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Corner Transfer Matrices for the Gaussian Model.

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Annalen der Physik. 7 . Folge, Band 48, Heft 1-3, 1991, S. 183-194
J. A. Barth, Lcipzig
Corner Transfer Matrices for the Gaussian Model
By I. PESCHEL
and T. T. TRUONG
Fachbereich Physik, Freie Universitiit Berlin, Germany
Zum3 200. Jahreskuy! der Annalen der Physik
A b s t r n c t . We study Baxter’s corner transfcr matrix for a Gaussian model on a strongly ;inisotropic square lattice of finite size. The problem is equivalent to finding t h e normal modes of a
vibrnt ing cliiiin with position-dependont mtisses and springs. The cigenralues are found analytically,
using Carlitz polynomials, and the predictions of conformcrl invnriiince ;ire verified for tho c r i t i d
system.
Eclien-Transformatria fur oin Gauss-Modell
I n l i a l t s i i b e r s i c ht . Wir untersuchen Baxters Ecken-Trnnsfermiltrix fur ein Gauss-Modell :huf
einem Qundriltgitter endlicher GrijBe mit stark anisotropen Kopplungen. Das Problem ist iiquivalont
zii dem, die Normalschwingungcn einer Kette mit ortsnbhiingigen Jlassen und Federn zu finden.
Mit Hilfe von Carlitz-Polynomcn lassen sich die Eigenwerte anulytisch bestimmen u ~ i ddie Vorhersagen der konformcn lnvarianz am kritisclien Piinkt iiberpriifen.
1. Introdrictioir
The corner transfer matrix (CTM) introduced by Baxter [ 11 has become a powerful
tool in two-dimensional statistical mechanics. It arises naturally if one divides a planar
system into wedge-shaped pieces. For the solvable models, this geometry leads to
the following remarkable property: Writing the CTM in the form A = esp(-&‘), the
eigenvalues of .@ are equidistant in the infinite system, with a splitting dependent on
the temperature. Partition functions and order parameters then take the form of simple
infinite products [2].
The properties of f i n i t e systems are especially interesting a t a critical point. Here,
conformal invariance again predicts equidistant, low-lying levels for .@ and a splitting
proportional to l/ln N where N measures the size of the system [3]. Therefore, the
CTM has been investigated in recent papers for finite Ising and vertex models, maiiily
in the anisotropic limit [4-81. It turned out that in these free-fermion problems Z?
can be diagonalized with the help of special polynomials introduced by Meixner, Pollaczek and Carlitz [9]. The spectrum was obtained for all temperatures and the conformal prediction could be verified.
I n this paper we study the Gaussian model. This plays a n important role in critical
phenomena since various other models can be related t o it [lo, 111. We treat again the
anisotropic (Hamiltonian) limit. The operator .@ can then be written down easily and
has a nice mechanical interpretation :it describes the quantum oscillations of a harmonic
Ann. Physik Leipzig 48 (1991) 1-3
186
chain where the inverse masses and the spring constants increase linearly along the chain.
We diagonalize 3 analytically, both a t the critical point and awayfromit. I n particular,
we determine its low-lying eigenvalues for a large, but finite system. The comparison
with conformal results is straightforward here since the continuum limit of t h e CTM
can easily be taken.
The diagonalization involves special polynomials, similar t o those in the Ising case.
This is not too surprising since the Gaussian model is the bosonic analogue of the Ising
model. (The correspondence for the row-to-row transfer matrices is outlined in the
Appendix). It is remarkable, however, that Carlitz in his 1960 paper on "Some orthogonal polynomials related t o elliptic functions" [12] studied exactly those four types of
polynomials which turn up in the CTM's of the two models. At the critical point they
all become Meisner polynomials of the second kind [9]. These were first studied in the
Thirties in connection with certain probability distributions [ 13, 141. I n the present
context they find an interesting physical application.
2. Corner Transfer Matrices
Wc consider a Gaussian model with variables @(-a < @ < a)a t each site of a
1
square lattice. The interactions between nearest neighbour sites are
Ja(@ - @')a
2
1
in the two directions (a = 1 , 2 ) and a mass term.-m2@2 is included. The CTM is the
-
2
partition function of lattice segments like those shown in Fig. 1, thus relating primed
and unprimed variables: A = A ( @ ,@'). I n the limit of large K , = PJ, and small K 2 =
pJa it can he written down by inspection. Let us first consider the lattice of Fig. la,
where t,he variables at the upper corner and along the lower boundary are fixed and
one obtains
set equal to zero. Writing A = exp(-K:H)
(2.1)
where h-; = l / K , is the dual coupling of K,, P = K,/h':, pa = /lma/K: and @, =
Qsfl= 0. The derivative operator arises from the horizontal bonds [15]. Note that the
last term in (2.1) includes only the contributions from one edge, as usually in a transfer
matrix.
The operator H obtained in this way can be viewed as the quantum Hamiltonian
of a vibrating chain with fixed ends. The inverse masses as well as the spring constants
in this chain inireme proportionally to n. There are internal and (for p p 0) external
springs. This is a rather unusual vibrational problem, but similar inhomogeneous systems
have been investigated occasionally [16, 171. We follow the standard procedure in lattice dynamics and solve the eigenvalue equation
(V - &V)?#
=0,
(2.2)
where T and V are the matrices of the classical kinetic and potential energy. This leads
t o the equations, for 1 5 n 5 N
-(2n - 1)2nyn-,
[4n2(2+ A 2 ) - w2]yn - 2n(2n l)lp,+, = 0, (2.3)
where A 2 = p2/iZa,w2 = e2/A2.We parametrize Aa in the form
+
Aa=(k+-- 1
k
2).
+
I. PESCHEL
and T.T.TRUOXG,
Corner Transfer Matrices
18i
Q”
Q “1
(bl
Fig. 1. Segments of n square lattice which lead to the coriicr transfer matrices in the test
so that k = 1 corresponds to bhe critical point (m = 0). Putting
Pn- 1
y n = p - y z n - I)!’
(2.5)
we obtain, for n 2 0
P, = [4n2(l k2) - ~ ‘ / c ] P , , -l (2% - 2)(2n - 1)22n* k2 * P,,-z.
(2.6)
This is exactly the recursion relation of the Carlitz polynomials g,, [12] with the identification
P,(o) = C . q , , ( - d k ; k ) .
(2.7)
The constant C is determined by the normalizationof y together with go = 1.The cigenvalues E, = ilw, follow from yNfl= 0, or
+
=0
(2.8)
and will be consideredin the next section. If qv denotes the amplitude of the normal mode
y(uv),H becomes
PN(0,)
(2.9)
Ann. Phyaik Leipzig 48 (1991)1-3
188
and in terms of boson operators b,, 'b the final expression is
N
(2.10)
"= 1
I n the same way the lattice of Fig. l b can be treated. Here, the upper and lower
variables are free. Compared to the previous case, the n-dependent coefficients are
interchanged and one obtains
H'
2
1
N
a
2
2
n=1
a@;
+
1
Tp2
1
- - 2 (2n - l ) - + T A z
A'
2
TI=
N-1
2
2n(@,+, - di,)'
n= 1
(2.11)
(2n - 1) q .
1
Equation (2.3) is then changed into
-(2n-2)(2n-
t)y;,-1+[(2n-
1)2(2+42)-d]y;-(2n-1)2ny;+l=0
(2.12)
and the substitution
(2.13)
leads to
Pk = [(2n - 1)2(1+ k2) - 0 ~ r l . ] P ~ - 1 (Sn - 3)(2n - 2)2(2n - 1)k2.PL-2.
(2.14)
This is the recursion relation of the Carlitz polynomials fn 1121 so that
PL(w)= C'
*
/,(-02k;
k).
(2.15)
The eigenvalues for free boundary conditions are determined by the condition y&, I =
yx. At the critical temperature, this relatioil can be expressed as
0
9 * g,v-1(-09;
1)= 0 ,
(2.16)
so that the gn enter again as in Equ. (2.8).
The recursion relations found here are very similar t o those appearing in the treatment of the Ising model CTM (compare Equs. (6) and (9) of Ref. [GI). We also note that
(2.6), (2.14) differ only by a shift of the integers and thus can be viewed as forming
one combined set [181. Physically, this would correspond t o superimposing the lattices
of Figs. l a , b . At the critical point, this idea can be used to simplify the recursion rela-
tions. Putting
yn =
*
fzn.
(2.17)
one can show that, for k = 1, (2.3) follows from
where the coefficients increase only linearly for large n. This equation is in turn very
similar to the corresponding relation for the CTM of the doubled Ising model and the
related X Y spin chain [51.
I. PESCHEL
and T. T. TRUONG,
Corner Transfer Matrices
181)
3. Eigenvalues
I n this section we determine the eigeiifrequencies 0, of the vibrating chain ;Lnd
thereby the spectrum of the CTM. For this we need closed espressions for the polynomials gn or f,,. As noted above, both can be treated together and a common integral
representation, using the generating function [ 181 and Cauchy’s theorem, reads
n!.+ cn(w, k) dn(w, k) .
R%(x;k) = 2Eb
-
Slln+l(w,
k)
elxv
dw .
(3.1)
Here sn, cn, dn are Jacobi elliptic functions and one has
R2,,+1(2;k) = C
~ f n ( - ~ ’ k);
; B z a + z ( ~k); = -x~~,,(-z’; k).
(3.2)
We evaluate (3.1) as in Ref. [GI by choosing a rectangular contour in the v-plane with
edges at Re w = &K and l m w = &K‘, where K = K ( k ) , K = K ( k ’ ) , k’ = (1 - k2
and K ( m ) is the complete elliptic integral of the first kind. Using the t,ransforrnation
proportics of the Jacobi functions we arrive at the following result for the even p l y noinials
I<
x sinh(zy) + k2’1sinIiK’x J’
- 1%-
I
dy cn(y, k)dn(y,k)~ I I ~ ” + ~ ( ?k)J sin(ay)
,
. (3.3)
With (2.7), (2.8) the eigenfrequeiicies of a chain with N sites follow from the zeros of
R1,,+?.
We distinguish three c<ases:
(A)
k + 0. Then the I-term in (2.1) is iiiiimportant and the masses oscillate indepen-
dently with
to,=A*2~;
V =
I , 2 ,..., N .
This result also follows from (3.3). For k
evaluated as
3
(3.4)
0 only the first term remains and can be
(3.5)
Thus, the zeros are strictly equidistant and lead again to (3.4).
(b) 0 < k < 1. I n this intermediate case one obtains an esplicit result only if tlie
system is sufficiently large. Then again only the first term i n (3.3) is important. I t s
zeros are determined by sin Kx = 0, so that
0, =
z
(r.K ( k ). v ;
Y =
I , 2 , ....
(3.6)
The levels are again equidistant, with a splitting dependent on k, i.e. on the distance
from the critical point, as in the CTM’s of other models. However, Equ. (3.6) only holds
for the low-lying levels. A silm rule demands that the higher LO,lie above t,he law (3.6).
A numerical calculation for A’ = 20 is shown in Fig. 2 and illustrates this feature.
For a fixed size of the system, the linear region becomes smaller as one approaches the
critical point. At the same time, the spacing vanishes like l/ln k.
Ann. Physik Leipzig 45 (1991)1-3
190
1
10
20
v Fig. 2. Spectrum of tlic operator €I, Equ. (2.1), for N = 20 sites and four vnliies of thc parmnctcr 11.
The curvcs are guides for tho eye
(c) k = 1. For the Gaussian model this corresponds to the critical point. For th(3
chain it means that the esternal springs are absent. The polynomials can then 'I.,(.
written
(7)/
+W
R,(x; 1) = i-n! sinh
G
".
=,tanhn-l(y).
dY
eizu.
(3.7)
-W
They are a spccid case of the Mcixner-Pollaczek polynomials [ O ] . Conipared to those
encountered in the fermionic systems [ 5 , 61 they have one additional power of cosli y
in the integrand.
For large m, one can einploy a saddle-point technique t o find the zeros [ 71. The stationary
points are ;c f with
1
taiih x h =
[ i z & 1/4(nZ - 1) - $1
2(n
1)
and R, takes the form
+
(3.S)
X
where y + is the phase a t z+.The calculation gives F+ = -In
2
of R2N+2
one obtains, for In N 9 1
w,
2n
=-1nN
';
Y
=1,2
4n and from the zeros
... .
This logarithmic dependence on N is also the prediction of conformal invariance and will
be derived in the continuum limit in the next sect,ion. Note that because of (2.16) the
0, are the same for the operator H ' . By contrast, the formula for the Ising model coiiinstead of Y [6, 71.
I. PESCEEL
and T. T. T ~ n o x aComer
,
Transfer Matrices
191
4. Continiiurn limit
-4t the critical point a continuum description should be valid for a large system.
Xow, i n the present problem the discrete eigenfunctions even for small OJ, vary considerably near n = 0 and the rontjnuum approsirnation is not very good there. This effect,
however, becomes less pronounced if one cuts some sites off the left end of t h c chain.
This is seen in Fig. 3 whcrc the lowest eigenfunction (Y = 1) is already quite smooth.
We tlicrcfore set p = nu, y,, = y(p) and consider a chain with fixed ends at r = M u
and R = Nu,where rc is the lattice constant. Expanding the quantities in Equ. (2.3)
we obtain the differential equation
A change of variables c, = r csp(;r/l), 1 = L/ln
(+)transforins this into
with ip(0) = y ( C ) = 0. The solutions arc sine frinctions
1-
2 .
yy(4= )I1- s1n(c/.r)
9
(4.3)
with q = m / L . Therefore
(4.4)
This formilla is the continullin version of Equ. (3.9) and shows again the well-krioivn
logarithmic dependence of the levels on thc size of the systcn-i [3].
Written in terms of p , the eigenfunctions are
For the parameters iised in Fig. 3, these continuum functions are practically the same
discrete frinctions shown there. Generally speaking, they resemble Airy functions,
hut t h e nmplitiide of the oscillations is the same everywhere.
The continuum limit can also be taken directly in H . One then obtains, after rescaling
the @'s
:is the
with
(1.7)
From the correlation lengths 5, = 1 / z / m near the critical point (c.f. Appendix) one
can see that 2AK: = 21/K2/K1is just the effective opening angle 0 at the corner if
the anisotropic system is made isotropic by a rescaling of the lattice [ 191. Therefore
Ann. Physik Leipzig 18 (l!l!)l)1-3
0 .L
0.3
0.2
0.1
0
- 0.1
-0.2
-0 3
- 0.L
Fig. :1. Eigenfiinctions ?pa, Eqii. (2.3), for the lowest threc rigenvalues w,. in
tn-ccn ?L = 2 and n = PO. The curves are guides for the eye
i~
cliain of 19 sitw bc-
(4.6), ( 4 . i ) correspond to the general formiiln [?O]
A = e-
@U,
for the CTM of an isotropic Gaussian continuum.
One can use the same substitution as above to transform H,:.
This corresponds to
R conformal mapping of the original, anriiilsr system i rito a, strip-like o w . Correspondingly, El(, becomes
(4.9)
and, i n this form, is the operator associated with the row-to-row transfer matrix of
the continurim system. I n the mechanical picture, it describes the longitudinal qiiantum
vibrations of a homogeneous string. It is diagonalized by the functions given in (4.3)
and thereby the same w, are obtained. The calculation of the universal (Casimir) term
i n the ground state energy of (4.7) is somewhat more involved [20].
6 . Concliision
In this paper we have studied the corner transfer matris for an anisotropic Gaussian
model of finite size. This was done in analogy to the corresponding calculations for the
Ising model. I n the latter case, this leads t o the study of inhomogeneous spin one-half
chains. Here, we were led to harmonic chains with a particular distribution of masses
and spring constants. We determined their spectrum esplicitly for large systems and
could verify the conformal prediction at the critical point. Compared t o the Ising case
somewhat different polynomials occur in the solution and this leads t o a shift of the
eigenvalues. This is analogous to the situation in the row-to-row transfer matrices and,
although looking like a small detail, actually reflects the different universality classes
1
(c = Falid c = 1 in the conformal classification of the two models.
I. PESCREL
sncl T.T.TRUOXG,
Corner Transfer Xntrices
193
Acknomledgemeiit. We thank Dr. Jf. Baake for his L-Iiiversity of Borin report,
(HE-86-22) with references on small oscillations and orthogonal polynomials.
Appendix
For a comparison, we present heit I~rieflythe row-to-yon- transfer matris of tlie
Gaussian model. Consider a sqiiare lattice i n the form of a strip in the direction nf J,,
with variables fised at the edges. Tlic symnirtrized transfer matiis theri is
(A.1)
(A.2)
with q = d / ( N
+ 1) (1 = 1, 1,... iv) lends t o
(A.5)
(A .I i)
where mq = 2 sin(q/./a).
I n the Hamiltoninii limit (K:, K, 1) the operator in the exponent describes n.
hoitiogeneous harmonic chain and one 111ere)y has to introduce hoson operators via GI ;=
(6,
b,+)/2Qn,, Qi = (0:
,n2/J2t o bring it into diagonal form. In general, howewr,
a further Bogoljrtbov transformation is necessary. 111the new operators, V becomes
<
+
+
(A.7)
with
eB = Parsinh
(
vq a).
K,SZ,
(A.S)
(A.7) is the analogiie of the corresponding formula in the k i n g model where fermion
operators appear [ 2 l , 221. By parametrizing the Gaussian niodel properly one can also
bring &Qinto the well-known Ising forin [23]. The allowed y-values, however, are not the
also different [24, 251. Froni
same in the two models and therefore the Casimir effect isthe gap in &p one obtains the correlation length El = VJ,/m near the critical point.
The sound velocity at the critical point is given by u = fK2/Klu
194
Ann. Pliysik Leipzig 48 (19!Jl) 1-3
One can also superimpose two independent Gaussian models and then use Ihster’s
vertex formulation. The corresponding diagonal transfer matrix has been disciissed hy
Babudzhyaii and Tetelmati [%I. In the anisotropic limit, it! is again related to R simple
homogeneous chain.
References
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[‘7] BAXTER,
R. J.: Exactly solved models in statistical meclinnics. London: Acirclemic Press 198-1.
[3] PESCHEL,
I. ; TRUOSG,T. T. : Z. Physik B 69 (1987) 385.
[A] PESCIIEL,I.: J. Phys. h ’11 (19W) L 185.
[5] TRUOSG,T. T.;PESCHEL,
I.: J . Phys. A I1(19S8) L 1029.
[61 TRnosc, T. T.; PESCHEL,
I.: Z. Physik B 55 (19S9) 119.
[i] DAVIES, B.; PEARCE,
P. 9.:
J . Pliys. 4,t.0 appear 1990.
[S] TRUOXG,
T. T.; PESCHEL,
I.: Int. J. Mod. Phys., to appear 1990.
[9] see CHIHARA,
T.s.: An introduction t.o ort.hogoniil polynomials, ?u’ewYork: Cordon and Bwrcli
195s.
[lo] KADAXOFF.
L. P.; Beows. A. C.: Ann. Phys. (S.Y.) 121 (1979) 315.
[ll] DEN Nus, 31. P. 11.: Phys. Rev. H I 3 (1981) 6111.
[l?] CARLKTZ,
L.: Duke 1Iut.h. J . ‘15 (1960) 44%
[13] MEISNER,J.: Ann. Physik (5) 30 (1937) 44.
[14] &rEISSER,J.; J . LOII~OII
1litth. SOC.!) (1934) 6.
[l.;] see e.g. KOGCT,
J . B.: Rev. Mod. Pliys. 51 (“3)
569.
[16] Various systems connected with Ltgucrre polynomials are discussed in: BOTTEx\, 0.:Jdireslwr.
Dent. Math. Ver. $2 (1953) $2.
(1985) 7 3 J .
[17] A recent study of a non-uniform string is: FULCHER, L. P.: Am. J. Phys.
[lS] CARLITZ,
L.: Duke JIatli. J . 2s (1961)107.
[19] BARBER,
11. K.;PESCHEL,
1.; PEARCE,
P. A.: J. Stat. Phys. 37 (1984) 497.
[30] CARDY,
J . L.; Pssram, I.: Rucl. Pliys. B SO0 [ F S 2 ] (19%) 377.
[ill SCHULTZ,
T. D.; JIATCKS,
D. C.; LIEB,
E. H.: Rev. Mod. Phys. 36 (I%;4) Y X .
[22] ABR.iII.Uf, D. B.: Stud. Appl. Mnt.h. .iO (1971) 71.
[93] SATO,
&I.; VIWA,
T.; Jnrso, 31.: Holonomic Qiiuntiim Fielcls 1- Pitbl. Kes. Inst. Miitli. Sci.
(Kyoto Univ.) 16 (1980) 351.
[2-4] BLOTE,H. W.J.; CARDP,J. L.; NIGHTIKCALE,
11. P.: Pliys. Rev. Lett. 56 (19SG) 7-42.
[%I AFFLECK.I.: Pliys. Rev. Lett. 56 (1986)74ti.
(1982) 484.
[%] BABUDZHYAS,
G. 31.; TETELJIAN,M. G.:Theor. JIath. Phgs.
Bei der Rednktion eingegangen am 5. April 1990.
Anschr. d. Verf.: Prof. Dr. I. PESCHEL,
Prof. Dr. T. T. TRUOKG
Fuchbereich Physik
Freie Universitat Berlin
Arnimallee 14
W-1000Berlin 33, Germany
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