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Corner transfer matrix study of an inhomogeneous Ising model.

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Ann. Physik 1 (1992) 125-133
Annalen
der Physik
0 Johann Ambrosius Barth 1992
Corner transfer matrix study of an inhomogeneous
Ising model
Ingo Peschel and Roland Wunderling
Fachbereich Physik, Freie Universitat Berlin, Arnimallee 14, W-1000 Berlin 33, Germany
Received 16 December 1991, accepted 16 January 1992
Abstract. We consider a planar Ising model with an extended defect described by couplings of the form
K ( r ) = K . (1 + A h ) . We determine the spectrum of the corner transfer matrix numerically and
analytically and use it to calculate the local magnetization at the defect and the corresponding, nonuniversal critical exponent.
Keywords: Ising model; Defect; Magnetization; Critical behaviour; Corner transfer matrix.
1 Introduction
Inhomogeneous two-dimensional Ising models have been the topic of quite a few studies,
not only because they are solvable, but also because they may display interesting
nonuniversal critical behaviour. This was found for models with line-like defects in the
bulk [ l , 21 or extended defects, either at a surface [3, 41 or in the bulk [ 5 ] , and can be
understood from general arguments [6, 71. Recently another type of model has been
added to this class [8, 91. The inhomogeneity in this case has rotational symmetry, i. e.
the couplings are of the form K ( r ) = K * (1 + A h ' ) , where r is the distance from a
centre. Physically, this could be the result of a lattice distortion. The critical behaviour
then is non-universal if 0 equals the thermal exponent yt. For the Ising case this means
u = 1. The critical exponent /3, of the local magnetization near the centre of the defect
then depends continuously on the amplitude A . It was calculated in Ref. [8] (in the
following referred to as BP) using a conformal mapping of the critical system onto a strip.
In the present paper we study this Ising model once more but with a completely
different technique. It is well-known that Baxter's corner transfer matrix (CTM) is the
ideal tool to obtain order parameters in solvable two-dimensional models [lo, 111. It
arises if one divides the system into wedge-shaped pieces and allows one to calculate the
order in the centre by fixing the variables along the outer boundary. So far, the CTM has
been used only for homogeneous systems but it is obviously also suited to any problem
with point symmetry. We apply it here to the model described above, treating the
anisotropic limit and rescaling it to the isotropic case. Since we have an Ising model, the
CTM can be expressed in terms of fermions and the basic quantities are the singlefermion eigenvalues. We determine them numerically and, at the critical point, analytically. With their help the centre magnetization mo and the critical exponent PI can be
calculated. For PI we recover the previous result and thereby confirm the validity of
conformal invariance in this inhomogeneous system.
Ann. Physik 1 (1 992)
126
The CTM spectrum is also of interest in its own right. In the hon ogeneous system it
has the striking property that the low-lying levels are equally spaced at all temperatures
[lo- 141. Our calculations show how this “linear” spectrum is deformed in the
inhomogeneous case. In particular, near the critical point it becomes a BCS-like spectrum
with a gap. This is again in accord with conformal considerations. The functional form
of the eigenvalues explains the continuously varying exponent and one can also see how
the CTM magnetization formula is related to the expression for the correlation length in
a strip, Thus, one obtains a coherent picture of the mathematical mechanisms behind the
physical behaviour of this system.
.--.-a
I
1
I
IW1)
...
...
2
0--.-0-0-0
..
.
...
...
:
.--.....--.-.....--.
~ 1
13-e-a..
1
I
I
.@--$-@.
I
I
,
Ik(iY-1)
.$-@-a
Fig. 1 Segment of the square lattice giving rise
.Q/,
and notation of
the couplings.
1 to the corner transfer matrix
N
2 Formulation
We consider the Ising model on a square lattice. Fig. 1 shows the segment giving rise to
the CTM. Of the two couplings K , , K 2 only the latter shall depend on the distance from
the centre, i. e. from the top spin in the figure. The spins along the lower boundary are
fixed. In the extreme anisotropic (Hamiltonian) limit K 2 ( n ) Q 1, K , s I the CTM can
be written down immediately [ 12, 131. One obtains ./ = exp{- 2KT Z ]where KT is the
dual coupling of K , , tanhKT = exp[-2K,), and -1
is given by
This is the Hamiltonian of an inhomogeneous transverse Ising spin chain. We choose
a
2n
+1
so that a measures the strength of the inhomogeneity. Upon introducing fermions
through the Jordan-Wigner transformation, 2’becomes
where I = K2/KT measures the temperature. The ordered phase corresponds to 1
A Bogoljubov transformation then brings Z‘into diagonal form
> 1.
I. Peschel. R. Wunderling, Corner transfer matrix study of an inhomogeneous.. .
with new Fermi operators I?,,, I?,?.Following Ref. [15] one obtains the squares
eigenvalues of a matrix ( A + B ) ( A - B ) which reads explicitely
Aopu,
127
as
-*..
'**.
2
AN-I
+
2
r((N-I
A N - 1 ,L(N
(5)
lN-1 P N
P2N
N a n d p N = 0. The diagonalization
with A,, = A(2n + 1 + a),pc, = 2n f o r 0 In
will be discussed in the next section. Because the end spins c$,0% are fixed, one
eigenvalue w o is exactly zero.
The operator .dwas introduced above as the 90" CTM of a very anisotropic system.
However, by rescaling the lattice constants by the correlation lengths one sees that it also
describes, near the critical temperature, an isotropic system with opening angle B = 4KT
4 1 [ 161. The inhomogeneity does not enter this relation since it appears only in the
coupling K 2 ( n ) .Thus we can write for an isotropic system
By considering the temperature variable t = 2(KT - K 2 ) one can also connect the
parameter a to the variable a = 8 n K 4 used in BP to describe the isotropic system.
They are related by a = a / n Finally, the centre magnetization mo is given by
mo =
z++ z-+
z+++ z - +
-
(7)
where 2 , + and Z - + are the partition functions with centre spin and outer spins fixed
to be parallel or antiparallel, respectively [lo, 1 I]. They are given by the trace of . d ( 2 n )
with even or odd number of fermionic excitations and one obtains
valid, because of the rescaling, in the vicinity of the critical point. At criticality, mo will
vanish. This can occur in various ways. In the case of a homogeneous system infinitely
many co, collapse towards zero. But it is sufficient that one eigenvalue becomes zero, or
that the o,accumulate at a non-zero value. This will, in fact, be found in the inhomogeneous system.
3 Numerical results
The eigenvalues w , were calculated using the Sturm sequencing property for tridiagonal
matrices [17]. The size Ncould be taken as large as 3 lo4 and was limited only by the
size of the matrix elements. The general features of the spectrum are
-
128
Ann. Physik 1 (1992)
For low temperatures (A
I ) the levels are given by w,,, = I ” = I ( 2 v + I + a).
This is a linear relation between o,and v.
With increasing temperatures the low-lying levels deviate from a linear law, if a =k 0.
This is illustrated in Fig. 2 for a = 1.
As one approaches the critical point A = 1, the o,accumulate at w = I a 1, as seen
in Fig. 3. In contrast to the homogeneous system there is a gap in the critical spectrum,
for a > 0.
Fig. 2 Lowest single-particle levels w , of 2,
equation (3), for a system of size N = 8 192,
a = 1 and three different temperatures.
Fig. 3 Lowest single-particle levels w , at the
critical point as a function of the inhomogeneity parameter a for a system of size N = 32768.
For a < 0, a number of additional levels appear below o
large N , they are given by
=
I a 1.
Asymptotically for
(9)
For the special values a = -(2m + 1) for which Am = 0 and the spin chain is cut
between sites m and m + 1 they are easily identified as the levels of the cut-off piece.
In addition there is, below these levels, a state for which o varies algebraically with
N at the critical point. It is a continuation of the lowest state for a = 0 for which w
cc I/ln N [12, 13, 16, 171.
These results will be rederived analytically and discussed further in section 4.
Finally, above the critical point one eigenvalue is asymptotically zero for all values of
a.
Inserting the o,into equation (8) gives m o ( I ,N ) . To determine the asymptotic value
rno(I, m ) = m o ( I ) sizes
,
N = 10 are sufficient if 1 2 2. Closer to the critical point one
needs larger systems, for examle N = 100 for I = 1.01. Magnetization curves obtained
in this way are shown in Fig. 4 for various values of a. As expected, an increase (decrease)
of the couplings raises (lowers) the whole magnetization curve.
The exponent P, can be determined directly from a plot of lnmo vs. ln(A - 1). It is
simpler, however, to work at the critical point and to use the finite-size formula
I. Peschel, R. Wunderling, Corner transfer matrix study of an inhomogeneous.. .
129
which can be derived from a scaling argument as for the homogeneous case. One uses
the fact that the perturbation does not change its form under renormalization. This is
also the reason for the non-universality of the exponent [6, 71.
Both methods give the same values for /3, = P,(a) and the resulting curve coincides
with the one obtained before by BP. The monotonous decrease of PI with a reflects the
increasing steepness of the magnetization curves for larger a, as seen in Fig. 4.
Fig. 4 Centre magnetization m,, equation (8),
as a function of temperature measured by 1 /A.
The curves from bottom to top correspond to
inhomogeneity parameter a = -0.9, -0.5, 0.0,
0.2 and 1 .O, respectively.
4
’.
0 6
07
08
09
,l/h
Continuum limit
At the critical point, the eigenvectors corresponding to the lowest eigenvalues of (5) are
slowly varying functions and a continuum limit can be taken. It is known, however, from
a similar problem [ 191 that one has to exclude the very centre of the system in this case.
Thus one considers a spin chain with its left end at n = k instead of n = 0.
To obtain the continuum limit, one first changes w,, + (- 1)’’ tynin the eigenvector y!
Writing p = ns, where s is the lattice spacing, w,, = w @ ) and expanding W @ k s) up
to second order one arrives at the differential equation
With
w@)
= y@)/fi
this simplifies to
The boundary conditions which follow from the discrete equations are
a
y’(r) = -y(r),
r
2r
y ( R ) = 0, R
=
=
Ns.
ks
(13)
(14)
We note that the fixing of the left boundary spin (which corresponds to pk = 0) is
essential for obtaining (13). For a free spin the condition would be y ( r ) = 0. The sign
of a enters only through the boundary condition at p = r.
Setting o2= a2 + lc2, the solution of (12), (14) reads
Ann. Physik 1 (1992)
130
The rapid variation of y @ ) for small p shows that one cannot extend it right into the
origin. The allowed K-values follow from Eq. (1 3). With the abbreviations
z
=
K
-1n-,
2
R
r
8
=
aln-
R
r
it can be rewritten as
Denoting the solutions by
0,=
z,, the eigenvalues are given by
Jyq
ln-
The dispersion thus is of the Dirac form with a mass term m = I a I or, equivalently, of
BCS form with a gap A = I a I. This is the lower bound found in the numerical
calculations. The zv have to be determined numerically. For large v they become
equidistant, I\,= (2v - 1)/7d2, while for small v one has deviations from these values.
Only for a = 0 they are completely regular.
For a < 0 there is an additional solution with K = ip so that w 2 < a2. Then
[ = p / 2 InR/r follows from the equation
and the eigenalue w = upis
wp =
JY.
ln-
This solution exists only for 8 < - 2 . For large negative 8 one finds
This is the algebraic dependence on the size mentioned in section 3. The additional
localized levels below o = 1 a 1 cannot be recovered in the continuum limit.
These results for the cc), could also have been obtained assuming conformal
invariance. Then one can relate2, which operates in a staircase geometry, to the simpler
of a homogeneous strip of width L with fixed boundary conditions and perturbed
away from the critical temperature by O ( l / L ) .But this relation can also be established
directly. The substitution [19]
<
I. Peschel, R. Wunderling, Corner transfer matrix study of an inhomogeneous.. .
131
transforms the differential equation (1 2) into
fi
+ Iy' - (-L)2]y
0,
dv2
=
where y = (1 / 2 L ) In (R/~)w.Together with the boundary conditions this is the equation
for the single-particle eigenvalues y of 27. This is the same situation as found in the
treatment of the homogeneous Gaussian model [19]. For 6 = 0, Eq. (22) even reduces
to the corresponding equation there. The boundary conditions, however, are different for
the two cases.
5 Magnetization formula
With the analytical result of the previous section we can calculate mo for a critical
system of finite size. Strictly speaking, one then treats a system with a small hole at the
centre around which the spins are rigidly coupled. However, the corresponding mo
should be essentially the same as for one central spin. We write (8) in the form
In mo =
a"
C In tanh -
2
V
and insert equation (17) for the w,. For large values of lnR/r the spacing of the values
K , vanishes and one can go over to an integral. The non-regularity of the lowest levels
plays no role and one finds, for a > 0
In m o = In r
Thus )novaries as R
p,=
--
1
"
-- 271
p
dKIn tanh
as expected. Changing to the variable w one has
W
where the BCS-like density of states u
gives
a2 "
=
2
7
1
0
In tanh 2 '
/
v
n appears. A partial integration then
sh2 u
,a> 0
d u sh(n a c h u)
With a = a / n this is exactly the expression of BP.
For the case a < 0 one has to write I a [ in Eq. (26) and to add the effect of the
eigenvalue up,Eq. (19). The latter gives an additional contribution I a I /2 = I a I / 2 n
to the exponent and this corresponds again to the result found before.
In BP the exponent PI was derived from the gap in a strip of N sites with periodic
boundary conditions. This gap, or inverse correlation length, is given by the difference
132
Ann. Physik 1 (1992)
f [ A ( NZ
(2n
+
1))
n= 1
i
-
A
(G
2n)l ,
m
where A ( k ) =
are the single-fermion eigenvalues. Compared with (23)
this expression looks quite different but, due to conformal invariance, both are actually
closely related.
It is also interesting to compare with the Baxter model which also shows a continuously
varying exponent [ 101. In that case the nature of the CTM spectrum is always the same
and the variation of p arises simply from the dependence of the level splitting on the fourspin coupling. In our case, on the other hand, p, varies because the functional form of
the ejgenvalues depends on the inhomogeneity.
6
Conclusion
We have studied an inhomogeneous planar king model and its corner transfer matrix.
We determined the general nature of the CTM spectrum and its analytical form at the
critical point. As a characteristic difference to homogeneous models we found that at
criticality the single-particle levels collapse not at zero, but at a finite value. The spectrum
was used to obtain the local magnetization at the defect and the conformal results were
recovered and thereby checked. One could also see how the expressions leading to PI in
the two approaches are related.
Inhomogeneous Ising models described by (T =k 1 can be investigated in the same way.
For example, numerical investigations show for enhanced couplings and (T < 1 that the
levels do not collapse at the critical point so that the local magnetization stays finite there
and then drops to zero discontinuously. This is analogous to the siutation at a surface
with a corresponding extended defect [4].
The present study presents another example where, for geometrical reasons, the CTM
is the natural quantity to work with. A system with an actual free corner has been treated
recently [20]. But one can imagine still other cases. One could treat the square analogue
of the present radially symmetric system, or models with star-like defect lines which, in
the Ising case, also show continuous exponents [21, 221. However, as experience shows,
one cannot expect the CTM approach to be as simple then as for the homogeneous
models where it has first been used.
References
[I]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
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[14] B. Davies, Physica A 154 (1988) 1
[I51 E. H. Lieb, T. D. Schultz, D. C. Mattis, Ann. Phys. (N. Y.) 16 (1961) 407
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[18] J. H. Wilkinson, The Algebraic Eigenvalue Problem Clarendon Press, Oxford 1977
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133
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