Criticality, Gaussian fields and exact block renormalization

Let $I\subseteq \mathbb{R}$ be a nonempty open interval of values of a scalar control parameter $\theta$, expressed in one fixed unit if it is dimensional. A function $g:I\to \mathbb{R}$ is real analytic if for every ${\theta }_0\in I$ there are $ϵ>0$ and real coefficients $(a_k{)}_{k\ge 0}$ such that $({\theta }_0-ϵ,{\theta }_0+ϵ)\subseteq I$, the series ${\sum }_{k=0}^{\mathrm{\infty }}{a}_k(\theta -{\theta }_0{)}^k$ converges there, and its sum equals $g(\theta )$ throughout that interval. Let $(N_n{)}_{n\ge 1}$ be positive integers with $N_n\to \mathrm{\infty }$, interpreted as finite system sizes such as site counts. A finite-volume partition-function sequence on $I$ is a sequence of real-analytic functions $Z_n:I\to (0,\mathrm{\infty })$. For each $n$, define its dimensionless log partition density per site by $p_n(\theta ):=N_n^{-1}\log Z_n(\theta )$, where $\log$ is the natural logarithm. If the pointwise limit $p(\theta ):={\lim }_{n\to \mathrm{\infty }}{p}_n(\theta )$ exists as a finite real number for every $\theta \in I$, call $p$ the infinite-volume pressure associated with this sequence. A point ${\theta }_c\in I$ is critical under the nonanalytic-pressure criterion exactly when there do not exist $ϵ>0$ with $({\theta }_c-ϵ,{\theta }_c+ϵ)\subseteq I$ and real coefficients $(a_k{)}_{k\ge 0}$ such that ${\sum }_{k=0}^{\mathrm{\infty }}{a}_k(\theta -{\theta }_c{)}^k$ converges and equals $p(\theta )$ throughout that interval.

This definition specifies only the nonanalytic-pressure criterion and makes no claim that a particular model satisfies it.

The natural logarithm as the inverse of the exponential function

Assume the Axiom of Countable Choice: every sequence $(X_n{)}_{n\in \mathbb{N}}$ of nonempty sets has a function $f$ with $f(n)\in X_n$ for every $n$. For $N\ge 1$, a real symmetric positive-definite matrix $K\in {\mathbb{R}}^{N\times N}$ (meaning $z^{\mathsf{T}}Kz>0$ for every nonzero $z\in {\mathbb{R}}^N$) defines the finite Gaussian probability ${\gamma }_K$ on ${\mathbb{R}}^N$ by

$Z_K:={\int }_{{\mathbb{R}}^N}{e}^{-z^{\mathsf{T}}Kz\mathrm{/}2}d{\lambda }_N(z),{\gamma }_K(dz):=Z_K^{-1}{e}^{-z^{\mathsf{T}}Kz\mathrm{/}2}d{\lambda }_N(z)\mathrm{.}$

Under the stated Axiom of Countable Choice, $Z_K=(2\pi {)}^{N\mathrm{/}2}(\det K{)}^{-1\mathrm{/}2}$. For the coordinate identity $X(z)=z$ on ${\mathbb{R}}^N$, define its mean vector by ${\mu }_i:={\int }_{{\mathbb{R}}^N}{z}_i{\gamma }_K(dz)$ and its covariance matrix by ${\mathrm{\Sigma }}_{ij}:={\int }_{{\mathbb{R}}^N}(z_i-{\mu }_i)(z_j-{\mu }_j){\gamma }_K(dz)$ for $1\le i,j\le N$. These integrals are finite under the Gaussian theorem Gaussian covariance for positive quadratic forms and a boundary counterexample, which gives $\mu =0$ and $\mathrm{\Sigma }=K^{-1}$. Thus ${\gamma }_K$ is centered, meaning its mean vector is zero. Here $K$ is called the precision matrix because it appears in the quadratic exponent; for this centered law its covariance matrix is its inverse. For $N=0$, define ${\mathbb{R}}^0$ to be the singleton containing the empty vector and define ${\gamma }_{\mathrm{\varnothing }}$ to put mass one on that point; its partition factor and empty determinant are both $1$.

The variables $z_i$ and exponent $z^{\mathsf{T}}Kz\mathrm{/}2$ are dimensionless in this mathematical definition. In a physical quadratic model, if each component of $\mathrm{\Phi }$ has declared field unit ${\mathrm{\Phi }}_{\ast }$ and $H(\mathrm{\Phi })=\frac{1}{2}{\mathrm{\Phi }}^{\mathsf{T}}{K}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{\Phi }$ is energy in joules, then each entry of $K_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ has units $\mathrm{J}\mathrm{/}({\mathrm{\Phi }}_{\ast }{)}^2$. With $\beta =(k_{B}T{)}^{-1}$, where in SI $k_B$ is Boltzmann's constant with units $\mathrm{J}{\mathrm{K}}^{-1}$ and $T>0$ is absolute temperature in kelvins, $\beta$ has units ${\mathrm{J}}^{-1}$. For $z_i={\mathrm{\Phi }}_i\mathrm{/}{\mathrm{\Phi }}_{\ast }$, the dimensionless precision is $K=\beta {\mathrm{\Phi }}_{\ast }^2{K}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$. This dimensional conversion does not assert an empirical model.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Gaussian covariance for positive quadratic forms and a boundary counterexample

Let $d,L\ge 1$ be integers and let ${\mathrm{\Lambda }}_{d,L}:=(\mathbb{Z}\mathrm{/}L\mathbb{Z}{)}^d$, where $\mathbb{Z}\mathrm{/}L\mathbb{Z}$ is the quotient set of integers modulo $L$. Use the quotient addition $[a{]}_L+[b{]}_L=[a+b{]}_L$ and additive inverse $-[a{]}_L=[-a{]}_L$, and define these operations coordinatewise on ${\mathrm{\Lambda }}_{d,L}$. For $x\in {\mathrm{\Lambda }}_{d,L}$, write $\tilde{x}=({\tilde{x}}_1,\dots ,{\tilde{x}}_d)\in \{0,\dots ,L-1{\}}^d$ for its unique coordinate representatives. For $1\le j\le d$, let $e_j\in {\mathrm{\Lambda }}_{d,L}$ have coordinate $[1{]}_L$ in position $j$ and $[0{]}_L$ elsewhere. Count the two terms $x+e_j$ and $x-e_j$ separately even when they coincide for $L=1$ or $L=2$.

For a real-valued function $f:{\mathrm{\Lambda }}_{d,L}\to \mathbb{R}$, define

$(-\mathrm{\Delta }f)(x):={\sum }_{j=1}^d(2f(x)-f(x+e_j)-f(x-e_j))\mathrm{.}$

For a complex-valued function, use the same formula for $-\mathrm{\Delta }$, which is its complex-linear extension. Write ${\mathbb{C}}^{{\mathrm{\Lambda }}_{d,L}}$ for the complex-valued functions on ${\mathrm{\Lambda }}_{d,L}$ with pointwise vector operations. For $r=(r_1,\dots ,r_d)\in \{0,\dots ,L-1{\}}^d$, define the function ${\chi }_r:{\mathrm{\Lambda }}_{d,L}\to \mathbb{C}$ by

${\chi }_r(x):=\exp \text{\hspace{0.17em}⁣}\left(\frac{2\pi i}{L}{\sum }_{j=1}^d{r}_j{\tilde{x}}_j\right)\mathrm{.}$

Use the inner product $⟨f,g⟩:=L^{-d}{\sum }_{x\in {\mathrm{\Lambda }}_{d,L}}f(x)\overline{g(x)}$. Identify real functions with coordinate vectors using the lexicographic order on the representatives $\tilde{x}$, and let $I_{{\mathrm{\Lambda }}_{d,L}}$ be the identity operator on those functions. A real matrix $A$ is symmetric when $A^{\mathsf{T}}=A$; a real symmetric matrix is positive semidefinite when $v^{\mathsf{T}}Av\ge 0$ for every real vector $v$, and positive definite when $v^{\mathsf{T}}Av>0$ for every nonzero real vector $v$. For all $r,s\in \{0,\dots ,L-1{\}}^d$, the inner products satisfy $⟨{\chi }_r,{\chi }_s⟩=1$ if $r=s$ and $0$ otherwise; every complex-valued function on ${\mathrm{\Lambda }}_{d,L}$ is a unique linear combination of the functions ${\chi }_r$.

For each $r$, the equation $-\mathrm{\Delta }{\chi }_r={\lambda }_r{\chi }_r$ holds with ${\lambda }_r:=4{\sum }_{j=1}^d{\sin }^2(\pi r_j\mathrm{/}L)$; this scalar is the eigenvalue associated with ${\chi }_r$.

Consequently $-\mathrm{\Delta }$ is real symmetric positive semidefinite, and for real $m^2>0$ and $\kappa >0$ the matrix $m^2{I}_{{\mathrm{\Lambda }}_{d,L}}+\kappa (-\mathrm{\Delta })$ is real symmetric positive definite with eigenvalues $m^2+4\kappa {\sum }_{j=1}^d{\sin }^2(\pi r_j\mathrm{/}L)>0$.

The congruence class $[a{]}_n$ and the quotient set $\mathbb{Z}\mathrm{/}n$

For $n\ge 1$, every class in $\mathbb{Z}\mathrm{/}n$ has one representative $r$ with $0\le r<n$, so $\mid \mathbb{Z}\mathrm{/}n\mid =n$; while $\mathbb{Z}\mathrm{/}0$ is in bijection with $\mathbb{Z}$

Addition and multiplication on $\mathbb{Z}\mathrm{/}n$ by $[a{]}_n+[b{]}_n=[a+b{]}_n$ and $[a{]}_n[b{]}_n=[ab{]}_n$

For every natural $n$, $(\mathbb{Z}\mathrm{/}n,+)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold

Orthogonality of the characters $x\mapsto e^{2\pi ikx\mathrm{/}N}$ on $\mathbb{Z}\mathrm{/}N\mathbb{Z}$

Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule

The complex exponential by its power series

$\exp (z+w)=\exp z\exp w$, and the complex exponential extends the real exponential

$\ker (\exp )=2\pi i\mathbb{Z}$, and $\exp z=\exp w$ exactly when $z-w\in 2\pi i\mathbb{Z}$

$\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$

Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive

The addition formulas for sine and cosine

Pythagorean and parity identities for all six trigonometric functions on their natural domains

For $\mathrm{\mid }u\mathrm{\mid }\le \pi \mathrm{/}2$, $2\mathrm{\mid }u\mathrm{\mid }\mathrm{/}\pi \le \mathrm{\mid }\sin u\mathrm{\mid }\le \mathrm{\mid }u\mathrm{\mid }$. For all real $u,v$, $\mathrm{\mid }\sin u-\sin v\mathrm{\mid }\le \mathrm{\mid }u-v\mathrm{\mid }$.

The derivatives of sine and cosine are cosine and minus sine

Let $r\ge 0$ and $s\ge 1$ be integers. For $n\ge 1$, a real matrix $M\in {\mathbb{R}}^{n\times n}$ is symmetric positive definite if $M^{\mathsf{T}}=M$ and $v^{\mathsf{T}}Mv>0$ for every nonzero $v\in {\mathbb{R}}^n$. Let $A\in {\mathbb{R}}^{r\times r}$, $B\in {\mathbb{R}}^{r\times s}$, and $C\in {\mathbb{R}}^{s\times s}$, and set

$K:=\left(\begin{array}{cc}A& B\\ B^{\mathsf{T}}& C\end{array}\right)\in {\mathbb{R}}^{(r+s)\times (r+s)}\mathrm{.}$

Assume $K$ is symmetric positive definite. Then $C$ is symmetric positive definite, $\det C>0$, and $C^{-1}$ exists. The Schur complement $S:=A-B{C}^{-1}{B}^{\mathsf{T}}$ is symmetric positive definite when $r\ge 1$; when $r=0$, $S$ is the empty $0\times 0$ matrix. For all $x\in {\mathbb{R}}^r$ and $y\in {\mathbb{R}}^s$,

${\left(\begin{array}{c}x\\ y\end{array}\right)}^{\mathsf{T}}K\left(\begin{array}{c}x\\ y\end{array}\right)=x^{\mathsf{T}}Sx+(y+C^{-1}{B}^{\mathsf{T}}x{)}^{\mathsf{T}}C(y+C^{-1}{B}^{\mathsf{T}}x)\mathrm{.}$

For $r=0$, the first term and $S$ are interpreted on the singleton space ${\mathbb{R}}^0$.

A matrix admits a Cholesky factorisation with positive diagonal exactly when it is Hermitian positive definite, and that factor is unique

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $d,L\ge 1$ be integers, let ${\mathrm{\Lambda }}_{d,L}=(\mathbb{Z}\mathrm{/}L\mathbb{Z}{)}^d$ have $N=L^d$ sites, and let $\kappa >0$. Define the zero-mass precision $K:=\kappa (-\mathrm{\Delta })$ on ${\mathbb{R}}^{{\mathrm{\Lambda }}_{d,L}}$, with the periodic Laplacian convention of Fourier spectrum and positivity of the periodic lattice Laplacian. Then the unrestricted Lebesgue normalizer diverges:

${\int }_{{\mathbb{R}}^{{\mathrm{\Lambda }}_{d,L}}}\exp \text{\hspace{0.17em}⁣}(-\frac{1}{2}{z}^{\mathsf{T}}Kz)d{\lambda }_N(z)=+\mathrm{\infty }\mathrm{.}$

Thus this quadratic weight does not define a probability density on the full coordinate space. A finite-volume massless Gaussian must first restrict the field, for example to $\{z:{\sum }_{x\in {\mathrm{\Lambda }}_{d,L}}{z}_x=0\}$, before inverting the Laplacian.

Fourier spectrum and positivity of the periodic lattice Laplacian

Assume the Axiom of Countable Choice from The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Let integers $d,L\ge 1$ specify the periodic lattice ${\mathrm{\Lambda }}_{d,L}=(\mathbb{Z}\mathrm{/}L\mathbb{Z}{)}^d$ and let $m^2>0$ and $\kappa >0$ be dimensionless real parameters in lattice units. On the real coordinate space ${\mathbb{R}}^{{\mathrm{\Lambda }}_{d,L}}$, let

$K_{d,L,m,\kappa }:=m^{2}I+\kappa (-\mathrm{\Delta }),$

where $-\mathrm{\Delta }$ is the periodic Laplacian of Fourier spectrum and positivity of the periodic lattice Laplacian. Its eigenvalues are

$m^2+4\kappa {\sum }_{j=1}^d{\sin }^2(\pi r_j\mathrm{/}L),r\in \{0,\dots ,L-1{\}}^d,$

so it is symmetric positive definite. The finite periodic Gaussian lattice field is the centered probability ${\gamma }_{K_{d,L,m,\kappa }}$ of Finite Gaussian probability with a positive precision matrix, with density proportional to $\exp (-z^{\mathsf{T}}{K}_{d,L,m,\kappa }z\mathrm{/}2)$ against Lebesgue measure on ${\mathbb{R}}^{L^d}$ and covariance $K_{d,L,m,\kappa }^{-1}$ by Gaussian covariance for positive quadratic forms and a boundary counterexample at $\beta =1$.

The lattice coordinates $z_x$ and the quadratic exponent are dimensionless. Under the physical quadratic convention of Finite Gaussian probability with a positive precision matrix, where $H(\mathrm{\Phi })=\frac{1}{2}{\mathrm{\Phi }}^{\mathsf{T}}{K}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{\Phi }$ and each field component has unit ${\mathrm{\Phi }}_{\ast }$, choosing $z_x={\mathrm{\Phi }}_x\mathrm{/}{\mathrm{\Phi }}_{\ast }$ gives dimensionless precision $K=\beta {\mathrm{\Phi }}_{\ast }^2{K}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ when the energy is divided by $k_{B}T$. Here $\beta =(k_{B}T{)}^{-1}$ with the SI units for $k_B$ and $T$ specified in Finite Gaussian probability with a positive precision matrix; equivalently the physical precision for this lattice model is $K_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=(k_{B}T\mathrm{/}{\mathrm{\Phi }}_{\ast }^2)K_{d,L,m,\kappa }$. This conversion makes no empirical claim. At $m^2=0$, the constant field is a zero mode and the unrestricted normalizer diverges by Unconstrained massless periodic Gaussian has infinite normalizer, so the full-space positive-precision definition does not apply. A finite-volume massless model must first specify a restriction such as the subspace ${\sum }_{x\in {\mathrm{\Lambda }}_{d,L}}{z}_x=0$ before inverting the Laplacian.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Fourier spectrum and positivity of the periodic lattice Laplacian

Unconstrained massless periodic Gaussian has infinite normalizer

Finite Gaussian probability with a positive precision matrix

Gaussian covariance for positive quadratic forms and a boundary counterexample

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$, meaning every sequence $(X_n{)}_{n\in \mathbb{N}}$ of nonempty sets has a function $f$ with $f(n)\in X_n$ for every $n$. Let integer $d\ge 1$, parameter $\kappa >0$, and $k=(k_1,\dots ,k_d)\in [-\pi ,\pi {]}^d$, and put

$D_d(k):=4{\sum }_{j=1}^d{\sin }^2(k_j\mathrm{/}2)\mathrm{.}$

Here ${\lambda }_d$ is $d$-dimensional Lebesgue measure restricted to the cube. For a measurable function $h:[-\pi ,\pi {]}^d\to \mathbb{R}$, write $h^{+}=\max \{h,0\}$ and $h^{-}=\max \{-h,0\}$; its signed Lebesgue integral is $\int h^{+}d{\lambda }_d-\int h^{-}d{\lambda }_d$ when at most one of these two integrals is infinite, and $h$ is integrable when $\int \mathrm{\mid }h\mathrm{\mid }d{\lambda }_d<+\mathrm{\infty }$. For $x=(x_1,\dots ,x_d)\in {\mathbb{Z}}^d$, define $k\cdot x:={\sum }_{j=1}^d{k}_j{x}_j$ and $\mathrm{\mid }k{\mathrm{\mid }}_2^2:={\sum }_{j=1}^d{k}_j^2$. Define the massless infinite-lattice Green integral when it is finite by

$G_{d,\kappa }(x):=(2\pi {)}^{-d}{\int }_{[-\pi ,\pi {]}^d}\frac{\cos (k\cdot x)}{\kappa D_d(k)}d{\lambda }_d(k),$

where the integrand is assigned value $0$ at $k=0$, a choice that does not affect the integral because a singleton has Lebesgue measure zero. If $d>2$, this integral is absolutely finite for every fixed $x\in {\mathbb{Z}}^d$. If $1\le d\le 2$, the nonnegative diagonal integral with $x=0$ is $+\mathrm{\infty }$. Thus the dimension threshold concerns this free Gaussian Fourier integral; it does not assert an interacting critical exponent or a continuum scaling limit.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Finite periodic Gaussian lattice field with positive mass

Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$

The nonnegative Lebesgue integral

Integrable real and complex functions, and their integrals

A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included