Volume 125. number 6,7 PHYSICS LETTERS A 23 November 1987

THE INFLUENCE OF THE ANISOTROPY ON THE PHASE DIAGRAM OF THE ONE-DIMENSIONAL N-VECTOR MODEL WITH A LONG-RANGE INTERACTION

S.A. BULGADAEV L.D. Landau Institute for Theoretical Physics, Kosygin Street 2, 117334 Mosco. USSR

Received 27 August 1987; accepted for publication 28 September 1987 Communicated by A.A. Maradudin

Instanton solutions of the model are constructed and it is shown that the anisotropy results in the appearence of a phase tran- sition. The corresponding critical behaviour is found.

After papers [1 ] two-dimensional nonlinear a- models have been intensively studied for more than a decade. They can be applied in various fields of modern physics; from string theory in quantum field theory [2] up to the theory of weak localisation in solid state physics [3]. The most interesting prop- erties of the a-model are its geometric nature, asymptotic freedom and the existence of classical, topologically nontrivial instanton solutions [ 1 ].

At the same time there exists a one-dimensional analogue of the nonlinear a-model which is the con- tinuous limit of the one-dimensional ferromagnet with long-range interaction J ( r ) ~ 1/r 2 [4]. It also has an asymptotic freedom [5] and different appli- cations: to the theory of the Josephson junction with dissipation [6] and probably, to open string theory.

We will show in this note that the one-dimensional a-model on the classical level is conformal invariant and has instanton solutions and we will also consider its quantum properties and the influence of the an- isotropy on them. We emphasize that there exists a remarkable similarity of properties of the one-di- mensional and two-dimensional nonlinear a-models.

The action S of the one-dimensional nonlinear O(N) a-model looks like

S 1 (n , . - n , . . ) 2 =~-df dxdx' (x_x,)2

_ 1 f d x d x ' (d,.n'd,-.n) l n l x - x ' l , 2c~ .)

n = ( n l ..... n x ) , n 2 = l . (1)

It is translational and conformal invariant (the non- locality of S pays for this). The latter follows from the invariance of (1) with respect ot the inversion x - - . £ = - l / x ,

( x _ x , ) 2 ~ ( 2 _ 2 , ) 2 . (2)

Here the field n(x) transforms as a scalar,

n(x)--,~(y:) =n(x). (3)

Let us write the equations of motion, corresponding to (1),

8 S 1 f cbc' ( n ' - n " ) - n " ( n ' - n " ' ) 2 8n(x) - ~ ( x _ x , ) 2 = 0 . (4)

The instantons must exist at N = d + 1, i.e. at N=2 , as a result of topologically nontrivial mappings of the circle obtained as compactification of the x-axis (the corresponding boundary condition n(oo) =n( - ~ ) is necessary for the convergence of S) into the circle S j 9 n ( x ) . Considering the x-axis the real axis of the complex plane z = x + i y and turning from n = (n~, n2) = (cos ~p, sin fp) to the complex function

w ( z ) = nl ( z ) +in2(z) = exp[i¢(z)] ,

Iwl~=x=l, (5)

we write down the action in the form

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V o l u m e 125, n u m b e r 6,7 P H Y S I C S L E T T E R S A 23 N o v e m b e r 1987

1 " d z d z ' / w ( z ) - w ( : ' ) 12 (6) S = e c e . ( z - : ' ) 2

Inl : - t)

conserving the conformal invariance. Variation of (6) gives

w ( z ) - w ( z ' ) = 0 (7) , d : ' ( z _ z , ) 2 ,

Im H

which for ~ bounded in the semiplane, w(z) : ~ . , <cons t , coincides with the condi t ion that w(z) is a boundary value of the function w(z) an- alytical in the semiplane. Thus, taking (5) into ac- count we can see that the solution (7) is a boundary value of any function analytical (to be more exact, meromorph ic ) in the semiplane and bounded at Iz[ -~m, mapping this semiplane into the unit circle: (1) in the case of the upper semiplane we have an instanton solution with the topological charge m,

w,.(z) = e '~ ( ] z - a ~

~=1 Z- - ( l~

~o,,,(x) ~ ~ a r c t g X - a ~ ' + , = _ ~ ( 8 ) . ,- I a2~

where aE, and a2, are arbi t rary parameters defining the posi t ion and width of the instantons, (2) in the case of the lower semiplane the mapping obta ined from (8) by complex conjugat ion gives the anti- instanton

w , , , ( z ) = e ''~ f l ~--d~

~=1 ~--Cl~ '

~o ,,, (x) = - ~, , , (x ) , ( 9 )

with the topological charge - m . Note that (8) and (9) are the boundary' values o f the corresponding in- s tantons of the two-dimensional a -mode l [ 1 ]. The number of arbi t rary parameters of the m-ins tanton (and ant i - ins tanton) is equal to 2 m + l and its ac- tion is equal to

S,,, = (2re)2m/2o~ . ( 1 0 )

It follows from (10) that the integral

I d z d z ' ] w ( z ) - w ( z ' ) 1 2 (11) I = ( 2 ~ ) 2 _ ( Z - - Z ' ) 2

hn z. : ( }

can be considered as a new expression for the top-

ological charge of the mapping of the axis into the circle.

The nonl inear O ( N ) a -mode l (1) allows difl 'erent general izat ions s imilar to the two-dimensional case as one can easily see from its gradient form. For in- stance, for the Lie groups G and homogeneous spaces G/H we get

S ' = ~ f c L v d x ' T r ( g ~d ,g 'g ~d, g) l n l x - x ' " 2~

g e G / H . 1 2 )

In this case the condi t ion for the existence of the in- stantons is reduced to the condit ion 7r ~ (G /H) ¢ I and implies the mul t iconnectedness of G/H. In relation with this we point out that the major i ty of the ho- mogeneous spaces considered in physics is singly connected. The U(N) and the SO(N) groups and also the real project ive space PR ~ and the Stiefel mani- fold R V.~.~ , (the sets of ( N - 1 )-repers in the N-di- mensional space) are an exception.

~z , (U(N)) = 7 t , ( S O ( 2 ) ) = Z .

R /r 1 ( S O ( N ) ) = To, ( P R x ) = / r , ( V x.x I ) : Z 2 . (13)

Now we consider the quantum propert ies of the model (1). First we note that (1) has two possible cont inuat ions: over the d imensional i ty of the space d = 1 + e (e-cont inuat ion) and over the exponent of the denomina to r 2 ~ 1 + a ( a -con t inua t ion ) [5,7]. Separately in each case a scale invariance is lost but if we continue simultaneously provided a = 1 + e then the model remains scale-invariant .

At e = 0 and 1 - a < < l renormal iza t ion of ( 1 ) i s s imilar to the two-dimensional case [ 1,5]. For the charge (~ at small 1 - a we get

d / c ~ = ( 1 - a ) a - - C ' ~ G e . ( ' ~ = ( N - 1 ) / 2 ~ : .

/ = l n ( x / x , , ) , (14)

-vo= 2~A ~ is a cut-offconstant . It follows from (14) that at a = 1 the theory is asymptot ica l ly free; here eq. (14) is appl ied up to l , = l n ( R J x < , ) where R,: is the correlat ion radius.

R ~ M - I ~ x . e I ' '~ ' (15)

At l < l,: the behaviour of the physical quanti t ies is defined by the logari thmic dependencies . For in-

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Volume 125. number 6,7 PHYSICS LETTERS A 23 November 1987

stance, for the correlation function G ( x ) and mag- netization ( n ~ ( x ) ) we get

G ( x ) = [ 1 - a C x log(x/xo)] N/tN- ~ ~ ,

( n ~ ( x ) ) ~ (1 - a C ~ v l ) ~/2 . (16)

At a < 1 there is a nontrivial fixed point and con- sequently, in the theory exists a second-order phase transition at a = a * ,

a * = ( l - a ) / C x . (17)

Its critical exponents in this approximation do not depend on N [5] (r/is obtained from (21))

u = ( 1 - a ) -~ , q = 2 - a , (18)

and they agree with the exponents of the spherical model [ 8 ] and the limit N - ~ . The other exponents are obtained from the known relationships between them [9]. These results are obtained in the frame- work of the perturbative renormalization group (RG) , applicable at small fluctuations of the field n(x),

n-- I

n(X) =n(X) [ 1 --~02(X)] 1,2 .~_ E e , ( x ) ~ o , ( x ) , a=l

e,'et,=~,~, , e , . n = O ,

(q~2)_ ( ~ ~ ) ~ a C x l n ( x / x o ) < < l . (19)

To take into account large fluctuations (for instance, instantons at a = 1, N = 2) it is necessary to use the quasiclassical approximation. It was shown in the two-dimensional case that taking into account these solutions and fluctuations on their background re- sults in the statistical sum of the instanton quark gas [ 10] (or merons [ 11 ]) with a logarithmic interac- tion and the effective temperature of this gas is such that it is in the plasma phase; the screening radius is the correlation radius of the model and it coincides with the one obtained in the framework o f R G [ 10]. A similar behaviour is expected in the one-dimen- sional case since it follows from eqs. (10), (15) that at N = 2, a = 1, Rc also coincides with that defined by instantons.

Now we consider the influence of the anisotropy on the possibility of the phase transition and critical properties o f the model. Only the anisotropies of the discrete type (for instance, of the crystallographic

type) are o f interest, since it follows from eqs. (14), (15) that at cr = 1 and N > 1 there are no phase tran- sitions. The potential of the anisotropy V(n) can be given as the expansion over spherical functions on the sphere S N- ~. At small fluctuations locally near a certain potential minimum, whose position is de-

2 _ 1 ( q = 1, ..., Q is the num- fined by the vector eq, eq - -

ber of equivalent minima of V(n ) ) , it can be represented as

S 1 .,.~-nkJ d,x ( e q ' ? l ) '~ , (20)

where k = 1, 2; k> 2 corresponds to the external con- stant magnetic fields, to the uniaxial anisotropy and to the higher anisotropies respectively. $1 introduces a dimensional parameter Hk into the theory. This pa- rameter defines a bare correlation radius Rc= ( a H k ) -~. At HA such that

(~02 ) ~ a C N ln(aHkxo) - l << 1

we get from RG the equation for Yk = HA.x,

dzyk = [ 1 -- ½k( k + N - 2 )a/2n2]y~ . (21)

The equation for a ( l ) remains unchanged in this ap- proximation. Here we should note that in the fol- lowing orders on a, y the ordinary kinetic term

Sk= ½m j dx (dxn) 2

is generated in S. In the lattice variant the interac- tion o f the nearest spin

½m Z (nx-n-,+~o) 2 x

corresponds to it. The behaviour of the model de- pends again on the value of 1 - a. At a < 1 as before we have PT of the second order in the same point (17) with dimensions A~ of the parameters Yk and dimensions A~kl= 1--dk of "magnetizat ions" ((eq-n) k) conjugated to them,

1-o" Ak= 1-- ½ k ( k + N - 2 ) N - 1 " (22)

At a = 1 eq. (21 ) is applicable up to l,. which is de- termined from the condition

ln[ a ( lc)yk( lc)] - i = l n ( R J x o )

and in the logarithmic approximation we have

l~ = ln ( Rc/xo) = ln ( a H , xo) -1 (23)

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Volume 125, number 6,7 PHYSICS LETTERS A 23 November 1987

At l > l, co r re la t ions o f the n (x) c o m p o n e n t s or thog-

onal to eq decrease and the sys tem is in a homoge -

neous state wi th " m a g n e t i z a t i o n " < (e,~.n) ~) along

e,, and with the e f fec t ive t e m p e r a t u r e (~(l~).

(~(l~) =(~[1 -c~C, , ln(R~/x, , )]--

( ( e , ~ . n ) ~ ) = [ l + e e C x l n ( R J x o ) ] ~ + ' :~.:,.~ ~,

(24)

This state exists only at ~/~ < c~* which is de f ined by

the cond i t i on c~¢~ - , ~

( ~ * = C ~ / l n ( c e * y ~ ) - ~ _ C ~ / l n ( e e H ~ x . ) ~ (25)

Express ions ( 2 1 ) - ( 2 5 ) are sel f -consis tent only at

small H,,

( l&~)C~ >>ln(cxH~x,,) ~ (26)

Thus , we have shown that a small an i so t ropy results

in the appea rence o f PT at r7 = 1 and that it influ-

ences weakly PT at a < 1.

At large an i so t ropy the cri t ical p roper t i es o f the

mode l (1) co inc ide with that o f the Dyson cha ins

wi th d iscre te spins which have been cons ide red in

ref. [ 12]. The lat ter are de t e rmined by the gas o f low-

t e m p e r a t u r e exc i ta t ions o f the chain, i.e. k inks which

have i so topic charges s , q = e , , - e , . In the gas o f these

kinks in te rac t ing by the law

- (S , , , . S , . ; ) [ (x ) ' ~ -- 11/(7(1 - - o r ) + O ( x ' ) ,

PT is possible at a~< I [ 12]. Here in the cons ide red

con t i nuous va r i an t o f the cha in (1) in to which ac-

t ion we add S~ for regularity, the analogue o f the kinks

are the solu t ions o f the equa t i on

md~-,.n(x) - d , V(n) = 0 ,

connec t ing d i f fe ren t m i n i m a o f V(n) . The i r char-

acter is t ic wid th is r(~~ ( H d m ) ,.,2 and the i r ac t ion

S ~ ( H d n ) ~ - . A p p r o x i m a t i n g the pa r t i t ion func t ion

o f the m o d e l by a rare dens i ty gas o f such solut ions

and using the fo rmula

, : , J

f d~vd_v' ( d , n . d , n ) = k ~ n ' A : ~ n .

A ~ n - = n ( x ) I i, ,

we get the a b o v e - m e n t i o n e d gas o f kinks which en-

sures the equ iva l ence o f the cri t ical p roper t ies o f the

d iscre te and con t inuous models . N o t e that ins tan ton

so lu t ions can be represen ted now as a superpos i t ion

o f kinks.

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