Microscopic approach to a class of 1D quantum critical models

0.32 MB·English

by  K. K. Kozlowski

Microscopic approach to a class of 1D quantum critical models
KarolK. Kozlowski1
IMB, UMR5584duCNRS, UniversitédeBourgogne, France.
Jean-Michel Maillet2
LaboratoiredePhysique,UMR5672du CNRS, ENSLyon,France.
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n Abstract
a
J

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Startingfromthefinitevolumeformfactorsoflocaloperators,weshowhowandun-
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derwhichhypothesisthec = 1freebosonconformalfieldtheoryintwo-dimensions

] emergesasaneffectivetheorygoverningthelarge-distanceregimeofmulti-pointcor-
h relation functions in a large class of one dimensionalmassless quantum Hamiltoni-
p
ans. In ourapproach,in the large-distancecriticalregime, thelocaloperatorsof the
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h initialmodelarerepresentedbywellsuitedvertexoperatorsassociatedtothefreebo-
t sonmodel. Thisprovidesaneffectivefieldtheoreticdescriptionofthelargedistance
a
m behaviourofcorrelationfunctionsin1Dquantumcriticalmodels.Wedevelopthisde-
scriptionstartingfromthefirstprinciplesanddirectlyatthemicroscopiclevel,namely
[
intermsofthepropertiesofthefinitevolumematrixelementsoflocaloperators.

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v
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Introduction
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5 It widely believed that the spectrum of a quantum Hamiltonian is intimately related to certain overall properties
1 of the large-distance asymptotic behaviour of its correlation functions. If, in the infinite volume L limit, a given
:
v model’sspectrum exhibitsagapbetweenafinitelydegenerated groundstateandthetowerofexcitedstatesabove
i
X it, than the correlation functions are expected to decay exponentially fast in the distance of separation between
the various operators involved in the correlator. This behaviour changes for gapless models, viz. those whose
r
a
ground state, in the L + limit, becomes directly connected to the continuum of excited states above it. For
→ ∞
those models, the correlation functions are expected to decay algebraically in the distance of separation. The
powers of the distance which arise in this algebraic decay are called critical exponents. Models having the same
values of their critical exponents are said to belong to the same universality class. In fact, it is believed that the
featuresdeterminingagivenuniversality classarerathersparseinthesensethatsolelycertainoverallsymmetries
of the model should fix it [17]. Unfortunately, the lack of any exactly solvable and truly interacting many-body
quantumcriticalmodelindimensionshigherthanonedidnotallow,sofar,foradirectandexplicitcheckofthese
properties. However, extensive numerical data and experimental results do speak in favour of such an interplay
[22].
1e-mail:[email protected]
2e-mail:[email protected]
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Thesituation improves drastically inthe caseofonespatial dimension ,whereaplethora ofexactly solvable
†
models arises: the Luttinger model (see [20] and references therein), 1+1 dimensional conformal field theories
[3,12]ormodelsthataresolvablebyoneoftheversionsoftheBetheAnsatz[6]suchasthecelebrated XXZspin-
1/2 chain or the 1D Bose gas at arbitrary repulsive coupling. The abundance of exact solutions turns these one-
dimensional modelsintoalaboratory allowingonetotesttheuniversality principle inmanyconcrete situations.
Itisworthremindingthat,inallthesemodels,twoscalescoexist: amicroscopicscaleδwhichisrelatedtothe
lattice spacing or, moregenerally proportional totheinverse oftheFermimomentum, andamacroscopic scale L
correspondingtothevolumeornumberofsitesofthemodel. Scaleinvariancecanonlyberealizedatdistances∆x
betweenlocaloperators thatrangebetweenthesetwoscalesandbutfarfromthem. Thisimposesthattheratioof
thetwoscales L/δisverylargeinsuchawaythatthereexistawholerangeofdistancesverifyingδ<< ∆x << L.
For∆xclosetoeitherthemicroscopicorthemacroscopicscalesthescalingpropertiesofthecorrelationfunctions
getmodifiedindrastic ways.
The very structure of a conformal field theory imposes a simple form for the transformation of its operators
under scaling. In its turn, this imposes that the correlation function exhibit an algebraic in the distance pre-
factor. Inparticular, two-point functions arepurely-algebraic. Theexponents driving thepower-law behaviour of
acorrelator arethen constructed intermsofthescaling dimensions oftheoperators that arebeing averaged. The
dataissuingfromconformalfieldtheoriescanthusbeusedsoastoprovideonewithquiteexplicitpredictions for
the large-distance decay of correlators in massless quantum models in one spatial dimension. In fact, there, one
canbeslightlymorepreciseinrespect tothetwopillarsonwhichthesepredictions build
inthecriticalregimeδ <<∆x << L,correlationfunctionsshouldexhibitconformalinvarianceasarguedby
•
Polyakov[43]. Thissuggeststhattheleadingcontributiontothelong-distanceasymptoticsshouldberepro-
duced by correlation functions of appropriate operators in a two-dimensional conformal field theory. Still,
theidentification ofwhichconformalfieldtheoryistobeusedandwhicharethe"appropriate" operators is
leftopenatthisstage.
The 1/L corrections to the model’s ground and low-lying excited state’s energies contain the information
•
on the central charge of the conformal field theory describing the asymptotics and the scaling dimensions
ofthetowerofoperators describing thecorrelation functions asarguedbyCardy[7,9].
The choice of"appropriate" operators, on the conformal field theory side, is done by advocating that these
•
should inheritthesymmetrythatissatisfiedbythecorrelation function onestartswith.
Torephrase theabove,Polyakov’sargumentjustifieswhyaconformalfieldtheoryshouldemergeasaneffec-
tive large-distance theory in the domain δ << ∆x << L of the model while Cardy’s observation permits one to
extract, fromtheknowledge ofthestructure ofamodel’s excitations, thequantities whichwouldcharacterise the
effectiveconformal fieldtheorydescribing themodel’slarge-distance regime.
AsimilarlineofthoughisfollowedbyexploitingtheLuttingerliquidmodelasatoolforprovidingthecritical
exponents. IsisarguedthatamodelbelongstotheLuttingerliquiduniversalityclassifithasthesameformofthe
low-lying excitations above its ground state. Theparameters describing these excitations fix the Luttinger liquid
model that is pertinent for the model of interest. The critical exponents of the model one starts with are then
deduced fromtheonescomputed explicitly fortheLuttingerliquidmodel[18,19,36].
Independently of which of the above methods one chooses to employ, one needs to access to the 1/L correc-
tions to the energy spectrum of a model’s Hamiltonian so as to be able to carry out effective predictions. The
extractionofsuchcorrections fromthespectraofvariousquantum integrablemodelswasinitiatedintheseriesof
works[10,11,29,30,31]andledtotheidentification ofthecentralchargeandscalingdimensions, fornumerous
Inthelanguageofclassicalstatisticalmechanics,thiscorrespondstothecaseofatwo-dimensionalmodel[48].
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quantum integrable models. It is worthy mentioning the work [15] where the 1/L corrections to the free energy
of certain non-integrable perturbations of the two-dimensional Ising model were obtained, on rigorous grounds
through constructive fieldtheorymethods. ByusingCardy’sformofthe1/Lcorrections, thisworkdemonstrated
that,forsufficiently smallperturbations, themodelstillhasthesamecentralcharge 1/2astheIsingmodel.
Although effective in the sense that producing explicit answers, the above techniques are more of a list of
prescriptions than a well argued from the first principles line of though that joins, argument after argument,
the structure of a given microscopic model with the resurgence of an effective field theoretic description in the
large-distance regime. Several attempts have been made in the literature to bring some elements of rigour or, at
least, some ab inicio justification of the principle. In a series of works [42, 44], by using the tools developed in
[40], Polyakov argued the arisal of a universal behaviour on the basis of perturbative field theoretic calculations.
Although constituting an important step forward, his reasonings did not allow for any control on the magnitude
of the contributions that he dropped from his calculations on the basis of some hand-waving arguments, without
mentioning the problems inherent to lacks of convergence in perturbative handling in quantum field theory. A
rigorous approach allowing one to prove the power-law decay of certain two-point functions in interacting one-
dimensional fermion models that are asufficiently small perturbation of afree fermion model wasfirstproposed
by Pinson and Spencer [41], further developed by Mastropietro [37] and then generalised in [14] to the case of
multi-point energy-energy correlation functions in certain non-integrable perturbations of the two-dimensional
Ising model. Anadaptation of this approach [4, 5]also allowed to establish the universality of certain properties
of Luttinger liquid type for a class of sufficently small perturbations of a free fermion model. The method relies
onthepossibility toprovidearigorous construction ofthepathintegralforsuchmodelsinfinitevolumeandthen
study its infinite volume and scaling limits through rigorous renormalisation group methods. Finally, we should
alsomentiontheapproach forprovingtheuniversality ofcertaintypesofbi-dimensional Isinglattcies -whichare
related to one-dimensional massless quantum Hamiltonians by means of the correspondence proposed in [48]-
thatwasdeveloped bySmirnovin[47].
Thepresentpaperintroducesaconvenientmicroscopicdescriptionforthespectrum,spaceofstatesandmatrix
elements of local operators in a class of one-dimensional massless quantum models. Our setting allows us to
construct, in the large-distance regime δ << ∆x << L, a one-to-one map (up to higher order corrections in
the distance) between the relevant to the large-distance regime sub-space of the model’s Hilbert space and the
free boson Hilbert space. The local operators of the original model are then represented in terms of a collection
of vertex operators acting in the free boson Hilbert space. The setting we introduce, as shown in the works
[23,25,35],isclearly verified foralarge-class ofquantum integrable models, theXXZspin-1/2 chain being the
most prominent example. However, we do trust that the overall hypothesis that we lay down for the structure of
themodel’s observable isquite universal andatleastencompasses severalinstances ofonedimensional quantum
liquids [16]. This is supported, e.g. by perturbative calculations around a free model [4, 5, 45]. Furthermore all
objectsthatwehandlearestandard withinthephenomenological approach tointeracting Fermisystems[16,38].
Goingslightlydeeperintothedetailsofthephysics’jargon,themainassumptionsonwhichoursettingbuilds
arethat
in the large-volume limit, the relevant part to the critical regime of the model’s spectrum is purely con-
•
structed intermsofparticle-hole excitations3;
the form factors -expectation values of local operators- taken between two states realised in terms of exci-
•
tations thatarealllocated inanimmediate vicinity oftheFermizone- admitastructure descending froma
3Inthefollowing,forsimplicity,weshallassumeapurelyparticle-holeexcitationspectrum.Thisallowsonetolightenthediscussions.
Westressthatthisassumptionisnotalimitationinthat,forinstance,theboundstatesareexpectedtosolelyproduceexponentiallysmall
correctionsinthedistance.
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local repulsion principle between the interacting momenta of the particle and hole building up the excited
state.
The main result of the paper can be phrased as follow. Let (x ),..., (x ) be local operators located at at
1 1 r r
O O
positions x ,...,x andassociated withsomeone-dimensional quantum modelinfinite(but large) volume Land
1 r
having the Hilbert space h . The operators is assumed to induce solely transitions between spin sectors
phys s
O
differing bysomefixedinteger o . Then, inthelarge δ x x Llimit, ther-point ground-to-ground state
s a b
≪ | − | ≪
expectation valueoftheseoperators satisfies
(x ) (x ) O (ω ) O (ω ) (0.1)
1 1 r r 1 1 r r
DO ···O Ehphys ≃ D ··· Eheff
where the sign is to be understood as an equality up to the first leading correction arising in each oscillating
≃
harmonics inthespacing difference. Theoperator O , s = 1,...,r,appearing intherhsof (0.1)actintheHilbert
s
spaceh associated tothefreebosonmodelwhiletheexpectation valueonthelhsof (0.1)istakenintheHilbert
eff
spaceh oftheinitialmodel;theyarebuiltupfromthefreebosonvertexoperators V(ν,κ;ω):
phys
Os(ωs) = Fκ Os · 2Lπ ρ(νs(q)−os+κ)+ρ(νs(−q)+κ)·e2ipFκxs·VL −νs(−q),−κ | ω−s VR νs(q)−os,κ−os | ω+s (0.2)
Xκ Z (cid:0) (cid:1) (cid:16) (cid:17) (cid:0) (cid:1) (cid:0) (cid:1)
∈
Inthisformula,
• ω±s = e±2iπxLsα± isthevariableattachedtotheleft(−)orright(+)endoftheFermizonewhichparametrises
the position oftheoperator onthefree boson model side. Thefactors α arecertain re-scaling coefficients
±
whichtakeintoaccountthevariationofthedressedmomentumattheendpointsoftheFermizone. Theare
expressed intermsofthephysicalobservables ofthemodel,cf. (2.22);
theindicesLandRrefertotheleftorrightcopyofafreebosonHilbertspacehonwhichthevertexoperator
•
acts,viz. h = h h ;
eff L R
⊗
V ν,κ ω is the vertex operator, cf. (1.27), which connects states belonging to sectors differing by the
• |
ch(cid:0)arge κ; (cid:1)
ν ( q) are the values of the so-called shift function relatively to the ground state, cf. (2.29), and evaluated
s
• ±
attheleft(-q)orright(q)endoftheFermizone. Wedostressthatitisafunction ofthechargeκandofthe
operator’s spino .
s
ρ(ν (q) o +κ)andρ(ν ( q)+κ)isthetowerofscaling dimensionsassociated withtheoperator ;
s s s s
• − − O
,κ Z,isthetoweroftypicalformfactorsoftheoperator takenbetweenthegroundstateandthe
κ s s
• F O ∈ O
low(cid:0)est(cid:1)-lying excitedstatebelonging totheκ-Umklappsector.
We refer the reader to Sub-Section 3.3.1 where explicit examples for the above-listed quantities are given in the
caseofthefundamentallocaloperatorsfortheXXZchain. InSub-Section3.3.2weshowhowthestructureofthe
Luttinger-liquid criticalexponents canberecoveredwithinourformalism.
What equation (0.1) states is that, in the weak sense -i.e. for expectation values- and for large distances of
separationalltheessenceofthephysicaloperator (x )actingintheHilbertspaceh isgraspedbytheeffective
a a phys
O
operator Os(ωs) acting in the free boson model’s Hilbert space heff. Formula (0.2) provides an effective map, in
the critical regime, between local operators of the physical models and vertex operators in the corresponding
conformal field theory. This map is completely determined by the typical form factors , κ Z, of the
κ s
F O ∈
operators a(xa)andthevaluesνs( q)takenbytheso-calledshiftfunctionontheleft/righten(cid:0)dpo(cid:1)intoftheFermi
O ±
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zone. A more detailed description of the correspondence and the constituents ofthe operator O can be found in
s
the core of the paper. We do stress that the expectation value appearing in the rhs of (0.1) is trivial to estimates
bymeansofthefreefermionexchangerelations, cf. (D.5)-(D.6). Wedostressthatthecorrespondence (0.1)-(0.2)
onlyprovides onewiththeleading ordertoeachoscillatory harmonics. Thisleading order onlyinvolves primary
operators. However, should one be interested in sub-leading corrections then one would need to supplement the
expansion (0.2) with additional terms that would involve descendants. Obtaining these terms is, in principle,
possible withinthemethoddeveloped inthepresent paper,although woulddemandmorecomputations.
Ontopofprovidingthemicroscopicoriginoftheappearance ofac= 1conformalfieldtheoryasaneffective
theory in the large-distance regime, our setting supports the interpretation of the universality hypothesis that has
been advocated by one of the authors in [34]. Namely, within our approach, the leading power-law contribution
associated with a given oscillating harmonic in the large-distance asymptotic expansion of a correlator stems
from a saddle-point like contribution that is extracted from the form factor series expansion of the correlator. In
the large-distance regime studied in the present paper, the saddle-point is located on the two ends of the Fermi
zone. In the time-dependent case, one should as well incorporate all the possible critical points of the dressed
momentum/energy combination xp tε,cf. [26]. Itisthesingular structure, viz. localbehaviour, ofanoperator’s
−
form factor in the vicinity of the saddle-point that fixes the value of the critical exponents. Thus, two models
belonging to the same universality class -in the sense that their appropriate correlators share the same critical
exponents-havetosharethesamesingularitystructureoftheirformfactors. Notethattheregularpartoftheform
factorscandifferfromonemodeltoanotherinthatthattheysolelyimpactthevalueoftheamplitudes,viz. thenon-
universal partofthelarge-distance asymptotic expansion. Recallthatthestandard formulation oftheuniversality
hypothesis states that two models sharing the same overall symmetries share the same critical exponents. In this
light, it would appear that the set of symmetries of a given model completely fixes the singularity structure of
its form factors. However, it is not clear for us at the moment how one could derive the singular structure of a
model’s form factors solely building on its symmetries. Nonetheless, it seems more satisfactory to us to think of
universality in terms of classes of models sharing the same singular structure of their form factors. Indeed, this
criterionispreciseandcanbechecked,incertaincases,byexplicitcalculations, sayinaperturbativeregime. The
matter is that there is, a priori, no criterion allowing one to say that one has identified all the symmetries of a
modelthatarepertinentforfixingitsuniversalityclass. Itcouldwellbethattwomodelshaveapparentlythesame
symmetry structure but that one has missed some important yet quite oblivious symmetry of one of the models
thatwouldimplythat,infact,themodelsbelongtotwodistinctuniversality classes. Itisworth,inthisrespect, to
remind that two apparently very similar models, the 2D Ising and the the eight-vertex model in its lattice model
formulation belongtofundamentally distinctuniversality classes.
The paper is organised as follows. The present section is the introduction. In Section 1 we recall the free
fermion based description of the space of states of the free boson model. We also present new formulae for the
expectation values of exponents of specific current operators. In Section 2 we present the general framework
-properties of the model’s spectrum and form factors- that allow us to derive the correspondence with the free
boson model. Finally, in Section 3, we establish the main result of the paper, namely the set of formulae (0.1)-
(0.2) that appeared in the introduction. The paper contains four appendices. In Appendix A, we briefly review
the special functions which arise in the course of our study. In Appendix B we compute some two-dimensional
integrals which are of interest to the probelm. In Appendix C, we prove Proposition 1.1. Finally, in Appendix
D, we show how one can recover, starting from the formalism developed in the present paper, the value of the
multi-point restricted sumsthatwerefirstintroduced in[27].
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1 A free fermion description of the free boson model
Inthepresentsection,werecallthefreefermionbasedconstructionofthespaceofstatesforthefreebosonmodel.
Our presentation basically follows the notations and conventions that can be found in the excellent review paper
[1]. The various results found in this review originate from a long series of developments which started with the
cornerstone workofKyoto’sschool (Jimbo,MiwaandSato)inthelate’70’sonholonomicquantum fields[21].
1.1 Overalldefinitions and generalities
1.1.1 Thespaceofstates
Weconsiderasetoffermionicoperators{ψn}n∈Zandtheir∗associates{ψ∗n}n∈Zwhichsatisfytheanti-commutation
relations
ψ ,ψ = ψ ,ψ = 0 and ψ ,ψ = δ , (1.1)
{ n m} { ∗n ∗m} { n ∗m} n,m
where δ isthe Kronecker symbol. Wealso assume the existence ofavacuum vector 0 which satisfies to the
n,m
|
properties (cid:11)
ψ 0 = 0 for n < 0 and ψ 0 = 0 for n 0 . (1.2)
n| ∗n| ≥
(cid:11) (cid:11)
Thedualvacuum 0 fulfilstheanalogous properties
|
(cid:10)
0 ψ = 0 for n < 0 and 0 ψ = 0 for n 0 . (1.3)
| ∗n | n ≥
(cid:10) (cid:10)
Starting fromthevacuum (resp. thedualvacuum)oneconstructs vectors (resp. dualvectors) through arepetitive
actionofthefermionoperators. Suchvectorsareparametrised, inanaturalway,bythesets
= p np ; h nh (1.4)
Jnp;nh { a}1 { a}1
n o
consisting oftwocollections ofordered integers1 p < < p and1 h < < h . Itisconvenient, fora
≤ 1 ··· np ≤ 1 ··· nh
deeperphysical insight, tothinkoftheintegers p np asalabelling oftheparticle-like excitations andtothinkof
theintegers h nh asalabelforthehole-like ex{citaa}t1ions. Tobemoreprecise, toeachset defined asabove,
{ a}1 Jnp;nh
oneassociates thevector
= ψ ψ ψ ψ 0 (1.5)
|Jnp;nh ∗−h1··· ∗−hnh · pnp−1··· p1−1|
(cid:11) (cid:11)
andthedualvector
= 0 ψ ψ ψ ψ . (1.6)
Jnp;nh| | ∗p1−1··· ∗pnp−1· −hnh ··· −h1
(cid:10) (cid:10)
TheHilbertspacehofthemodelisthendefinedasthespanofthevectorsintroduced above
1 p < < p
h = span(cid:26)|Jnp;nh(cid:11)with np,nh ∈ N and 1≤≤ h11 <······ < hnnhp pa,ha ∈ N∗(cid:27). (1.7)
Note that, whenthe number ofparticle and hole-like integers coincide, i.e. n = n = n, one canidentify the
p h
set withaYoungdiagram. Theone-to-onemapisobtainedbyinterpretingtheintegers p n ; h n asthe
Jnp;nh { a}1 { a}1
n o
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Frobeniuscoordinates oftheYoungdiagram . Suchanidentificationisalsopossiblewhenn , n . Thebijection
† p h
is, however, more complicated. We refer the interested reader to the proof of Lemma 1.1 where one can find all
of the necessary details for constructing such a bijection. Note that such a bijection was constructed for the first
time,although indirectly, inAppendixA.3of[24].
TheHilbertspace halsoenjoys ofanother basis Y;ℓ whichmakestheconnection withaYoungdiagram Y
|
explicit. Thisnewbasissinglesoutthecollectionoftheso-(cid:11)callvacuumanddualvacuumstateshavingaprescribed
charge ℓ:
ψ ψ 0 ℓ > 0 0 ψ ψ ℓ > 0
|ℓ =  ψℓ−1··ψ· 0|0 (cid:11) ℓ < 0 and ℓ| =  (cid:10) 0|ψ∗0··· ∗ℓψ−1 ℓ < 0 . (1.8)
(cid:11)  ∗ℓ··· ∗−1| (cid:10)  | −1··· ℓ
 (cid:11)  (cid:10)
General states of this basis are build asequal in number particle-hole excitations overthe vaccua ℓ , resp. their
|
dualvaccua ℓ . Let (cid:11)
|
(cid:10)
1 α < < α
Y = α n : β n with ≤ 1 ··· n (1.9)
{ a}1 { a}1  1 β < < β
n o  ≤ 1 ··· n
betheFrobeniuscoordinates oftheYoungdiagram Y. Thesecond basiswementionedabovetakestheform
Y;ℓ = ψ ψ ψ ψ ℓ
| ∗ℓ−β1··· ∗ℓ−βd ℓ+αd−1··· ℓ+α1−1| . (1.10)
Y;ℓ(cid:11) = ℓ ψ ψ ψ ψ (cid:11)
| | ∗ℓ+α1−1··· ∗ℓ+αd−1 ℓ−βd ··· ℓ−β1
(cid:10) (cid:10)
Infact,onecanevenconsider mixturesofthebasis Y;ℓ and ,namelythebasis
| |Jnp;nh
(cid:11) (cid:11)
;ℓ = ψ ψ ψ ψ ℓ . (1.11)
|Jnp;nh ∗ℓ−h1··· ∗ℓ−hnh · pnp+ℓ−1··· p1+ℓ−1|
(cid:11) (cid:11)
Itiseasytoconvince oneselfthat
Span Y;ℓ+r : Y Young diagram = Span : sets n n = ℓ+r
| |Jnp;nh Jnp;nh p− h
n (cid:11) o n (cid:11) o
= Span ;ℓ : sets n n = r . (1.12)
|Jnp;nh Jnp;nh p − h
n (cid:11) o
1.1.2 Thespaceofoperators
The fermionic operators ψ and ψ can be thought of as building bricks allowing one to construct more general
j ∗j
operatorsontheHilbertspaceh. Thesimplestclassofoperatorisobtainedasalinearcombinationinthefermions:
v = V ψ resp. w = W ψ , (1.13)
m m ∗ m ∗m
Xm Xm
whereV ,W arebounded sequences. Thefieldandconjugated fieldoperators
m m
Ψ(z) = ψ zj and Ψ (z) = ψ z j (1.14)
j ∗ ∗j −
Xj Z Xj Z
∈ ∈
areanarchetype ofsuchoperators .
weadopttheslightlyunusualconventionwheretheoriginoftheFrobeniuscoordinatesisthediagonalsothatthesestartfrom1.
†
7
Thesecondimportantclassofoperators isobtained bymeansofabilinearpairingofthefermionsrealisedby
aninfinitematrix Ahavingbounded entries:
A ψψ . (1.15)
ij i ∗j
iX,j Z
∈
Expressions of the type (1.13) or (1.15) may look a bit formal in particular due to convergence issues of the
sums. However, all the sums of interest truncate as soon as one computes matrix elements taken between the
fundamental systemofbasisvectorslabelledbythesets . Itisinthissensethatalloftheaboveexpressions
Jnp;nh
shouldbeunderstood.
The most fundamental example of an operator built through a bilinear pairing is the charge operator which
takestheexplicitform
J = ψ ψ ψ ψ . (1.16)
0 k ∗k − ∗k k
Xk 0 Xk<0
≥
It isreadily seen that the vector isassociated with the eigenvalue n n ofthe charge operator J . We
|Jnp;nh p − h 0
willsometimessaythatthevector|Jnp;(cid:11)nh hascharge np−nh.
In other words, the charge operator in(cid:11)duces a grading of the Hilbert space h into the direct sum of spaces hℓ
having afixedchargeℓ
h = h with h = span : n n = ℓ . (1.17)
ℓ ℓ |Jnp;nh p− h
Mℓ Z n (cid:11) o
∈
The current operators provide one with another example of important operators that are bilinear in the fermions.
Toeachhalf-infinite sequence t = tk k N oneassociates thecurrentoperators
{ } ∈ ∗
J t = t J with J = ψ ψ for k , 0 . (1.18)
±(cid:0) (cid:1) Xk 1 k · ±k k Xj Z j ∗j+k
≥ ∈
Itisreadilychecked thatthecomponents ofthecurrentoperator satisfythealgebra
J ,J = kδ . (1.19)
k ℓ k, ℓ
−
(cid:2) (cid:3)
These commutation relations allow one to interpret the currents J as bosonic operator modes (see e.g. [1]).
k
Furthermoresince, fork N ,onehas
∗
∈
J 0 = 0 and 0 J = 0 (1.20)
k k
| | −
(cid:11) (cid:10)
itfollowsthat onecaninterpret Jk k N asbosonic creation operators while J k k N asthe bosonic annihilation
{ } ∈ ∗ { − } ∈ ∗
operators. It ison the basis of such an observation that the construction ofthe free boson field theory is made in
thefreefermionmodelwehaveintroduced sofar.
Itisalsostraightforward tocheckthecommutation relations
J ,ψ = ψ , J ,ψ = ψ . (1.21)
k ℓ ℓ−k k ∗ℓ − ℓ+k
(cid:2) (cid:3) (cid:2) (cid:3)
The above commutation relations can be conveniently encoded on the level of the field and conjugated field
operators. Asamatteroffact,then,theytaketheformofthefollowingexchange relations
Ψ(z)·eJ±(t) = e−ξ(t,z±1)·eJ±(t)·Ψ(z) and Ψ∗(z)·eJ±(t) = eξ(t,z±1)·eJ±(t)·Ψ∗(z) (1.22)
8
whereξ(t,z) = t zk. Givenageneric sequence t,this isthe mostcompact form oftheexchange relation
k 1 k ·
sincethesumdefiPnin≥gξ cannotbecomputed explicitly. However,forthespecificchoice t = t where
±
νω k νωk
t = − and t = (1.23)
+ k − k − k k
(cid:0) (cid:1) (cid:0) (cid:1)
thesumreducestotheTaylorexpansionofthelogarithm. Itisthe t = t choicethatplaysaparticularlyimportant
±
role in our analysis in that it is associated with the construction of vertex operators. We therefore introduce a
specialnotation forthecurrentoperators associated withtheparameters t ,namely
±
ω k
J (ν,ω) J (t ) = ν ∓ J . (1.24)
k
± ≡ ± ± ∓ Xk 1 k · ±
≥
Inthecaseoftheoperators J (ν,ω),theexchange relations (1.22)particularise to
±
z 1 ν z 1 ν
Ψ(z) eJ (ν,ω) = 1 ± ∓ eJ (ν,ω) Ψ(z) and Ψ (z) eJ (ν,ω) = 1 ± ± eJ (ν,ω) Ψ (z) . (1.25)
· ± − ω ± · ∗ · ± − ω ± · ∗
(cid:16) (cid:16) (cid:17) (cid:17) (cid:16) (cid:16) (cid:17) (cid:17)
Thereisalsoanotheroperatorthatplaysaroleinoursetting,theso-calledshiftoperatorwhichweshalldenote
aseP. Theshiftoperatormapsh ontoh . Infact,theoperatorParisingintheexponentistheconjugateoperator
ℓ ℓ+1
to J . Althoughtheshiftoperatorhasnosimpleexpression intermsofthefermions, ittakesaparticularly simple
0
forminthebasisofhsubordinatetoYoungdiagrams,cf. (1.10). Indeed,anyintegerpowererPoftheshiftoperator
satisfieserP Y;ℓ = Y;ℓ+r ,viz.
| |
(cid:11) (cid:11)
erP = Y;ℓ+r Y;ℓ (1.26)
| |
XY,ℓ (cid:11)(cid:10)
wherethesumrunsoverallintegersℓandallYoungdiagrammsY(see[1]formoredetails).
Weendthis sub-section byintroducing ther-shifted bosonic vertexoperators which playacentral roleinthe
correspondence:
V (ν,r ω) = eJ (ν+r,ω) eJ+(ν+r,ω) erP . (1.27)
−
| · ·
1.1.3 TheWicktheorem
Inthis sub-section wereview thestatement ofWick’s theorem in thecase ofan insertion of group-like elements.
ThisnamereferstoaclassofoperatorsG satisfying totheso-called basicbilinearcondition:
G ψ G ψ = ψ G ψ G . (1.28)
· j ⊗ · ∗j j· ⊗ ∗j ·
Xj Z(cid:16) (cid:17) (cid:16) (cid:17) Xj Z(cid:16) (cid:17) (cid:16) (cid:17)
∈ ∈
Group-like elements single out, among other things, because they allow one for a powerful generalisation of the
Wicktheorem whichweshallrecallattheendofthissub-section.
The operators we have introduced so far provide several examples of group-like elements. For instance, it is
easily checked that the operators eJ (t) in general, eJ (ν,ω) in particular, and erP all satisfy to the basic bilinear
± ±
condition. It is likewise trivial to establish that, ifG,G are group-like elements, then so is their product G G .
′ ′
·
Thesefactsensurethatthatther-shifted vertexoperators V(ν,r ω)aregroup-like elements.
|
We stress that group-like elements need not to be invertible. For instance, given any group element G and a
collection v ,...,v ,resp. w ,...,w ,oflinearcombinations ofthebasicfermionoperators ψ ,resp. ψ ,viz.
1 n ∗1 ∗n j ∗j
v = V ψ resp. w = W ψ , (1.29)
k km m ∗k km ∗m
Xm Xm
9
theoperators
G = w w v v G and G = G w w v v (1.30)
v;w ∗1··· ∗n n··· 1· v;w · ∗1··· ∗n n··· 1
e
arealsoagroup-like. Weshallnotenterdeeperintothedetailhereandreferthereaderto[1]formoredetails.
Group-like elementsallowforthefollowinggeneralisation ofWick’stheorem.
Theorem1.1 Letv ,resp. w , j= 1,...,nbeanyoperatorslinearinthefermionsandG,G group-likeelements.
j ∗j ′
Thentheassociated expectation valueadmitsadeterminant representation:
0 Gw w v v G 0 0 Gw v G 0
| ∗1··· ∗n n··· 1 ′| = det | ∗p k ′| . (1.31)
(cid:10) 0 G G 0 (cid:11) n"(cid:10) 0 G G 0 (cid:11)#
| · ′| | · ′|
(cid:10) (cid:11) (cid:10) (cid:11)
1.2 Expectation values ofexponents ofcurrent operators
In the present sub-section, we compute the expectation value of the operators eJ (ν,ω)eJ+(ν,ω) between any two
−
states ofthebasis. Thecorrespondingresultwillprovideafirstbricktowardsthecorrespondence
|Jnp,nh |Jnk,nt
withthefreeb(cid:11)osonmo(cid:11)delthatwillbebuiltinSection3.
Priortostating theresult, wefirstneedtointroduce afewhandynotations. Giventwosetsofintegers
= p np ; h nh and = k nk ; t nt (1.32)
Jnp,nh { a}1 { a}1 Jnk,nt { a}1 { a}1
n o n o
onedefinesthesetfunctions
nk nt
nh 1−kb−ha+ν np pa+tb+ν−1
b=1 b=1
̟ ; ν = Q (cid:0) (cid:1) Q (cid:0) (cid:1) . (1.33)
Jnp;nh Jnk;nt | ( nt )· ( nk )
(cid:16) (cid:17) Ya=1 tb ha+ν Ya=1 pa kb+ν
b=1 − b=1 −
Q (cid:0) (cid:1) Q (cid:0) (cid:1)
and
D(cid:16)Jnp;nh | ν,ω(cid:17) = (cid:18)sinπ[πν](cid:19)nh ·Yan=p1(ωpa−1Γ pap+a ν !)·Yan=h1(ωhaΓ hah−a ν !)
np nh
(p p ) (h h )
b a b a
− · −
a>b a>b
Q Q . (1.34)
× np nh
(p +h 1)
a b
a=1b=1 −
Q Q
Thesefunctions appearasthebuilding blocksoftheso-called discreteformfactor
F(cid:16)Jnp;nh;Jnk;nt | ν,ω(cid:17) = (−1)np+nt ·(−1)(np−nh)(2np−nh+1) ·(cid:18)sinπ[πν](cid:19)np−nh ·D(cid:16)Jnp;nh |ν,ω(cid:17)
×D Jnk;nt | −ν,ω−1 ·̟ Jnp;nh;Jnk;nt | ν . (1.35)
(cid:16) (cid:17) (cid:16) (cid:17)
In(1.35)wemadeuseofthehypergeometric-like notations forratiosofΓ-functions, cf. AppendixA.
It is the above discrete form factor that enters in the description of the expectation values of the current
operators eJ (ν,ω)eJ+(ν,ω) takenbetweenthestates and .
− |Jnk;nt Jnp;nh|
(cid:11) (cid:10)
10