数理解析研究所講究録
第 1904 巻 2014 年 13-30

13Domain Wall Renormalization Group Approach to
the $2d$ Ising Model with External Magnetic Field
Ken-Ichi Aoki,1, Yasuhiro Fuji i 1, Tamao Kobayashi, 2,
Daisuke Sat 0l,\S and Hiroshi Tomita1,1
1 lnstitutefor Theoretical Physics, Kanazawa University
2
Department of Liberal Arts, Yonago National College of Technology$*$$\ddagger$$\dagger$$|$

Abstract
We represent the spin configuration of 2-dimensional Ising model by the domain wall configuration defined on the dual link and formulate the domain wall
renormalization group (DWRG) according to the tensor network renormalization
technique. In this report, DWRG is extended to include the external magnetic
field by introducing the oriented domain wall variables. The renormalization grouptransformation is obtained in 6 dimensional parameter space completely analytically and its eigenvalues around the non-trivial fixed point is calculated to give themagnetic susceptibility exponent.

1

Introduction

The 2-dimensional Ising model is one of the best work benches for non-perturbative renormalization group approach [1], because it has non-trivial spontaneous magnetization andalso the Onsager’s exact solution as the leading guide. In fact, various Renormalization
Group (RG) approaches for the model has been already proposed.
The domain wall renormalization group (DWRG) [5] is a new type of renormalizationgroup method adopting the tensor renormalization group (TRG) [2] technique. TRG
has been applied to various models, particularly classical and quantum spin models in
2-dimension [3, 4].
In domain wall representation, there has been a remaining non-trivial problem of how
to calculate magnetic exponents. Our aim here is to propose a simple extension of our
method [5] to include the external magnetic field.
-mail address: aoki@hep.s.kanazawa-u.ac.jp-mail address: -fujii@hep. .kanazawa-u.ac.jp-mail address: kobayasi@yonago-k.ac.jp-mail address: satodai@hep.s.kanazawa-u.ac.jp-mail address: h-tomita@hep.s.kanazawa-u.ac.jp$*e$$\dagger_{e}$$y$$\ddagger_{e}$$\S_{e}$$\eta_{e}$

$s$

14

2

Domain Wall Representation

Domain walls are the boundary of spin up or down domains. The domain wall variables
are best put on the dual links defined in Fig.1, where the solid lines represent the originalsquare lattice and the dashed lines represent the dual lattice which is also square. The
dual links correspond to the boundary of every neighboring spins.

Figure 1: Dual lattice of the $2d$ square lattice.

The domain wall variables thus reside on the dual links as shown in Fig.2. Spin
, and it may
and domain wall variables are defined byvariables aretake two values: $\alpha=+1$ represents no domain wall and $\alpha=-1$ represents the existenceof the domain wall.
$\sigma_{i}=\pm 1$$\alpha_{ij}=\sigma_{i}\sigma_{j}$Figure 2: Definition of domain walls.

Any spin configuration can be represented by a domain wall configuration. As shown in
Fig.3, the domain wall variables on the four links sharing a single dual site are determinedby four spins around it. We regard the final part of Fig.3 as the domain wall vertex definedon the dual site.
However, the correspondence between spins and domain walls are not one to one. Tow
are mapped to the single domain wall configuration
spin configurations$\{\sigma_{i}\},$$\{-\sigma_{i}\}$$\{\alpha_{ij}\}$15
and this is two to one mapping. Therefore, although there seems to be no spontaneous
symmetry breaking in the domain wall representation, its partition function or the freeenergy has the exactly same singularities as the original spin partition function.

Figure 3: Domain wall representation of spin configuration.
The domain walls constitute the boundary of spin up or down domains, and therefore
the domain wall is conservative and makes topologically loop objects. Then the nonconservative odd vertices as shown in Fig.4 are prohibited.
$\prime$1
1$–rightarrow$
1
1$\bullet/$Figure 4: Prohibited non-conservative vertices.
Then we define the vertex tensor at each dual site, which is a function of four domain
wall variables around the dual site as shown in Fig.5. Since it has four suﬃces, it is called
the tensor.
$\alpha$$T$$T_{\alpha\beta\gamma\delta} \beta---|---\delta|$$\gamma$Figure 5: Vertex tensor.

The value of the vertex is determined so that the total product and the total summation
with respect to the domain wall variables give the partition function of the original system
as shown in Fig.6. The factor 2 at the top represents the two to one mapping nature ofdomain wall configuration versus spin configuration,

16$\beta$$Z=2$$\sum T_{\alpha\beta\delta\gamma}T_{\delta\mu\nu\rho}\cdots$$\alpha\beta\delta\gamma$

Figure 6: Partition function.

Therefore, the vertex value is proportional to the square root of the local statisticalweight of the configuration. In case of non-conservative vertex, the statistical weight musttensor are only 8 elements
vanish. Then the non-vanishing elements of thisas shown in Fig.7$2\cross 2\cross 2\cross 2$

1$—+–\cdot T_{++++}$

1/’$ro\iota z\iota j\circ nsT_{+-+-}$$—\ulcorner_{and_{I}csro\iota at/ons}T_{++--}$and its

Figure 7: Non-vanishing elements of tensor.

3

Domain wall renormalization group.

In this section we discuss how to define the renormalization group transformation in thedomain wall representation, that is, renormalization of its vertex tensor . The basic
strategy to define the renormalization group is flow-charted step by step in Fig.8.
According to this chart, first we define coarse graining of the dual lattice. We define
the coarse grained original lattice a la Wilson [1], and then the coarse grained dual lattice
is automatically defined as in Fig.9.
Note here that the coarse graining of original lattice keeps a half of their sites commonly, while the coarse graining of dual lattice has no common site nor link. In terms ofdecimation, the coarse graining of lattice is obtained by decimation of sites, whereas thecoarse graining of dual lattice is obtained by decimation of domains (dual plackets).
Then we define the coarse grained domain status as shown in Fig.10. Note that thedecimation of domains here is equivalent to the decimation of spins on the original lattice.
$T$

17

Definition of coarse grained dual lattice
Micro domain$Configuratio\cap l$mapping rule

1

$-$

Macro

$wa\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$

1.

$doma/nwa\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$

state

state

’ $ntroduc/ng$ macro variables

$1$

$-$

integrate out micro variables

Inherit the statistical weight:
Eﬀective interactions given by a function of macro variables

Figure 8: Domain wall renormalization group.

$arrow$

Decimating sites$arrow$

Decimating domains

Figure 9: Coarse graining lattice and dual lattice.

18
The coarse grained domain status is just equal to the micro domain status at the centerof it.

Figure 10: Definition of coarse grained domains.

Next, the coarse grained domain walls are defined by the boundary of the coarse
grained domains. Thus we get the mapping function from micro domain wall configuration to the macro domain wall configuration. In Fig.11 we show the mapping function,
where the blue dual link denotes the micro domain wall and the red coarse grained duallink denotes the macro coarse grained domain wall. The 4 configurations in the upper
row correspond to non-existence of the macro domain wall and those in the lower row
correspond to existence of the macro domain wall respectively. It is the eight to two
reduction mapping and this is the coarse graining of domain walls.

Figure 11: Macro domain walls and the mapping function.

It should be noted here that the macro coarse grained domain walls are boundary of
the coarse grained domains, and accordingly the macro domain walls are also conservedby themselves.
An example of the coarse graining of domain walls is shown in Fig.12. The left small
island survives the coarse graining whereas the right one disappears.
Now the most non-trivial part comes up. We have defined the domain wall mapping
function between micro and macro. Thus the macro domain wall configuration must havesome proper statistical weight to be maximally consistent with that given by the micro

19

Figure 12: Coarse graining example.

domain wall configuration. And also the macro domain wall weight must be represented
by the renormalized tensor vertex. To make this procedure we learn from the tensor
network renormalization technique.
First of all, we split the micro domain wa114-vertex into a product of 3-vertices
as shown in Fig.13, which is something like transforming the four-fermi interactions into
a pair of yukawa interactions by introducing auxiliary scalar fields. This decompositionis performed by the following matrix diagonalization,
$T$

$S$

(1)

$T=UDV$,

where tensor is regarded as a matrix by grouping the domain wall indices as
andare unitary matrices. The resultant matrix $D$ is a diagonal matrix of singularvalueswhich are all positive semi-definite. Then we can define as$T$$T_{\{\alpha\beta\}\{\gamma\delta\}}$$U,$ $V$$S$$\lambda_{M}$$T_{\alpha\beta\gamma\delta}= \sum_{M}U_{\alpha\beta M}\sqrt{\lambda_{M}}\sqrt{\lambda_{M}}V_{M\gamma\delta}=S_{1\alpha\beta M}S_{2\gamma\delta M}$$S_{1\alpha\beta\Lambda I}=U_{\alpha\beta l1f}\sqrt{\lambda_{M}}, S_{2\gamma\delta M}\sqrt{\lambda_{M}}V_{M\gamma\delta}$

(2)

,

.

(3)

Actually our
matrix is a symmetric real matrix and it can be diagonalized by theorthogonal transformation, that is, we have $S_{1}=S_{2}.$$T$$\alpha^{1}$$\alpha|$$-\beta+_{\wedge}\tau^{\delta}-(D^{rightarrow\beta\backslash _{M}^{S}}\backslash_{\gamma_{I}}s_{\delta-}$Figure 13: Splitting vertex

$T$

.

–
Figure 14: Renormalization of$T.$

We like to consider the scalar particle index $M$ as the macro domain wall variable.
Here we have to do some approximation. Since the renormalized domain wall has only two

20
states naively, while the index $M$ has 4 degrees of freedom to satisfy the above equation.
Then we adopt the approximation that only the relatively important channels should be
kept, that is, only two indices $M$ are taken among four. This is the approximation we
have to do to define the domain wall renormalization group.
and we get the
Next we integrate out all original domain wall indiceseﬀective weight function of $M$ at each coarse grained (renormalized) dual link, where $M$
is regarded as the renormalized domain wall variable. The eﬀective weight function is
written by the renormalized vertex tensor $T’$ defined by$\alpha,$$T_{MNKL}’= \sum_{\alpha\beta\gamma\delta}S_{\alpha\beta M}S_{\beta\gamma N}S_{\gamma\delta K}S_{\delta\alpha L}$$\beta,$$\cdot$$\gamma,$.

(4)This process is understandable as integrating out original fermion fields to leave eﬀective
interactions among scalar fields, and it is graphically expressed in Fig.14.
The global landscape is drawn in Figs.15 and 16. After the renormalization procedure,times larger spacing. This transformation can be
the renormalized dual lattice has
repeated and we have completed to define the domain wall renormalization group in thelowest order approximation.
$\sqrt{2}$Figure 15: Global landscape 1.

Figure 16: Global landscape 2.

The detailed analysis of this domain wall renormalization group is found in [5]. There,
the renormalization transformation is written down completely analytically and it isproved that the physical selection of channels to interpret the renormalized indices as
the macro domain wall variables is perfectly consistent with the optimized selection oflargest singular values. Also the fixed point structure of the renormalization group flow
and the eigenvalues of the renormalization transformation around the fixed point arecalculated to give good values compared with the Onsager’s exact solution.

4

Introducing the external magnetic field

So far, there is no external field imposed, and no way of calculating magnetic quantities.
In this section we propose a simple extension of the domain wall renormalization groupto accommodate the external magnetic field.

21
The domain wall representation is a degenerate description of the spin configurations,. To
that is, it does not discriminate
inverted spin configurations,anddiscriminate these
inversion we introduce orientation of the domain wall as shown in
Fig.17. Our rule here is that the up spin domain is on the right hand side of the oriented
domain wall when moving along the arrow. Simultaneously we double the no-domain wallstate and call them as up domain and down domain as in Fig.18. Now our domain wall
variable has four states at every dual link.
$\{\sigma\}$$Z_{2}$$\{-\sigma\}$$Z_{2}$

Figure 17: Oriented domain walls.

Figure 18: Up domain and down domain.

$\#$

$\S$

$arrow\prime IlI$

$trightarrow\}$

up domain

Figure 19: Four domain wall states.

Figure 20:$Z_{2}$

symmetry transformation.

The statistical weight is given by,(5)$\exp(K\sum_{<ij>}\sigma_{i}\sigma_{j}+h\sum_{i}\sigma_{i})$wheremeans the nearest neighbor pair of spins, $K$ is the coupling constant and is
the external magnetic field. The four states of the domain wall variable on the dual link
is diagrammatically expressed as in Fig.19. The
conjugate transformation is definedfor these states as in Fig.20, which is important to understand the symmetry constraintof the vertex tensor and its renormalization transformation.
Now the indices $a,$of tensor takes 4 states each as in Fig.19, and tensor
has originally $4^{4}=256$ elements. Only 16 elements are non-vanishing including
rotation because of the domain wall conservation law with orientation, which are listed
in Fig.21. In Fig.21, the barred variable means
conjugate, while $a=a,$are self$h$$\langle ij\rangle$$Z_{2}$$\beta,$$\delta$$\gamma,$$T$$T_{\alpha\beta\gamma\delta}$$Z_{2}$$c=\overline{c}$conjugate.
This domain wall representation has 6 dimensional parameter space $\{N, \overline{N}, b, b, a, c\}$and one to one correspondence to spin interactions $\{C_{0}, K, K_{D}, K_{4}, h, h_{3}\}$ , whose definition is given by Eq.(6) and Fig.22,$Z_{2}$

22$I_{I}\underline{\overline{b}}I^{--}I^{\cross 4}$$.–a^{1\underline{N}},^{--}1l$

Figure 21: Non-vanishing 16 elements of the vertex tensor.

$- \beta\lrcorner^{\alpha}\ulcorner\delta$$+_{\sigma_{2}\sigma_{3}}^{\sigma_{1}\sigma_{4}}$$\gamma$$\gamma$Figure 22: Spin configuration.

Figure 23: 2-dimensional matrix.

$\exp[C_{0}+K(\sigma_{1}\sigma_{2}+\sigma_{2}\sigma_{3}+\sigma_{3}\sigma_{4}+\sigma_{4}\sigma_{1})+K_{D}(\sigma_{1}\sigma_{3}+\sigma_{2}\sigma_{4})+K_{4}(\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4})$(6)$+h(\sigma_{1}+\sigma_{2}+\sigma_{3}+\sigma_{4})+h_{3}(\sigma_{1}\sigma_{2}\sigma_{3}+\sigma_{2}\sigma_{3}\sigma_{4}+\sigma_{3}\sigma_{4}\sigma_{1}+\sigma_{4}\sigma_{1}\sigma_{2})]$We use 2-dimensional matrix representationof
as in Fig.23, and matrix
is $16\cross 16$ dimensional. Actually, non-vanishing indicesor
are verymuch and limited as shown in Fig.24. Only 8 elements are non-vanishing among 16
elements, and therefore
non-vanishing block.
has only$T_{\{\alpha\beta\}\{\gamma\delta\}}$$T_{\alpha\beta\gamma\delta}$$\{\alpha\beta\}$$T_{\{\alpha\beta\}\{\gamma\delta\}}$$\{\gamma\delta\}$$8\cross 8$$T_{\{\alpha\beta\}\{\gamma\delta\}}$$z_{2}\downarrow \frac{t|}{---1\underline{|l}}I \lrcorner| \lrcorner t I^{\theta}$$–\underline{S}_{r}| \underline{\mathfrak{g}_{I}^{I}l} \mathscr{Q}$

Figure 24: Non-vanishing indices.
Considering the oriented domain wall conservation law, possible combination of indicesandis divided into theare summarized in Fig.25. Thus the matrix
even section and odd section and their conjugate sections. Each section consists ofstructure. Odd section and odd section are also constrained by the rotational symmetry.
Accordingly, the total image of
takes the form shown in Fig.26. The matrix
is block diagonal consisting of 4 blocks ofeach.
Now we set up the oriented domain wall renormalization group. The renormalized
domain wall states are defined by renormalized domain states. Then each sector corresponds to a unique macro domain wall state as shown in Fig.27. To keep the domain$\{\alpha\beta\}$$\{\gamma\delta\}$$T_{\{\alpha\beta\}\{\gamma\delta\}}$$2\cross 2$$Z_{2}$$T_{\{\alpha\beta\}\{\gamma\delta\}}$$T_{\{\alpha\beta\}\{\gamma\delta\}}$$2\cross 2$

23
Even section$\overline{Even}$section–$Z_{2}$

Odd section$rightarrow$$Z_{2}$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$–
rotation

Figure 25: Even and odd section.

wall interpretation through the renormalization procedure, we have to pick up one macrochannel each from every section.

Figure 26: Block diagonal matrix $T.$

We diagonalize
as follows. Cosine and sine functions constituting the orthogonal matrices are denoted by. Diagonalization of even sector is given by,$T_{\{\alpha\beta\}\{\gamma\delta\}}$$c_{a},$$s_{a}$$(\begin{array}{ll}N bb c\end{array})=(\begin{array}{ll}c_{1} -s_{1}s_{1} c_{1}\end{array})(\begin{array}{ll}\lambda_{1} 00 \lambda_{2}\end{array})(\begin{array}{ll}c_{1} s_{1}-s_{1} c_{1}\end{array})$$\overline{Even}$section is expressed by using the$Z_{2}$(7)

conjugate variables,$(\begin{array}{l}\overline{N}\overline{b}\overline{b}\overline{c}\end{array})= (\overline{\frac{c_{1}}{s_{1}}} -\overline{\frac{S}{c_{1}}1})(\overline{\lambda_{1}0}\frac{0}{\lambda_{2}})(\begin{array}{ll}\overline{c_{1}} \overline{s_{1}}-\overline{s_{1}}\overline{c_{1}} \end{array})$(8)

Odd and odd sections are parametrized by,$(ab \frac{a}{b})=(\begin{array}{ll}c_{2} -\mathcal{S}_{2}s_{2} c_{2}\end{array}) (\begin{array}{ll}\lambda_{3} 00 \lambda_{4}\end{array})(\begin{array}{ll}c_{2} s_{2}-s_{2} c_{2}\end{array})$

where we adopt the following$Z_{2}$(9)

conjugation rules,$\overline{\lambda_{3}}=\lambda_{3}, \overline{c_{2}}=s_{2}, \overline{s_{2}}=c_{2}$

.

(10)

24
$\overline{Even}$.

section$-$$\ddagger$$\dot{c}_{\backslash }\backslash$$rightarrow$$\backslash \backslash$$\backslash \backslash$$Z_{l}$$\backslash \backslash \theta\cdot\blacksquare\cdot\cdot$$\epsilon$:$-$

Macro Up domain

Macro Down domain

Odd section

Odd section$|$$Z_{2}$$rightarrow$rotation

Macro Domain Wall

Figure 27: Macro domain wall states.

Here larger eigenvalues (selected channels) in each section are taken to be
and they are also the 4largest singular values of a118 singular values, which assures ourphysical selection is consistent with the optimized singular value decomposition policy.
The Feynman rules which couples micro and macro domain walls are listed in Fig.28.
Domain walls are conserved in total for micro and macro. It should be noted that they are
oriented Feynman rules and therefore they are irreversible, that is, the example diagrams
at the bottom of Fig.28 do not exist.
$\lambda_{1},$$\overline{\lambda_{1}},$$\lambda_{3},$$\overline{\lambda_{3}},$$rightarrow^{Z_{2}}$$\sqrt{\lambda_{1}}c_{1}$$\sqrt{\lambda_{1}}s_{1}$$\sqrt{\lambda_{3}}c_{2}$$\sqrt{\lambda_{3}}s_{2}$$===\Rightarrow$$===\ll$$\sqrt{\overline{\lambda_{1}}}\overline{c_{1}}$$\sqrt{\overline{\lambda_{1}}}\overline{s_{1}}$$\sqrt{\lambda_{3}}s_{2}$$\sqrt{\lambda_{3}}c_{2}$e.g.
Figure 28: Feynman rules between micro and macro.
Renormalization transformation is carried out by calculating the one-loop diagrams
conjugation.

as in Fig.29. Note that all transformation rules are consistent with the$Z_{2}$

25
For example and keep
self conjugateness.
From the statistical weight defined in Eq.(5), the physical initial values of these 6parameters are given by,
$a’$

$c’$

$Z_{2}$$N = e^{2K+h}\ovalbox{\tt\small REJECT}\overline{N}=e^{2K-h}$$b = e^{h/2}rightarrow\overline{b}=e^{-h/2}$(11)$a = 1, c=e^{-2K}$

where left and right handed sides $ofrightarrow areZ_{2}$ conjugate to each other.
Now we write down the renormalization group transformation explicitly and analytically as follows,$\lambda_{1} = \frac{1}{2}(N+C+\sqrt{(N-C)^{2}+4b^{2}})$$\overline{\lambda_{1}}= \frac{1}{2}(\overline{N}+C+\sqrt{(\overline{N}-C)^{2}+4\overline{b}^{2}})$$\lambda_{3} = \frac{1}{2}(b-\overline{b}+\sqrt{(b-\overline{b})^{2}+4a^{2}})$$t_{1} = \frac{1}{b}(\lambda_{1}-N) , t_{1}=\frac{1}{\overline{b}}(\overline{\lambda_{1}}-\overline{N}) , t_{2}=\frac{\lambda_{3}-b}{a},$$c_{a}^{2}$$=$$\frac{1}{1+t_{a}^{2}}$$s_{a}^{2}=1-c_{a}^{2}$,for$a=\{1, \overline{1}, 2\}$(12)

Due to the conjugate property of the total system, there exists a 4-dimensional subspaceofinvariant space,$Z_{2}$$Z_{2}$$N=\overline{N}, b=\overline{b}, a, c$

(13)

,

in which the renormalization group flows are confined. Therefore this subspace is renormalization group invariant subspace. The renormalization group transformation in thisinvariant subspace is exactly equal to what we have obtained previously [5].
We have to find the non-trivial fixed point to evaluate critical properties of the spontaneous magnetization. Actually, however, there is no fixed point in the transformationin Eq.(12), since the vertex tensor $T$ continues to increase due to the volume eﬀect of
the renormalization procedure which compensates the decimation of half of the degrees
of freedom at each renormalization step. We will set some normalization condition of $T$
and multiply a constant factor to all elements in $T$ to satisfy it at each renormalization
step. This is called the gauge fixing since such constant multiplication does not change
any physical quantities of density type objects, such as, magnetization per site etc.
The position of a fixed point itself depends on the gauge fixing condition adopted,
called the gauge simply. However, the eigenvalues around the fixed point does not dependon the gauge. Here we show a proof of it briefly. We have two gauge fixing conditions,and coordinates on the gauge fixed manifolds are denoted by and respectively. There
are gauge transformation path (fiber bundle) in the total space on which $T$ is physicallyequivalent. This gauge transformation defines a relation between and $y,$$Z_{2}$

$x$

$y$

$x$

$y=g(x)$ ,

(14)

26

$+$

$=\lambda_{1}^{2}(c_{1}^{4}+s_{1}^{4})$$Z_{2}\downarrow$

$a’=$

$+$

Figure 29: Renormalization transformation.

27
where

$g$

is a non-singular function. Now the renormalization transformation are written

as
(15)

$x’=R_{1}(x) , y’=R_{2}(y)$ ,and they have a fixed point at$x^{*}$or$y^{*}$

satisfying,$x^{*}=R_{1}(x^{*}) , y^{*}=R_{2}(y^{*}) , y^{*}=g(x^{*})$The eigenvalues for and spaces are denoted by
calculated to coincide with each other as follows,
$x$

$y$

$\lambda_{1}$and$\lambda_{2}$

.

(16)

respectively, and they are$\lambda_{1} = R_{1}’(x^{*})=\frac{d}{dx}[g^{-1}(R_{2}(g(x^{*})))]=g^{-1/}(R_{2}(g(x^{*})))R_{2}’(g(x^{*}))g’(x^{*})$(17)$= g^{-1/}(y^{*})R_{2}’(y^{*})g’(x^{*})=R_{2}’(y^{*})=\lambda_{2}$After the gauge fixing, the eﬀective dimension of the total parameter space is five. In
theinvariant 3-dimensional subspace we have found a non-trivial fixed point [5]. This
fixed point is also a fixed point in the tota15-dimensiona1 space. The eigenvalues aroundthe fixed point consist of threeeven eigenvalues and two
odd eigenvalues.
We show transformation of variables between $\{C_{0}, K, K_{D}, K_{4}, h, h_{3}\}$ and domain
representations $\{N, \overline{N}, b, b, a, c\}$ in Fig.30, where letters in red are contributed by
odd interactions.
$Z_{2}$$Z_{2}$$Z_{2}$$Z_{2}$$Z_{2}\uparrow+C:\ N=\exp[C_{0}+2K+2K_{D}+K_{4}+h+4h_{3}]$$\bullet\bullet\bullet 4\bullet \bullet\blacksquare \overline{N}=\exp[C_{0}+2K+2K_{D}+K_{4}-l\iota-4h_{3}]$$i:[$$+_{r*}^{11}$$b= \exp[C_{0}^{\gamma}-K_{4}+\frac{h}{2}-2h_{3}]$$Z_{2}\uparrow$$\overline{b}=\exp[C_{0}-K_{4}-\frac{h}{2}+2h_{3}]$$a=\exp[C_{0}-2K_{D}+K_{4}]$$c=\exp[C_{0}-2K+2K_{D}+K_{4}]$Figure 30: Transformation between spin and domain wall interactions.

The inverse transformation from domain wall interactions $\{N, \overline{N}, b, b, a, c\}$ to spin

28
interactions $\{C_{0}, K, K_{D}, K_{4}, h, h_{3}\}$ are given by,$C_{0} = \frac{1}{16}\log[N\overline{N}a^{4}b^{4}\overline{b}^{4}c^{2}] K=\frac{1}{8}\log[N\overline{N}/c^{2}]$$K_{D} = \frac{1}{16}\log[N\overline{N}c^{2}/a^{4}] , K_{4}=\frac{1}{16}\log[N\overline{N}a^{4}c^{2}/b^{4}\overline{b}^{4}]$(18)$h = \frac{1}{4}\log[Nb^{2}/\overline{N}^{2}\overline{b}^{2}] h_{3}=\frac{1}{16}\log[N\overline{b}^{2}/\overline{N}^{2}b^{2}]$In the actual analysis of the renormalization group flow, we set a gauge fixing condition,$C_{0}=0$ , that is, the spin independent factor is discarded.

5

ResultsThe
even quantities such as the free energy, the specific heat and the correlation length
odd magnetic properties
exponent have been reported previously [5]. Here we show
only.
We plot typical high and low temperature flows in terms of the spin interaction parameters in Fig.31 and Fig.32, where is the number of renormalization steps. In bothcases only the single spin magnetic interaction survives at the infrared (macro) limit,
since we have to work with finite correlation length situation. The diﬀerence between
high and low temperatures is seen by the final enhancement factor of$Z_{2}$$Z_{2}$

$n$

$h$

$h.$

$\subset 0$$\overline{\alpha}$$\cup 0\supset$$0 5101520253035$$n$Figure 31: High temperature flow.

In Fig.33, we plotted the magnetization versus the initial nearest neighbor couplingfor various initial external magnetic fields. We can see the second
constantorder phase transition of the spontaneous magnetization around $\alpha=0.37036.$
Finally we evaluate renormalization group eigenvalues around the non-trivial fixed
point. We have three -even eigenvalues,$(\alpha\equiv e^{-2K})$$Z_{2}$$\{1.4224, -0.2908, 0\}$ ,

(19)

29

$\circ t$$\underline{c}$$\overline{\alpha}$$\mathring{\cup}\supset$$0 5101520253035$$n$Figure 32: Low temperature flow.

and two$Z_{2}$

odd eigenvalues,{2.0214, 1.0547}.
The relevant$Z_{2}$(20)-even eigenvalue gives the correlation length critical exponent,$\nu=\frac{\log\sqrt{2}}{\log 1.4224}=0.984$

,

(21)

which looks perfect compared to the exact value of 1. The peculiar zero eigenvalue appears
because the renormalization group transformation in theeven 3-dimensional subspaceis actually constrained to be 2-dimensional. This seems an occasional constraint and nophysical significance is expected.
As for -odd eigenvalues, we encounter a serious deficit; there are two relevant operators. So far we have no good idea about how to cure this problem in this lowest orderformulation. Some normalization issue owing to the singularity of decimation type renormalization group may be related. If we use the larger eigenvalue to evaluate the magnetic
susceptibility exponent,we have$Z_{2}$$Z_{2}$$\gamma=2.028$

,

(22)

which looks not so good compared to the exact value of 7/4.
We have another idea of including the external magnetic field in the dual representationof the domain wall representation. Analysis of this dual renormalization group will bereported elsewhere [6].

Acknowledgments
We thank Yusuke Yoshimura for fruitful and critical discussion and comments, and appreciate very much for inspiring correspondences by Prof. Leo Kadanoﬀ.

30$\mathring{\underline{=c\varpi N}}$$\dot{=oc\Phi\varpi}’$

Figure 33: Magnetization versus temperature.

References[1] L.P. Kadanoﬀ, Physics. 2 (1966) 263.

K. G. Wilson, Rev. Mod. Phys. 47 (1975) 773[2] M. Levin and C. P. Nave, Phys. Rev. Lett. 99 (2007) 120601.
[3] M. Hinczewski and A. N. Berker, Phys. Rev. E77 (2008) 011104.
[4] Z.-C. Gu, M. Levin and X.G. Wen, Phys. Rev. B78 (2008) 205116 and arXiv:0806.3509[cond-mat.str-el].
[5] K.-I. Aoki, Tamao Kobayashi and H. Tomita, Int.J.M.Phys.B23 (2009) 3739.
[6] K.-I. Aoki, Yasuhiro Fujii, Tamao Kobayashi, Daisuke Sato, Hiroshi Tomita andYusuke Yoshimura, in preparation.