We study the correlation function of the 2d SU(2) principal chiral model on
the lattice. By rewriting the model in terms of Z(2) degrees of freedom coupled
to SO(3) vortices we show that the vortices play a crucial role in disordering
the correlations at low temperature. Using a series of exact transformations we
prove that, if satisfied, certain inequalities between vortex correlations
imply exponential fall-off of the correlation function at arbitrarily low
temperatures. We also present some Monte Carlo evidence that these correlation
inequalities are indeed satisfied. Our method can be easily translated to the
language of 4d SU(2) gauge theory to establish the role of corresponding SO(3)
monopoles in maintaining confinement at small couplings.Comment: 13 pages LaTe

Kovács, Tamás G.

Tomboulis, E. T.

Physics Letters B

English

arXiv

We study the correlation function of the 2d SU(2) principal chiral model on the lattice. By rewriting the model in terms of Z(2) degrees of freedom coupled to SO(3) vortices we show that the vortices play a crucial role in disordering the correlations at low temperature. Using a series of exact transformations we prove that, if satisfied, certain inequalities between vortex correlations imply exponential fall-off of the correlation function at arbitrarily low temperatures. We also present some Monte Carlo evidence that these correlation inequalities are indeed satisfied. Our method can be easily translated to the language of 4d SU(2) gauge theory to establish the role of corresponding SO(3) monopoles in maintaining confinement at small couplings

Kovács, T G

Tomboulis, E T

CERN Document Server

UCLA/95/TEP/25
hep-lat/9508010
SO(3) vortices and disorder in the 2d SU(2) chiral
model
1
Tamas G. Kovacs
2
and E. T. Tomboulis
3
Department of Physics
University of California, Los Angeles
Los Angeles, California 90024-1547
Abstract
We study the correlation function of the 2d SU(2) principal chiral model on the lat-
tice. By rewriting the model in terms of Z(2) degrees of freedom coupled to SO(3)
vortices we show that the vortices play a crucial role in disordering the correlations at
low temperature. Using a series of exact transformations we prove that, if satised,
certain inequalities between vortex correlations imply exponential fall-o of the cor-
relation function at arbitrarily low temperatures. We also present some Monte Carlo
evidence that these correlation inequalities are indeed satised. Our method can be
easily translated to the language of 4d SU(2) gauge theory to establish the role of
corresponding SO(3) monopoles in maintaining connement at small couplings.
1
Research supported in part by NSF Grant PHY89-15286
2
e-mail address: kovacs@physics.ucla.edu
3
e-mail address:tomboulis@uclaph.bitnet
1
It is commonly believed that two-dimensional lattice spin models with a continu-
ous nonabelian symmetry exhibit no phase transitions at nite temperature. Indeed,
according to the Mermin-Wagner theorem, these models do not have an ordered low
temperature phase with spontaneous breakdown of the continuous symmetry. How-
ever, this does not in itself rule out the possibility of a Kosterlitz-Thouless (KT) type
phase transition. The KT transition is characterised by power-law decay of the corre-
lation function in the low temperature phase as opposed to an exponential decay above
the critical temperature. The absence of this KT-type phase transition has not been
proved rigorously for nonabelian spin models.
Two dimensional spin models are well known to have properties analogous to that
of four dimensional gauge theories. In 4d nonabelian gauge theories the corresponding
problem is whether the system remains conning down to arbitrarily small nite cou-
plings. A 4d SU(2) lattice gauge theory can be rewritten in terms of SO(3) and Z(2)
variables. This exhibits SO(3) monopoles in the measure, and recently a program has
been developed for establishing the presence of these monopoles, and their associated
strings, as a sucient mechanism for connement at arbitrarily small positive coupling
[1]. Completely analogous considerations can be developed for the SU(2)SU(2) chiral
spin model in 2d. In a previous paper [2] we have rewritten the partition function and
the two-point correlation function of this model in terms of SO(3) and Z(2) variables,
and presented some arguments supporting the role that SO(3) vortices [3] (the ana-
logues of gauge monopoles) and strings connecting them can play in disordering the
system.
In the present paper we use this picture to derive an exponentially falling upper
bound on the correlation function of the SU(2)SU(2) model in terms of correlations
between SO(3) \stringy" vortices. The main idea is the following. After rewriting the
SU(2) degrees of freedom in terms of Z(2) and SO(3) variables and performing a duality
transformation on the Z(2) variables the resulting system is an Ising model on the dual
lattice coupled to SO(3) vortices. The eect of the vortices on the Ising spins is similar
to that of an external magnetic eld, provided that appropriate expectations of vortices
2
pairwise connected by long strings winding around the lattice satisfy certain positivity
conditions and remain nonvanishing in the large lattice limit. It is crucial that these
expectations are dened with respect to an SO(3) (rather than an SU(2)) measure
(see below). One thus may bound the correlation function of the original SU(2) model
from above by a corresponding expectation in an eective Ising model with a nonzero
external magnetic eld. This correlation in turn is well known to decay exponentially
thus giving an exponentially falling upper bound to the two-point function of the SU(2)
principal chiral model.
We shall work on a nite two dimensional square lattice  with periodic boundary
conditions. The degrees of freedom (\spins") are SU(2) group elements U
s
attached to
lattice sites with nearest neighbour ferromagnetic couplings. The partition function of
the system is
Z =
Y
s2
Z
dU
s
exp
2
4

X
l2
trU
l
3
5
; (1)
where  is the inverse temperature, sites, links and plaquettes are labelled with s,
l, p respectively, and U
l
= U
y
s
U
s
0
with [ss
0
] being @l, the ordered boundary of the
link l. Throughout the paper all group integrations (discrete and continuous) will be
performed using the Haar-measure normalised to unity.
Let f
1
; 
2
g be a pair of Z(2) elements, Z(2)=f1g being the centre of the symmetry
group SU(2). We can introduce a twist 
1
in the `1' direction and 
2
in the other and
denote by Z(
1
; 
2
) the partition sum in the presence of these twists. By a twist in
the direction i we mean that on a stack of links winding around the lattice along the
i direction the couplings are changed from  to 
i
. Physically 
i
=  1 means that
a topologically nontrivial \domain-wall" was created along the aected links. This
domain-wall winds around the lattice, it is closed but not a boundary of any region.
The order parameter that we shall consider is the expectation of such a twist dened
as
G(L) =
R
d
1
R
d
2

2
Z(
1
; 
2
)
R
d
1
R
d
2
Z(
1
; 
2
)
=
Z
+
(L)  Z
 
(L)
Z
+
(L) + Z
 
(L)
; (2)
where Z

(L) =
R
d
2
Z(1; 
2
) and L is the linear size of the lattice. (2) is the spin
3
analog of the electric-ux free-energy order parameter of gauge theory [5]. If by sending
the lattice size to innity, G(L) goes to zero exponentially, the system is essentially
disordered and it does not \feel" the presence of the enforced domain-wall even when
the lattice becomes innitely large.
G(L) has qualitatively the same asymptotic behaviour as the spin-spin correlation
function, and it can indeed be rigorously proved [4] that the two-point correlation
function for separation L is bounded from above by a constant times G(L). This means
that the exponential fall-o of G(L) would imply the same asymptotic behaviour of
the spin-spin correlation function.
For a quantitative study of G(L) we rewrite the theory (1) by means of a decompo-
sition into Z(2) and SU(2)/Z2  SO(3) variables as was done in [2], to which we refer
for details. After performing an additional duality transformation on the Z(2) degrees
of freedom the order parameter assumes the form
G(L) =
1
Z
+
(L) + Z
 
(L)
Y
s2
Z
dU
s
exp
0
@
X
l2
M(U
l
)
1
A

Y
p2
Z
d!
p

d
p
(!
p
) 
C
exp
2
4
X
l62C
K(U
l
)!
l
 
X
l2C
K(U
l
)!
l
3
5
; (3)
where 
l
= sign trU
l
, d
p
=
Q
l2@p

l
, the 's are characters of Z(2) and 
C
=
Q
l2C

l
with C being a loop winding around the lattice in the direction perpendicular to the
twist. The functions M(U
l
) and K(U
l
) are given by
K(U
l
) =
1
2
ln cothjtrU
l
j; M(U
l
) =
1
2
ln (coshjtrU
l
j sinh jtrU
l
j) (4)
Now the integrand in (3) depends on the SU(2) variables U
s
only through the SU(2)/Z(2)
cosets since it is clearly invariant under a local transformation of the U
s
's by elements
of Z(2). Thus the integration is eectively over SO(3) degrees of freedom. In compen-
sation, (3) also contains Z(2) spins !
p
attached to plaquettes. Spins on neighbouring
plaquettes sharing the link l interact via the uctuating coupling K(U
l
). The Z(2)
part of the system is now essentially a ferromagnetic Ising model but with couplings
depending on the SO(3) degrees of freedom. Because of the duality transformation, low
4
temperature in the original model corresponds to high temperature i.e. small K(U
l
)'s
in this Ising model. The Ising and the SO(3) variables are further coupled through the
Z(2) characters. By denition 
d
p
(!
p
) = 1 if d
p
= 1: and 
d
p
(!
p
) = !
p
if d
p
=  1,
which means that there is an SO(3) vortex on the plaquette p. The Z(2) characters
thus couple the Ising spins to SO(3) vortices. It is also obvious from the construction
that vortices only occur pairwise connected by "-strings", i.e. stacks of links on which

l
=  1 (Fig. 1). This terminology is motivated by the analogy with Dirac strings in
gauge theories. The strings are not gauge invariant but their endpoints, the vortices
(monopoles in gauge theories), are. Finally, the M(U
l
) part of the action depends only
on the SO(3) variables.
In this representation G(L) involves a twist along C in the dual Ising model of !
spins. Notice that while in the original system the twist was along a stack of links
winding around the lattice in the \1" direction, in the dual system the twisted links
form a loop C going around the lattice in the direction perpendicular to \1". Fur-
thermore, there are two additional crucial factors in the measure: 
C
, and the product
of Z(2) characters that couple SO(3) vortices to Z(2) spins. For a xed SO(3) con-
guration with vortices at (p
1
; :::p
2n
) and S -strings crossing C, these give an overall
factor of ( 1)
S
Q
2n
i=1
!
p
i
. Were it not for this additional factor depending on the SO(3)
conguration, the system would essentially be an ordinary ferromagnetic Ising model
at high temperature with unbroken Z(2) symmetry and G(L) going to unity on large
lattices.
Let us then look at the Z(2) characters in the measure. In a xed \background"
SO(3) conguration containing 2n vortices, the sum over the Ising congurations gives
a 2n-point function of the Ising system with couplings K(U
l
). When nally the SO(3)
variables are integrated out, we get a sum over all possible (even) numbers and locations
of vortices which translates into a sum of all possible correlations in the Ising system
taken with additional weights coming from the SO(3) part of the measure. This is
very much reminescent of the expansion of the partition function of an Ising model
with respect to an external magnetic eld, where the same type of sum appears. This
5
motivates the following physical picture. The high temperature Ising system of ! spins
would be in its symmetric phase by itself but the coupling to the vortices breaks the
symmetry by creating an eective external magnetic eld. In this broken phase the
free energy of the twist (3) along C grows exponentially with the lattice size implying
exponential decay for G(L).
To make this argument quantitative we proceed as follows. We rst insert a delta
function in the measure in (3) to constrain 
C
to unity. Let G(L;C
+
) denote G(L)
computed in the presence of this constraint. It can be rigorously proven that G(L) 
G(L;C
+
), so it is enough to verify exponential fall-o for G(L;C
+
). We next compare
G(L;C
+
) to the quantity
G
e
(h;L) =
1
Z
e
(h;L)
Y
p2
Z
d!
p
exp
2
4
X
l62C
K(1)!
l
 
X
l2C
K(1)!
l
+
X
p2
h!
p
3
5
; (5)
i.e. the expectation of a twist along C in an eective Ising system of ! spins coupled
to a magnetic eld h. Now using a method similar to Ginibre's proof of Griths-type
inequalities [6], the dierence G
e
(h;L) G(L;C
+
) can be expressed as a sum of terms,
each term being of the form
 
L;


[P
1
; P
2
; :::; P
n
]  h
n
Y
i=1
(
 
P
i
 


+
P
i
)
Y
p6=fP
i
g

+
p
i
L
; (6)
times positive numerical coecients. Here, P
1
; :::; P
n
is a set of n vortex pairs, 

P
i
=


p
i1


p
i2
, where p
i1
; p
i2
denote the locations of the two vortices of the i-th pair, and


p
constrains d
p
to be 1. h   i
L
means integration with respect only to a pure
SO(3) measure dened simply by the action M(U
l
), 4, and including the constraint

C
= 1. Also, we dened

  tanh h. Since the integration measure is positive, at

 = 0 expression (6) is positive, hence G
e
(0; L)  G(L). By continuity the same then
holds in a neighbourhood of

 = 0 (h = 0). If this neighbourhood does not shrink to
zero when the lattice size is sent to innity, G(L) must obey exponential decay since
G
e
(h;L) does for any non-zero h, the mass-gap being proportional to tanh h. We
have thus rigorously reduced the existence of a mass gap to a condition on vortex pair
correlations.
6
Next consider the assertion
 
L;


(P
1
; P
2
; :::; P
n
)  const.  
L;


(P
1
) :::  
L;


(P
n
); (7)
i.e. that the correlators   of vortex pairs (6) are bounded by products of vortex pair
correlations. This (highly nontrivial) inequality can be rigorously proven for

 = 0 by
an argument that reduces (7) to an application of the FKG inequalities [7]. It is then
very plausible that it also holds for suciently small values of

. Indeed, preliminary
results indicate that (7) holds for

 such that the r.h.s. remains nonnegative, i.e. for

  k


0
, where k is a numerical constant of order unity, and 0 


0
such that
 
L;


(P ) = h
 
p
1

 
p
2
Y
p6=p
i

+
p
i
L
 


0
h
Y
p2

+
p
i
L
 0 (8)
holds on arbitrarily large lattices regardless the location (p
1
and p
2
) of the two members
of the vortex pair P . This is equivalent to the statement that the free energy cost of a
pair of vortices
F
L
(p
1
; p
2
) =  
1

ln
h
 
p
1

 
p
2
Q
p6=p
i

+
p
i
L
h
Q
p2

+
p
i
L
(9)
is bounded for any lattice size. The only case when this free energy can in principle
be sensitive to the lattice size is when p
1
and p
2
are on opposite sides of C and the
constraint 
C
= 1 forces the -string connecting them to go around the lattice (Fig. 1).
Now in two dimensions the energy of such an -string stays nite as L!1 in the
semiclassical approximation, where one obtains F
L
(P ) = constant, whereas it diverges
with L in higher dimensions. This is due to ux spreading [8] that allows the cost of
the string creation to be spread laterally in the direction perpendicular to the string.
As explained above, constant F
L
(P ) implies h and hence a mass gap proportional
to exp (const). In the remainder of the paper we shall briey present the results
of a Monte Carlo measurement of F
L
(p
1
; p
2
) that indicate that this behavior of F
L
(P )
holds for the exact expectation (9) . Details of this Monte Carlo calculation will appear
elsewhere.
We measured the quantity exp( F
L
(p
1
; p
2
)) by Monte Carlo using the SO(3) ac-
tion jtrU
l
j appearing in (9). The simulations were performed on lattices 5  L  13
7
at  = 2:0. As can be seen from the location of the specic heat peak, this is already
on the weak coupling side of the crossover in the SO(3) model. The measurement
was done by simply counting in a long Monte Carlo run the number of congurations
having exactly two vortices at the xed locations p
1
and p
2
and in addition satisfying
the constraint 
C
= 1 (Fig. 1). This constraint forces the eta string connecting the
two nearby vortices to run all the way around the lattice. Finally the number of these
congurations was divided by the number of congurations containing no vortices at
all. Our results are summarised in Figure 2. We can see that the probabilty of having
a vortex pair at a given location with a long eta string decreases on small lattices until
it stabilises on moderate sized lattices (L  8  9) at a nonzero constant value. Recall
that for our purposes it is enough that this quantity remains nonzero in the L ! 1
limit. Figure 3 illustrates the pronounced eect on (9) of ux spreading in the lateral
direction.
To summarise, we have seen how vortices can disorder the correlation function in the
two dimensional SU(2)SU(2) chiral spin model at arbitrarily low temperatures. By
an exact rewriting of the original model to separate SO(3) and Z(2) degrees of freedom,
we derived sucient conditions on certain SO(3) vortex correlations for the existence of
an exponentiallyfalling upper bound to the order paramete. To complete the argument
we made two interrelated assumptions concerning the behavior of the correlations (7),
(9). We presented the results of a Monte Carlo calculation that conrm the expected
behavior of (9). Clearly it would be very worthwile to nd an analytic proof of these
two assumptions thus completing a rigorous demonstration of the absence of a KT-type
phase transition in this model. It would be also interesting to extend these arguments
to similar models with dierent symmetry groups. There are two obvious possible ways
of generalisation. One is to SU(n) principal chiral models and the other to O(n) vector
models by noting that the SU(2) principal chiral model is equivalent to the O(4) vector
model.
8
Acknowledgements
We thank the Department of Theoretical Physics, Kossuth Lajos University, Debrecen,
Hungary for granting us part of the computer time used for the Monte Carlo simulation.
References
[1] E.T. Tomboulis, Phys. Lett. B303 (1993) 103; E.T. Tomboulis, Nucl. Phys. Proc.
Suppl. B34 (1994) 192
[2] T.G. Kovacs and E.T. Tomboulis, Phys. Lett. B321 (1994) 75
[3] J. Kogut, M. Snow and M. Stone, Nucl. Phys. B215 [FS7] (1983) 45
[4] E.T. Tomboulis, L.G. Yae; Commun. Math. Phys. 100 (1985) 313
[5] G. 't Hooft. Nucl. Phys. 153 (1979) 141
[6] J. Ginibre, Commun. Math. Phys. 16 (1970) 310
[7] C.M. Fortuin, P.W. Kasteleyn, and J. Ginibre, Com. Math. Phys. 22 (1971) 89
[8] L.G. Yae, Phys. Rev. D21 (1979) 1574
9
Figure captions
Figure 1 Two vortices connected by an -string winding around the
lattice.
Figure 2 The probability of the vortex pair shown in Figure 1 as a
function of the lattice size at inverse temperature  = 2:0.
Figure 3 The probability of a vortex pair with -string going around
the lattice at  = 2:0. The length of the side of the lattice
parallel to the string is xed to 9 and only the \transverse"
size (perpendicular to the string) is changed.
10
l
=  1
Figure 1
C





 vortex
11
2e-07
4e-07
6e-07
8e-07
1e-06
1.2e-06
1.4e-06
4 6 8 10 12 14
lattice size
3
3
3
3
3
3
3
3
3
12
01e-06
2e-06
3e-06
4e-06
5e-06
8 9 10 11 12
lattice size
3
3
3
3
3
13


SO(3) vortices and disorder in the 2d SU(2) chiral model

DEA University of Debrecen Electronic Archive

arXiv:hep-lat/9508010v1  7 Aug 1995UCLA/95/TEP/25hep-lat/9508010SO(3) vortices and disorder in the 2d SU(2) chiralmodel1Tama´s G. Kova´cs2 and E. T. Tomboulis3Department of PhysicsUniversity of California, Los AngelesLos Angeles, California 90024-1547AbstractWe study the correlation function of the 2d SU(2) principal chiral model on the lat-tice. By rewriting the model in terms of Z(2) degrees of freedom coupled to SO(3)vortices we show that the vortices play a crucial role in disordering the correlations atlow temperature. Using a series of exact transformations we prove that, if satisfied,certain inequalities between vortex correlations imply exponential fall-off of the cor-relation function at arbitrarily low temperatures. We also present some Monte Carloevidence that these correlation inequalities are indeed satisfied. Our method can beeasily translated to the language of 4d SU(2) gauge theory to establish the role ofcorresponding SO(3) monopoles in maintaining confinement at small couplings.1Research supported in part by NSF Grant PHY89-152862e-mail address: kovacs@physics.ucla.edu3e-mail address:tomboulis@uclaph.bitnet1It is commonly believed that two-dimensional lattice spin models with a continu-ous nonabelian symmetry exhibit no phase transitions at finite temperature. Indeed,according to the Mermin-Wagner theorem, these models do not have an ordered lowtemperature phase with spontaneous breakdown of the continuous symmetry. How-ever, this does not in itself rule out the possibility of a Kosterlitz-Thouless (KT) typephase transition. The KT transition is characterised by power-law decay of the corre-lation function in the low temperature phase as opposed to an exponential decay abovethe critical temperature. The absence of this KT-type phase transition has not beenproved rigorously for nonabelian spin models.Two dimensional spin models are well known to have properties analogous to thatof four dimensional gauge theories. In 4d nonabelian gauge theories the correspondingproblem is whether the system remains confining down to arbitrarily small finite cou-plings. A 4d SU(2) lattice gauge theory can be rewritten in terms of SO(3) and Z(2)variables. This exhibits SO(3) monopoles in the measure, and recently a program hasbeen developed for establishing the presence of these monopoles, and their associatedstrings, as a sufficient mechanism for confinement at arbitrarily small positive coupling[1]. Completely analogous considerations can be developed for the SU(2)×SU(2) chiralspin model in 2d. In a previous paper [2] we have rewritten the partition function andthe two-point correlation function of this model in terms of SO(3) and Z(2) variables,and presented some arguments supporting the role that SO(3) vortices [3] (the ana-logues of gauge monopoles) and strings connecting them can play in disordering thesystem.In the present paper we use this picture to derive an exponentially falling upperbound on the correlation function of the SU(2)×SU(2) model in terms of correlationsbetween SO(3) “stringy” vortices. The main idea is the following. After rewriting theSU(2) degrees of freedom in terms of Z(2) and SO(3) variables and performing a dualitytransformation on the Z(2) variables the resulting system is an Ising model on the duallattice coupled to SO(3) vortices. The effect of the vortices on the Ising spins is similarto that of an external magnetic field, provided that appropriate expectations of vortices2pairwise connected by long strings winding around the lattice satisfy certain positivityconditions and remain nonvanishing in the large lattice limit. It is crucial that theseexpectations are defined with respect to an SO(3) (rather than an SU(2)) measure(see below). One thus may bound the correlation function of the original SU(2) modelfrom above by a corresponding expectation in an effective Ising model with a nonzeroexternal magnetic field. This correlation in turn is well known to decay exponentiallythus giving an exponentially falling upper bound to the two-point function of the SU(2)principal chiral model.We shall work on a finite two dimensional square lattice Λ with periodic boundaryconditions. The degrees of freedom (“spins”) are SU(2) group elements Us attached tolattice sites with nearest neighbour ferromagnetic couplings. The partition function ofthe system isZ =∏s∈Λ∫dUs expβ∑l∈ΛtrUl , (1)where β is the inverse temperature, sites, links and plaquettes are labelled with s,l, p respectively, and Ul = U†sUs′ with [ss′] being ∂l, the ordered boundary of thelink l. Throughout the paper all group integrations (discrete and continuous) will beperformed using the Haar-measure normalised to unity.Let {τ1, τ2} be a pair of Z(2) elements, Z(2)={±1} being the centre of the symmetrygroup SU(2). We can introduce a twist τ1 in the ‘1’ direction and τ2 in the other anddenote by Z(τ1, τ2) the partition sum in the presence of these twists. By a twist inthe direction i we mean that on a stack of links winding around the lattice along thei direction the couplings are changed from β to τiβ. Physically τi = −1 means thata topologically nontrivial “domain-wall” was created along the affected links. Thisdomain-wall winds around the lattice, it is closed but not a boundary of any region.The order parameter that we shall consider is the expectation of such a twist definedasG(L) =∫dτ1∫dτ2 τ2Z(τ1, τ2)∫dτ1∫dτ2 Z(τ1, τ2)=Z+(L)− Z−(L)Z+(L) + Z−(L), (2)where Z±(L) =∫dτ2 Z(±1, τ2) and L is the linear size of the lattice. (2) is the spin3analog of the electric-flux free-energy order parameter of gauge theory [5]. If by sendingthe lattice size to infinity, G(L) goes to zero exponentially, the system is essentiallydisordered and it does not “feel” the presence of the enforced domain-wall even whenthe lattice becomes infinitely large.G(L) has qualitatively the same asymptotic behaviour as the spin-spin correlationfunction, and it can indeed be rigorously proved [4] that the two-point correlationfunction for separation L is bounded from above by a constant times G(L). This meansthat the exponential fall-off of G(L) would imply the same asymptotic behaviour ofthe spin-spin correlation function.For a quantitative study of G(L) we rewrite the theory (1) by means of a decompo-sition into Z(2) and SU(2)/Z2 ≈ SO(3) variables as was done in [2], to which we referfor details. After performing an additional duality transformation on the Z(2) degreesof freedom the order parameter assumes the formG(L) =1Z+(L) + Z−(L)∏s∈Λ∫dUs exp∑l∈ΛM(Ul)×∏p∈Λ∫dωp χdηp(ωp) ηC exp∑l 6∈CK(Ul)δωl −∑l∈CK(Ul)δωl , (3)where ηl = sign trUl, dηp =∏l∈∂p ηl, the χ’s are characters of Z(2) and ηC =∏l∈C ηlwith C being a loop winding around the lattice in the direction perpendicular to thetwist. The functions M(Ul) and K(Ul) are given byK(Ul) =12ln cothβ|trUl|, M(Ul) =12ln (cosh β|trUl| sinh β|trUl|) (4)Now the integrand in (3) depends on the SU(2) variables Us only through the SU(2)/Z(2)cosets since it is clearly invariant under a local transformation of the Us’s by elementsof Z(2). Thus the integration is effectively over SO(3) degrees of freedom. In compen-sation, (3) also contains Z(2) spins ωp attached to plaquettes. Spins on neighbouringplaquettes sharing the link l interact via the fluctuating coupling K(Ul). The Z(2)part of the system is now essentially a ferromagnetic Ising model but with couplingsdepending on the SO(3) degrees of freedom. Because of the duality transformation, low4temperature in the original model corresponds to high temperature i.e. small K(Ul)’sin this Ising model. The Ising and the SO(3) variables are further coupled through theZ(2) characters. By definition χdηp(ωp) = 1 if dηp = 1: and χdηp(ωp) = ωp if dηp = −1,which means that there is an SO(3) vortex on the plaquette p. The Z(2) charactersthus couple the Ising spins to SO(3) vortices. It is also obvious from the constructionthat vortices only occur pairwise connected by ”η-strings”, i.e. stacks of links on whichηl = −1 (Fig. 1). This terminology is motivated by the analogy with Dirac strings ingauge theories. The strings are not gauge invariant but their endpoints, the vortices(monopoles in gauge theories), are. Finally, the M(Ul) part of the action depends onlyon the SO(3) variables.In this representation G(L) involves a twist along C in the dual Ising model of ωspins. Notice that while in the original system the twist was along a stack of linkswinding around the lattice in the “1” direction, in the dual system the twisted linksform a loop C going around the lattice in the direction perpendicular to “1”. Fur-thermore, there are two additional crucial factors in the measure: ηC , and the productof Z(2) characters that couple SO(3) vortices to Z(2) spins. For a fixed SO(3) con-figuration with vortices at (p1, ...p2n) and S η-strings crossing C, these give an overallfactor of (−1)S∏2ni=1 ωpi. Were it not for this additional factor depending on the SO(3)configuration, the system would essentially be an ordinary ferromagnetic Ising modelat high temperature with unbroken Z(2) symmetry and G(L) going to unity on largelattices.Let us then look at the Z(2) characters in the measure. In a fixed “background”SO(3) configuration containing 2n vortices, the sum over the Ising configurations givesa 2n-point function of the Ising system with couplings K(Ul). When finally the SO(3)variables are integrated out, we get a sum over all possible (even) numbers and locationsof vortices which translates into a sum of all possible correlations in the Ising systemtaken with additional weights coming from the SO(3) part of the measure. This isvery much reminescent of the expansion of the partition function of an Ising modelwith respect to an external magnetic field, where the same type of sum appears. This5motivates the following physical picture. The high temperature Ising system of ω spinswould be in its symmetric phase by itself but the coupling to the vortices breaks thesymmetry by creating an effective external magnetic field. In this broken phase thefree energy of the twist (3) along C grows exponentially with the lattice size implyingexponential decay for G(L).To make this argument quantitative we proceed as follows. We first insert a deltafunction in the measure in (3) to constrain ηC to unity. Let G(L,C+) denote G(L)computed in the presence of this constraint. It can be rigorously proven that G(L) ≤G(L,C+), so it is enough to verify exponential fall-off for G(L,C+). We next compareG(L,C+) to the quantityGeff(h, L) =1Zeff(h, L)∏p∈Λ∫dωp exp∑l 6∈CK(1)δωl −∑l∈CK(1)δωl +∑p∈Λhωp , (5)i.e. the expectation of a twist along C in an effective Ising system of ω spins coupledto a magnetic field h. Now using a method similar to Ginibre’s proof of Griffiths-typeinequalities [6], the difference Geff(h, L)−G(L,C+) can be expressed as a sum of terms,each term being of the formΓL,θ¯[P1, P2, ..., Pn] ≡ 〈n∏i=1(θ−Pi − θ¯θ+Pi)∏p 6={Pi}θ+p 〉L, (6)times positive numerical coefficients. Here, P1, ..., Pn is a set of n vortex pairs, θ±Pi=θ±pi1θ±pi2, where pi1, pi2 denote the locations of the two vortices of the i-th pair, andθ±p constrains dηp to be ±1. 〈 − 〉L means integration with respect only to a pureSO(3) measure defined simply by the action M(Ul), 4, and including the constraintηC = 1. Also, we defined θ¯ ≡ tanhh. Since the integration measure is positive, atθ¯ = 0 expression (6) is positive, hence Geff(0, L) ≤ G(L). By continuity the same thenholds in a neighbourhood of θ¯ = 0 (h = 0). If this neighbourhood does not shrink tozero when the lattice size is sent to infinity, G(L) must obey exponential decay sinceGeff(h, L) does for any non-zero h, the mass-gap being proportional to tanhh. Wehave thus rigorously reduced the existence of a mass gap to a condition on vortex paircorrelations.6Next consider the assertionΓL,θ¯(P1, P2, ..., Pn) ≥ const.× ΓL,θ¯(P1)× ...× ΓL,θ¯(Pn), (7)i.e. that the correlators Γ of vortex pairs (6) are bounded by products of vortex paircorrelations. This (highly nontrivial) inequality can be rigorously proven for θ¯ = 0 byan argument that reduces (7) to an application of the FKG inequalities [7]. It is thenvery plausible that it also holds for sufficiently small values of θ¯. Indeed, preliminaryresults indicate that (7) holds for θ¯ such that the r.h.s. remains nonnegative, i.e. forθ¯ ≤ kθ¯0, where k is a numerical constant of order unity, and 0 ≤ θ¯0 such thatΓL,θ¯(P ) = 〈θ−p1θ−p2∏p 6=piθ+p 〉L − θ¯0〈∏p∈Λθ+p 〉L ≥ 0 (8)holds on arbitrarily large lattices regardless the location (p1 and p2) of the two membersof the vortex pair P . This is equivalent to the statement that the free energy cost of apair of vorticesFL(p1, p2) = −1βln〈θ−p1θ−p2∏p 6=pi θ+p 〉L〈∏p∈Λ θ+p 〉L(9)is bounded for any lattice size. The only case when this free energy can in principlebe sensitive to the lattice size is when p1 and p2 are on opposite sides of C and theconstraint ηC = 1 forces the η-string connecting them to go around the lattice (Fig. 1).Now in two dimensions the energy of such an η-string stays finite as L→∞ in thesemiclassical approximation, where one obtains FL(P ) = constant, whereas it divergeswith L in higher dimensions. This is due to flux spreading [8] that allows the cost ofthe string creation to be spread laterally in the direction perpendicular to the string.As explained above, constant FL(P ) implies h and hence a mass gap proportionalto exp−(const)β. In the remainder of the paper we shall briefly present the resultsof a Monte Carlo measurement of FL(p1, p2) that indicate that this behavior of FL(P )holds for the exact expectation (9) . Details of this Monte Carlo calculation will appearelsewhere.We measured the quantity exp(−βFL(p1, p2)) by Monte Carlo using the SO(3) ac-tion |trUl| appearing in (9). The simulations were performed on lattices 5 ≤ L ≤ 137at β = 2.0. As can be seen from the location of the specific heat peak, this is alreadyon the weak coupling side of the crossover in the SO(3) model. The measurementwas done by simply counting in a long Monte Carlo run the number of configurationshaving exactly two vortices at the fixed locations p1 and p2 and in addition satisfyingthe constraint ηC = 1 (Fig. 1). This constraint forces the eta string connecting thetwo nearby vortices to run all the way around the lattice. Finally the number of theseconfigurations was divided by the number of configurations containing no vortices atall. Our results are summarised in Figure 2. We can see that the probabilty of havinga vortex pair at a given location with a long eta string decreases on small lattices untilit stabilises on moderate sized lattices (L ≈ 8− 9) at a nonzero constant value. Recallthat for our purposes it is enough that this quantity remains nonzero in the L → ∞limit. Figure 3 illustrates the pronounced effect on (9) of flux spreading in the lateraldirection.To summarise, we have seen how vortices can disorder the correlation function in thetwo dimensional SU(2)×SU(2) chiral spin model at arbitrarily low temperatures. Byan exact rewriting of the original model to separate SO(3) and Z(2) degrees of freedom,we derived sufficient conditions on certain SO(3) vortex correlations for the existence ofan exponentiallyfalling upper bound to the order paramete. To complete the argumentwe made two interrelated assumptions concerning the behavior of the correlations (7),(9). We presented the results of a Monte Carlo calculation that confirm the expectedbehavior of (9). Clearly it would be very worthwile to find an analytic proof of thesetwo assumptions thus completing a rigorous demonstration of the absence of a KT-typephase transition in this model. It would be also interesting to extend these argumentsto similar models with different symmetry groups. There are two obvious possible waysof generalisation. One is to SU(n) principal chiral models and the other to O(n) vectormodels by noting that the SU(2) principal chiral model is equivalent to the O(4) vectormodel.8AcknowledgementsWe thank the Department of Theoretical Physics, Kossuth Lajos University, Debrecen,Hungary for granting us part of the computer time used for the Monte Carlo simulation.References[1] E.T. Tomboulis, Phys. Lett. B303 (1993) 103; E.T. Tomboulis, Nucl. Phys. Proc.Suppl. B34 (1994) 192[2] T.G. Kova´cs and E.T. Tomboulis, Phys. Lett. B321 (1994) 75[3] J. Kogut, M. Snow and M. Stone, Nucl. Phys. B215 [FS7] (1983) 45[4] E.T. Tomboulis, L.G. Yaffe; Commun. Math. Phys. 100 (1985) 313[5] G. ’t Hooft. Nucl. Phys. 153 (1979) 141[6] J. Ginibre, Commun. Math. Phys. 16 (1970) 310[7] C.M. Fortuin, P.W. Kasteleyn, and J. Ginibre, Com. Math. Phys. 22 (1971) 89[8] L.G. Yaffe, Phys. Rev. D21 (1979) 15749Figure captionsFigure 1 Two vortices connected by an η-string winding around thelattice.Figure 2 The probability of the vortex pair shown in Figure 1 as afunction of the lattice size at inverse temperature β = 2.0.Figure 3 The probability of a vortex pair with η-string going aroundthe lattice at β = 2.0. The length of the side of the latticeparallel to the string is fixed to 9 and only the “transverse”size (perpendicular to the string) is changed.10ηl = −1Figure 1C⊗⊗⊗ vortex112e-074e-076e-078e-071e-061.2e-061.4e-064 6 8 10 12 14lattice size33333 3 3 3 3Figure 21201e-062e-063e-064e-065e-068 9 10 11 12lattice size33333Figure 313

SO(3) vortices and disorder in the 2d SU(2) chiral model

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