Explaining universality: infinite limit systems in the renormalization group method

Abstract

I analyze the role of infinite idealizations used in the renormalization group (RG hereafter) method in explaining universality across microscopically different physical systems in critical phenomena. I argue that despite the reference to infinite limit systems such as systems with infinite correlation lengths during the RG process, the key to explaining universality in critical phenomena need not involve infinite limit systems. I develop my argument by introducing what I regard as the explanatorily relevant property in RG explanations: linearization* property; I then motivate and prove a proposition about the linearization* property in support of my view. As a result, infinite limit systems in RG explanations are dispensable.

Appendices

Appendix A: The linearization* process

Through the linearization* process, we accomplish two main feats: first, we categorize the eigenvectors of the RG transformation matrix into relevant, irrelevant, and marginal eigenvectors, and define a basin of attraction spanned by the irrelevant eigenvectors in which all Hamiltonians flow to the nontrivial fixed point; second, we retrieve the critical exponents that characterize the universality class associated with the nontrivial fixed point. In this appendix, I illustrate how the two main feats are achieved through the linearization* process in detail. Note that this explication should be read in a heuristic way, since discussions about the exact mathematical structures of the space of Hamiltonians are still underway.³⁴

Following Goldenfeld (1992) and Simons (1997) , we start the linearization* process by taking $S=S^*+\delta S$, a point in the vicinity of the nontrivial fixed point. We then linearize the RG transformation on S,

$$\begin{aligned} R_b(S^*+\delta S)=R_bS^*+R_b\delta S=S^*+R_b\delta S. \end{aligned}$$

For $S_i$, the i-th parameter of the Hamiltonian, we have

$$\begin{aligned} S_i=S^*_i+\sum _j[R_b]_{ij}\delta S_j. \end{aligned}$$

We assume that $[R_b]_{ij}$ is a real-valued symmetric matrix.³⁵ The semi-group property of $R_b$ gives us the relation

$$\begin{aligned} R_bR_{b'}O_i=\nonumber \\ \lambda _i(b)\lambda _i(b')O_i=\lambda _i(bb')O_i\nonumber \\ =R_{bb'}O_i, \end{aligned}$$

(1)

where $O_i$ is an eigenvector of $R_b$, $R_{b'}$, and $R_{bb'}$, and $\lambda _i(b)$, $\lambda _i(b')$, and $\lambda _i(bb')$ are the respective eigenvalues. Now, we differentiate Eq. 1 with respect to $b'$,

$$\begin{aligned} \lambda _i(b)\frac{d\lambda _i(b')}{db'}O_i=b\frac{d\lambda _i(bb')}{dbb'}O_i. \end{aligned}$$

(2)

We can solve Eq. 2 for $\lambda _i(b)$ with the initial condition $\lambda _i(1)=1$, and obtain

$$\begin{aligned} \lambda _i(b)=b^{y_i}, \end{aligned}$$

(3)

with $y_i$ independent of b.

We can expand $\delta S$ in terms of the eigenvectors of $R_b$,

$$\begin{aligned} \delta S=\sum _i g^iO_i. \end{aligned}$$

Therefore,

$$\begin{aligned} R_b(\delta S)=R_b\sum _i g^iO_i=\sum g^i\lambda _i(b)O_i=\sum g^ib^{y_i}O_i=\sum g'^iO_i. \end{aligned}$$

We then distinguish the following three cases based on the values of $y_i$:

• 1. $y_i>0$. In this case, $g'^i$ grows after RG transformations.

• 2. $y_i<0$. In this case, $g'^i$ shrinks after RG transformations.

• 3. $y_i=0$. In this case, $g'^i$ does not change after RG transformations.

The importance of this categorization is that after iterations of RG transformations, the components of $\delta S$ along directions $O_i$ for which case 1 holds are the only relevant components to drive the Hamiltonian away from the non-trivial fixed point. The components of $\delta S$ in other directions either shrink to 0, driving the Hamiltonian towards the nontrivial fixed point (as in case 2), or stay the same (as in case 3). Because of this, we give the corresponding eigenvalues $\lambda _i (b)$ and eigenvectors $O_i$ the following names:

• 1. $\lambda _i (b)$ and $O_i$ are relevant eigenvalues/eigenvectors if $y_i>0$.

• 2. $\lambda _i (b)$ and $O_i$ are irrelevant eigenvalues/eigenvectors if $y_i<0$.

• 3. $\lambda _i (b)$ and $O_i$ are marginal eigenvalues/eigenvectors if $y_i=0$.

The irrelevant eigenvectors span the basin of attraction. Because $g^i$ would shrink to 0 after iterations of RG transformations if $O_i$ is an irrelevant eigenvector, we know that Hamiltonians that start in the basin of attraction would eventually flow to the nontrivial fixed point. Hamiltonians that start off the basin of attraction would eventually drive away from the basin of attraction towards a trivial fixed point by dint of relevant eigenvalues.

Through the linearization* process, we can also retrieve the critical exponents. To illustrate that, let’s consider a simple system with only one parameter, which can be taken to be the temperature, and consider the critical exponent v in $\xi \sim t^{-v}$. Other critical exponents and for more complex systems can be similarly calculated.

For this simple system, we have $T'=R_b(T)$ and at the critical temperature $T^*$, $T^*=R_b(T^*)$. $T=0$ and $T=\infty$ are the two trivial fixed points. Linearizing in the vicinity of the fixed point, we have

$$\begin{aligned} T'-T^*&=R_b(T)-R_b(T^*)\simeq \lambda _i(b)(T-T^*)+corrections, \end{aligned}$$

(4)

$$\begin{aligned} \lambda _t(b)&=b^{y_t}, \end{aligned}$$

(5)

where Eq. 5 is from Eq. 3. We define the reduced temperature

$$\begin{aligned} t=\frac{T-T^*}{T^*}, \end{aligned}$$

then together with Eqs. 4 and 5 we have

$$\begin{aligned} t'=tb^{y_t}. \end{aligned}$$

³⁶ Iterating the RG transformation n times, we have

$$\begin{aligned} t^n=(b^{y_t})^nt. \end{aligned}$$

Now we consider how the correlation length transforms after n RG transformations. Because $\xi '=\frac{\xi }{b}$ after one transformations, after n RG transformations we have

$$\begin{aligned} \xi (t)=b^n\xi (t^n)=b^n\xi ((b^{y_t})^nt) \end{aligned}$$

(6)

We can choose b arbitrary so that it satisfies

$$\begin{aligned} b^n=\left( {\frac{u}{t}}\right) ^{\frac{1}{y_t}}, \end{aligned}$$

(7)

with u being an arbitrary positive number much larger than unity. Equations 6 and 7 together give us

$$\begin{aligned} \xi (t)=(u^{-1}t)^{-\frac{1}{y_t}}\xi (u) \text {as } t\rightarrow 0. \end{aligned}$$

(8)

Because u is much larger than unity, $\xi (u)$ is the correlation length for temperatures well above the critical temperature, where fluctuations are small, we can use standard approximation methods to estimate its value. Therefore, if we compare Eq. 8 with the power relation between the correlation length and reduced temperature $\xi \sim |t|^{-v}$, we can read off

$$\begin{aligned} v=\frac{1}{y_t}. \end{aligned}$$

Finally, we know from Eqs. 4 and 5 that

$$\begin{aligned} y_t=\frac{1}{b}\log \lambda _t(b). \end{aligned}$$

So it suffices to know $R_b$, or a good approximation of it, in order to calculate $\lambda _t(b)$, $y_t$, and hence v, a critical exponent.

Appendix B: Proofs of propositions

Proposition 1

Proof

Suppose that a system ${\mathcal {S}}$ is in the universality class delimited by a nontrivial fixed point $S^*$. Given an arbitrary linearization*-adequate neighborhood, from Definition 2 we know that there exists $m\in {\mathbb {N}}$ such that for any arbitrary natural number n, there is a Hamiltonian S with $S\not \in BoA$ such that for all Hamiltonians $S'$ with $0<\Vert S'-S_c\Vert \le \Vert S-S_c\Vert$ and $S' \not \in BoA$, we have $R^{m+i}_{b}(S)\in B_\epsilon (S^*)$ for all $0\le i<n$ and $i\in {\mathbb {N}}$. Now fix an arbitrary natural number n, we have a Hamiltonian S with $S\not \in BoA$ such that for all Hamiltonians $S'$ with $0<\Vert S'-S_c\Vert \le \Vert S-S_c\Vert$ and $S'\not \in BoA$, $R^{m+i}_{b}(S)\in B_\epsilon (S^*)$ for all $0\le i<n$ and $i\in {\mathbb {N}}$. That is to say, all $S'$ exhibits the linearization* property.³⁷$\square$

Proposition 2

Proof

Suppose that Conditions 1, 2, 3, and 4 all hold for a physical system ${\mathcal {S}}$.

$\Longrightarrow$:

Suppose further that $R_b^n(S_c)\rightarrow S^*$ as $n\rightarrow \infty$.

Choose an arbitrary linearization*-adequate neighborhood $B_\epsilon (S_c)$. From Condition 3, there exists an open set V in the basin of attraction such that (1) $S^*\in V$, and (2) for all surfaces $R_b^n$ such that $R_b^n\cap V\ne \emptyset$ and for all $P\in (R_b^n\cap V)$, there exists $r\in {\mathbb {R}}^+$ such that $Q\in B_\epsilon (S^*)$ for all $Q\in R_b^n$, $Q\not \in BoA$, and $\Vert Q-P\Vert _b^n<r$.

Because $R_b^n(S_c)\rightarrow S^*$ as $n\rightarrow \infty$, we can find $m\in {\mathbb {N}}$ such that for all $m'>m$, we have $R_b^{m'}(S_c)\in V$. Now choose an arbitrary natural number n. We have $R_b^{m+i}(S_c)\in V$ for all $0\le i<n$.

For each $i\in {\mathbb {N}}$ with $0\le i<n$, since $R_b^{m+i}(S_c)\in (R_b^{m+i}\cap V)$, we have some $r_i>0$ such that for all points $Q'$ with $Q'\in R_b^{m+i}$, $Q'\not \in BoA$, and $\Vert Q'-R_b^{m+i}(S_c)\Vert _b^{m+i}<r_i$, we have $Q'\in B_\epsilon (S^*)$. From Condition 1, we know that there is a point $S^i$ with $S^i\not \in BoA$ on the original unrenormalized surface, such that

$$\begin{aligned} 0<\Vert R_{b^{m+i}}(S^i)-R_{b^{m+i}}(S_c)\Vert _{b^{m+i}}^1<r_i. \end{aligned}$$

(9)

By the semi-group property of the RG transformation, $R_bR_{b'}(P)=R_{bb'}(P)$ for all P in the space of Hamiltonians, Equation 9 is equivalent to

$$\begin{aligned} 0<\Vert R_b^{m+i}(S^i)-R_b^{m+i}(S_c)\Vert _b^{m+i}<r_i, \end{aligned}$$

(10)

³⁸ from which we have $R_b^{m+i}(S^i)\in B_\epsilon (S^*)$.

Now, let S be an unrenormalized point with $S\not \in BoA$ such that $0<\Vert S-S_c\Vert \le \Vert S^i-S_c\Vert$ for all $i<n$. It follows from Condition 2 that

$$\begin{aligned} 0<\Vert R_{b^{m+i}}(S)-R_{b^{m+i}}(S_c)\Vert _{b^{m+i}}^1\le \Vert R_{b^{m+i}}(S^i)-R_{b^{m+i}}(S_c)\Vert _{b^{m+i}}^1<r_i, \end{aligned}$$

(11)

for all $0\le i<n$. By the semi-group property of the RG transformation, Eq. 11 is equivalent to

$$\begin{aligned} 0<\Vert R_b^{m+i}(S)-R_b^{m+i}(S_c)\Vert _b^{m+i}<r_i, \end{aligned}$$

(12)

for all $0\le i<n$. Therefore, $R_b^{m+i}(S)\in B_\epsilon (S^*)$.

Similarly, from Condition 2, we have, for all $S'$ with $S'\not \in BoA$ such that $0<\Vert S'-S_c\Vert \le \Vert S-S_c\Vert$,

$$\begin{aligned} 0<\Vert R_{b^{m+i}}(S')-R_{b^{m+i}}(S_c)\Vert _{b^{m+i}}^1\le \Vert R_{b^{m+i}}(S) -R_{b^{m+i}}(S_c)\Vert _{b^{m+i}}^1<r_i, \end{aligned}$$

(13)

for all $0\le i<n$. By the semi-group property of the RG transformation,

$$\begin{aligned} 0<\Vert R_b^{m+i}(S')-R_b^{m+i}(S_c)\Vert _b^{m+i}<r_i \end{aligned}$$

(14)

for all $0\le i<n$. Therefore, $R_b^{m+i}(S')\in B_\epsilon (S^*)$ for all $S'$ with $0<\Vert S'-S_c\Vert \le \Vert S-S_c\Vert$ and $S'\not \in BoA$. The system ${\mathcal {S}}$ is in the universality class delimited by $S^*$.

$\Longleftarrow$:

Suppose further that ${\mathcal {S}}$ is in the universality class delimited by $S^*$ under Definition 2.

Given an arbitrary open set V in the basin of attraction such that $S^*\in V$. By Condition 4, we have a linearization*-adequate neighborhood $B_\epsilon (S^*)$ such that $(\partial B_\epsilon (S^*)\cap BoA) \subseteq V$.

Because ${\mathcal {S}}$ is in the universality class delimited by $S^*$ under Definition 2, there exists $m\in {\mathbb {N}}$ such that for any arbitrary $n\in {\mathbb {N}}$, there exists S with $S\not \in BoA$ on the unrenormalized surface such that for all $S'$ with $0<\Vert S'-S_c\Vert <\Vert S-S_c\Vert$ and $S'\not \in BoA$ we have $R_b^{m+i}(S')\in B_\epsilon (S^*)$ for all $0\le i<n$ and $i\in {\mathbb {N}}$.

I will first show that $R_b^{m'}(S_c)\in \partial B_\epsilon (S^*)$ for all $m'>m$. To do so, we choose any arbitrary $m'>m$ and any arbitrary neighborhood $N(R_b^{m'}(S_c), r)=\{Q|\ \Vert R_b^{m'}(S_c)-Q\Vert _b^{m'}<r\}$ on the $R_b^{m'}$ surface. We want to show that $N(R_b^{m'}(S_c), r)$ contains at least one point in $B_\epsilon (S^*)$, and at least one point not in $B_\epsilon (S^*)$. The latter part is immediately satisfied, since $R_b^{m'}(S_c)\in N(R_b^{m'}(S_c), r)$ and $R_b^{m'}(S_c) \not \in B_\epsilon (S^*)$ because it is in the basin of attraction. We’re left to show that $N(R_b^{m'}(S_c), r)$ contains at least one point in $B_\epsilon (S^*)$.

From Condition 1, we have a point $S''$ with $S''\not \in BoA$ on the unrenormalized surface such that $0<\Vert R_{b^{m'}}(S'')-R_{b^{m'}}(S_c)\Vert _{b^{m'}}^1<r$. By the semi-group property of the RG transformations, we have $\Vert R_b^{m'}(S'')-R_b^{m'}(S_c)\Vert _b^{m'}<r$. This means that $R_b^{m'}(S'')\in N(R_b^{m'}(S_c), r)$.

Now choose any $n>(m'-m)$, we know that there exists S with $S\not \in BoA$ on the unrenormalized surface such that for all $S'$ with $0<\Vert S'-S_c\Vert <\Vert S-S_c\Vert$ and $S'\not \in BoA$ we have $R_b^{m+i}(S')\in B_\epsilon (S^*)$ for all $0\le i<n$ and $i\in {\mathbb {N}}$, and in particular, $R_b^{m'}(S')\in B_\epsilon (S^*)$.

If $\Vert S''-S_c\Vert <\Vert S-S_c\Vert$, then from the above discussion, we have $R_b^{m'}(S'')\in B_\epsilon (S^*)$.

If $\Vert S''-S_c\Vert \ge \Vert S-S_c\Vert$, then for all $S'$ with $0<\Vert S'-S_c\Vert <\Vert S-S_c\Vert \le \Vert S''-S_c\Vert$, we have $\Vert R_{b^{m'}}(S')-R_{b^{m'}}(S_c)\Vert _{b^{m'}}^1<\Vert R_{b^{m'}}(S'') -R_{b^{m'}}(S_c)\Vert _{b^{m'}}^1<r$, by Condition 2. By the semi-group property of the RG transformations, we have $\Vert R_b^{m'}(S')-R_b^{m'}(S_c)\Vert _b^{m'}<r$. Therefore, $R_b^{m'}(S')\in B_\epsilon (S^*)$ and $R_b^{m'}(S')\in N(R_b^{m'}(S_c), r)$.

In both cases above, $N(R_b^{m'}(S_c), r)$ contains at least one point in $B_\epsilon (S^*)$. Therefore, $R_b^{m'}(S_c)\in \partial B_\epsilon (S^*)$.

It is easy to show that $R_b^{m'}(S_c)$ is in the basin of attraction, because $S_c$ represents the system at its critical point. Therefore, $R_b^{m'}(S_c)\in (\partial B_\epsilon (S^*)\cap BoA)\subseteq V$ for all $m'>m$. Because V is arbitrary, we have $R_b^n(S_c)\rightarrow S^*$ as $n\rightarrow \infty$. $\square$

Appendix C Glossary of technical terms

• Space of Hamiltonians: the space of “Hamiltonians” strictly speaking is a misnomer. In fact, it’s the space of coupling constants that serve as parameters in Hamiltonians of some generic form (see, e.g. Palacios (2019))). Every point in the space of Hamiltonians represent a set of values for the coupling constants, and the set of values then correspond to a Hamiltonian.

• ${\mathcal {S}}$: this refers to physical systems such as the water-steam system. The system ${\mathcal {S}}$ can be represented by points on the unrenormalized surface.

• $S, S', S''$: these refer to points on the unrenormalized surface that represent the physical system ${\mathcal {S}}$.

• $R_b^n(S)$: this refers to a point on the space of Hamiltonians which is the result of the n-th iteration of the RG transformation of length b applied to the point S.

• The $R_b^n$ surface: this refers to a surface in the space of Hamiltonians that is the co-domain of $R_b^n(S)$, for all S on the unrenormalized surface.

• $S^*$: this is a non-trivial fixed point.

• $S_c$: this is the Hamiltonian on the unrenormalized surface that represents the system ${\mathcal {S}}$ at its critical point.

• $P, Q, Q'$: these are points in the space of Hamiltonians, not necessarily representing any system.

• Basin of Attraction (BoA): the basin of attraction is a surface in the space of Hamiltonians that is spanned by the irrelevant parameters. $S^*$ and $S_c$ are both in the basin of attraction. Among other things, the correlation length for points on the basin of attraction is infinite.

• $B_\epsilon (S^*)$: this is the linearization*-adequate neighborhood around the nontrivial fixed point $S^*$.