From Defects to Boundaries

0.23 MB·

by  Z. Bajnok

FROM DEFECTS TO BOUNDARIES
1 2
Z. Bajnok and A. George
1
HAS, Theoreti
al Physi
s Resear
h Group,
Institute for Theoreti
al Physi
s, Eötvös University
5
0 H-1117 Budapest, Pázmány s. 1/A, Hungary.
2
0
Department of Mathemati
s, The University of York
2
Heslington, York, YO10 5DD, England.
n
a
J
Abstra
t. In this paper we des
ribe how relativisti
(cid:28)eld theories
ontaining defe
ts

1
are equivalent to a
lass of boundary (cid:28)eld theories. As a
onsequen
e previously derived
1
results for boundaries
an be dire
tly applied to defe
ts, these results in
lude redu
tion

3 formulas, the Coleman-Thun me
hanism and Cut
osky rules. For integrable theories the
v defe
t
rossingunitarityequation
anbederivedanddefe
t operatorfound. Forageneri
9 purely transmitting impurity we use the boundary bootstrap method to obtain solutions
9
of the defe
t Yang-Baxter equation. The groundstate energy on the strip with defe
ts is
1
also
al
ulated.
4
0
4
1. Introdu
tion
0
/
h Over the last de
ade there has been a growing theoreti
al and experimental interest in
t
- systems
ontaining defe
ts or impurities. Although experimental investigations, su
h as
p
e into quantum dots, require a more general des
ription the theoreti
al fo
us has largely
h been on the integrable aspe
ts of defe
t systems. The defe
t Yang Baxter equations (or
:
v RT relations), unitarity and defe
t
rossing unitarity relations were formulated in [1, 2℄,
i
X however no derivation of defe
t
rossing unitarity was presented, it was simply
onje
tured
r by analogy to the bulk theory. They showed the absen
e of simultaneous transmission and
a
re(cid:29)e
tion from an integrable defe
t for theories with diagonal bulk s
attering matri
es,
this was generalized for non-diagonal theories with very general, even dynami
al, defe
ts
in [3℄. An analysis of RT algebras [4, 5℄ and a proposal to avoid this `no go theorem'
for simultaneous transmission and re(cid:29)e
tion
an be found in [6℄. Re
ently a Lagrangian
des
ription of integrable defe
ts has been introdu
ed in [7, 8℄, in
luding a dis
ussion on
how the boundary Lax pair approa
h may be adapted for defe
t systems. For a quantum
me
hani
al treatment of the defe
t problem we refer to [9℄, while for higher dimensional

onformal defe
ts to [10, 11, 12℄.
In this paper, by extending the `folding tri
k' (formulated previously for free [13, 14℄
or
onformal [11, 15℄ theories), we show that any relativisti
defe
t theory, integrable or
not,
an be des
ribed as a
ertain type of boundary theory. As a
onsequen
e previous
1
bajnokafavant.elte.hu
2
ag160gar(cid:28)eld.elte.hu
1
FROM DEFECTS TO BOUNDARIES 2
results developed for boundary theories, su
h as perturbation theory, redu
tion formulas,
Coleman-Thun me
hanism and Cutkosky rules [16, 17℄,
an be dire
tly applied to defe
ts.
1+1
For simpli
ity we restri
t the s
ope of this letter to real relativisti
s
alar (cid:28)elds in
dimensionsinordernotto
ompli
atethenotationsand
on
entrateontheprin
ipalissues.
Similarlywerestri
tourselvestostrong,i.e. parityinvariant,bulktheories. Generalizations
of these results, to higher dimensions, or to more (cid:28)elds are straightforward.
By applying the defe
t-boundary
orresponden
e for integrable theories we
an use the
results from integrable boundary theories, su
h as the derivation of boundary
rossing
unitarity [18℄, the de(cid:28)nition of the boundary state, boundary bootstrap, et
. for defe
ts.
Sin
e the defe
t theory and the proposed boundary theories are equivalent we
an use
results from the study of defe
ts, su
h as the no go theorem, to make statements about
the behaviour of these boundary theories. We show how one
an
onstru
t solutions of
the boundary Yang-Baxter equation by bootstrapping on a known solution. We apply this
method for purely transmitting impurity by applying the defe
t-boundary
orresponden
e
and obtain non-trivial solutions for any bulk theory. These solutions have a generi
form
and
orrespond to parti
les with imaginary rapidities `trapped' at the defe
t. Finally
we use the defe
t operator, obtained from the boundary state via the defe
t-boundary

orresponden
e, to perform a thermodynami
Bethe ansatz
al
ulation of the ground state
energy of the system de(cid:28)ned on the strip with defe
ts.
2. Defe
t-boundary
orresponden
e
ΦL x ≤ 0 x = 0
If a (cid:28)eld, , is restri
ted onto left half line by a boundary lo
ated at the
bulk and boundary intera
tions are des
ribed by the Lagrangian
1 1
L = Θ(−x) (∂tΦL)2 − (∂xΦL)2 −VL(ΦL) −δ(x)U(ΦL) .
(cid:18)2 2 (cid:19) (1)
U(ΦL)
Here the boundary potential is understood to only depend on the (cid:28)eld through the
ΦL(0,t)
value of the (cid:28)eld at the boundary . The
orresponding equation of motion and
boundary
ondition are
∂VL ∂U
(−∂t2 +∂x2)ΦL(x,t) = ∂ΦL ; x ≤ 0 , ∂xΦL(x,t)|x=0 = −∂ΦL . (2)
The Diri
hlet boundary
ondition
an be des
ribed either as a limit of the above one, or
U = 0 ΦL(0,t) = ϕ

an be obtained dire
tly from the Lagrangian with but demanding ,
ϕ
where is a
onstant.
P : x ↔ −x
The parity trxan≥sfo0rm, , of this theoryLis another theory, whi
h livRes on the
right half line . We
all the original theory and its parity transform , whi
h
satis(cid:28)es the equation of motion and boundary
ondition
∂VR ∂U
(−∂t2 +∂x2)ΦR(x,t) = ∂ΦR ; x ≥ 0 , ∂xΦR(x,t)|x=0 = ∂ΦR . (3)
VR = VL
Thetwotheoriesareequivalent: andthesolutionsofthetwRotheoriesare
Lonne
ted
bytheparitytransformation. Similarlytheparitytransformofthe theoryisthe theory.
FROM DEFECTS TO BOUNDARIES 3
Observe that this is not a symmetry of one theory but merely two di(cid:27)erent des
riptions of
the same theory.
The quantum version of these theories are also equivalent, whi
h
an be seen as follows.
The quantum theories are de(cid:28)ned following [16℄: The intera
tion is supposed to swit
hed
on adiabati
ally in the remote past and swit
hed o(cid:27) in the remote future. The asymptoti
Hilbert spa
es are then the free Hilbert spa
es and, being equivalent, are related by a uni-
R
tary transformation
alled the -matrix. This matrix is
onne
ted to the Greens fun
tions
via the boundary redu
tion formula, and
an be
omputed in Lagrangian perturbation
theory. L
Now let us see how the identi(cid:28)
ation of the two theories goes: The theory has as-
|iniL ni ki > 0
ymptoti
`in' states, ,
ontaining free parti
les with real momentum , (i.e.
x = −∞
traveling toward the boundary), lo
ated at in the order of de
reasing momentum,
|outiL
highest momentum furthest to the left. The asymptoti
`out' states, , are
omposed
n k < 0
f f
of free parti
les with real momentum , i.e. traveling away form the boundary,
x = −∞
also lo
ated at R in the order of in|i
nreiRasing momentum, most nkegia<tiv0e furthest to
the left. In the theory the `in' states, ,
ontain parti
les with , while `out'
|outiR kf > 0
states, , with momentum , and both the parti
les of the `in' and `out' states
x = ∞
are lo
ated at and ordered su
h that the fastest moving parti
les are furthest to the
right. Clearly the parti
les of both theories behave in the same way, (cid:28)rst traveLling towards
theRboundary, then, after re(cid:29)e
ting, away from the boundary. The states oPf and those
of are equivalent and are mapped to ea
h other by the parity operator , whi
h does
nothing but
hanges the sign of the parti
les momenta.
R
The -matrix is a unitary operator whi
h
onneR
tLs the aRsRymptoti
`in' stateLs to thRe
`out' states of ea
h theories. They are written as and respe
tively for and
and both
an be
omputed from the
orresponding boundary redu
tion formulae. These
ΦL ΦR
formulae
ontain the
orrelation fun
tions of the (cid:28)elds and and
an be
omputed
using perturbation theory derived from (2) and (3), respe
tively. Any term in the two
perturbative expansions (cid:21) and su
h a way in the
orrelation fun
tions (cid:21) are related by the
P
parity operator, , moreoverRthe redu
tion formulae are the Ppa:riRtyLt↔ranRsRformed oLf ea
h
Rother. As a
onsequen
e the -matri
es are related by parity, . Thus and
are equivalent theories on theLquantum level. R
Defe
t theories are simply an and a (possibly di(cid:27)erent) type theory
onne
ted at
the origin by the defe
t, they
an be des
ribed by the Lagrangian
1 1
L = Θ(−x) (∂tΦL)2 − (∂xΦL)2 −VL(ΦL)
(cid:18)2 2 (cid:19)
1 1
+ Θ(x) (∂tΨR)2 − (∂xΨR)2 −WR(ΨR) −δ(x)U(ΦL,ΨR) .
(cid:18)2 2 (cid:19) (4)
ΦL ΨR
If the (cid:28)elds and are not the same then the equation of motion and boundary
ΦL ΨR ΨR ΦR

onditions for are given by (2) and those for by (3) with repla
ing . Note
that, for a general defe
t, there is no requirement that the values of the two (cid:28)elds at the
defe
t be the same. If the two (cid:28)elds are in fa
t the same we
all the defe
t system an
FROM DEFECTS TO BOUNDARIES 4
`impurity' system. It is des
ribed by the Lagrangian
1 1
L = (∂ φ)2 − (∂ φ)2 −V(φ) −δ(x)U(φ) ,
t x
(cid:18)2 2 (cid:19) (5)
φ
where the (cid:28)eld exists on the whole line. This system has the usual equation of motion
and an `impurity
ondition'
∂U
lim(∂ φ(x,t)| −∂ φ(x,t)| ) = .
x x=ǫ x x=−ǫ
ǫ→0 ∂φ (6)
Note that the left hand side of (6) does not redu
e to zero as solutions for these (cid:28)elds
generally have dis
ontinuities in the gradient at the impurity. The impurityLagrangian(5)
φ = ΦL V(φ) = VL(ΦL)

an be written in the form of a defe
t Lagrangian (4) if we write ,
x ≤ 0 φ = ΨR V(φ) = WL(ΨR) x ≥ 0
for and , for , and impose the requirement that
ΦL|x=0 = ΨR|x=0 =: φ0
. The boundary
onditions for the (cid:28)elds of the defe
t Lagrangian
now be
omes equivalent to the impurity
ondition
∂U
∂xΨR|x=0 −∂xΦL|x=0 = .
∂φ (7)
0
We will generally assume that we are dealing with a defe
t system, with the understanding
that impurities
an be dealt with in a similar fashion by applying the impurity
ondi-
tions instead of defe
t
onditions. When there are signi(cid:28)
ant di(cid:27)eren
es between how the
impurity and defe
t theories need approa
hingRwe will expΨliR
it→ly dΨisL
uss ea
h separately.
By making a parity transformation on the theory, , we
an equivalently
des
ribe the defe
t theory as
1 1
L = Θ(−x) (∂tΦL)2 − (∂xΦL)2 −VL(ΦL)
(cid:18)2 2 (cid:19)
1 1
+ Θ(−x) (∂tΨL)2 − (∂xΨL)2 −WL(ΨL) −δ(x)U(ΦL,ΨL) .
(cid:18)2 2 (cid:19) (8)
ΨL ΨR
The equations of motion and boundary
ondition for and are related like (2) and
ΦL
(3). The equation of motion and boundary
ondition for remain un
hanged. For an
impurity theory the same transformation
an be made after it is written as a defe
t theory
as des
ribed above. The boundary
onditions be
ome
∂U
∂xΨL|x=0+∂xΦL|x=0 = − ; ΦL|x=0 = ΨL|x=0 =: φ0 .
∂φ (9)
0
The Lagrangian (8) des
ribes a boundary (cid:28)eld theory with two self intera
ting (cid:28)elds, ea
h
restri
ted to the left half line, these (cid:28)elds only intera
t with ea
h other at the boundary.
The two theories (4) and (8) are
lassi
ally equivalent, the solutions
an be mapped to
P : ΨL ↔ ΨR
ea
h other by a
ting on one (cid:28)eld with the parity operator .
Theyarealsoequivalentatthequantumlevel,whi
h
anbeseensimilarlytotheprevious

asLe by de(cid:28)ning them in the frameworkΦoLf [16℄. Tkhie>`in0' states in the defe
t theory
onsist
of typeRparti
les, belonging to (cid:28)eld Ψ,Rwith ki < 0approa
hing the defe
t from the
left and parti
les, belonging to (cid:28)eld , with approa
hing from the right, as
FROM DEFECTS TO BOUNDARIES 5
before the order of these parti
les is su
h that |tihneiLfa⊕st|ienstiRmoving parti
les are furthesLt
fromRthe defe
t. The `in' state
an be written as k < 0 . Inkth>e `o0ut' state we have
f f
and parti
les traveling away from the defe
t with and , and ea
h on their
|outiL⊕|outiR
respe
tive sides of the defe
t, and ordered as before. This
an be written as .
There exRisTts a unitary opLerLator whi
h
onne
ts the `out' and `in' asymptotRi
Lstates,

alled the -matrix. The ( ) element of|tihniisLmatrix|oiustitLhe re(cid:29)e
tion matrRixR whi
h
gives the amplitudes for pro
ess
onne
ting with . Similarly the ( ) element
RR |iniR
i|soutthieRre(cid:29)e
tioLnRmatrix whi
h gives the amplitTudLes of pro
ess|einsi
Lonne
t|ionugtiR with
RL . The ( ) element is the transitTiRon matrix |
ionniRne
ting|outiLwith and the
( ) element is the transition matrix
onne
ting with .
The adiabati
des
ription of (8) has `in' states whi
h
ontains parti
les with momentum
ki < 0 ΦL ΨL x = −∞
of both types and , approa
hing the boundary from with the usual
|ini ⊕|ini
Φ Ψ
ordering. These states
an be written as . The `out' states have the same type
k < 0 |outi ⊕|outi
f Φ Ψ
of parti
les but with , with the same ordering and written as . We
R R,
have the usual -matrix, whi
h
onne
ts the `in' and `out' states. It has four elements
(ΦΦ) (ΨΨ) |ini |outi |ini |outi
Φ Φ Ψ Ψ
the and elements
onne
t the states to and to states,
R R (ΦΨ) (ΨΦ) R
ΦΦ ΨΨ ΦΨ
respe
tively, and are denoted and . The and elements, denoted
R |ini |outi
ΨΦ Φ Ψ
and , respe
tively give the amplitudes of pro
esses
onne
ting with and
|ini |outi
Ψ Φ
to . We
an de(cid:28)ne a parity operator that a
ts on one of the (cid:28)elds of the
PΨ : ΨL ↔ ΨR Φ
boundary system only, , but leaves the (cid:28)eld un
hanged. This operator
maps the asymptoti
states of the defe
t and of the boundary theory to ea
h other. To
R RT
see how the and matri
es are related one has to relate them to the
orresponding

orrelation fun
tions via the redu
tion formula, whi
h (cid:28)nally has to be
omputed from
the Lagrangian. Analyzing the a
tion of the parity operator at ea
h step we obtain the
following identi(cid:28)
ations:
P : R = RΦΦ RΨΦ ↔ RT = RL TR
Ψ (cid:18) RΦΨ RΨΨ (cid:19) (cid:18) TL RR (cid:19) (10)
Now we
an exploit this
orresponden
e. All that we have learned already for boundary
theories
an be applied straightforwardly to defe
t theories: we
an present perturba-
tion theory, redu
tion formulas and in the view of [17℄, the derivation of Coleman-Thun
me
hanism and Cutkosky rules for defe
t theories is also straightforward. Note that the
1
generalization for higher dimensions with a
odimension defe
t is obvious, just as is
∂ Φ ∂ Ψ
t t
the a

ommodation of more (cid:28)elds. Also and dependent potentials or boundary
Φ,∂ Φ Ψ,∂ Ψ
x x

onditions that relate linearly to
an be in
orporated.
3. Integrable aspe
ts,
onsequen
es
We will now fo
us on the integrable aspe
ts of the
orresponden
e between defe
ts and
boundaries. Classi
al integrability in the presen
e of boundaries has been the subje
t of
several years of study. All the methods developed for boundary integrable systems are now
available for the study of integrable defe
t or impurity theories. For example Skylanin's
formalism [19℄ whi
h allows the
onstru
tion of the
onserved
harges for the boundary
FROM DEFECTS TO BOUNDARIES 6
(cid:28)eld, and
an be used to (cid:28)nd the integrable boundary
onditions [20℄
an now be used on
defe
t or impurity systems. The Lax pair for defe
t systems is dis
ussed in [7, 8℄ and is
equivalent to the boundary Lax pair approa
h des
ribed in [21℄.
At the quantum level the existen
e of in(cid:28)nitely many
onserved
harges give severe
R
restri
tion on the allowed pro
esses. As a
onsequen
e the -matrix of a boundary system
S R

an be de
omposed into a produ
t of two parti
le -matri
es and one parti
le -matri
es.
The former entirely des
ribes the s
attering of two parti
les in the bulk of the (cid:28)eld, the
latter des
ribes the re(cid:29)e
ting of a single parti
le from the boundary. Similarly for defe
t
RT
systems the -matrix
an be de
omposed into a produ
t of the, possibly di(cid:27)erent, two
S RT
parti
le -matri
es of the two (cid:28)elds either side of the defe
t and one parti
le -matri
es
des
ribing the re(cid:29)e
tion and transmission of individual parti
les through the defe
t. The
R RT
integrable one parti
le -matrix and -matrix depend on the parti
le energy through
θ S
the absolute value of the rapidity, , of the in
ident parti
le. The two parti
le -matri
es
depend on the absolute value of the di(cid:27)eren
e between the rapidities of the s
attering
parti
les. The asymptoti
states of the integrable boundary and defe
t systems have the
same des
ription as in the non-integrable
ases, with the additional
ondition that, in both

ases, the number ofparti
les inthe `out' statemust be the sameas thenumber of parti
les
in the `in' state and the
onserved
harges of the two asymptoti
states must
oin
ide. The
R RT
integrable and -matri
es are unitarity operators whi
h map one parti
le `in' states
to one parti
le `out' states and are related as des
ribed in (10).
S SL SR
The integrable defe
t theory haLs two R-matri
es, and , respe
tively des
ribing
the two parti
le s
attering in the and (cid:28)elds. The integrable boundary theory has a
S
single -matrix des
ribing the two parti
le s
attering of the bulk (cid:28)eld. If the bulk (cid:28)eld is
Φ Ψ

omposed of two nonintera
ting (cid:28)elds, and , as in the
ase des
ribed as equivalent to a
S S, S
ΦΦ
defe
t theory, then the -matrix, has a blo
k diagonalform with diagonalelements ,
S S S S
ΦΨ ΨΦ ΨΨ ΦΦ
, and . The element des
ribes the s
attering of two parti
les both of the
Φ S Ψ S S
ΨΨ ΦΨ ΨΦ
(cid:28)eld , similarly des
ribes the s
attering of two parti
les of (cid:28)eld , and both
des
ribe the s
attering between two parti
les where one belongs to ea
h of the separate
Φ Ψ S
(cid:28)elds and . Sin
e these -matri
es
an be
omputed from the bulk Lagrangian, (where
Φ Ψ
the and parti
les intera
t only with themselves) we know, that
SΨΦ = SΦΨ = 1 ; SΦΦ ≡ SL ; SΨΨ ≡ SR .
(11)
S
(A -matrix of a di(cid:27)erent form has been used in [22, 23℄ to des
ribe an integrable defe
t
of a non-relativisti
theory.)
Fa
torizability of the defe
t system gives the defe
t Yang-Baxter, or re(cid:29)e
tion transmis-
sion equations, whi
h was des
ribed by several authors [1, 2, 3, 6℄. Here we just present
one of them and the
orresponding equation
FROM DEFECTS TO BOUNDARIES 7
γ f γ f
c θ1 β
c e dθ2 e
b θ β b
2
d
θ α
1
a a
α
TLβαda(θ1)TLγβcb(θ2)SRfdce(θ2 −θ1) = SLcadb(θ2 −θ1)TLβαed(θ2)TLγβfc(θ1) .
(12)
P
Ψ
We
annow a
ton the defe
t system withthe operator . Diagrammati
allythis(cid:29)ips the
right side of the defe
t Yang-Baxter equation onto the left of the boundary. We see this is
a spe
ial
ase of the boundary Yang-Baxter equation where both re(cid:29)e
tions
ause a
hange
in the spe
ies of the re(cid:29)e
ting parti
les. This
orresponds to how, as we already know,
PΨ TL RΦΨ R
maps the transmission matrix into the element of the -matrix. Di(cid:27)erent
re(cid:29)e
tion transmission equations
orrespond to other spe
ial
ases of the boundary Yang-
Baxter equation. The boundary Yang-Baxter diagram that is equivalent to (12) and the

orresponding equation is
f
f γ γ
c θ β
e e 1
e c θ
c 2
d
b d b θ2 β b
d
θ α
1
a
α
a
R βd(θ )S bd(θ +θ )R γc(θ )S fe(θ −θ )
ΦΨαa 1 ΦΨbd 1 2 ΦΨβb 2 ΨΨdc 2 1
= S cd(θ −θ )R βe(θ )S ce(θ +θ )R γf(θ ) ,
ΦΦab 2 1 ΦΨαd 2 ΨΦce 1 2 ΦΨβc 1 (13)
a b Φ e f
where the `in' states, labeled and , belong to the (cid:28)eld and the `out' states and
Ψ
belong to .
The re(cid:29)e
tion and transmission amplitudes of the defe
t satisfy the unitarity equations
RLβαba(θ)RLγβcb(−θ)+TLβαba(θ)TRγβcb(−θ) = δacδαγ ,
TRbaβα(θ)TLcbβγ(−θ)+RRbaβα(θ)RRcbβγ(−θ) = δacδαγ ,
RLβαba(θ)TLcbβγ(−θ)+TLβαba(θ)RRγβcb(−θ) = 0 ,
TRβαba(θ)RLγβcb(−θ)+RRβαba(θ)TRγβcb(−θ) = 0 .
(14)
R Rbβ(θ)Rcγ(−θ) = δcδγ
aα bβ a α
The -matrix has to satisfy the unitarityequation , whi
h, in terms
R
ofthedefe
tvariables,readsasequation(14). The -matrixmustalsosatisfytheboundary
FROM DEFECTS TO BOUNDARIES 8
Rβb(θ) = Scb(2θ)Rβc¯(iπ −θ)

rossing unitarity equation [18℄, αa ad αd¯ , in terms of the defe
t this
gives
RLβαba(θ) = SLcabd(2θ)RLαβcd¯¯(iπ −θ) , TLβαba(θ) = TRαβ¯ab¯(iπ −θ) ,
RRβαba(θ) = SRcabd(2θ)RRαβcd¯¯(iπ −θ) , TRβαba(θ) = TLαβ¯ab¯(iπ −θ) , (15)
observe, however, that these equations are derived and not just
onje
tured as in [1, 2℄.
Now
onsider the
onsequen
es of these equations. Sin
ethey arethe sameas in[1, 2, 3℄ we
S
have analogous
onsequen
es; for rapidity dependent bulk -matri
es, with no degrees of
freedom at the defe
t, there is no simultaneous transmission and re(cid:29)e
tion. Equivalently,
for the boundary system des
ribed earlier, re(cid:29)e
tions are either spe
ies
hanging or spe
ies
preserving.
Let us now derive the defe
t operator using the defe
t-boundary
orresponden
e for the
model
ontainingone self-
onjugatedparti
leasde(cid:28)ned in(4). Consider (cid:28)rst the boundary
system as des
ribed in (8). If we
hange the roles of spa
e and time the boundary be
omes
B |Bi = B|0i
an initial state
reated by a boundary operator from the va
uum . This
operator, using the Zamolod
hikov-Faddeev
reation annihilation operators, is given by
[18℄ as follows:
∞ dθ iπ iπ
B = exp R −θ A+ (−θ)A+ (θ)+R −θ A+ (−θ)A+ (θ)
(cid:26)Z 2π (cid:18) ΦΦ(cid:18) 2 (cid:19) ΦL ΦL ΦΨ(cid:18) 2 (cid:19) ΦL ΨL
0
iπ iπ
+R −θ A+ (−θ)A+ (θ) +R −θ A+ (−θ)A+(θ)
ΨΦ(cid:18) 2 (cid:19) ΨL ΦL (cid:19) ΨΨ(cid:18) 2 (cid:19) ΨL Ψ (cid:27)
Now if we unfold the system the time boundary be
omes a defe
t line, where the defe
t
A+ (θ) → A (−θ)
operator is inserted. Using the relation (10) and making the
hange ΨL ΨR
we get the defe
t operator
∞ dθ iπ iπ
D = exp(cid:26)Z 2π (cid:18)RL(cid:18) 2 −θ(cid:19)A+ΦL(−θ)A+ΦL(θ)+TL(cid:18) 2 −θ(cid:19)A+ΦL(−θ)AΨR(−θ)
0
iπ iπ
+TR(cid:18) 2 −θ(cid:19)AΨR(θ)A+ΦL(θ)(cid:19)+RR(cid:18) 2 −θ(cid:19)AΨR(θ)AΨR(−θ)(cid:27)
A similar operator was proposed in [27℄ but without any derivation. Our derivation
on-
Φ Ψ A (θ )A+ (θ ) =
(cid:28)rmstheirresult. Observethatsin
ethe and parti
lesdonotintera
t ΨR 1 ΦL 2
A+ (θ )A (θ )
ΦL 2 ΨR 1 and no normal ordering is needed. We note also that as a
onsequen
e of
TL iπ +θ = TR iπ −θ
equation (15) 2 2 so the defe
t operator in the purely transmitting
(cid:0) (cid:1) (cid:0) (cid:1)

ase simpli(cid:28)es to:
∞ dθ iπ
D = exp(cid:26)Z 2π (cid:18)TR(cid:18) 2 −θ(cid:19)AΨR(θ)A+ΦL(θ)(cid:19)(cid:27) (16)
−∞
4. Purely transmitting defe
ts
As integrable defe
t theories with nontrivial S-matrix
an only have purely transmitting
or re(cid:29)e
ting defe
ts we
an
onsider how to (cid:28)nd solutions in these two
ases separately.
FROM DEFECTS TO BOUNDARIES 9
L R
If a defe
t is purely re(cid:29)e
ting then we
an solve the and parts independently, these
equations arenot
oupled
ontrary towhat is
laimedin[3℄. Thus let us
on
entrate on the
purely transmittingdefe
ts, in the boundary language this means that the refe
tion matrix
is purely o(cid:27) diagonal. One
an obtain nontrivial solutions of the boundary Yang-Baxter
equation by the fusion method as follows.
Reβ(θ),{α},{β}
cα
Suppose we have found a family of solutions, of the boundary Yang-
Baxter (13), unitarity and boundary
rossing unitarity equations. Now we
an produ
e
another solution, for an ex
ited boundary, via the boundary bootstrap equation [18℄
γ γ
c c g iu β
f
d θ
e
b θ β b
iu α
a
α
a
gβ Rcγ(θ) = gγ Sde(θ−iu)Rfβ(θ)Seg(θ+iu) .
aα bβ gβ ab eα fd (17)
Rcγ(θ)
This new solution, bβ , automati
ally satis(cid:28)es the required equations. The proof of
the boundary Yang-Baxter equation or fa
torization
an be shown (cid:28)rst by `shifting' the
traje
tories from the ex
ited boundary to the ground state boundary, then using the proof
on this boundary and shifting ba
k to the ex
ited boundary. The unitarity and boundary
S

rossing unitarity is a
onsequen
e of the similar equations for the bulk -matrix. If the
R
original -matrix does not have any pole in the physi
al strip we
an always add a CDD
fa
tor to ensure its existen
e. By applying the folding method to transmitting defe
t
systems as des
ribed in this letter we
an apply this method to (cid:28)nd non-trivial solutions
of defe
t systems.
S S
ΦΦ ΨΨ
S Focdr(θe)xa=mpSle ldect(θ)us assume that and R ar=e pRarity=tr1ansforms of ea
h other
ΦΦab ΨΨba ΦΨ ΨΦ
. In this
ase the solution , whi
h
orresponds to
trivial transmission, satis(cid:28)es all the required equations. By adding a CDD fa
tor to this
θ = iu
solution without spoilingits ni
e properties we
an ensure the existen
e of a poleat ,
0 < u < π/2 0 = α
. Let us label this unex
ited boundary solution by . Now if a parti
le
a a = β
with label is bound to the boundary the ex
ited boundary state will have label .
b c
A parti
le of type
an re(cid:29)e
t from the boundary to parti
le by
hanging the boundary
a γ = d
state from to as shown on the previous (cid:28)gure. By using (17) we
an (cid:28)nd an ex-
R
pression for the re(cid:29)e
tion matrix for this ex
ited boundary. The -matrix for the ex
ited
boundary now depends upon the spe
ies of re(cid:29)e
ting parti
le,
RΦΨcbad(θ) = TLcbad = SΦΦcadb(θ−iu) ; RΨΦcbad(θ) = TRcbad = SΨΨdbac(θ+iu) .
(18)
S
Clearly they satisfy (12), sin
e the bulk -matrix satis(cid:28)es the bulk Yang-Baxter equation.
They alsosatisfytheunitarityand
rossunitarityrelationssin
eitredu
es totheanalogous
S
equations of the bulk -matrix. Observe that equation (12) is similarto the Lax equations
FROM DEFECTS TO BOUNDARIES 10
u = 0
of integrable latti
e models and so the bulk S matrix always solves it. If the solution
des
ribes a standing parti
le. Bootstrapping further on this ex
ited defe
t we
an bind
either di(cid:27)erent type, or more parti
les to the defe
t. It is similar to how one
an (cid:28)nd
higher spin solutions of the Lax equation in latti
e models.
We note that anotherspe
ialsolutionforpurely transmittingimpurityisfound by Konik
and LeClair, for the sine-Gordon model using the perturbed CFT approa
h [24℄.
Purely transmitting defe
ts are of parti
ular interest in the study of boundary systems
asTtLhbaeβαy(θm)ay beTuRbasβαed(θt)o
onstru
t new solutions to the re(cid:29)e
tion equation. If there exists
a and whi
h satis(cid:28)es the transmission equation (12), unitarity (14) and
S Scd(θ) R bβ(θ)
ab 0aα

rossing unitarity (15) for some -matrix, , and a that satis(cid:28)es the re(cid:29)e
tion
S
equation (13) for the same -matrix then the matrix
R1baβα(θ) = TLcaγα11(θ)R0dcαβ22(θ)TRbdβγ11(−θ) , (19)
α = α ⊗ α β = β ⊗ β
1 2 1 2
where and , also satis(cid:28)es the re(cid:29)e
tion equation [25℄. This is
diagrammati
ally represented below.
(cid:0)(cid:1)β(cid:0)(cid:1) β1 β2
(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)
b (cid:0)(cid:1)(cid:0)(cid:1) b (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) −θ d(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)R1 γ (cid:0)(cid:1)(cid:0)(cid:1)R0
(cid:0)(cid:1)θ(cid:0)(cid:1) 1 (cid:0)(cid:1)θ(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1) c (cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1) θ (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)
a (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) a (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
(cid:0)(cid:1)α(cid:0)(cid:1) α1 (cid:0)(cid:1)α(cid:0)(cid:1)2
R R R
1
The -matri
es obtained in this way are
alled dressed -matri
es in [26℄. If is blo
k
α β R
diagonal in the , basis then the -matrix
an be proje
ted into one of the irredu
ible
blo
ks, the defe
t is then said to be fused to the boundary. Solutions developed in this
manner, using (18), are equivalent to performing the boundary bootstrap pro
edure.
5. Appli
ation: defe
t TBA
Let us now
onsider one appli
ation of the boundary-defe
t
orresponden
e developed
in this paper; suppose we would like to
al
ulate the groundstate energy of an integrable
m S
system,
ontaining one parti
le type with mass , and bulk s
attering matrix , in a
L
(cid:28)nite geometry. The system is de(cid:28)ned on an interval of size with integrable boundary
n ith

onditions at both ends, and integrable purely transmitting defe
ts with the defe
t
x
i
lo
alized at as shown in the following (cid:28)gure
1 1 2 2 i i n n
R T T T T T T T T R
0 L R L R L R L R n+1
... ...
0 x x x x x =L
1 2 i n n+1