correspondsto(12),soonewantstoapplythedecimation(5)tothe
sum of the quadraticparts of bond actions of theform
π Ω2
m(υ) ξ˙,ζ˙ +m(τ) ζ˙,ξ˙′ u(υ)(ξ,ζ)+u(τ) ζ,ξ′ .
Ω 4π2 −
(cid:26) h (cid:16) (cid:17) (cid:16) (cid:17)i h (cid:0) (cid:1)i(cid:27)
Because this sum is quadratic in the particle to eliminate, ζ, the
corresponding functional integral can be calculated (for example
by Gaussian integration of the Fourier transformed sum of action
bonds[27])andthesemiclassicalapproximationisexactinthiscase.
To first order in Ω2 theresult is2
eℏiπΩ 4Ωπ22[m(υ)(ξ˙,ζ˙)+m(τ)(ζ˙,ξ˙′)]−[u(υ)(ξ,ζ)+u(τ)(ζ,ξ′)]
(cid:26) (cid:27) ≃
eℏiπΩ 4Ωπ22m˜(τ)(ξ˙,ξ˙′)−u˜(τ)(ξ,ξ′) ,
≃ (cid:26) (cid:27)
where m˜(τ) is a new symmetric quadratic form with components
given by
m˜(τ) = m(υ)+ 2m(1υ2)u(1υ2) + m(2υ2)+m(1τ1) u(1υ2)2
11 11 u(2υ2)+u(1τ1) (cid:16) u(2υ2)+u(1τ1(cid:17)) 2
(cid:16) (cid:17)
m˜(τ) = m(1υ2)u1(τ2)+m(1τ2)u(1υ2) + m(2υ2)+m(1τ1) u(1υ2)u(1τ2) (14)
12 u(2υ2)+u(1τ1) (cid:16) u(2υ2)+u(cid:17)(1τ1) 2
m˜(τ) = m(τ)+ 2m(1τ2)u(1τ2) + m(2υ2)(cid:16)+m(1τ1) u(1τ2(cid:17))2
22 22 u(2υ2)+u(1τ1) (cid:16) u(2υ2)+u(1τ1(cid:17)) 2
(cid:16) (cid:17)
and thenew form u˜(τ) has components
u(υ)2
u˜(τ) = u(υ) 12
11 11 − u(υ)+u(τ)
22 11
u(υ)u(τ)
u˜(τ) = 12 12 (15)
12 u(υ)+u(τ)
22 11
u(τ)2
u˜(τ) = u(τ) 12 .
22 22 − u(υ)+u(τ)
22 11
Withthesenewquadraticforms,thedecimation(5)becomessimply
(υ τ) x,x′;Ω = (c.g.s. decimation)+ (16)
⊕
(cid:0) (cid:1) + 1 1Ω2m˜(τ) ξ,ξ′ u˜(τ) ξ,ξ′ dt,
Ω −
Z0
2There is possibly also a logarithmic term due(cid:0)to th(cid:1)e integra(cid:0)tion(cid:1)measure which
correspondstoaredefinitionofthegroundstateenergy.
8
where ‘c.g.s decimation’ is the decimation (11). For the renormal-
ization(6)onealsoneedsthe‘undecimated’bondactionswhichcor-
respondtoisolated bondactionsoftypeτ,sooneshoulddefinealso
m˜(υ)(ξ,ξ′)=m(τ)(ξ,ξ′)
(17)
u˜(υ)(ξ,ξ′)=u(τ)(ξ,ξ′).
ForaGFKmodel(3)atthestartofrenormalization,foreithertypeτ
orυofbondsthequadraticformsarem =0,m +m =1,u =
12 11 22 jj
v (x,x′)forj =1,2andu = v (x,x′),wherethesubscriptsin
,jj 12 − ,12
vdenotedifferentiationwithrespecttothefirstandsecondvariables
and(x,x′)isasegmentofaclassicalgroundstate. Becauseuisthen
dependenton (x,x′), after iterating theabove transformations (14),
(15) and (17) one endsup with a set of asymptotic quadratic forms
depending on the ground state, m¯(τ) m¯(υ) , u¯(τ) and u¯(υ) which
x,x′ x,x′ x,x′ x,x′
scale by factors3 ω = α2ε2/ 1.255071 for the mass forms and
J ≃
/α2 for thepotential forms [19], i.e.
J
m˜¯(τ) α2ε2m¯(τ)
x,x′ ≃ J x,x′
 m˜¯(xυ,x)′ ≃ α2Jε2m¯(xτ,x)′ . (18)
 u˜¯(xτ,x)′ ≃ αJ2u¯(xτ,x)′
u˜¯(υ) α2u¯(υ)
x,x′ ≃ J x,x′
Finally, using(14–18),and defining
π
τ¯ x,x′;Ω := v¯(τ) x,x′
Ω −
(cid:0) (cid:1) 1h Ω2(cid:0) (cid:1)
+ m¯(τ) ξ,ξ′ u¯(τ) ξ,ξ′ dt
4π2 x,x′ − x,x′
Z0 (cid:26) (cid:27) (cid:21)
π (cid:0) (cid:1) (cid:0) (cid:1)
υ¯ x,x′;Ω := v¯(υ) x,x′
Ω −
(cid:0) (cid:1) 1h Ω2 (cid:0) (cid:1)
+ m¯(τ) ξ,ξ′ u¯(τ) ξ,ξ′ dt ,
4π2 x,x′ − x,x′
Z0 (cid:26) (cid:27) (cid:21)
(cid:0) (cid:1) (cid:0) (cid:1)
the renormalization (7) at (τ¯,υ¯,ℏ) becomes
τ¯(x,x′;Ω) τ¯(x,x′;Ω)
υ¯(x,x′;Ω) υ¯(x,x′;Ω) .
R ℏ  ≃  κℏ 
   
Thus,byincludingthescalingoffrequenciesε 1.649415(seefoot-
≃
note 3) in the renormalization (6), the point (τ¯,υ¯,0) becomes an
3 In [19] the scaling is actually for the quantities an := u(2n2−1)+u(1n1), bn := u(1n2),
cn :=m(2n2−1)+m(1n1) and dn :=m(1n2) with scalingconstants ω≃1.255071 foran and
bn and β/α for cn and dn, so the first and third equations in (18) are defined up to
a quadratic coboundary. Here we use the scale factors for frequency ε:= ωJ/α2 ≃
1.649415andenergyJ =αβ instead. Theoriginofω isstillamystery.
p
9
approximate fixed point for small Ω (actually a line of fixed points
parametrised by Ω). In particular the phonon spectrum is asymp-
totically self-similar underscaling by ε in thedirection of Ω.
Finally, note that fixing ε also determines the eigenvalue in the
ℏ-direction4 as κ= /ε 2.630716.
J ≃
2.3 Scaling of the kernel
The results above imply that the effect of including both the fre-
quencies and quantum directions, close to the TBA fixed point (for
∆u := u−uc, ∆P := P −Pγ−1(uc), Ω and ℏ small), is that the
following asymptoticrelation ofbondactions(regardedasfunctions
in parameter space) holds
(τ,υ)(∆u,∆P,Ω,ℏ) J (τ,υ)(δ∆u,η∆P,εΩ,κℏ). (19)
≃ ε
If K′ is the trace of the kernel for a chain of size F , the mth
J,Fm m
Fibonacci number,in the discretised form with J ‘time steps’ (such
that KF′m =limJ→∞KJ′,Fm), then
√ JFm−1
K′ (∆u,∆P,Ω,ℏ) K′ (δ∆u,η∆P,εΩ,κℏ) J .
J,Fm ≃ J,Fm−1 α ε
(cid:18)| | (cid:19)
Here the multiplying factor comes from the functional integration
measure due to the change of coordinates (here with diagonal mass
components µ(n) =m¯(n−1)+m¯(n))
22 11
x J−1 µ(n)ΩJ dx(j)
′ n (Ω,ℏ) = n
DJ α 4π2iℏ α
Yn (cid:16) (cid:17) Yn jY=0r | |
√ J
= αJε DJ′x˜n (εΩ,κℏ),
n (cid:18)| | (cid:19)
Y (cid:0) (cid:1)
where x˜ are thepositions of therenormalized particles.
n
2.4 Non-zero temperature scaling
Thequantumpartition function at temperatureΘcan beeasily ob-
tained from the trace of the kernel by Wick rotation, i.e. doing
2π/Ω = iβℏ, where β = 1/Θ. The quantum partition function is
−
then
(βℏ,ℏ)=K′( iβℏ,ℏ)= ′xe−βSE(x;βℏ),
Z − D
Z
4Thisresultwaspublishedin[12]containingamistake: k=Jεinsteadofthecorrect
valueκ=J/ε.
10