数理解析研究所講究録 1275 巻 2002 年 175-178

175
Universal natures and rich structures in infinite-0rder phase transitions
C. Itoi (Nihon University)

Introduction

1

The second order derivative of the free energy with respect to aenvironmental parameter
diverges at the critical point, when an ordinary second-0rder phase transition occurs. The
correlation length of the system has asingularity at the critical point$g$$g_{\mathrm{c}}$$\xi\sim|g-g_{c}|^{-\nu}$.

(1)

In the renormalization group (RG) method, the critical point is given as afixed point of the. The maximal eigenvalue in the linearized RG flow near the fixed point gives the inverseof the critical exponent$b_{1}$$\mathrm{R}\mathrm{G}$$b_{1}=1/\nu$.

Here, we do not have to solve any recursion relation or differential equation explicitly to obtain. On
critical exponents, one has to only diagonarize the scaling matrix at the fixed point of
the other hand, in an infinite-0rder phase transition, the free energy has an essential singularity,
and any order derivative of the free energy does not diverge. The correlation length shows
strong divergence at the critical point with$\mathrm{R}\mathrm{G}$$\xi\sim\exp A|g-g_{c}|^{-\tilde{\nu}}$.

In this case, athermodynamic quantity scaled with apositive power of the correlation length
does not diverge at any order derivative, such as afree energy, while that with anegative power
diverges. The Kosterlitz-Thouless (KT) transition is the well-known example as an infinite-0rder
phase transitions. This transition appears in $c=1$ conformal field theories with amarginal, or 1/2 universally. In this case,perturbation. In this case, the critical exponent is
the scaling matrix at the critical point vanishes, and then the renormalization group equation
is obtained bybecomes nonlinear differential equation. Commonly the critical exponent
integrating the differential equation of the renormalization group explicitly. In general situation,
however, the renormalization group equation cannot be integrated explicitly. In this talk, I, which enables us to extract the universal critical exponent
present amethod of RG forfrom the nonlinear differential equation in an algebraic way [1]. It will be shown that theis given by the maximal eigenvalue of the scaling matrix ininverse of the critical exponent. In section 2, Idescribe the method of RG for RG briefly. In section 3,the linearized RG for
Igive several non-trivial examples of quantum spin systems which differs ffom the universality
class of the KT transition.
$\tilde{\nu}=1$$\tilde{\nu}$$\mathrm{R}\mathrm{G}$$\tilde{\nu}$$1/\tilde{\nu}$$\mathrm{R}\mathrm{G}$176
2Renormalization group for renormalization group
Here, Istudy asystem with coupling constants g(gl,g2,parameterg) obeys the following RG differential equation$=$$\cdots,g_{n})$. The running coupling$\mathrm{x}(t,$(2)$\frac{d\mathrm{x}}{dt}=\mathrm{V}(\mathrm{x})$with an initial condition. The real parameter is logarithm of ascale parameterin the RG transformation. Here Icall time. The vector field
is sometimes called betafunction. Let the origin be afixed point of this. The correlation length 4in thesystem is considered as the scale determined by the time when the solution spends near thefixed point. If the beta function is expanded in :at the fixed point,$\mathrm{x}(0,\mathrm{g})=\mathrm{g}$

$t$

$t$

$\mathrm{V}(\mathrm{x})$$\mathrm{R}\mathrm{G}\mathrm{V}(0)=0$$\mathrm{x}$$x$$V_{\dot{1}}( \mathrm{x})=\sum_{j}A_{\dot{1}}^{j}x_{j}+\sum_{jk}C_{\dot{1}}^{jk}x_{j}x_{k}+\cdots$

the maximal eigenvalue of the scaling matrix
gives the inverse of the critical exponent. This well-known fact implies
does not have to integrate the differential equation
explicitly in order to obtain the leading behavior in critical phenomena. Here, Iconsider thecase that the first derivative of the beta function vanishes at the fixed point.
situationyields infinite-0rder phase transition. For example in the KT transition which is famous as an
infinite-0rder transition, the RG equation of the KT transition is$A_{\dot{1}}^{j}$$1/\nu$$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}\cdot \mathrm{o}\mathrm{n}\mathrm{e}$$\mathrm{T}\underline{\mathrm{h}\mathrm{i}}\mathrm{s}$$\frac{dx_{1}}{dt}$$=$$-x_{2}^{2}$$\frac{dx_{2}}{dt}$$=$$-x_{1}x_{2}$,(3)which can be integrated explicitly. The spending time of the running coupling near the&ed
point is evaluated and the characteristic length of the system is obtained as afunction of theinitial data$\xi$$\xi\sim\exp A|\mathrm{g}-\mathrm{g}_{\mathrm{c}}.|^{-1/2}$.

Since the RG equation cannot be solved explicitly in general, Iapply a RG method to the RG
nonlinear differential equation. Here, we consider the RG differential equation$\frac{dx}{d}i=\sum_{jk}C_{\dot{1}}^{jk}x_{j}x_{k}$.

If the function
is asolution of this equation,
becomes asolution of thisequation. On the basis of this scaling relation, Idefine arenormalization group transformation
for the initial parameter. First, fix asurface in the coupling constant space and consider theproblem with an initial parameter on this surface. Let us define atransformation:
for an arbitrary real parameter$\mathrm{x}(t, \mathrm{g})$$e^{\tau}\mathrm{x}(e^{\tau}t,\mathrm{g})$$\mathrm{S}$$R_{\tau}$$\tau$$R_{\tau}(\mathrm{g})=e^{\tau}\mathrm{x}(s(\tau),\mathrm{g})$,$\mathrm{S}arrow \mathrm{S}$177g) is on the surface
is determined for agiven in such away that the point
RG transformation for RG. Ishow the following properties of RG for RG.
S. Here, Icall

where$s(\tau)$$e^{\tau}\mathrm{x}(s(\tau),$$\tau$$R_{\tau}$

1. Aone parameter semi group property of RG for RG$R_{\tau_{2}}R_{\tau_{1}}=R_{\tau_{1}+\tau_{2}}$.

2. Astraight flow line in the original RG corresponds to afixed point of this RG for$\mathrm{R}\mathrm{G}$.

3. The maximal eigenvalue of the scaling matrix in the RG for RG gives the inverse of the.
critical exponent$1/\tilde{\nu}$Therefore, one can obtains the critical exponent
explicitly.

$\tilde{\nu}$without solving the differential equation

3Examples
In two parameter systems, by solving the RG equation explicitly, one can check the method ofin the KT transition. Here, Ipresent three other, such as acritical exponent
RG fornontrivial examples of one dimensional quantum spin systems, aspin 1bilinear-biquadraticmodel [3], aspin-0rbital model [4] and azigzag chain model [5], which shows infinite-0rder
transitions different from the KT universality class. The phase diagram of each model has arich structure. Aspin 1bilinear-biquadratic model is well-known as asystem with the Haldanegap. ABethe ansatz solvable point is a critical point, where the system is described in
Wess-ZuminO-Witten (WZW) model with $c=2$ . This system shows an infinite-0rder transition
from the Haldane gap phase to agapless phase at this critical point. The critical exponent
is obtained both in integrating the RG equation and the RG for RG method. In a one
dimensional spin-0rbital model, one non-trivial critical point is aBethe ansatz solvable point
WZW model with $c=3$ . There are an extended
where the system is described in thegapless phase and dimer gap phase, where the transition between two phases is infinite-0rder.
or 1are obtained in both ways. In the zigzag chain model, anThe critical exponent
interesting new phenomenon is discovered recently. In the Hamiltonian of the spin 1/2 zigzag
chain model(4),$\tilde{\nu}=1/2$$\mathrm{R}\mathrm{G}$$\mathrm{S}\mathrm{U}(3)$$\tilde{\nu}=3/5$$\mathrm{S}\mathrm{U}(4)$$\tilde{\nu}=2/3$$H= \sum_{i}(J_{1}\vec{S}_{i}\cdot\vec{S}_{\dot{l}+1}+J_{2}\vec{S}_{\dot{l}}\cdot\vec{S}_{\dot{\iota}+2})$, $J_{1}=4.149\cdots J_{1}$ , and $J_{1}=0$ . The transition at
there are three critical points $J_{1}=$$J_{1}=-4J_{2}$ corresponds to the ferromagnetic transition that is first order. The next one at$J_{1}=4.149\cdots J_{2}$ is the transition from antiferromagnetic gapless phase to the dimer gapped. The transition at $J_{1}=0$ is anon-KT tyPe
phase which is the KT type transition withobtained by RG for RG method. Around this point, theinfinite-0rder transition with
with five marginal perturbations. The RG equation of thissystem is described in$4\mathrm{J}2$$\tilde{\nu}=1$$\tilde{\nu}=2/3$$(c=1\mathrm{C}\mathrm{F}\mathrm{T})^{2}$

178
system in the one-loop approximation is$l \frac{dx_{1}}{dl}=x_{1}^{2}-x_{3}x_{4}-x_{4}^{2}$,$l \frac{dx_{2}}{dl}=x_{2}^{2}+x_{3}x_{4}+x_{3}^{2}$,$l \frac{dx_{3}}{dl}=-\frac{1}{2}x_{1}x_{3}+\frac{3}{2}x_{2}x_{3}+x_{2}x_{4}$$l \frac{dx_{4}}{dl}=x_{1}x_{3}+\frac{3}{2}x_{1}x_{4}-\frac{1}{2}x_{2}x_{4}$$l \frac{dx_{5}}{dl}=\frac{1}{2}x_{3}x_{4}$,,,(5)where the initial values of this equation are given in certain functions ofand . The RG
equation indicates the instability of the critical point $J_{1}=0$ for the perturbation. Indeedin the antiferromagnetic region $J_{1}>0$ , the system is dimerized where the translational symmetry of this model is broken. In afield theory description, the corresponding chiral symmetry$J_{1}$$J_{2}$$J_{1}\neq 0$

breaking occurs. The numerical calculation shows the finite correlation length, dimerization
order parameter and the energy gap [2]. The gap scaling formula eq.(l) withfits thedata surprisingly
even for relatively large . In the ferromagnetic region $J_{1}<0$ , however,
the gap has never been observed in numerical calculation. This fact is puzzling because theferromagnetic perturbation seems to yield the same instability as in the antiferromagnetic one.
Now, Iunderstand this puzzle as follows [5]. This RG has afixed lne$\tilde{\nu}=2/3$$\mathrm{w}\mathrm{e}\mathrm{L}$$J_{1}$$x_{1}=x_{2}=0$, $x_{3}+x_{4}=0$.

(6)

and the eigenvalues of the scaling matrix on this fixed line all vanish. Studying the flow near
this fixed line, all perturbations is found to be marginaly relevant. The flow becomes quite
slow near this fixed line, however, finally the flow runs away from the fixed line. Since therunning coupling
spends long time near the fixed line, the characteristic length scale ofthe system becomes always an astronomical length scale. Therefore, the correlation length isfinite but quite long in an extended region. At the sme time the energy gap is finite, but very
tiny without fine-tuning of the coupling . The scaling formula eq.(l) of the correlation length
holds only for small. This spin model is arare example of astrong scale reduction withoutfine-tuning of the coupling constant.
$\mathrm{x}(t)$$J_{1}$$|J_{1}|$References[1] C. Itoi and H. Mukaida, Phys.Rev.E 60, 3688 (1999)[2] S. R. White and I. Affleck, Phys. Rev. B54,9862 (1996).
[3] C. Itoi and M. -H. Kato, Phys. Rev. B55,8295 (1997).

[4] C. Itoi, S. Qin and I. Affleck, Phys. Rev. B61,6747 (2000).
[5] C. Itoi and S. Qin, Phys. Rev. B63224423 (2001)