Corner Transfer Matrix Renormalization Group Method Applied to the Ising
  Model on the Hyperbolic Plane
arXiv:0704.1949v1  [cond-mat.stat-mech]  16 Apr 2007
typeset using JPSJ.sty <ver.1.0b>
Corner Transfer Matrix Renormalization Group Method Applied to the
Ising Model on the Hyperbolic Plane
Kouji Ueda1), Roman Krcmar2), Andrej Gendiar2), and Tomotoshi Nishino1)
1Department of Physics, Graduate School of Science, Kobe University, Kobe 657-8501, Japan
2Institute of Electrical Engineering, Slovak Academy of Sciences, SK-841 04 Bratislava, Slovakia
(Received )
Critical behavior of the Ising model is investigated at the center of large scale finite size systems,
where the lattice is represented as the tiling of pentagons. The system is on the hyperbolic plane,
and the recursive structure of the lattice makes it possible to apply the corner transfer matrix
renormalization group method. From the calculated nearest neighbor spin correlation function
and the spontaneous magnetization, it is concluded that the phase transition of this model is
mean-field like. One parameter deformation of the corner Hamiltonian on the hyperbolic plane
is discussed.
KEYWORDS: CTMRG, Ising Model, Scaling, Curvature
§1. Introduction
Baxter’s method of corner transfer matrix (CTM) has
been known as one of the representative tool for analyti-
cal stydy of statistical models in two dimension (2D).1–3)
The method is also of use for numerical calculations of
one point functions, such as the local energy and the
magnetization.3) This numerical application is a kind of
numerical renormalization group (RG) method, where
the block spin transformation is obtained from the di-
agonalization of CTMs. Such a RG scheme has many
aspects in common with the density matrix renormal-
ization group (DMRG) method,4–7) expecially when the
method is applied to 2D classical lattice models.8)
Introducing the flexibility in the system extension pro-
cess of the DMRG method to the Baxter’s method of
CTM, the authors developed the corner transfer matrix
renormalization group (CTMRG) method.9–12) In this
article we report a modification of the CTMRG method,
for the purpose of applying the method to classical lat-
tice models on the hyperbolic plane. Using the recursive
structure of the lattice, we obtain one point functions at
the center of sufficiently large finite size systems.
Quite recently Hasegawa, Sakaniwa, and Shima re-
ported deviations of critical indices of the Ising model on
the hyperbolic plane from the well known Ising univer-
sality classes in two dimension.14, 15) They predicted that
phase transition of such systems would be mean-field like.
To confirm their prediction, in the next section we con-
sider the Ising model on a lattice, which is represented
as the tiling of pentagons. The necessary modification of
the CTMRG method on this lattice is explained in §3.
We calculate the nearest neighbor spin correlation func-
tion and the spontaneous magnetization at the center of
large scale finite size systems. Critical indices for these
one point functions are studied in §4, and we confirme the
mean-field like properties of the phase transition. Con-
clusions are summarized in the last section. We discuss
a possible deformation of corner Hamiltonian in the hy-
perbolic plane.
§2. Ising Model on the Tiling of Pentagons
Let us consider the hyperbolic plane, which is the two
dimensional surface with constant negative curvature.
Figure 1 shows a part of a sufficiently large regular lat-
tice on the plane, where the lattice is constructed as the
tiling of pentagons.13) All the arcs are geodesics, which
divides the lattice into two parts of similar structure.
Each lattice point represented by an open circle is the
crossing points of two geodesics.
Fig. 1. Ising model on a regular lattice on the hyperbolic plane.
Open circles represent Ising spins on the lattice point, and
) is the local Boltzmann weight defined in
Eq. (2.2). Two geodesics drawn by thick arcs divide the system
into four quadrants, which are called as the corner.
We investigate the ferromagnetic Ising model on this
lattice. The Hamiltonian of the system is defined as the
sum of nearest neighbor Ising interactions
H = −J
〈i,j〉
σiσj , (2.1)
where 〈i, j〉 represents pair of neighboring sites, and
σi = ±1 and σj = ±1 are the Ising spins on the lattice
http://arxiv.org/abs/0704.1949v1
2 Kouji Ueda, Roman Krcmar, Andrej Gendiar, and Tomotoshi Nishino
points. Throughout this article we assume the absence
of external magnetic field. For the latter conveniences,
we represent the system as the ‘interaction round a face
(IRF)’ model. The local Boltzmann weight for each face
of pentagonal shape — the IRF weight — is given by
W (σa, σb, σc, σd, σe) (2.2)
= exp
(σaσb + σbσc + σcσd + σdσe + σeσa)
where β = 1/kBT is the inverse temperature, and σa, σb,
σc, σd, and σe are the spin variables around the face as
shwon in Fig. 1.
The partition function of a finite size system (with suf-
ficiently large diameter) is formally written as the con-
figuration sum of the Boltzmann weight of the whole
system
all the
spins
all the
faces
W , (2.3)
where we are interested in the thermodynamic limit of
this system. Note that it is rather hard to investigate the
system by use of the Monte Carlo simulations, since the
number of sites contained in a cluster blows up exponen-
tially with respect to its diameter. As a complemental
numerical tool, we employ the CTMRG method.
§3. Corner Transfer Matrix Renormalization
Group Method
The two geodesics shown by thick arcs in Fig. 1 divide
the system into four parts, which are called as corners.3)
Figure 2 shows the structure of a corner. We label the
spins on a cut as {σ1, σ2, σ3, . . .}, and those on another
cut as {σ′1, σ
3, . . .}, where σ1 is equivalent to σ
1. The
corner transfer matrix is the Boltzmann weight with re-
spect to a corner, which is calculated as a partial sum of
the product of IRF weights in the corner
C(σ′1, σ
3, . . . |σ1, σ2, σ3, . . .)
spins inside
the quadrant
faces in
the quadrant
W . (3.1)
The configuration sum is taken over spins ‘inside’
the corner, leaving those spins on the cuts. Con-
ventionally the matrix C is interpreted as block di-
agonal with respect to σ1 and σ
1, and the element
C(σ′1, σ
3, . . . |σ1, σ2, σ3, . . .) for those cases σ1 6= σ
is set to zero. The CTM thus defined is symmetric in
the case of the pentagonal lattice under consideration.
As shown in Fig. 2 a corner has the structure where
three parts labeled by P̄ and two parts labeled by C̄
are joined to a face W . The ‘fusion’ relation can be
represented by a formal equation9, 10)
C = W · P̄ C̄P̄ C̄P̄ C̄ . (3.2)
Note that C̄ is a corner of smaller size.
For convenience, let us observe the structure of the
part of the system shown in Fig. 3, where two P̄ , and C̄
are joined to W . Labeling the shown part by P , we can
Fig. 2. Recursive structure of a corner C.
formally write the fusion relation in the same manner
P = W · P̄ C̄P̄ . (3.3)
For a conventional reason we call P as the ‘half-row’,
although P is not a row on the hyperbolic plane. It is
easily understood that P̄ is a half-row of smaller size.
We have thus obtained recursive structure of the corner
C and the half-row P .
Fig. 3. Recursive structure of the ‘half-raw’ P .
As we have defined CTM for a corner, let us express
the Boltzmann weight with respect to the half-row
P (σ′1, σ
3, . . . |σ1, σ2, σ3, . . .)
spins inside
the half-row
faces in
the half-row
W (3.4)
in the matrix form, where the positions of spins
{σ1, σ2, σ3, . . .} and {σ
3, . . .} are shown in Fig. 3.
We call the weight in the matrix form as the half-row
transfer matrix (HRTM).
The 4-th power of the CTM
ρ = C4 (3.5)
is a kind of density matrix, since its trace gives the par-
tition function
Z = Tr ρ = TrC4 (3.6)
of a finite size cluster that consists of four corners. The
CTMRG Method Applied to the Ising Model on the Hyperbolic Plane 3
matrix dimension of CTM, and also that of the HRTM,
increases exponentially with respect to the system size.
In order to obtain Z numerically up to sufficiently large
systems, we introduce the block spin transformation that
is created from the diagonalization of the density matrix
ρ.4, 5)
Instead of directly diagonalizing ρ in Eq. (3.5), we first
create its contraction
ρ′(σ′2, σ
3 . . . |σ2, σ3, . . .)
ρ(σ′1, σ
3 . . . |σ1, σ2, σ3, . . .) (3.7)
and then diagonalize it
ρ′(σ′2, σ
3 . . . |σ2, σ3, . . .)
A(σ′2, σ
3 . . . | ξ)λξ A(σ2, σ3, . . . | ξ) , (3.8)
where the eigenvalue λξ is non-negative. Following the
convention in DMRG, we assume the decreasing order
for λ
. The orthogonal matrix A(σ2, σ3, . . . | ξ) repre-
sents the block spin transformation from the ‘row-spin’
{σ2, σ3, . . .} to the effective spin variable ξ. We keep m
numbers of representative states, which correspond to
major eigenvalues, for the block spin variable ξ. Apply-
ing the matrix A to the CTM and the HRTM, we obtain
‘renormalized matrices’ of 2m-dimension
C(σ′1, σ
2, . . . |σ1, σ2, . . .) → C(σ
′, ξ′ |σ, ξ)
P (σ′1, σ
2, . . . |σ1, σ2, . . .) → P (σ
′, ξ′ |σ, ξ) , (3.9)
where we have dropped the indices from σ1 and σ
9, 10)
Combining the recursive structures in Eqs. (3.2) and
(3.3), and the renormalization scheme in Eqs. (3.7)-(3.9),
we can obtain the CTM and HRTM in the renormalized
form for arbitrary system size by way of successive ex-
tension of the system.
Fig. 4. Extension pocess of (a) CTM and (b) HRTM in the renor-
malized expression.
Suppose that we have C(σ′, ξ′ |σ, ξ) and P (σ′, ξ′ |σ, ξ)
for a finite size cluster. In order to explain the extension
process, let us rewrite these matrices as C̄(s′ , ζ′| s, ζ) and
P̄ (s′ , ζ′| s, ζ).
(1) Substitute C̄(s′ , ζ′| s, ζ) and P̄ (s′ , ζ′| s, ζ) into the
fusion process in Eqs. (3.2) and (3.3). Figure 4
shows these fusion processes among W , C̄, and P̄ ,
where rectangles correspond to the block spin vari-
ables. The spin variables that are contracted out are
shown by black marks. As a result, we obtain the
extended CTM C(σ′ , s′ , ζ′ |σ, s, ζ) and the extended
HRTM P (σ′ , s′ , ζ′|σ, s, ζ).
(2) From the extended CTM C(σ′ , s′ , ζ′|σ, s, ζ) obtain
the density matrix ρ(σ′ , s′ , ζ′|σ, s, ζ) by Eq. (3.5).
Contracting out the spin at the center, obtain
ρ′ (s′ , ζ′| s, ζ) as Eq. (3.7), and diagonalizing it to ob-
tain the block spin transformation matrix A(s, ζ| ξ)
from Eq. (3.8).
(3) Applying A(s, ζ| ξ) to both C(σ′ , s′ , ζ′|σ, s, ζ) and
P (σ′ , s′ , ζ′|σ, s, ζ), obtain the extended CTM
C(σ′ , ξ′ |σ, ξ) in the initial form, and the same for
HRTM to obtain P (σ′ , ξ′ |σ, ξ).
(4) return to the first step.
The system size, which is the length of the longest
geodesics in the system, increases by 2 for each itera-
tion.16) In order to start the above extension process, we
set the initial condition
C̄(σ′ |σ) = P̄ (σ′ |σ) = δ(σ′ | 1) δ(σ| 1) (3.10)
that represents ferromagnetic boundary, where δ(a| b) =
δa,b is the Cronecker’s delta.
During the iteration we can obtain one point functions
at the center of the system. For example, the sponta-
neous magnetization is calculated as
〈σ〉 =
σ,s,ζ
σρ(σ, s, ζ|σ, s, ζ)
σ,s,ζ
ρ(σ, s, ζ|σ, s, ζ)
. (3.11)
In the same manner we obtain the nearest neighbor spin
correlation function 〈σs〉. It should be noted that one
point functions thus calculated at the center do not al-
ways represent the averaged property of the whole sys-
tem even in the thermodynamic limit, since the area near
the boundary has non-negligible weight in the hyperbolic
plane.
§4. Numerical Result Compared With the
Bethe Approximation
Let us calculate the spontaneous magnetization 〈σ〉,
and spin correlation function 〈σs〉 for the nearest spin
pair. We regard the Ising interaction strength J as the
energy unit, and use the temperature where the Boltz-
mann constant kB is equal to unity. Most of the numer-
ical calculations are performed keeping m = 40 states.
The dumping of the density matrix eigenvalues is very
fast, and actually the calculated results with m = 10 do
not differ from those obtained with m = 40 even at the
critical temperature TC. The iteration number required
for the numerical convergence is at most 400000 for the
calculated data points.
Figure 5 shows the calculated results. The square of
the spontaneous magnetization 〈σ〉2 is a linear function
of temperature in the neighborhood of TC. From the
behavior we estimate the transition temperature TC =
2.799. The nearest neighbor spin correlation function
4 Kouji Ueda, Roman Krcmar, Andrej Gendiar, and Tomotoshi Nishino
2.775 2.780 2.785 2.790 2.795 2.800 2.805
2.775 2.780 2.785 2.790 2.795 2.800 2.805
Fig. 5. Square of the spontaneous magnetization 〈σ〉2 (upper)
and the nearest neighbor spin correlation function 〈σs〉 (lower)
with respect to the temperature T .
〈σs〉 has a kink at TC, and is linear in T around there.
These calculated results support the existence of mean-
field like transition, that is subject to the critical indices
β = 1/2 and α = 0. We thus confirmed the prediction
by Hasegawa, Sakaniwa, and Shima.14, 15)
Compared with the transition temperature of the
square lattice Ising model T
Square
C = 2.269, the calcu-
lated TC is fairly higher and is close to the transition
temperature calculated from the Bethe approximation
TBetheC = 2.885.
17, 18) The result suggest that neglection
of the ‘loop back effect’ is not so conspicuous in the hy-
perbolic plane.
§5. Conclusion and discussion
We have calculated the spontaneous magnetization
and the nearest neighbor spin correlation function of the
Ising model on a pentagonal lattice on the hyperbolic
plane. The numerical algorithm of the CTMRG method
is modified for this purpose. The calculated critical tem-
perature is TC = 2.799, and we observe the mean-field
like phase transition.
The modified CTMRG method we have developped is
applicable to regular lattices that consists of geodesics
on the hyperbolic plane. For those lattices that does
not contain geodesics, one has to either treating asym-
metric density matrix or to draw geodesics by use of
transformation such as duality transformation and the
star-triangle relation. Generalization of the modified
CTMRG method to the vertex model is straight for-
ward. It may be interesting to classify ordered states
of eight-vertex model on a variety of regular lattices in
the hyperbolic plane.
An interest is in the eigenvalue structure of the den-
sity matrix at the transition temperature. Its analytic
form is not well defined in the thermodynamic limit of
classical lattice models on the flat 2D plane.19, 20) The
rapid eigenvalue dumping observed on the hyperbolic
plane suggests that there would be a way of regulariz-
ing the CTM at the criticality. The classical-quantum
correspondence from such a view point is worth consid-
ering. Formally speaking the corner transfer matrix C
can be written as the exponential of the corner Hamil-
tonian HC. When the lattice is on the flat plane, the
simplest example of the corner Hamiltonian is written in
the sum of local operators
HC = h(σ1, σ2) + 2h(σ2, σ3) + 3h(σ3, σ4) + . . .
= h1 + 2h2 + 3h3 + 4h4 + . . . , (5.1)
where hi = h(σi , σi+1) is the local hamiltonian that acs
between neighboring sites. A possible one parameter de-
formation of the corner Hamiltonian to the hyperbolic
geometry may given by
HC(Λ) = h1 +
sinh 2Λ
sinhΛ
sinh 3Λ
sinhΛ
h3 + . . . , (5.2)
which is reduced to HC in Eq. (5.1) in the limit Λ → 0.
The deformed Hamiltonian satisfies the recursive struc-
HC(Λ) = coshΛ
sinh 2Λ
sinhΛ
sinh 3Λ
sinhΛ
h4 + . . .
+ h1 + coshΛ h2 + cosh 2Λ h3 + . . . (5.3)
discussed in Okunishi’s RG scheme on the corner Hamil-
tonian.21) Such a deformation has similar regularization
effect proposed by Okunishi quite recently.22)
T. N. and A. G is partially supported by a Grant-
in-Aid for Scientific Research from the Ministry of Ed-
ucation, Science, Sports and Culture. T. N. thank to
Okunishi for valuable discussions.
1) R.J. Baxter, J. Math. Phys. 9 (1968) 650.
2) R.J. Baxter, J. Stat. Phys. 19 (1978) 461.
3) R.J. Baxter, Exactly Solved Models in Statistical Mechanics
(Academic Press, London, 1982) p. 363.
4) S. R. White: Phys. Rev. Lett. 69 (1992) 2863.
5) S. R. White: Phys. Rev. B 48 (1993) 10345.
6) Density-Matrix Renormalization - A new numerical method
in physics -, eds. I. Peschel, X. Wang, M. Kaulke and K. Hall-
berg, (Springer Berlin, 1999), and references there in.
7) U. Schollwöck: Rev. Mod. Phys. 77 (2005) 259 and references
there in.
8) T. Nishino, J. Phys. Soc. Jpn. 64 (1995) 3598.
9) T. Nishino, J. Phys. Soc. Jpn. 65 (1996) 891.
10) T. Nishino, J. Phys. Soc. Jpn. 66 (1997) 3040.
11) A. Gendiar and T. Nishino, Phys. Rev. E 65 (2002) 046702.
12) K. Ueda and T. Nishino, K. Ueda, R. Otani, Y. Nishio,
A. Gendiar, and T. Nishino, J. Phys. Soc. Jpn. 74 (2005)
supplement p.111.
13) A number of examples of the regular lattices on
the hyperbolic plane is listed in the following URL;
http://www2u.biglobe.ne.jp/˜hsaka/mandara/index.html.
14) H. Shima and Y. Sakaniwa: cond-mat/0511539.
15) I. Hasegawa, Y. Sakaniwa, and H. Shima: cond-mat/0612509.
16) Precisely speaking, such a way of system extension does not
guarantee the cluster shape where every spins at the boundary
is equally distant from the center.
17) H.A. Bethe, Proc. Roy. Soc. London A 150 (1935) 552-75.
18) Since the configuration number of the lattice is z = 4, the
mean-field approximation applied to the Ising model on the
pentagonal lattice under consideration gives the same transi-
tion temperature as the approximation applied to the square
lattice Ising model. Such an equivalence holds up to the Bethe
approximation.
19) I. Peschel, J. Stat. Mech. (2004) P06004 and references there
CTMRG Method Applied to the Ising Model on the Hyperbolic Plane 5
20) K. Okunishi, Y. Hieida, Y. Akutsu, Phys. Rev. E 59 (1999)
R6227.
21) k. Okunishi, J. Phys. Soc. Jpn. 74 (2005) 3186.
22) K. Okunishi, to appear in J. Phys. Soc. Jpn, preprint
arXiv:cond-mat/0702581.
