Abstract:
We consider a two-dimensional hierarchical lattice in which the vertices
of a square represent an elementary cell. In the generalized hierarchical model,
the distance between opposite vertices of a square differs from that
between adjacent vertices and is a parameter of the new model.
The Gaussian part of the Hamiltonian
of the 4-component generalized fermionic hierarchical model
is invariant
under the block-spin renormalization group transformation.
The transformation of the renormalization group in the space of coefficients,
which specify the Grassmann-valued density of the free measure, is explicitly
calculated as a homogeneous mapping of degree four in the two-dimensional
projective space.
The work was carried out at the expense of the Strategic Academic Leadership
Program of the Kazan (Volga Region) Federal University (PRIORITET-2030) and
dedicated to the centenary of Academician Vasily Sergeevich Vladimirov.
The renormalization group method is one of the central methods of modern statistical physics and quantum field theory. The main mathematical problem in the rigorous justification of the renormalization group (RG) theory is the non-locality of the transformation of the RG. The latter means that even if the initial Hamiltonian has a simple form and depends on a finite number of coupling constants, the iterations of the RG transformation generate more and more complex interactions of spins in the Hamiltonian, which makes it impossible to mathematically analyze an infinite-dimensional dynamical system. The models of statistical physics on a hierarchical lattice, which were introduced in mathematical physics by Dyson [1], proved useful for rigorous mathematical study of the RG theory. The RG transformation in bosonic hierarchical models is local and is reduced to a non-linear integral operator in the free measure density space [2]. Later, it was shown that, for certain values of the number of points in a unit cell, the hierarchical lattice can be realized as a lattice in $p$-adic space, and $p$-models are continuous versions of hierarchical models (see [3]). Volovich [4] proposed a $p$-adic analog of the string amplitude. The results of first studies in various areas of $p$-adic mathematical physics are presented in the monograph by Vladimirov, Volovich and Zelenov [5]. More recent results are discussed in the survey [6]. The paper [7] gives an overview of research on the fermionic hierarchical model.
Unlike the bosonic case, the RG transformation in fermionic hierarchical model can be calculated exactly and is represented as a birational mapping in the two-dimensional space of model coupling constants. Note that the RG in Euclidean models is studied only in lower orders of perturbation theory in the neighbourhood of a Gaussian fixed point. The unique property of the hierarchical fermionic model is that it is capable of describing the global flow of the RG in the entire plane of coupling constants [8]. The explicit description of the properties of the RG within the framework of the hierarchical fermionic model generates a number of non-trivial conjectures for hierarchical and Euclidean bosonic models [9].
A generalization of the fermionic model was proposed in [10]. A two-dimensional lattice is considered, in which the elementary cell is represented by the vertices of a square. In the standard hierarchical model, the distances between the vertices of the square are equal. In the generalized model, the distance between opposite vertices differs from that between neighbouring vertices and is actually a parameter of the new model. In the present paper, we explicitly compute the transformation of the RG in the density space of a free measure and describe some of its properties.
Let $T=\{0,1,\dots\}$ and $V^s_k=\{j\in T\colon k\,{\cdot}\, 2^s\leqslant j<(k+1)\,{\cdot}\, 2^s\}$, where $k\in T$, $s\in N=\{1,2,\dots\}$. The hierarchical distance $d_2(i,j)$, $i,j\in T$, $i\ne j$, is defined as $d_2(i,j)=2^{s(i,j)}$, where
$$
\begin{equation*}
s(i,j)=\min\{ s\colon \text{is } k\in T \text{ such that } i,j\in V^s_k\}.
\end{equation*}
\notag
$$
For any $k=(k_1,k_2)\in T^2$, $l=(l_1,l_2)\in T^2$, $k\ne l$, we define $s(k,l)=\max(s(k_1,l_1)$, $s(k_2,l_2))$. The hierarchical distance on $T^2$ is defined as $d_2(k,l)=2^{s(k,l)}$. Consider the 4-component fermionic field
where the components are generators of a Grassman algebra. Let us redenote $V^N_0$ by $\Lambda_N$. Let $\Gamma_N$ be the Grassmann subalgebra generated by $4\,{\cdot}\,4^N$ generators $\overline\psi_1(i)$, $\psi_1(i)$, $\overline\psi_2(i)$, $\psi_2(i)$, $i \in \Lambda_N$.
The action of the RG transformation $R(\alpha,\delta)$ on $\psi^*$ is defined by
is defined on the whole lattice as a quasistate (expectation value) $\langle\,{\cdot}\, \rangle$ on the algebra of all monomials such that $\langle F\rangle$ for an even degree monomial $F$ is calculated according to the Wick rules and $\langle F\rangle=0$ for any odd degree monomial, $\delta_{n,m}$ is the Kronecker delta. Consider the following functions on $T^2$:
where $\lambda>0$ is a real parameter. Let $b(k,l;\lambda;\alpha)=f(k,l;\lambda;\alpha-4)$. It was shown in [10] that a zero-mean Gaussian fermionic field with binary correlation function
is invariant under the transformation of the RG with parameter $\alpha$ (see (1.1)). Let us denote the corresponding Gaussian quasi-state as $\rho_0(\lambda;\alpha)$.
To construct non-Gaussian states, we will use the Gibbs description of the field. Consider the restriction of the Gaussian field $\psi^*$ to the volume $\Lambda_N$. Let $B_N(\lambda;\alpha)=(b(k,l;\lambda;\alpha))_{k,l\in\Lambda_N}$ be the correlation matrix of this restriction. We also set $H_N(\lambda,\mu;\alpha)=(h_N(k,l;\lambda,\mu;\alpha))_{k,l\in\Lambda_N}$, where
$g(\lambda,\mu;\alpha)$ and $C(N;\lambda,\mu;\alpha)$ are some normalizing functions [10]. It was proved in [10] that if $\alpha>2$, $\alpha\ne 4$, then, for all $\lambda$ satisfying
Consider the restriction of the Gaussian state $\rho_0(\lambda;\alpha)$ to the volume $\Gamma_N$. According to Theorem 1 in [10], for any $F(\psi^*)\in\Gamma_N$, we have
We denote by $R(\alpha)u$ the transformation in the density space defined by the right-hand side of (1.7). This formula implies that the RG transformation $R(\alpha)$ is local and independent of $N$.
§ 2. Calculation of the renormalization group transformation
In this section, instead of regular densities of the form
The triples $c=(c_0,c_1,c_2)$, which define the densities $u(\psi^*)$, will be considered as points in the two-dimensional real projective space, because two sets that differ by a non-zero factor represent the same Gibbs state. The coefficients of the quadratic form $Q_0$ are as follows:
Proof. Let the vertices $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$ of the square $V_0^1$ be denoted by $1$, $2$, $3$, $4$, respectively. We also write $\eta^*(1) = \eta^*(0,0)$, $\eta^*(2) = \eta^*(1,0)$, $\eta^*(3)=\eta^*(1,1)$, $\eta^*(4)=\eta^*(0,1)$ for $\eta^*(0,0)$, $\eta^*(1,0)$, $\eta^*(1,1)$, $\eta^*(0,1)$, respectively.
Note that in the form $A_k^1$ all the terms depend on the variables $\overline\zeta_k(1)$ or $\zeta_k(1)$, while the form $A_k^2$ does not involve such terms. We also set
Since the set of coefficients ($c_0$, $c_1$, $c_2$) defining the general density is determined up to a non-zero factor, the RG transformation can be written as
Note that formulas (2.2)–(2.4), which define the mapping, involve the coefficients $t_2,\delta$, which, in turn, depend on $\alpha$. For further purposes, we denote the mapping of the renomalization group $R(\alpha)$ in the $c$-space by
It is easy to see that the mapping of the transformation $R(\alpha;\delta, t_2)$ in the projective space is conjugate to the transformation $R(\alpha;\delta, 1)$.
So, in what follows, we will consider the mapping $R(\alpha;\delta, 1)$, which we denote by $R(\alpha;\delta)$.
Note that if $\lambda=1$, then $\mu=1$, and in this case $s_2=s_3$, and so $\delta= t_2/t_1=2$, which corresponds to the usual hierarchical lattice. The case $\lambda=1$ corresponds to the standard hierarchical lattice. From the expressions for $f_0$, $f_1$, $f_2$ we can find a common factor $(c_0-2c_1+c_2)^2$, and the transformation $R(\alpha;2)$ coincides with the RG transformation in the fermionic model on the usual hierarchical lattice [10]. Next, consider the representation of our model in the $(r,g)$-coordinates. Let
Corollary. The domains $\{(r,g)\colon g >0\}$ (the upper half-plane), $\{(r,g)\colon g < 0\}$ (the lower half-plane) and $\{(r,g)\colon g=0\}$ are invariant domains of the mapping $R(\alpha; \delta)$. The action of $R(\alpha,\delta)$ on the line $\{(r,g):g=0\}$ is linear:
The points $A_0$ and $A_1$ are fixed points of $R(\alpha,\delta)$ for all $\alpha$ and $\delta$. The point $A_2$ is a singular point of the mapping $R(\alpha;\delta)$ for all $\alpha$ and $\delta$:
In the $(r,g)$-coordinates, the point $A_0$ is $(0,0)$, and so we say that $A_0$ is a Gaussian fixed point. The point $A_1$ does not lie in the regular plane $\{(r,g)\}$, and to it there corresponds the density $u(\psi^*)=\overline\psi_1\psi\overline\psi_2\psi_2$ (the Grassmann delta function).
Theorem 2. The fixed point $A_0$ is repulsive for $\alpha>3$, is a saddle point for $2<\alpha<3$, and is an attracting point for $\alpha<2$. The fixed point $A_1$ is attracting for $\alpha>2$, a saddle point for $1< \alpha< 2$, and a repulsive point for $\alpha<1$.
The degree of any monomial in the polynomials $P_1(x,y)$ and $P_3(x,y)$ exceeds $1$, and the degree of any monomial in $P_2(x,y)$ is positive. Therefore, the differential of the mapping $R(\alpha;\delta)$ at the point $x=0$, $y=0$ has the form
The eigenvalues of the differential $D_0$ are $\gamma_1=2^{\alpha - 2}$, $\gamma_2= 2^{2\alpha - 6}$. So if $\alpha>3$, then $\gamma_1>1$, $\gamma_2>1$ and the fixed point $A_0$ is repulsive. For $2<\alpha<3$, $A_0$ is a saddle point, and for $\alpha<2$, $A_0$ is an attracting point. The value $\alpha=3$ is a bifurcation value, and, therefore, a new (non-Gaussian) branch of fixed points exists near the point $A_0$.
In a neighbourhood of the point $A_1=(0,0,1)$, we introduce the local coordinates $u=c_1/c_2$, $v=c_0/c_2$.
The degree of any monomial in the polynomials polynomials $P_4(u,v)$ and $P_6(u,v)$ exceeds 1, and the degree of any monomial in the polynomial $P_5(u,v)$ is positive. Therefore, the differential of the mapping $R(\alpha,\delta)$ at the point $u=0$, $v=0$ has the form
The eigenvalues of the differential $D_1$ are $\lambda_1=2^{-(\alpha - 2)}$, $\lambda_2= 2^{-2\alpha+2}$.
Therefore, $A_1$ is attracting for $\alpha>2$, is a saddle fixed point for $ 1<\alpha<2$, and is a repulsive point for $\alpha<1$. The value $\alpha =1$ is a bifurcation value, and, therefore, there is a new branch of fixed points near the point $A_1$. Theorem 2 is proved.
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Citation:
M. D. Missarov, D. A. Khajrullin, “The renormalization group transformation in the generalized fermionic hierarchical model”, Izv. Math., 87:5 (2023), 1011–1023
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This publication is cited in the following 1 articles:
M. D. Missarov, D. A. Khairullin, “Nepodvizhnye tochki preobrazovaniya renormalizatsionnoi gruppy v obobschennoi fermionnoi ierarkhicheskoi modeli”, Izv. vuzov. Matem., 2024, no. 12, 101–108