Random field Ising systems on a general hierarchical lattice: Rigorous inequalities

13 Pages·0.14 MB·English

by  Avishay Efrat

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Random field Ising systems on a general hierarchical lattice:
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J Avishay Efrat and Moshe Schwartz

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School of Physics and Astronomy, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
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Random Ising systems on a general hierarchical lattice with both, random
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values of the Jacobian renormalization matrix at the pure fixed point are ob-
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tained. These inequalities lead to upper bounds on the crossover exponents
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PACS numbers: 05.50.+q, 64.60.Cn, 75.10.Nr, 75.10.Hk
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Typeset using REVTEX
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Despite the many years of research and the large number of researchers working on
the subject, the study of critical behavior of random systems has led to only few exact
results. On the other hand, some of these results [1,2] played an important role, especially
in the context of the random field problem. In a recent study [3] we considered a random
bond Ising system on a general hierarchical lattice, where the renormalization group (RG)
transformation is exact [4], and obtained inequalities concerning the eigenvalues {λ } of
i
the Jacobian renormalization matrix, at the pure fixed point. The purpose of the present
study is to show that similar inequalities can be obtained if random fields are included. In
contrast to the case of random bonds and zero fields, correlations are now generated by
the renormalization flow. Nevertheless, it appears that these correlations are, first, confined
to the fields so that the distribution of bonds is left uncorrelated, and second, restricted
to nearest-neighbor (nn) correlations. It is important to emphasize that these short-ranged
field correlationsaregenerated bythe RGtransformationeven if oneassumes no correlations
to begin with, and that the range of correlations does not increase under the transformation.
Our results are relevant to real lattices, since some approximate RG schemes on real lattices
are in fact exact RG schemes on hierarchical lattices (Migdal-Kadanoff [5,6] and others [7])
and since it is believed that the critical behavior of an Ising system on a real lattice can be
mimicked by that behavior on a properly chosen hierarchical lattice [8,9,10,11].
We consider a general hierarchical lattice described schematically in Fig. 1. The shaded
area shown in (a) consists of a set of lattice points where some of the pairs are joined. In (b),
a typical shaded area is represented. The solid lines are bonds to be iterated in constructing
the lattice while the dashed ones are not to be iterated. The bold lines represent the
possibility for some bonds to be strengthened, namely, multiplied by some constant. The
random Ising system is represented by the dimensionless Hamiltonian
−H = J σ σ + h σ , (1)
ij i j i i
(Xi,j) Xi
where (i,j) refers to connected pairs only. All three types of bonds of Fig. 1(a) then carry a
coupling J12 (for the bonds joining sites α and β), while each site carries a field h12. (Note
αβ α
2
that one of the members of the pair αβ may be either 1 or 2.)
The renormalized couplings and renormalized fields are given by
J˜ = f {J12,h12} (2a)
12 J αβ α
and
z˜i
h˜ = h + f {J12,h12} i = 1,2 (2b)
i i h αβ α
j=1
X
respectively, while z˜ is the coordination number of the site i on the renormalized lattice.
i
Both, f and f , depend only on couplings and fields within the rescaling volume associated
J h
with the pair of sites (1,2) (the shaded area in Fig. 1(a)). Eq. (2a) implies that J˜ and J˜
ij lm
are not correlated if the pairs (i,j) and (l,m) are not identical. This does not hold for the
renormalized fields. Due to the sum in Eq. (2b) over nn sites on the renormalized lattice,
it is clear that even if there are no correlations to begin with, correlations are generated by
the RG transformation, between fields on nn sites and fields and couplings on a site and a
˜ ˜
bond attached to it. (For example, in Fig. 1(b), the following pairs are correlated: (h ,h ),
1 2
(J˜ ,˜h ) and (J˜ ,˜h ).) It is easier to deal with such correlations by considering a bond-field
12 1 12 2
Hamiltonian of the form
−H = [J σ σ +h (σ +σ )], (3)
ij i j ij i j
(Xi,j)
in which the random variables are the couplings J and the bond-fields h . The set of RG
ij ij
transformation equations is now given by
J˜ = f {J12,h12} (4a)
12 J αβ αβ
and
h˜ = f {J12,h12}. (4b)
12 h αβ αβ
Eqs. (4), therefore imply that non of the two couplings J˜ or h˜ , is correlated with any of
ij ij
the two couplings J˜ or h˜ , if the pairs (i,j) and (l,m) are not identical.
lm lm
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In terms of the bond-fields, the site-fields are given by
1
h = h . (5)
i ij
2
Xj(i)
where j(i) indicates that the sum is over all sites j connected to i. A similar bond-field
Hamiltonian (3) was already used in the past [12,13,14], only with the additional term
h† (σ −σ ). Note that it is necessary to include the dagger fields only if one assumes
(i,j) ij i j
P
that the site-fields are initially uncorrelated. Since, however, the initial state of non corre-
latedsite-fields isnotpreserved byRGtransformationandnncorrelationsbetween site-fields
are developed, there is no reason to start with uncorrelated fields on the sites.
We assume the existence of a ferromagnetic fixed point at {J } = J∗ and {h } = 0.
αβ αβ
We denote the departure of J from J∗ by δJ and define the moments as
αβ αβ
Γ = h(δJ )i(h )ji. (6)
ij αβ αβ
Clearly at the fixed point Γ∗ = 0. We will be interested in the recursion relations of the
ij
moments near the pure fixed point,
Γ˜ = G {Γ }. (7)
ij ij lm
Therecursionrelationsaboveareobtainedfromtherecursionrelationsforthelocalcouplings
∂f ∗ 1 ∂2f ∗ 1 ∂2f ∗
δJ˜ = J δJ + J (δJ2 )+ J h2 +... (8a)
12 (Xα,β) ∂Jαβ! αβ 2 (Xα,β) ∂Jα2β! αβ 2 (Xα,β) ∂h2αβ! αβ
and
∂f ∗ ∂2f ∗
˜ h h
h = h + δJ h +..., (8b)
12 αβ αβ αβ
(Xα,β) ∂hαβ! (Xα,β) ∂Jαβ∂hαβ!
where (...)∗ denotes evaluation at the pure fixed point. Note that althoughwe areinterested
only in the expansion of Γ˜ to first order in Γ , we still need, in principle, orders higher
ij lm
than linear in Eqs. (8) above. Note also the terms missing in the expansions due to the
different parities of J and h. The renormalized moments are given by
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∗ i ∗ j
∂f ∂f
˜ J h ij
Γ = Γ + A Γ , (9)
ij (Xα,β)" ∂Jαβ! # " ∂hαβ! # ij (Xα,β)Xl,m lm lm
where clearly, in the last sum on the right end side of the above, l + m > i +j. Also the
ij
parity of m in the sum must equal the parity of j. The A ’s with l + m > i + j always
lm
involve derivatives higher than the first of at least one of the f’s. We arrange next the Γ ’s
ij
using a single index
G = Γ , (10)
k ij
with
(i+j)(i+j +1)
k = +j +1. (11)
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This brings Eq. (9) into the standard matrix notation
G˜ = A G . (12)
m mn n
It is not difficult to show that if k corresponds to (i,j) and k to (l,m) then l+m > i+j
1 2
implies k > k . This means that the matrix A is block-triangular (Fig. 2). Consider next
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one of the blocks along the diagonal of A. From the expansions (8) and (9) it follows directly
that the only contribution to (δJ˜)i(h˜)j in (δJ)l(h)m such that l+m = i+j, is the one with
l = i and m = j. The final conclusion thus is that the matrix A is triangular, so that
its eigenvalues are just its diagonal terms. The eigenvalues of the Jacobian transformation
matrix are thus
∗ i ∗ j
∂f ∂f
J h
λ = . (13)
ij
(Xα,β)" ∂Jαβ! # " ∂hαβ! #
This leads now to a number of interesting inequalities:
(a) All eigenvalues are positive,
λ ≥ 0. (14a)
ij
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(b) All eigenvalues are ordered,
λ < λ and λ < λ . (14b)
i+1,j ij i,j+1 ij
(c) All eigenvalues obey a convexity condition,
λ λ ≥ λ and λ λ ≥ λ . (14c)
ij kj i+k,j ij ik i,j+k
(d) All eigenvalues obey
(λ )2 ≤ λ λ , (14d)
ij i+k,j+l i−k,j−l
where k = −i,...,i and l = −j,...,j.
Proof: In a recent paper [3] we have considered random bond Ising systems for which
the subset {λ } is considered. There, we have already proven properties (a)-(c) and our
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proof here will follow the same line.
Properties (a) and (c) are proven by showing that
∂f
J (J∗,0) ≥ 0 (15a)
∂J
αβ
and
∂f
h (J∗,0) ≥ 0. (15b)
∂h
αβ
We have to consider then the specific transformations generated by
−H˜ = lntr′e−H, (16)
where tr′ represents trace only over the subset of spins {σ } internal to the rescaling volume,
α
not including the external spins σ and σ . The renormalized couplings and fields given by
1 2
Eqs. (4), can now be written in the forms
1
J˜ = − tr σ σ H˜ , (17a)
12 12 1 2
4
h i
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1
˜ ˜
h = − tr (σ +σ )H , (17b)
12 12 1 2
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h i
where tr indicates trace over the two external spins σ and σ . The derivatives of J˜ with
12 1 2 12
˜
respect to J and h with respect to h are thus given by
αβ 12 αβ
∂J˜ 1
12(J∗,0) = tr (σ σ )hσ σ i (18a)
12 1 2 α β 12
∂J 4
αβ
and
˜
∂h 1
12(J∗,0) = tr (σ +σ )hσ +σ i , (18b)
12 1 2 α β 12
∂h 4
αβ
where h...i is the average with respect to H with σ and σ held fixed. In calculating the
12 1 2
above derivatives at the pure fixed point, we use the following symmetry properties of the
system,
∗ ∗
hσ σ i = hσ σ i , (19a)
α β ++ α β −−
∗ ∗
hσ σ i = hσ σ i , (19b)
α β +− α β −+
∗ ∗
hσ i = −hσ i , (19c)
α ++ α −−
∗ ∗
hσ i = −hσ i , (19d)
α +− α −+
to obtain
∂J˜ 1
12(J∗,0) = [hσ σ i∗ −hσ σ i∗ ] (20a)
∂J 2 α β ++ α β +−
αβ
and
˜
∂h 1
12(J∗,0) = [hσ +σ i∗ ]. (20b)
∂h 2 α β ++
αβ
Here the sign indices specifically indicate the state of the spins σ and σ and the ∗ indicates
1 2
that the average is with respect to the pure fixed point Hamiltonian
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−H∗ = J∗ σ σ . (21)
i j
(Xi,j)
Now, according to the GKS inequalities [15,16], if all the many-spin couplings J =
A
h ,J ,... in a general Ising system are positive, all the many-spin correlations hσ i =
α αβ A
hσ i,hσ σ i,... must obey hσ i ≥ 0. Using Eqs. (20), the averages are taken with respect
α α β A
to the pure ferromagnetic Hamiltonian (21), where the two external spins of each of the
rescaling volumes, which are held fixed, serve effectively as local fields. When these effective
fields are held both positive, the GKS inequalities hold, so that
hσ i∗ ≥ 0 and hσ σ i∗ ≥ 0, (22)
α ++ α β ++
which is enough already to prove Ineq. (15b). Ineq. (15a) can be easily shown to hold
using, in addition to the GKS inequalities, other rigorous inequalities, just recently proven
[17], also concerning the many-spin correlations in general Ising systems. It states that if
all the many-spin couplings J are positive again, the absolute value of all the many-spin
A
correlations hσ i does not increase when the value of any of the couplings is reduced, taking
A
any value in the interval [−J ,J ]. According to this, it is clear that under reversal of the
A A
+1 state of any of the two external spins, the many-spin correlations can not increase. So,
we arrive at the conclusion that
hσ σ i∗ ≥ hσ σ i∗ (23)
α β ++ α β +−
and Ineq. (15a) is also proven. This completes the proof of properties (a) and (c).
We turn now to property (b). here we need to show that
∂f
J (J∗,0) < 1 (24a)
∂J
αβ
and
∂f
h (J∗,0) < 1. (24b)
∂h
αβ
at any finite temperature. But, referring to Eqs. (20) again, it is clear that
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1 1
[hσ σ i∗ −hσ σ i∗ ] ≤ [|hσ σ i∗ |+|hσ σ i∗ |] ≤ 1 (25a)
2 α β ++ α β +− 2 α β ++ α β +−
and that
1 1
[hσ i∗ +hσ i∗ ] ≤ [|hσ i∗ |+|hσ i∗ |] ≤ 1, (25b)
2 α ++ β ++ 2 α ++ β ++
while the equality sign can hold only at zero temperature. This proves property (b).
We are left now with property (d). Here we use the more general definition of a scalar
product, (u,v) ≡ w u∗v where ∀i, w ≥ 0 and the corresponding Schwartz inequality,
i i i i i
which reads ( wPu∗v )2 ≤ w |u |2 w |v |2 (here is the only place where the * repre-
i i i i i i i j j j
P P P
sents complex conjugate). We replace, next, the sum over the single index i with the double
index (αβ) and identify
∗ r ∗ s
∂f ∂f
J h
u ≡ (26a)
αβ
" ∂Jαβ! # " ∂hαβ! #
with r = 0,...,i and s = 0,...,j,
∗ p ∗ q
∂f ∂f
J h
v ≡ (26b)
αβ
" ∂Jαβ! # " ∂hαβ! #
with p = 0,...,i−r and q = 0,...,j −l, and
∗ i−r−p ∗ j−s−q
∂f ∂f
J h
w ≡ , (26c)
αβ
" ∂Jαβ! # " ∂hαβ! #
to obtain
(λ )2 ≤ λ λ , (27)
ij i+r−p,j+s−q i−r+p,j−s+q
where we have used the fact that all partial derivatives are real and positive. All is left now
is to denote r −p = k with k = −i,...,i and, similarly, s−q = l with l = −j,...,j, which
completes the proof of property (d).
In addition to that, denoting by m < 1 and m < 1, the maximal values of ∂J˜ /∂J
J h ij αβ
˜
and ∂h /∂h respectively, we obtain
ij αβ
λ ≤ λ mi−kmj−l, with k = 0,...,i and l = 0,...,j, (28)
ij kl J h
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so that we have also proven that the number of relevant interactions at the pure fixed point
is finite. The only case of which the equality sign hold is the diamond hierarchical lattice
(DHL) [4,18], where all bonds are equivalent. From Ineq. (28) follows an inequality for the
crossover exponents:
(i−1)lnm +jlnm
J h
φ < 1+ < 1 i+j = 2,3,..., (29)
ij
lnλ
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whereφ = y /y , y = lnλ /lnbandbistherescaling factor. Theconditionforcriticality
ij ij 10 ij ij
of the pure fixed point is max(λ ,λ ,λ ) < 1, while else, we expect a random critical
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point with a different set of critical exponents. It is interesting to note that it was just
recently shown [19] that even for the random bond Ising system, the Harris criterion for
pure criticality [20], α < 0 (α being the specific heat exponent) is equivalent to the
p p
obvious requirement, λ < 1, only in the special case of the DHL. In the more general case,
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it was shown that α ≤ φ so that the Harris criterion is only a necessary condition for pure
p 20
criticality to hold and counter examples where α < 0 and φ > 0 have been presented. The
p 20
analogous result for the random system is γ ≤ φ (γ being the susceptibility exponent of
p 02 p
the pure system) but since γ turns out always to be positive, the random field is always
p
relevant at the pure critical point.
We wish to conclude by emphasizing that the inequalities proven here hold not only for
exact RG transformations on HL’s but also for all other renormalization schemes (such as
MK scheme [5,6]) in which the renormalized couplings or fields are not correlated or even
in cases where it is clear that the correlations are not important [21].
ACKNOWLEDGMENTS
We wish to thank D. Andelman for drawing our attention to Refs. [12,13,14].
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