Some solutions to the eigenstate equation of free scalar quantum field theory
in the Schrödinger representation
T. A. Bolokhov

St. Petersburg Department of V. A. Steklov Mathematical Institute

Russian Academy of Sciences

27 Fontanka, St. Petersburg, Russia 191023

###### Abstract

Using extensions of the quadratic form in the potential term we construct
Gaussian eigenstates of the free scalar quantum Hamiltonian operator acting in
nontrivial functional space. Admissible positive closed extensions are
generated at least by two external sources, the distance between them being
limited by the extension parameter. The constructed functionals, in contrast
to the ground state of the free theory, correspond to different boundary
conditions and can be interpreted as eigenstates of some renormalized
asymptotically free quantum Hamiltonian.

Keywords: extensions of closed semi-bounded quadratic forms, square root of
operator, quantum field theory in the Schrödinger representation, ground state
functional.

## Introduction

The Hamilton operator of the free scalar quantum field in the Schrödinger
representation is defined by the following expression [1]

$\mathscr{H}=\int_{\mathbb{R}^{3}}d^{3}x\,\bigl{(}-\frac{\delta^{2}}{\delta
A(\boldsymbol{x})^{2}}+(\partial_{m}A)^{2}\bigr{)}.$ (1)

This operator acts on the set of two times differentiable functionals defined
on the domain of its potential term – the quadratic form of the Laplace
operator in the 3-dimensional space

$\mathfrak{l}(A)=\int_{\mathbb{R}^{3}}(\partial_{m}A(\boldsymbol{x}))^{2}\,d^{3}x.$
(2)

In order to construct the functional of the ground state of $\mathscr{H}$ one
need to defined the Laplacian as a self-adjoint operator

$L:\quad A(\boldsymbol{x})\to-\frac{\partial^{2}A(\boldsymbol{x})}{\partial
x_{m}^{2}}$

with respect to the scalar product

$(f(\boldsymbol{x}),g(\boldsymbol{x}))=\int_{\mathbb{R}^{3}}d^{3}x\,\overline{f(\boldsymbol{x})}g(\boldsymbol{x}).$
(3)

Then the functional of the ground state of (1) is given by the Gaussian of the
quadratic form $\omega(A)$ corresponding to the square root of $L$

$\Omega(A)=\exp\\{-\frac{1}{2}\omega(A)\\}=\exp\\{-\frac{1}{2}(A,L^{1/2}A)\\}.$

Due to the equation

$\frac{1}{4}\int_{\mathbb{R}^{3}}d^{3}x\,\bigl{(}\frac{\delta}{\delta
A(\boldsymbol{x})}\omega(A)\bigr{)}^{2}=(L^{1/2}A,L^{1/2}A)=\mathfrak{l}(A),\quad
A\in\mathcal{D}(\mathfrak{l}),$ (4)

the action of $\mathscr{H}$ on the functional $\Omega(A)$ is reduced to
multiplication by the scalar $\operatorname{Tr}L^{1/2}$

$\mathscr{H}\,\Omega(A)=\int_{\mathbb{R}^{3}}d^{3}x\,\bigl{(}-\frac{\delta^{2}}{\delta
A(\boldsymbol{x})^{2}}+(\partial_{m}A)^{2}\bigr{)}\,\exp\\{-\frac{1}{2}\omega(A)\\}=\\\
=\frac{1}{2}\int_{\mathbb{R}^{3}}d^{3}x\,\bigl{(}\frac{\delta^{2}}{\delta
A(\boldsymbol{x})^{2}}\omega(A)\bigr{)}\,\exp\\{-\frac{1}{2}\omega(A)\\}=(\operatorname{Tr}L^{1/2})\,\Omega(A).$

One may note that the flat integral of the product of variational derivatives
in the first term of $\mathscr{H}$ leads to the flat scalar product in the
quadratic form $\omega(A)$.

Outside of the domain of the quadratic form $\mathcal{D}(\mathfrak{l})$, that
is in the region where potential term is infinite, the Hamiltonian has
singularities. The nearest in some sense functions to
$\mathcal{D}(\mathfrak{l})$ are the ones with behaviour

$A(\boldsymbol{x})=\frac{C_{n}}{|\boldsymbol{x}-\boldsymbol{x}_{n}|^{1/2}},\quad\boldsymbol{x}\to\boldsymbol{x_{n}}$
(5)

in a number of points $\boldsymbol{x}_{n}\in\mathbb{R}^{3}$. Indeed, for any
$\epsilon>0$, if $A(\boldsymbol{x})$ is expanded in $\boldsymbol{x}_{n}$ as

$A(\boldsymbol{x})=\frac{C_{n}}{|\boldsymbol{x}-\boldsymbol{x}_{n}|^{1/2-\epsilon}},\quad\boldsymbol{x}\to\boldsymbol{x}_{n},$

then the integrand in (2) has singularity of the order

$(\partial_{m}A(\boldsymbol{x}))^{2}=\frac{C_{n}^{2}}{|\boldsymbol{x}-\boldsymbol{x}_{n}|^{3-2\epsilon}},\quad\boldsymbol{x}\to\boldsymbol{x}_{n},$

and so is locally integrable in $\mathbb{R}^{3}$. Meanwhile in the case
$\epsilon=0$ the integral in the quadratic form diverges logarithmically
around the points $\boldsymbol{x}_{n}$ and the Hamiltonian $\mathscr{H}$ has
singularities in the vicinities of functions with behaviour (5). As long as
the domain of $\omega$ is wider than $\mathcal{D}(\mathfrak{l})$ the
functional $\Omega(A)$ in the vicinities of (5) is well defined and has finite
boundary values.

Now let us assume that in the scalar product (3) there exists a closed
positive extension $\mathfrak{l}_{\kappa}(A)$ of the quadratic form
$\mathfrak{l}(A)$ to some function with the behaviour (5)

$\displaystyle\mathfrak{l}_{\kappa}(A)$ $\displaystyle=\mathfrak{l}(A),\quad
A\in\mathcal{D}(\mathfrak{l}),$ (6) $\displaystyle\mathfrak{l}_{\kappa}(B)$
$\displaystyle<\infty,\quad
B\stackrel{{\scriptstyle\boldsymbol{x}\to\boldsymbol{x}_{n}}}{{=}}\frac{C_{n}}{|x-x_{n}|^{1/2}}$
(7)

(on the extensions of closed semi-bounded quadratic forms see [2]). Then one
may substitute the quadratic form $\omega(A)$ with the form of the square root
of $L_{\kappa}$ corresponding to $\mathfrak{l}_{\kappa}(A)$ and observe that
the Gaussian

$\Omega_{\kappa}(A)=\exp\\{-\frac{1}{2}\omega_{\kappa}(A)\\}=\exp\\{-\frac{1}{2}(A,L_{\kappa}^{1/2}A)\\},$
(8)

also satisfies the eigenstate equation of $\mathscr{H}$

$\mathscr{H}\,\Omega_{\kappa}(A)=\frac{1}{2}\int_{\mathbb{R}^{3}}d^{3}x\,\bigl{(}\frac{\delta^{2}}{\delta
A(\boldsymbol{x})^{2}}\omega_{\kappa}(A)\bigr{)}\,\exp\\{-\frac{1}{2}\omega_{\kappa}(A)\\}=(\operatorname{Tr}L_{\kappa}^{1/2})\,\Omega_{\kappa}(A),$
(9)

provided that the following equation holds

$\frac{1}{4}\int_{\mathbb{R}^{3}}d^{3}x\,\bigl{(}\frac{\delta}{\delta
A(\boldsymbol{x})}\omega_{\kappa}(A)\bigr{)}^{2}=(L_{\kappa}^{1/2}A,L_{\kappa}^{1/2}A)=\mathfrak{l}_{\kappa}(A)=\mathfrak{l}(A),\quad
A\in\mathcal{D}(\mathfrak{l}).$ (10)

In order to verify equations (9) and (10) one need to fix the set w. r. to
which the variational derivative is operating. The form $\omega(A)$ in (4) is
defined by means of Fourier transform and can be reconstructed by variation w.
r. to the functions from the domain of the quadratic form $\mathfrak{l}(A)$.
This condition can be written as

$\overline{\omega|_{\mathcal{D}(\mathfrak{l})}}=\omega,$

where LHS is the closure of restriction of the form $\omega(A)$ to the set
$\mathcal{D}(\mathfrak{l})$. That is for any $A\in\mathcal{D}(\omega)$ there
exists a sequence $A_{n}\in\mathcal{D}(\mathfrak{l})$ such that

$||A_{n}-A||_{L_{2}(\mathbb{R}^{3})}\stackrel{{\scriptstyle
n\to\infty}}{{\to}}0,\quad\omega(A_{n}-A)\stackrel{{\scriptstyle
n\to\infty}}{{\to}}0.$

Thus the same set $\mathcal{D}(\mathfrak{l})$ should be used for the variation
in eigenstate equation (9), which means that we need to verify that the
following equation holds

$\overline{\omega_{\kappa}|_{\mathcal{D}(\mathfrak{l})}}=\omega_{\kappa}$ (11)

for the quadratic form $\omega_{\kappa}(A)$. As we will see in what follows
the proof of this equation requires some efforts and is not possible for any
extension $\mathfrak{l}_{\kappa}$ of the quadratic form of Laplace operator.

Another essential requirement of the presented construction is the positivity
of the form $\mathfrak{l}_{\kappa}$. Otherwise the operator $L_{\kappa}$
possesses negative eigenspace and the quadratic form $\omega_{\kappa}$ goes
imaginary thereon. This in turn contradicts the self-adjointness of
$\mathscr{H}$ and does not allow to introduce a scalar product on the set of
functionals with finite normalization for $\Omega_{\kappa}$.

The construction of solutions for equation of the ground state of the free
scalar quantum field by means of the extensions of quadratic form in the
potential term of the Hamiltonian was proposed in [3]. There were used the
extensions generated by $\delta$-potential, that is extensions to the
functions with behavior

$A(\boldsymbol{x})=\frac{C_{n}}{|\boldsymbol{x}-\boldsymbol{x}_{n}|},\quad\boldsymbol{x}\to\boldsymbol{x_{n}}.$

However the proof of the possibility to reconstruct the quadratic form of the
square root by its action on the domain of the quadratic form of Laplacian in
that preprint was not provided.

In this work we develop the scheme of construction of eigenfunctionals of the
free Hamiltonian by extending the quadratic form of Laplacian to the functions
with behaviour (5). We will show that positive extensions require that
behaviour at least in 2 points and will derive the admissible distance between
the latter as the function of the extension parameter. Then we will verify
that the extensions in question satisfy equation (11) and discuss the problem
arising for the extensions generated by $\delta$-potential.

Eigen-functionals $\Omega_{\kappa}(A)$ in contrast to the functional of the
ground state of the free theory $\Omega(A)$ correspond to other boundary
conditions but to the same action of Hamiltonian and can describe renormalized
to zero interaction with external antisymmetric (Fermi) sources.

##  1 Extensions of the quadratic form

Let us turn to the detailed description of extensions of quadratic form of the
Laplace operator (6), (7). We restrict to the case of the scalar field
$A(\boldsymbol{x})$, but will be keeping in mind that, f. e. extensions of the
quadratic form of Laplace operator in the space of functions in the Coulomb
gauge (solenoidal functions), might have their own peculiarities. We will work
in the momentum representation (in the space of Fourier transforms) of the
field $A$

$\hat{A}(\boldsymbol{p})=\frac{1}{(2\pi)^{3/2}}\int_{\mathbb{R}^{3}}A(\boldsymbol{x})e^{-i\boldsymbol{p}\cdot{\boldsymbol{x}}}\,d^{3}x$

and for simplicity will omit the Fourier hat in the notation of
$A(\boldsymbol{p})$. Action of $\mathscr{H}$ in this representation is as
follows

$\mathscr{H}\,\Omega(A)=\int_{\mathbb{R}^{3}}d^{3}p\,\bigl{(}-\frac{\delta^{2}}{\delta
A(\boldsymbol{p})^{2}}+p^{2}A^{2}(\boldsymbol{p})\bigr{)}\,\Omega(A).$

Quadratic forms $\mathfrak{l}(A)$ and $\omega(A)$ are now diagonalized

$\displaystyle\mathfrak{l}(A)$
$\displaystyle=\int_{\mathbb{R}^{3}}p^{2}|A(\boldsymbol{p})|^{2}\,d^{3}p,$
$\displaystyle L:$ $\displaystyle\,A(\boldsymbol{p})\to
p^{2}A(\boldsymbol{p}),$ $\displaystyle\omega(A)$
$\displaystyle=\int_{\mathbb{R}^{3}}p|A(\boldsymbol{p})|^{2}\,d^{3}p,$
$\displaystyle L^{1/2}:$ $\displaystyle\,A(\boldsymbol{p})\to
pA(\boldsymbol{p}),$

while the scalar product connecting quadratic forms with their operators is
still flat

$(f(\boldsymbol{p}),g(\boldsymbol{p}))=\int_{\mathbb{R}^{3}}d^{3}p\,\overline{f(\boldsymbol{p})}g(\boldsymbol{p}).$
(12)

The key objects for constructing extensions of the quadratic form
$\mathfrak{l}(A)$ are the resolvent of the self-adjoint operator $L$

$R_{\mu}=(L-\mu)^{-1}:\quad
A(\boldsymbol{p})\to\frac{1}{p^{2}-\mu}A(\boldsymbol{p}),\quad\mu\in\mathbb{C}\setminus\mathbb{R}^{+}\cup
0$

and the singular potential $v(\boldsymbol{p})$ – some generalized function
obeying the following conditions

$\displaystyle R_{\mu}v\in L_{2}(\mathbb{R}^{3})\text{ or
}(R_{\mu}v,R_{\nu}v)<\infty$ (13)
$\displaystyle(v,R(\mu)v)=\int_{\mathbb{R}^{3}}\frac{|v(p)|^{2}}{p^{2}-\mu}d^{3}p=\infty.$
(14)

As candidates for $v(\boldsymbol{p})$ we will consider powers of modulus of
$\boldsymbol{p}$

$v_{\alpha}(\boldsymbol{p})=p^{\alpha}.$

These potentials are somewhat localized in the coordinate representation

$\frac{1}{(2\pi)^{3/2}}\int_{\mathbb{R}^{3}}v_{\alpha}(\boldsymbol{p})e^{i\boldsymbol{p}\cdot\boldsymbol{x}}\,d^{3}p=\begin{cases}(2\pi)^{3/2}\delta(\boldsymbol{x}),&\alpha=0,\\\
\frac{2^{3/2+\alpha}\Gamma(\frac{3+\alpha}{2})}{\Gamma(-\frac{\alpha}{2})}x^{-\alpha-3},&\alpha\neq
0,\end{cases}$

so it makes sense to include into consideration their shifts in coordinate
space

$v_{\alpha,\boldsymbol{x}}(\boldsymbol{p})=e^{i\boldsymbol{p}\cdot\boldsymbol{x}}v_{\alpha}(\boldsymbol{p}).$

The Birman-Krein-Vishik theory [4] describes the domain
$\mathcal{D}(\mathfrak{l}_{\kappa})$ of extensions of quadratic form
$\mathfrak{l}$ as the sum of $\mathcal{D}(\mathfrak{l})$ and linear span of
the vector $R_{\mu}v$

$\mathcal{D}(\mathfrak{l}_{\kappa})=\mathcal{D}(\mathfrak{l})\dotplus\\{\xi
R_{\mu}v\\},\quad\mu\in\mathbb{C}\setminus\mathbb{R}^{+}\cup
0,\quad\xi\in\mathbb{C}.$

As shown in the work [5] if the potential $v(\boldsymbol{p})$ fails to satisfy
(14) then the vector $R_{\mu}v$ belongs to the domain
$\mathcal{D}(\mathfrak{l})$, that is extension changes the action of
$\mathfrak{l}$ and so changes the action of $\mathscr{H}$ which we intend to
avoid. In the term of $\alpha$ the divergence of integral in (14) results in
the inequality

$-\frac{1}{2}\leq\alpha.$

It is not hard to see that condition (13) imposes restriction on $\alpha$ from
the other side, namely the strict inequality

$\alpha<\frac{1}{2}.$

Domain of the self-adjoint operator $L_{\kappa}$ corresponding to the
quadratic form $\mathfrak{l}_{\kappa}$ includes linear combinations of vectors
$R_{\rho}v$, $R_{\rho}^{2}v$ of the following type

$R_{\rho}v+\xi(\kappa,\rho)R_{\rho}^{2}v,\quad\rho<0,\quad\xi\in\mathbb{C},$

with some $\xi$ defined by the boundary conditions of $L_{\kappa}$. But we
will not study this, neither we will not study the action of
$\mathfrak{l}_{\kappa}$, the only useful object for the construction of powers
of $L_{\kappa}$ is its resolvent given by the Krein formula

$R_{\mu}^{\kappa}=(L_{\kappa}-\mu)^{-1}=R_{\mu}+\frac{R_{\mu}v(R_{\bar{\mu}}v,\,\cdot\,)}{\kappa-\sigma(\mu)}.$
(15)

Here $\kappa$ is the real extension parameter and $\sigma(\mu)$ is the
Nevanlinna (Herglotz) function [6], defined by the equation

$\sigma(\mu)-\sigma(\nu)=(\mu-\nu)(R_{\bar{\mu}}v,R_{\nu}v)=(v,(R_{\mu}-R_{\nu})v).$

Note that in the case $\kappa=\infty$ the second term in the RHS of (15)
vanishes and we arrive at the resolvent $R_{\mu}$ of $L$

$R_{\mu}^{\infty}=R_{\mu}.$

In what follows we will restrict ourselves to potentials $v(\boldsymbol{p})$
of the kind $p^{-1/2}$, such that vectors $R_{\mu}v$ reside in the region
described by (5). Indeed, $R_{\mu}v$ in coordinate representation is given by
the integral

$(R_{\mu}v)(\boldsymbol{x})=\frac{1}{(2\pi)^{3/2}}\int_{\mathbb{R}^{3}}\frac{e^{i\boldsymbol{p}\cdot\boldsymbol{x}}\,d^{3}p}{(p^{2}-\mu)p^{1/2}}=\frac{4\pi}{(2\pi)^{3/2}x}\int_{0}^{\infty}\frac{\sin
px}{p^{2}-\mu}p^{1/2}\,dp,$

which after the change of variable turns into

$(R_{\mu}v)(\boldsymbol{x})=\frac{4\pi}{(2\pi)^{3/2}x^{1/2}}\int_{0}^{\infty}\frac{\sin
s}{s^{2}-\mu x^{2}}s^{1/2}\,ds.$ (16)

The integration by $s$ converges absolutely, is bounded for $x\to 0$ and is
not equal to zero in this limit. Then one may conclude that singularity of
$(R_{\mu}v)(\boldsymbol{x})$ is defined by the function in front of the
integral (16) and so is described by (5).

The Nevanlinna function for $v(\boldsymbol{p})=p^{-1/2}$ is given by the
following equation

$\sigma(\mu)-\sigma(\lambda)=\int_{\mathbb{R}^{3}}\bigl{(}\frac{1}{p^{2}-\mu}-\frac{1}{p^{2}-\lambda}\bigr{)}\frac{d^{3}p}{p}=2\pi(\ln(-\lambda)-\ln(-\mu)).$
(17)

Here we choose the main branch of the logarithm with the cut along the
negative semi-axis in such a way that RHS is defined for all complex $\mu$ and
$\lambda$ outside of the positive semi-axis plus zero. This yields that
denominator in (15) can be chosen as

$\kappa-\sigma(\mu)=2\pi\ln(-\frac{\mu}{\tilde{\kappa}}),\quad
0<\tilde{\kappa},$ (18)

where $\tilde{\kappa}$ is the positive value combining dimensional constant to
correctly define the logarithm and the extension parameter $\kappa$.

It is evident that function (18) decreases when $\mu$ goes from $-\infty$ to
zero and changes its sign for any value of $\kappa$. Thus resolvent (15) has a
pole in the point $\mu=-\tilde{\kappa}$ and corresponding self-adjoint
operator and quadratic form possess negative eigenspace. This contradicts the
requirement of positivity of the extension $\mathfrak{l}_{\kappa}$ and shows
that nontrivial solutions of the eigenstate equation of $\mathscr{H}$ produced
by a single localized external source are absent.

It is worth to note that logarithmic Nevanlinna function (17) and analysis of
the extensions of the quadratic form coincide with the corresponding objects
in the theory of singular perturbations of 2-dimensional Laplace operator by
the $\delta$-potential [7].

##  2 Interaction with two sources

In order to correct the result of the previous section let us turn to a more
complex singular potential. In the original work [8] Krein equation was
derived for the interaction with multiple sources $v_{m}(\boldsymbol{p})$

$R^{\kappa}_{\mu}=R_{\mu}+R_{\mu}v_{m}(\kappa-\sigma(\mu))_{ml}^{-1}(R_{\bar{\mu}}v_{l},\,\cdot\,),$

where $\kappa_{jk}$ is a real symmetric matrix and the matrix $\sigma_{jk}$ is
defined by the equation

$\sigma_{jk}(\mu)-\sigma_{jk}(\nu)=(v_{j},(R_{\mu}-R_{\nu})v_{k}).$

In our case it is natural to add to the resolvent the potential

$v_{\boldsymbol{x}}(\boldsymbol{p})=e^{i\boldsymbol{p}\cdot\boldsymbol{x}}p^{-1/2}$

shifted by the vector $\boldsymbol{x}$ in the coordinate space. In order to
avoid the treatment of zeros of the matrix $\kappa_{jk}-\sigma_{jk}(\lambda)$
one may turn to the basis

$\displaystyle v_{-}(\boldsymbol{p})$
$\displaystyle=\frac{1}{2}(v(\boldsymbol{p})-v_{\boldsymbol{x}}(\boldsymbol{p}))=\frac{1}{2}p^{-1/2}(1-e^{i\boldsymbol{p}\cdot\boldsymbol{x}}),$
$\displaystyle v_{+}(\boldsymbol{p})$
$\displaystyle=\frac{1}{2}(v(\boldsymbol{p})+v_{\boldsymbol{x}}(\boldsymbol{p}))=\frac{1}{2}p^{-1/2}(1+e^{i\boldsymbol{p}\cdot\boldsymbol{x}})$

and restrict to Krein resolvents with nontrivial second term generated by the
antisymmetric potential $v_{-}(\boldsymbol{p})$

$R_{\mu}^{\kappa}=R_{\mu}+\frac{R_{\mu}v_{-}(R_{\bar{\mu}}v_{-},\,\cdot\,)}{\kappa-\sigma(\mu)}.$
(19)

In this case we have the following equation for $\sigma(\mu)$

$\sigma(\mu)-\sigma(\lambda)=\int_{\mathbb{R}^{3}}\frac{1-\cos(\boldsymbol{p}\cdot\boldsymbol{x})}{p}\bigl{(}\frac{1}{p^{2}-\mu}-\frac{1}{p^{2}-\lambda}\bigr{)}\,d^{3}p=\\\
=4\pi\int_{0}^{\infty}\bigl{(}p-\frac{\sin
px}{x}\bigr{)}\bigl{(}\frac{1}{p^{2}-\mu}-\frac{1}{p^{2}-\lambda}\bigr{)}dp.$
(20)

As in (18) let us introduce the dimensional parameter $\tilde{\kappa}$
incorporating the extension parameter $\kappa$ and define $\sigma(\mu)$ as the
integral

$\sigma(\mu)-\kappa=4\pi\int_{0}^{\infty}\bigl{(}\frac{p}{p^{2}-\mu}-\frac{\sin
px}{x(p^{2}-\mu)}-\frac{p}{p^{2}+\tilde{\kappa}}\bigr{)}dp=\\\
=2\pi\ln\frac{\tilde{\kappa}}{-\mu}-4\pi\int_{0}^{\infty}\frac{\sin
px}{x}\frac{dp}{p^{2}-\mu}.$ (21)

This expression is valid for all $\mu\in\mathbb{C}\setminus\mathbb{R}^{+}\cup
0$ and satisfies the equation (20).

Let us consider the function $\sigma(\mu)-\kappa$ on the negative semi-axis.
For $\mu\to-\infty$ the second term in (21) has the following expansion

$4\pi\int_{0}^{\infty}\frac{\sin px}{x}\frac{dp}{p^{2}-\mu}=\frac{4\pi}{\mu
x^{2}}-\frac{8\pi}{\mu^{2}x^{4}}+\mathcal{O}(\frac{1}{\mu^{3}x^{6}}),$ (22)

and thus the whole sum (21) goes logarithmically to $-\infty$

$\sigma(\mu)-\kappa=2\pi\ln\frac{\tilde{\kappa}}{-\mu}-\frac{4\pi}{\mu
x^{2}}+\mathcal{O}(\mu^{-2})\to-\infty,\quad\mu\to-\infty.$

Derivative of the function $\sigma(\mu)$ for $\mu<0$ is strictly positive

$\sigma^{\prime}_{-}(\mu)=4\pi\int_{0}^{\infty}\bigl{(}p-\frac{\sin
px}{x}\bigr{)}\frac{dp}{(p^{2}-\mu)^{2}}>0,$ (23)

which means that $\sigma(\mu)$ monotonously increases with the increase of
$\mu$. In order to calculate the value $\sigma(0)$ let us write the
oscillating integral for finite $\mu$, $x$ in the terms of the integral sine
and cosine

$\displaystyle 4\pi\int_{0}^{\infty}\frac{\sin
px}{x}\frac{dp}{p^{2}-\mu}=\frac{4\pi}{\sqrt{\mu}x}$
$\displaystyle\bigl{(}\cos\sqrt{\mu}x\,\operatorname{Si}\sqrt{\mu}x-\sin\sqrt{\mu}x\,\operatorname{Ci}\sqrt{\mu}x\bigr{)}=$
(24) $\displaystyle\stackrel{{\scriptstyle\mu x^{2}\to
0}}{{=}}4\pi(1-\boldsymbol{\gamma}-\frac{1}{2}\ln(-\mu x^{2})+\mathcal{O}(\mu
x^{2})).$ (25)

Here $\operatorname{Si}(\sqrt{\mu}x)$ and $\operatorname{Ci}(\sqrt{\mu}x)$ are
defined as

$\displaystyle\operatorname{Si}(\sqrt{\mu}x)=\int_{0}^{\sqrt{\mu}x}\frac{\sin
t}{t}dt,$ (26)
$\displaystyle\operatorname{Ci}(\sqrt{\mu}x)=\boldsymbol{\gamma}+\frac{1}{2}\ln(-\mu
x^{2})-\int_{0}^{\sqrt{\mu}x}\frac{1-\cos t}{t}dt,$ (27)

and $\boldsymbol{\gamma}$ is the Euler constant. One may note that RHS in (24)
is the difference of products of Taylor-expandable even functions and the
logarithm. That is the branching by $\mu$ in (24) is defined solely by the
logarithm from the integral cosine (27). Substituting (25) into (21) one
derive the following expansion of $\sigma(\mu)-\kappa$

$\sigma(\mu)-\kappa=2\pi\ln\tilde{\kappa}x^{2}+4\pi(\boldsymbol{\gamma}-1)+\mathcal{O}(\mu
x^{2}),\quad\mu\to 0.$ (28)

This equation shows that one can always choose $\tilde{\kappa}$ in such a way
that RHS of (28) is negative as $\mu$ tends to zero. The latter means, due to
the positivity of derivative (23), that $\sigma(\mu)-\kappa$ does not turn to
zero on the negative semi-axis and that for such $\tilde{\kappa}$ resolvent
(19) defines positive operator $L_{\kappa}$ and real-valued quadratic form
$\omega_{\kappa}(A)$.

Figure 1: Functions $\sigma(\mu,x_{1})$, $\sigma(\mu,x_{2})$ compared to
$2\pi\ln\frac{\tilde{\kappa}}{-\mu}$

Let us look at (28) from the other side. Suppose that for chosen values of
$\tilde{\kappa}$ and $x=x_{1}$ the limit of RHS of (28) is negative and we can
correctly define the functional $\Omega_{\kappa}(A)$. When increasing the
distance $x$ the limit of RHS of (28) is increasing as well and for some
$x=x_{2}$ it turns to zero (see Fig. 1)

$\sigma(\mu,x_{2})|_{\mu=0}-\kappa=0.$

For $x>x_{2}$ the function $\sigma(\mu,x)-\kappa$ has zero in the interval
$-\infty<\mu<0$ and the corresponding self-adjoint operator $L_{\kappa}$
possesses negative eigenvalue. That is for any chosen extension parameter
$\tilde{\kappa}$ admissible relative positions of the sources always exist and
are always limited by the sphere with radius proportional to inverse square
root of $\tilde{\kappa}$.

##  3 The operator $L_{\kappa}^{1/2}$

The square root of some self-adjoined operator can be written in terms of its
resolvent by means of the spectral representation in the following way

$L_{\kappa}^{1/2}=\lim_{\epsilon\to 0}\frac{1}{2\pi
i}\int_{0}^{\infty}(R_{\lambda+i\epsilon}^{\kappa}-R_{\lambda-i\epsilon}^{\kappa})\lambda^{1/2}\,d\lambda,\quad\lambda^{1/2}>0.$

Substituting the expression for $R_{\lambda}^{\kappa}$ and using the notation
$\pm i0$ for the limit of $\epsilon$ we can write the square root as

$L_{\kappa}^{1/2}=\frac{1}{2\pi
i}\int_{0}^{\infty}\bigl{(}R_{\lambda+i0}-R_{\lambda-i0}+\\\
+\frac{R_{\lambda+i0}v_{-}(R_{\lambda-i0}v_{-},\,\cdot\,)}{\kappa-\sigma(\lambda+i0)}-\frac{R_{\lambda-i0}v_{-}(R_{\lambda+i0}v_{-},\,\cdot\,)}{\kappa-\sigma(\lambda-i0)}\bigr{)}\lambda^{1/2}\,d\lambda=\\\
=p+\frac{1}{2\pi
i}\int_{0}^{\infty}\bigl{(}\frac{R_{\lambda+i0}v_{-}(R_{\lambda-i0}v_{-},\,\cdot\,)}{\kappa-\sigma(\lambda+i0)}-\frac{R_{\lambda-i0}v_{-}(R_{\lambda+i0}v_{-},\,\cdot\,)}{\kappa-\sigma(\lambda-i0)}\bigr{)}\lambda^{1/2}\,d\lambda,$
(29)

where $p$ is the operation of multiplication by the modulus of momentum
variable $\boldsymbol{p}$. The function $R_{\lambda\pm i0}$ is singular in the
variable $p$ for $p^{2}=\lambda$, meanwhile functions $\sigma(\lambda\pm i0)$
differ by the gap

$\sigma(\lambda+i0)-\sigma(\lambda-i0)=2\pi\ln\frac{\tilde{\kappa}}{-\lambda-i0}-2\pi\ln\frac{\tilde{\kappa}}{-\lambda+i0}-\\\
-4\pi\int_{0}^{\infty}\frac{\sin
px}{x}(\frac{1}{p^{2}-\lambda-i0}-\frac{1}{p^{2}-\lambda+i0})\,dp=4\pi^{2}i-4\pi^{2}i\frac{\sin\sqrt{\lambda}x}{\sqrt{\lambda}x}.$
(30)

We will use expressions (29) and (30) in what follows as the tools to
calculate the quadratic form of $L_{\kappa}^{1/2}$.

##  4 Reconstruction of the quadratic form of $L_{\kappa}^{1/2}$

As was stated in the Introduction in order to construct eigenfunctionals of
operation $\mathscr{H}$ one need to prove that the quadratic form of
$L_{\kappa}^{1/2}$ can be reconstructed by its action on the set
$\mathcal{D}(\mathfrak{l})$ everywhere dense in the space
$L_{2}(\mathbb{R}^{3})$. Let the function $A(\boldsymbol{p})$ to be from
$\mathcal{D}(\omega_{\kappa})$, then there exists a sequence
$A_{n}\in\mathcal{D}(L_{\kappa}^{1/2})$ converging to $A$ in the norm induced
by the form $\omega_{\kappa}$

$||A-A_{n}||_{L_{2}(\mathbb{R}^{3})}\to 0,\quad\omega_{\kappa}(A-A_{n})\to
0,\quad n\to\infty.$

Thus it is sufficient to prove that any function $A_{n}(p)$ from
$\mathcal{D}(L_{\kappa}^{1/2})$ can be approximated in the norm of
$\omega_{\kappa}$ by functions from $\mathcal{D}(\mathfrak{l})$. If $A_{n}$
belongs to $\mathcal{D}(L_{\kappa}^{1/2})$ then it belongs to the domain of
quadratic form $\mathfrak{l}_{\kappa}$ of the operator $L_{\kappa}$

$\mathfrak{l}_{\kappa}(A_{n})=(L_{\kappa}^{1/2}A_{n},L_{\kappa}^{1/2}A_{n})<\infty.$

As long as $\mathcal{D}(\mathfrak{l}_{\kappa})$ is the direct sum of
$\mathcal{D}(\mathfrak{l})$ and the linear set generated by some vector
$R_{\rho}v_{-}$, $\rho<0$, then this $A_{n}$ is uniquely decomposing into the
sum

$A_{n}=a_{n}+\chi_{n}R_{\rho}v_{-},\quad
a_{n}\in\mathcal{D}(\mathfrak{l}),\quad\chi_{n}\in\mathbb{C},\quad\rho<0.$

Thus it is sufficient to show that function $R_{\rho}v_{-}$ for some $\rho<0$
can be approximated by functions from $\mathcal{D}(\mathfrak{l})$ in the norm
of quadratic form $\omega_{\kappa}$.

In order to verify the above proposition let us use expression (29) for the
operator $L_{\kappa}^{1/2}$ and write down its quadratic form

$\omega_{\kappa}(R_{\rho}v_{-})=\omega(R_{\rho}v_{-})+\frac{1}{2\pi
i}\int_{0}^{\infty}\bigl{(}\frac{(R_{\rho}v_{-},R_{\lambda+i0}v_{-})(R_{\lambda-i0}v_{-},R_{\rho}v_{-})}{\kappa-\sigma(\lambda+i0)}-\\\
-\frac{(R_{\rho}v_{-},R_{\lambda-i0}v_{-})(R_{\lambda+i0}v_{-},R_{\rho}v_{-})}{\kappa-\sigma(\lambda-i0)}\bigr{)}\lambda^{1/2}\,d\lambda.$
(31)

Here each fraction, up to some vanishing values, can be transformed as

$\frac{(R_{\rho}v_{-},R_{\lambda\pm i0}v_{-})(R_{\lambda\mp
i0}v_{-},R_{\rho}v_{-})}{\kappa-\sigma(\lambda\pm
i0)}=\frac{(\sigma(\lambda\pm
i0)-\sigma(\rho))^{2}}{(\lambda-\rho)^{2}(\kappa-\sigma(\lambda\pm i0))}=\\\
=\frac{2\sigma(\rho)-\kappa-\sigma(\lambda\pm
i0)}{(\lambda-\rho)^{2}}+\frac{(\kappa-\sigma(\rho))^{2}}{(\lambda-\rho)^{2}(\kappa-\sigma(\lambda\pm
i0))},$ (32)

and as the result

$\omega_{\kappa}(R_{\rho}v_{-})=4\pi\int_{0}^{\infty}\frac{p^{2}\,dp}{(p^{2}-\rho)^{2}}-\frac{1}{2\pi
i}\int_{0}^{\infty}\frac{\sigma(\lambda+i0)-\sigma(\lambda-i0)}{(\lambda-\rho)^{2}}\lambda^{1/2}d\lambda\\\
+\frac{1}{2\pi
i}(\kappa-\sigma(\rho))^{2}\int_{0}^{\infty}\frac{\sigma(\lambda+i0)-\sigma(\lambda-i0)}{(\lambda-\rho)^{2}(\kappa-\sigma(\lambda+i0))(\kappa-\sigma(\lambda-i0))}\lambda^{1/2}d\lambda.$
(33)

The gap of $\sigma(\lambda)$ calculated in (30) is limited in modulus, thus
all three integrals in (33) converge absolutely. Then in order to approximate
$R_{\rho}v_{-}$ by the function
$(\Lambda+\rho)R_{-\Lambda}R_{\rho}v_{-}\in\mathcal{D}(\mathfrak{l})$ one may
use the limit in the norm $L_{2}(\mathbb{R}^{3})$

$R_{\rho}v_{-}-(\Lambda+\rho)R_{-\Lambda}R_{\rho}v_{-}=R_{-\Lambda}v_{-}=\frac{1}{(p^{2}+\Lambda)p^{1/2}}\stackrel{{\scriptstyle
L_{2}(\mathbb{R}^{3})}}{{\to}}0,\quad\Lambda\to\infty.$

The same limit is valid in the norm of the form $\omega$

$\omega(R_{-\Lambda}v_{-})=\int_{\mathbb{R}_{3}}\frac{d^{3}p}{(p^{2}+\Lambda)^{2}}\to
0,\quad\Lambda\to\infty,$

thus in order to prove that

$\omega_{\kappa}(R_{-\Lambda}v_{-})\to 0,\quad\Lambda\to\infty,$

one needs to show that the second and the third integrals in (33) after the
substitution of $\rho$ with $-\Lambda$ tend to zero for $\Lambda\to\infty$. By
the change of variable $\tilde{\lambda}=\lambda/\Lambda$ second and third
terms in (33) can be rewritten as

$\displaystyle-\frac{1}{2\pi
i\Lambda^{1/2}}\int_{0}^{\infty}\frac{\sigma(\tilde{\lambda}\Lambda+i0)-\sigma(\tilde{\lambda}\Lambda-i0)}{(\tilde{\lambda}+1)^{2}}\tilde{\lambda}^{1/2}d\tilde{\lambda}$
$\displaystyle\frac{(\kappa-\sigma(-\Lambda))^{2}}{2\pi
i\Lambda^{1/2}}\int_{0}^{\infty}\frac{\sigma(\tilde{\lambda}\Lambda+i0)-\sigma(\tilde{\lambda}\Lambda-i0)}{(\tilde{\lambda}+1)^{2}(\kappa-\sigma(\tilde{\lambda}\Lambda+i0))(\kappa-\sigma(\tilde{\lambda}\Lambda-i0))}\tilde{\lambda}^{1/2}d\tilde{\lambda},$

correspondingly. Now both of the integrals converge absolutely and due to the
limited gap of $\sigma(\lambda)$, their values are bounded from above for any
$\Lambda$. Taking into account the vanishing coefficients $\Lambda^{-1/2}$ one
may conclude that quadratic form $\omega_{\kappa}(R_{-\Lambda}v_{-})$ vanishes
for $\Lambda\to\infty$.

Summarizing the above calculation we have shown that the quadratic form of the
operator $L_{\kappa}^{1/2}$ can be reconstructed by its action on the set
$\mathcal{D}(\mathfrak{l})$.

##  5 Reconstruction of the quadratic form for interaction with
$\delta$-potential

Let us explore how the quadratic form $\omega_{\kappa}$ is reconstructed by
its action on the set $\mathcal{D}(\mathfrak{l})$ for the extensions of the
quadratic form of Laplace operator generated by $\delta$-function [9]. This
singular potential in the momentum representation is a constant

$v_{c}(\boldsymbol{p})=\frac{1}{(2\pi)^{3/2}}\int_{\mathbb{R}^{3}}\delta(\boldsymbol{x})e^{-i\boldsymbol{p}\cdot\boldsymbol{x}}\,d^{3}x=\frac{1}{(2\pi)^{3/2}}.$

Corresponding analytical deficiency vector has the form

$R_{\mu}v_{c}=\frac{(2\pi)^{-3/2}}{k^{2}-\mu},$

and the Nevanlinna function is proportional to square root of the spectral
parameter with values in the upper half-plane

$\sigma_{c}(\mu)-\sigma_{c}(\lambda)=(v,(R_{\mu}-R_{\lambda}))=\frac{1}{(2\pi)^{3}}\int_{\mathbb{R}^{3}}\bigl{(}\frac{1}{p^{2}-\mu}-\frac{1}{p^{2}-\lambda}\bigr{)}\,d^{3}p=\\\
=\frac{i\sqrt{\mu}}{4\pi}-\frac{i\sqrt{\lambda}}{4\pi},\quad
0<\arg\sqrt{\mu},\arg\sqrt{\lambda}<\pi.$

Self-adjoint operators corresponding to extensions of quadratic form of
Laplacian are defined by resolvents of the following type

$R^{\kappa}_{\mu}=R_{\mu}+4\pi\frac{R_{\mu}v_{c}(R_{\bar{\mu}}v_{c},\,\cdot\,)}{4\pi\kappa-i\sqrt{\mu}}.$

In order for the denominator of the second term to be positive for $\mu<0$ the
following condition should hold

$\kappa\geq 0.$

Action of the form $\omega$ on the vector $R_{-\Lambda}v_{c}$ diverges
logarithmically

$\omega(R_{-\Lambda}v_{c})=\int_{\mathbb{R}^{3}}\frac{p\,d^{3}p}{(p^{2}+\Lambda)^{2}}=\infty,\quad\Lambda>0,$
(34)

while the action of the form $\omega_{\kappa}$ on the same vector is finite.
In order to see the latter one needs to introduce a regularization. In the
case of extension generated by the $\delta$-potential the expression for
$\omega_{\kappa}$ has the form of sum (33) with first term substituted with
(34). And the gap of $\sigma_{c}(\lambda)$ through the positive semi-axis is
equal to

$\sigma_{c}(\lambda+i0)-\sigma_{c}(\lambda-i0)=\frac{2i}{4\pi}\lambda^{1/2},\quad\lambda\in\mathbb{R}^{+}.$

To regularize the resulting expression in (33) one may change the upper limit
of (34) to $P$, and the upper limit of the second integral in (33) to $P^{2}$,
in correspondence to their dimensions, and then write

$\omega_{\kappa}(R_{-\Lambda}v_{c})=\lim_{P\to\infty}\bigl{(}\frac{4\pi}{(2\pi)^{3}}\int_{0}^{P}\frac{p^{3}\,dp}{(p^{2}+\Lambda)^{2}}-\frac{1}{(2\pi
i)}\frac{i}{4\pi}\int_{0}^{P^{2}}\frac{2\lambda\,d\lambda}{(\lambda+\Lambda)^{2}}\bigr{)}+\\\
+\frac{1}{2\pi
i}(4\pi\kappa-i\sqrt{-\Lambda})^{2}\int_{0}^{\infty}\frac{2i\lambda\,d\lambda}{(\lambda+\Lambda)^{2}(4\pi\kappa-i\sqrt{\lambda})(4\pi\kappa+i\sqrt{\lambda})}.$

The first two integrals cancel each other, dependence on $P$ disappears and
the remaining equation can be transformed as

$\omega_{\kappa}(R_{-\Lambda}v_{c})=\frac{1}{\pi}(4\pi\kappa-i\sqrt{-\Lambda})^{2}\int_{0}^{\infty}\frac{\lambda\,d\lambda}{(\lambda+\Lambda)^{2}(16\pi^{2}\kappa^{2}+\lambda)}=\\\
=\frac{(4\pi\kappa-i\sqrt{-\Lambda})^{2}}{\pi\Lambda}\int_{0}^{\infty}\frac{\tilde{\lambda}\,d\tilde{\lambda}}{(\tilde{\lambda}+1)^{2}(16\pi^{2}\kappa^{2}/\Lambda+\tilde{\lambda})},\quad\tilde{\lambda}=\frac{\lambda}{\Lambda}.$
(35)

Here RHS tends to $\frac{1}{\pi}$ for $\Lambda\to\infty$ but not to zero as we
expect for the value of $\omega_{\kappa}$ on the vectors of the type of

$R_{\rho}v_{c}-(\rho+\Lambda)R_{\rho}R_{-\Lambda}v_{c}=R_{-\Lambda}v_{c},\quad(\rho+\Lambda)R_{\rho}R_{-\Lambda}v_{c}\in\mathcal{D}(\mathfrak{l}).$

Moreover one can show that the value of the limit in (35) depend on the
sequence of vanishing vectors. However the author still has no proof of the
fact that there is no vanishing limit of the form $\omega_{\kappa}$ on any
sequence of difference of $R_{\rho}v_{c}$ and vectors from
$\mathcal{D}(\mathfrak{l})$.

Thus we assume that the closure of the quadratic form $\omega_{\kappa}$ acting
on the set $\mathcal{D}(\mathfrak{l})$ represents a restriction of the form
$\omega_{\kappa}$

$\overline{\omega_{\kappa}|_{\mathcal{D}(\mathfrak{l})}}=\tilde{\omega}_{\kappa}\subset\omega_{\kappa}\quad\tilde{\omega}_{\kappa}\neq\omega_{\kappa}.$

This restriction corresponds to a self-adjoint operator different from
$L_{\kappa}^{1/2}$ such that quadratic form of its square is not an extension
of the quadratic form of Laplacian. And this means that equation (10) does not
hold.

## Conclusion and discussion

We have shown that in order to construct solutions for the eigenstate equation
of the free scalar quantum theory the singular potential
$(1-e^{i\boldsymbol{p}\cdot\boldsymbol{x}})p^{-1/2}$ incorporating at least
two connected singularities has to be used. The distance between the
singularities is proved to be limited by the extension parameter. We
demonstrate for the potential in question that the quadratic form of the
square root is reconstructed by its action on the domain of quadratic form of
Laplacian. This requirement is essential to ensure that the equation (9) holds
for eigenfunctionals (8).

Further development of Gaussians $\Omega_{\kappa}$ involves introduction of
the scalar product and in particular investigation of the possibility to
normalize the functionals with respect to the ground state of the free theory
$\Omega$. Such a normalization can be performed by means of functional
integration, the result is proportional to

$\frac{\int\exp\\{-\frac{1}{2}\omega_{\kappa}(A)\\}\,\prod_{\boldsymbol{x}}\delta
A(\boldsymbol{x})}{\int\exp\\{-\frac{1}{2}\omega(A)\\}\,\prod_{\boldsymbol{x}}\delta
A(\boldsymbol{x})}=\exp\\{\frac{1}{4}\operatorname{Tr}(\ln L-\ln
L_{\kappa})\\}.$

Using the equation

$(R_{\mu}v,R_{\mu}v)=\sigma^{\prime}(\mu),$

it is not hard to see that the trace of the difference of logarithms can be
transformed to the integral

$\operatorname{Tr}(\ln L_{\kappa}-\ln L)=\frac{1}{2\pi
i}\int_{0}^{\infty}\bigl{(}\frac{\sigma^{\prime}(\lambda+i0)}{\kappa-\sigma(\lambda+i0)}-\frac{\sigma^{\prime}(\lambda-i0)}{\kappa-\sigma(\lambda-i0)}\bigr{)}\ln\lambda\,d\lambda,$
(36)

which diverges as dilogarithm of the upper limit. This means that constructed
eigenstates of $\mathscr{H}$ reside in their own functional space not
connected to the one of the free theory. Thus the action of $\mathscr{H}$ on
$\Omega_{\kappa}$ cannot be interpreted as a perturbation of the free quantum
field Hamiltonian in the sense of the theory of singular perturbations of
differential operators [10]. It is a separate object acting in its own space
by the laws of the free theory but leading to a different dynamics.

Another important subject of study is the eigenvalue of $\mathscr{H}$ on the
state $\Omega_{\kappa}$, which is the trace
$\operatorname{Tr}L_{\kappa}^{1/2}$. Taking into account the divergence of
weaker integral (36), one may conclude that normalization w. r. to the
eigenvalue of the free theory also does not hold

$\operatorname{Tr}L_{\kappa}^{1/2}-\operatorname{Tr}L^{1/2}=\frac{1}{2\pi
i}\int_{0}^{\infty}\bigl{(}\frac{\sigma^{\prime}(\lambda+i0)}{\kappa-\sigma(\lambda+i0)}-\frac{\sigma^{\prime}(\lambda-i0)}{\kappa-\sigma(\lambda-i0)}\bigr{)}\lambda^{1/2}\,d\lambda=\infty.$
(37)

But we would like to have the possibility to compare eigenvalues of
functionals $\Omega_{\kappa}$ for fixed extension parameter and different
distances between the sources $v(\boldsymbol{p})$ and
$v_{\boldsymbol{x}}(\boldsymbol{p})$. It turns out that the scalar theory does
not allow such comparison, the derivative of integral (37) with respect to $x$
diverges as well. The way out of this is the transition to a more natural
object of the interaction with the Fermi-sources, namely to the
electromagnetic or gauge field. It can be shown that for the quantum field in
the Coulomb gauge there exist extensions of the quadratic form of Laplacian
whose derivative of $\sigma(\mu)$ with respect to the distance $x$ decreases
fast enough in the spectral parameter for the derivative of (37) to be finite.

## References

  * [1] B. Hatfield, _Quantum Field Theory of Point Particles and Strings._ , Addison Wesley Longman, 1992. Chapt. 10 “Free Fields in the Schrdinger Representation”.
  * [2] V. Koshmanenko, _Singular Quadratic Forms in Perturbation Theory_ , Springer Dordrecht, 1999. Chapt. 2 “Quadratic Forms and Linear Operators”.
  * [3] T. A. Bolokhov, “Singular perturbations of a free quantum field Hamiltonian”, arXiv:1912.01458 (math-ph).
  * [4] A. Alonso, B. Simon, “The Birman - Krein - Vishik theory of selfadjoint extensions of semibounded operators”, J. Operator Theory, 4 (1980), 251–270.
  * [5] A. Kiselev, B. Simon, “Rank one perturbations with infinitesimal coupling”, J. Funct. Anal. 130 (1995), 345–356.
  * [6] F. Gesztesy, E. Tsekanovskii, “On Matrix-Valued Herglotz Functions”, arXiv:funct-an/9712004.
  * [7] L. D. Faddeev, “Notes on divergences and dimensional transmutation in Yang-Mills theory,” Theor. Math. Phys. 148 (2006) 986 [Teor. Mat. Fiz. 148 (2006) 133].
  * [8] M. G. Krein, “The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications.”, Rec. Math. (Mat. Sbornik) N.S., 20 (62), 1947, 431–495.
  * [9] F. A. Berezin, L. D. Faddeev, “A Remark on Schrodinger’s equation with a singular potential,” Sov. Math. Dokl. 2 (1961) 372 [Dokl. Akad. Nauk Ser. Fiz. 137 (1961) 1011].
  * [10] S. Albeverio, P. Kurasov, _Singular Perturbation of Differential Operators. Solvable Schrödinger type Operators_ , Cambridge University Press, 2000.


Also at: ]Comprehensive Heart Failure Center, Am Schwarzenberg 15, A15,
D-97080 Würzburg, Germany

# Geometrically frustrated systems which are as singles hotter than in company

Wolfgang Rudolf Bauer<EMAIL_ADDRESS>University/University Hospital of
Würzburg (ZIM), Oberdürrbacher Straße 6, D-97080 Würzburg, Germany, Tel.
+49-931-201-39011  [

###### Abstract

We show that a set of thermally weakly coupled geometrically frustrated
systems (GFSs), each of which is constraint to reside at negative Boltzmann
temperatures, is in equilibrium cooler than its constituents. It may even
exhibit positive temperatures at low energies. The challenge for the second
law of thermodynamics arising from potential heat flow related to the gradient
of temperatures between a GFS and its environment is resolved by considering
the energy fluctuations above the ground state. They are comprised in the
canonical temperature, derived from information theory. Whereas the gradient
of Boltzmann temperatures gives the direction of the stochastic drift of the
most probable state of a GFS within its environment, the canonical temperature
gradient defines that of heat flow.

PACS numbers

5.70.Ln, 5.20.Gg

##  I Introduction

Our everyday experience is that systems of same temperature, which are brought
from initial isolation to thermal coupling, maintain their temperature, which
is equivalent to that of the combined system. Absence of temperature gradients
implies no heat flow, and the combined system is said to be in thermal
equilibrium. Any other scenario would be in conflict with our notion of
thermal stability and the second law of thermodynamics in Clausius’s version
Clausius (1872)111Exceptions of this thermal stability exist for intensive
parameters as magnetization, where subsystems, after brought in weak thermal
contact, evolve towards different phases Ramírez-Hernández  _et al._ (2008)..
This base for thermometry derives in phenomenological thermodynamics from the
additive behavior of the state variables energy and entropy, which makes
temperature of the combined system an intensive parameter, i.e. it is
equivalent to that of its subsystems. Entropy of the combined system has in
this state its maximum Callen (1985), which ensures a stable thermodynamic
equilibrium. In statistical mechanics Boltzmann’s concept of entropy predicts
this state as the most probable one, which in “normal” macroscopic systems
Kubo (1965) clearly dominates over any other state, and ensures that entropy
becomes additive and temperature intensive. This concept may be consistently
extended to systems capable of having negative temperatures Baldovin  _et al._
(2021).

We recently described generic geometrically frustrated spin systems (GFSs), in
which the ground state exhibits maximum degeneracy and Boltzmann entropy
almost declines linearly with energy which makes the corresponding Boltzmann
spin temperature constantly negative throughout Bauer (2022). When brought in
thermal contact with another system, the framework of Boltzmann entropy and
temperature with reference to thermodynamic equilibrium of the combined
system, i.e. principle of equal weight of microstates and detailed balance,
may be straightforwardly transferred. The temperature gradient between a GFS
and another system acts as a thermodynamic driving force which directs the
combined system towards a more probables state. Boltzmann entropy of a single
GFS declines almost linearly with energy, which implies a constant negative
temperature. This linear dependence of entropy on energy implies that within a
set of thermally weakly coupled GFSs, all combinations of energy, under the
constraint of energy conservation, become equally probable. This indifferent
rather than the stable equilibrium in“normal” systems, implies that entropy of
the assembly of GFSs is strongly superadditive, and, consequently, its
temperature should differ from that of its constituents. At a first glance,
this scenario would be in conflict with the second law of thermodynamics,
which relates heat flow to temperature gradients. In this article we
systematically address how temperature of a set of thermally weakly coupled
GFSs is related to that of its constituents, and how the notion that
temperature gradients drive heat flow must be specified, to maintain the
second law of thermodynamics.

##  II Weakly coupled GFSs as microcanonical ensemble

###  II.1 Single GFS

The GFS we have in focus is a special Ising model of $N$ spins, with equal
mutual anti-ferromagnetic interaction of strength $J$. The paradigm is that of
primitive geometrically frustrated systems forming spin-ice Moessner (2001);
Moessner and Ramirez (2006). Within the GFS, the spins can be thought to be
located at the $N$ vertices of a regular $N-1$-dimensional simplex. With
orientation $s_{i}=\pm 1$ of the $i$-th spin, the microstate of a GFS becomes
$\sigma=(s_{1},\ldots,s_{N})^{T}$, and the Hamiltonian reads

$\displaystyle H(\sigma)$ $\displaystyle=$
$\displaystyle\frac{J}{2}\sum_{i,j\;i\neq
j}^{N}s_{i}s_{j}=\frac{J}{2}\;\left(m^{2}-N\right)\;,\text{with}$ (1)
$\displaystyle m$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{N}s_{i}\;.$ (2)

as magnetic dipole moment. The offset $-J/2\;N$ is omitted furtheron. The set
of microstates, $\Sigma_{m}$, with magnetic moment $m$ has

$|\Sigma_{m}|=\binom{N}{\tfrac{N+m}{2}}\approx\tfrac{1}{\sqrt{2\pi}}\;\frac{2^{N}}{\sqrt{N/4}}\;e^{-\tfrac{1}{2}\;\left(\nicefrac{{m^{2}}}{{N}}\right)}\;\;$
(3)

elements, where the Gaussian approximation holds for large values of $N$, and
dipole moments $m$ not too close to $N$, i.e. not in the tails of the
distribution. With energy $E(m)=J/2\;m^{2}$, the Boltzmann entropy
$S_{GFS}(E)=\;\ln(|\Sigma_{m}|)$ 222though $\pm m$ contribute to the same
energy $E$, the domains of states of $+m$ and $-m$ are within a single GFS not
connected, i.e. with respect to ergodicity these two branches are separated.
Hence, only one branch must be considered for entropy. and inverse temperature
$\kappa=\partial_{E}S_{GFS}(E)$ become with normalization of Boltzmann
constant $k_{\text{B}}=1$

$\displaystyle S_{GFS}(E)$ $\displaystyle=$
$\displaystyle-\tfrac{1}{JN}E+S_{GFS}(0)\;\;,$ (4) $\displaystyle\kappa$
$\displaystyle=$ $\displaystyle-\tfrac{1}{JN}\;.\ $ (5)

###  II.2 Assembly of thermally weakly coupled GFSs

Whereas the thermal interaction of a single GFS with another system was in
detail considered previously Bauer (2022), the focus of this manuscript lies
on the characterization of a collective of thermally weakly coupled GFSs.

The microstate of a group of $M$ GFSs, as introduced above, derives from
microstates of the individual GFSs,
$\boldsymbol{\sigma}=(\sigma_{1},\ldots,\sigma_{M})^{T}$. Weak mutual thermal
coupling between the single GFSs makes the Hamiltonian of the group the sum of
the individual Hamiltonians,
$H(\boldsymbol{\sigma})=\sum_{\nu=1}^{M}H^{(\nu)}(\sigma_{\nu})$. The energy
then depends on the magnetic moments
$\boldsymbol{m}=(m^{(1)},\ldots,m^{(M)})^{T}$ as

$E(\boldsymbol{m})=\frac{J}{2}\sum_{\nu=1}^{M}\left(m^{(\nu)}\right)^{2}=\frac{J}{2}\,\|\boldsymbol{m}\|^{2}\;,$
(6)

with the norm $\|\boldsymbol{m}\|=\sqrt{\boldsymbol{m}^{T}\boldsymbol{m}}$.
The number of microstates with magnetic moments, $\boldsymbol{m}$, derives
from Eqs. (3,6) as

$|\Sigma_{\boldsymbol{m}}|=\prod_{\nu=1}^{M}|\Sigma_{m^{(\nu)}}|=\left(\tfrac{1}{\sqrt{2\pi
N}}\;2^{(N+1)}\right)^{M}\;\underbrace{e^{-\frac{1}{N}\;\frac{1}{2}\|\boldsymbol{m}\|^{2}}}_{=e^{\kappa
E(\boldsymbol{m})}}\;.$ (7)

The number of realizations of magnetic moments with a certain energy $E$ is
according to Eqs. (6) proportional to the surface of the $M$-dimensional
hypersphere with radius $m_{M}(E)=\sqrt{2E/J}$. With Eq. (7) the number of
microstates with energy $E$ is (see appendix A)

$\displaystyle|\Sigma_{E}|$ $\displaystyle=$
$\displaystyle|\Sigma_{\|\boldsymbol{m}\|=\sqrt{2E/J}}|\;c\;{\cal{S}}_{M}(2E/J)^{\frac{M-1}{2}}$
(8) $\displaystyle=$ $\displaystyle\sqrt{2}\;\left(\tfrac{2^{N}}{\sqrt{\pi
N}}\right)^{M}e^{\kappa E}\;{\cal{S}}_{M}\;\;(E/J)^{\frac{M-1}{2}}\;,$ (9)

with ${\cal{S}}_{M}=2\pi^{M/2}/\Gamma(M/2)$ as surface area of the
$M$-dimensional unit sphere, $c$ some proportionality factor, and $\Gamma$ as
Gamma function. Boltzmann entropy is

$\displaystyle S(E)$ $\displaystyle=$ $\displaystyle\ln|\Sigma_{E}|$ (10)
$\displaystyle=$
$\displaystyle\ln|\Sigma_{\|\boldsymbol{m}\|=\sqrt{2E/J}}|+\ln\bigg{(}c{\cal{S}}_{M}\left(\tfrac{2E}{J}\right)^{\frac{M-1}{2}}\bigg{)}$
(11) $\displaystyle=$ $\displaystyle\kappa\;E+\tfrac{M-1}{2}\ln(E/J)+C\;,$
(12)

with $C$ as the logarithm of the remaining, energy independent factors of Eq.
(9), which scales almost linearly with $M$ 333Note that the terms related to
the surface of the hypersphere in Eqs. (8,11) do not vanish in the ground
state $E=0$ but become unity.. The validity of this Gaussian based
approximation with entropy obtained from direct counting of states is shown in
appendix A. In contrast to normal systems with short range interaction, the
entropy in Eq. (12) at energy $E$ does not emerge as the sum of entropies of
its constituents, the GFSs, each at same energy $E/M$, but instead one has

$\Delta S(E)>\kappa
E=\sum_{\nu=1}^{M}\underbrace{\kappa\;E/M}_{\begin{subarray}{c}\Delta\text{entropy
of a single}\\\ \text{ GFS at $\varepsilon=E/M$ }\end{subarray}}\;,$ (13)

This means that additivity of entropy does not hold here. In normal systems,
this additivity results from the sharp peak which the probability of energy
configurations exhibits around its maximum. This maximum is located at the
configuration where energies of the constituents are all equal, i.e. in our
case this would be $E/M$. This sharp peak of maximum probability implies that
in normal systems this state is clearly dominating and stable. However, the
degeneracy of temperature of a single GFS, i.e. the fact that all its energy
states exhibit the same temperature, implies that any energy configuration of
thermally weakly coupled GFSs, which conserves overall energy, occurs with the
same probability. Hence, one has not a stable dominating equilibrium state,
but an indifferent equilibrium with the consequence of supra additivity of
entropy in Eqs. (11,12).

The inverse temperature of $M$ thermally weakly coupled GFSs derives as

$\beta_{M}=\partial_{E}S(E)=\underbrace{\tfrac{1}{2}\;\bar{\varepsilon}^{\;-1}+\kappa}_{=\beta_{\infty}}-\tfrac{1}{2M}\bar{\varepsilon}^{\;-1}\;,$
(14)

with $\bar{\varepsilon}=E/M$ as the specific energy, and $\beta_{\infty}$ as
the asymptotic inverse temperature, which a very large number ($M\to\infty$)
of weakly coupled GFSs has. The latter is sum of an inverse temperature
resembling that of an ideal 1-D gas and that of a single GFS, $\kappa$.

Equation (14) implies that, with reference to Boltzmann temperature, within a
group of thermally weakly coupled GFSs, the single GFS is always hotter than
the group as a whole, which is counter intuitive. At smaller energies, the
inverse temperature of the group of GFSs is positive, $\beta_{M}>0$. It
monotonically decreases with increasing energy, eventually vanishes
($\beta_{M}\to 0$) at

$\bar{\varepsilon}_{0}=-\tfrac{M-1}{M}\frac{1}{2\kappa}\approx
J\;\frac{N}{2}\;\text{for large $M$},$ (15)

and stays thereafter in the negative temperature regime. In this context it
should also be mentioned that the above considerations about a single system
being hotter than an assembly of its thermally weakly coupled copies would
also hold if one switches the anti-ferromagnetic interaction to a
ferromagnetic one, i.e. $J<0$. In this case, however, all effects would be
restricted to the positive temperature regime.

The emergence of positive temperatures out of a group of thermally weakly
coupled GFSs, each of them being at the same constant negative temperature,
makes it attractive for thermometry. Whereas, due to the zeroth law of
thermodynamics, rather sophisticated concepts are required to make a single
GFS a thermometer for systems at positive temperature Bauer (2022), an
assembly of GFSs should accomplish this straightforwardly.

###  II.3 Scaling properties and limiting cases

In context with the above, it is worth to discuss some scaling properties of
the GFS assembly. The specific energy, i.e. average energy of $M$ GFSs, may be
rewritten as

$\displaystyle\bar{\varepsilon}$ $\displaystyle=$
$\displaystyle\frac{1}{M}\sum_{\nu=1}^{M}\frac{J}{2}\left(m^{(\nu)}\right)^{2}=N\frac{JN}{2}\frac{1}{M}\sum_{\nu=1}^{M}\left(\frac{m^{(\nu)}}{N}\right)^{2}$
(16) $\displaystyle=$
$\displaystyle-\frac{N}{2\kappa}\;\overline{\mathfrak{m}^{2}}\;$ (18)

where $\mathfrak{m}=m/N$ is the fraction of the magnetic moment related to the
number of spins, i.e. the magnetization of a GFS. Insertion into Eq. (14) then
yields

$\beta_{M}=\kappa\left(1-\left(1-\frac{1}{M}\right)\;\frac{1}{N\;\overline{\mathfrak{m}^{2}}}\right)\;.$
(19)

This implies that the difference of temperature of a GFS and its assembly is
negligible if

$\overline{\mathfrak{m}^{2}}\gg\frac{1}{N}\;,$ (20)

i.e. vice versa it is relevant in the low energy regime, which we have in
focus in this manuscript. With increasing magnetization/energy outside this
regime the temperature of the GFS and its assembly converge.

It should be mentioned that the way we formulate the spin interaction of the
GFS on the geometry of a $N-1$-dimensional simplex, which is analogue to the
primitive elements forming spin ice Moessner and Ramirez (2006); Moessner
(2001), is not an extensive system, i.e. within a single GFS $\kappa$ is not
an intensive parameter as it depends on the size of the system $\sim N^{-1}$.
However this may be accomplished by renormalization of the interaction
constant $J\to J^{\prime}/N$, an approach used to describe
(anti)ferromagnetism Campa  _et al._ (2009). The magnetization
$\mathfrak{m}=m/N$ is considered as an intensive parameter, making the single
system Hamiltonian in Eq. (1), $H=1/2\;J^{\prime}\mathfrak{m}^{2}\;N$, and
entropy in Eq. (4) becomes, $\sim-1/2\;\mathfrak{m}^{2}\;N$. Hence, by scaling
with $N$ the latter both become extensive parameters, which makes temperature
of the single system an intensive parameter
$\kappa\to\kappa^{\prime}=-1/J^{\prime}$, and the thermodynamic limit
$N\to\infty$ exists. Because Eq. (19) stays invariant under this rescaling,
Eq. (20) predicts that $\beta_{M}^{\prime}$ becomes identical with
$\kappa^{\prime}$ in this limit. In other words: for a constant, non-vanishing
magnetization $\mathfrak{m}=m/N$, temperatures of assembly and constituents
become equivalent in the thermodynamic limit. However in this manuscript we
stay at finite size GFSs in which the range of magnetization holds
$\overline{\mathfrak{m}^{2}}\leq\frac{1}{N}$, i.e. the magnetic moment
fulfills $\overline{m^{2}}\leq N$, where temperatures differences in Eq. (19)
are of relevance.

##  III GFSs as canonical ensemble

The in-equivalence of Boltzmann temperatures of an ensemble and its
constituents requires reconsideration of the canonical ensemble, and the way
its distribution is derived.

In the microcanonical based version Baldovin  _et al._ (2021); Puglisi  _et
al._ (2017); Cerino  _et al._ (2015), the system and its bath, as two weakly
coupled partners, form a microcanonical ensemble with energy $E$. The system
is in some state $\sigma$ with energy $H(\sigma)=\varepsilon$, and it is
considered to be small compared to its bath. The latter’s entropy may then be
approximated by $S_{bath}(E-\varepsilon)=S_{bath}(E)-\beta_{bath}\varepsilon$,
with $\beta_{bath}=\partial_{E}S_{bath}(E)$ as its inverse Boltzmann
temperature. Because the bath itself may be built from many weakly coupled
GFSs, Eq. (14) implies

$\beta_{bath}=\beta_{\infty}\;.$ (21)

The probability that the GFS is in the state $\sigma$ is given by the bath’s
entropy yielding the Boltzmann factor,

$\displaystyle p_{GFS}(\mathbf{\sigma})$ $\displaystyle\sim$ $\displaystyle
e^{S_{bath}(E-\varepsilon)}\sim e^{-\beta_{\infty}\;\varepsilon}\;.$ (22)

The information theory based ansatz focuses on the distribution function
$\varpi(\sigma)$ of microstates $\sigma$ of a group of many weakly coupled
GFSs. With $M$ GFSs this function underlies the constraint of energy
conservation

$g[\varpi]=\sum_{\sigma}\varpi(\sigma)H(\sigma)-E/M\overset{!}{=}0\;.$ (23)

The probability to find the distribution $\varpi(\sigma)$ is given by the
Shannon entropy
$S_{Sh}[\varpi]=-\sum_{\sigma}\varpi(\sigma)\ln(\varpi(\sigma))$ (see appendix
B) as

$\displaystyle w[\varpi]$ $\displaystyle\sim$
$\displaystyle\exp(MS_{Sh}(\varpi))\;,$ (24)

i.e. its scales according to the large deviation principle Touchette (2009)
with the Shannon entropy as rate function. The most probable distribution is
that with maximum Shannon entropy under the constraint of energy conservation,
which is obtained with the Lagrange multiplier $\beta^{(c)}$ from

$\partial_{\varpi({\sigma})}\big{(}S_{Sh}[\varpi]-\beta^{(c)}g[\varpi]\big{)}=0$
(25)

as

$\varpi_{max}(\sigma)\sim e^{-\beta^{(c)}H(\sigma)}\;.$ (26)

The Lagrange multiplier $\beta^{(c)}$ is by definition the inverse canonical
temperature. The advantage of the information theory based derivation is that
the canonical temperature solely derives from properties of the single system,
i.e. the GFS, and its specific energy $\bar{\varepsilon}=E/M$ in Eq. (23).
Because the assembly of GFSs has as a bath many constituents ($M$ large), the
large deviation scaling of the probability $w[\varpi]$ in Eq. (24) predicts
that it peaks sharply around its maximum at $\varpi_{max}$. Therefore the
states of the assembly are almost solely distributed according to
$\varpi_{max}$, which implies the identity

$\varpi_{max}(\sigma)=p_{GFS}(\sigma)\;.$ (27)

Consequently, the Boltzmann temperature of a bath, assembled from GFSs, is the
same as the canonical temperature of a GFS, introduced as a Lagrange
multiplier to maintain energy conservation, i.e. with Eqs. (21,22)

$\beta^{(c)}=\beta_{\infty}=\beta_{bath}\;.$ (28)

In this context it should be noted that the Boltzmann temperature of a large
set of $M^{\prime}\gg 1$ GFSs within a bath of $M\gg M^{\prime}$ GFSs becomes
identical with the canonical temperature $\beta^{(c)}$, in agreement with our
usual notion of thermodynamics.

Boltzmann and canonical temperature of a GFS differ, as Eqs. (5, 14, 28)
predict

$\kappa<\beta^{(c)}\;.$ (29)

This in-equivalence is related to the special equilibrium probability
distribution of energy which a GFS exhibits in the canonical ensemble. The
probability to find a GFS with energy $\varepsilon$ in one branch of magnetic
moment, $m(\varepsilon)=\pm\sqrt{2J\varepsilon}$, 444note that only one branch
must be considered for determination of entropy because the domains with
equivalent energy $\Sigma_{m},\;\Sigma_{-m}$ are disjoint. (see Eqs. (4, 5,
22, 28)) is,

$\displaystyle p_{GFS}(\varepsilon_{\pm})$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}{\sigma|H(\sigma)=\varepsilon}\\\
{\operatorname{sign}(m(\sigma)=\pm 1}\end{subarray}}p_{GFS}(\sigma)$ (30)
$\displaystyle\sim$
$\displaystyle\exp(S_{GFS}(\varepsilon)-\beta^{(c)}\varepsilon)$ (32)
$\displaystyle=$ $\displaystyle
e^{S_{GFS}(0)}\;\exp((\kappa-\beta^{(c)})\varepsilon)\;.$ (34)

This equilibrium distribution is also confirmed by simulations of spin
dynamics of a GFS in a heat bath (see appendix C). As $\kappa-\beta^{(c)}<0$,
the most probable state of a GFS is its degenerated ground state
($\varepsilon=0$), and, with reference to energy, this is a global but not a
local maximum, i.e. temperatures of GFSs $\kappa$ and bath $\beta^{(c)}$
differ here. In contrast to a GFS, normal additive, i.e. extensive systems
with a short ranged interaction, $(\bullet)_{ext}$, have a concave entropy
function $S_{ext}(\varepsilon)$, the slopes of which, i.e. inverse
temperatures, include that of the bath. This implies the existence of a local
maximum of
$p_{ext}(\varepsilon)\sim\exp(S_{ext}(\varepsilon)-\beta^{(c)}\varepsilon)$ at
some $\varepsilon^{*}(\beta^{(c)})$, i.e.

$\underbrace{\partial_{\varepsilon}\left.S_{ext}(\varepsilon)\right|_{\varepsilon^{*}}}_{=\beta_{ext}}-\beta^{(c)}=0\;,$
(35)

where canonical - and Boltzmann temperature become equivalent. This is
independent from sign of temperature, and provoked the premature statement,
that Boltzmann and canonical temperature are always equivalent in the negative
range as well Buonsante  _et al._ (2016); Abraham and Penrose (2017), which
was shown here not to hold in general. The existence of a local maximum of
probability of energy at some $\varepsilon^{*}(\beta^{(c)})$ in normal systems
is the base for the Legendre transformation to the Massieu function
$S_{ext}\to\tilde{S}_{ext}(\beta^{(c)})=S_{ext}(\varepsilon^{*}(\beta^{(c)})-\beta^{(c)}\varepsilon^{*}(\beta^{(c)})=-\beta^{(c)}F(\beta^{(c)})$,
with the Helmholtz free energy $F$ Swendsen (2017). The inversion of this
Legendre transformation reveals the most probable energy from the Massieu
function,
$\varepsilon^{*}(\beta^{(c)})=-\partial_{\beta^{(c)}}\tilde{S}_{ext}(\beta{(c)})$,
and the canonical entropy from free energy
$S_{ext}(\beta^{(c)})=-\partial_{T^{(c)}}F$, with temperature $T=1/\beta$
Matty  _et al._ (2017).

The in-equivalence of temperature of a single GFS and that of its bath
consisting of many weakly thermally coupled GFSs (Eq. (29)) prohibits the
existence of a local maximum of the probability to find a GFS at some energy,
and by this the construction of a Legendre transformation or free energy in
the usual way. However, the fluctuations above the ground state given by Eq.
(34) suggest to construct a partition function to derive the mean energy,

$\displaystyle Z(\beta^{(c)})$ $\displaystyle=$
$\displaystyle\sum_{\sigma}\exp(-\beta^{(c)}\varepsilon(\sigma))$ (36)
$\displaystyle=$
$\displaystyle\sum_{m}\exp(S(\varepsilon(m))\exp(-\beta^{(c)}\varepsilon(m))\;,$
(38) and with Eq. (3) $\displaystyle\approx$
$\displaystyle\tfrac{1}{\sqrt{2\pi}}\;\frac{2^{N}}{\sqrt{N/4}}\sum_{m}\exp\big{(}(\kappa-\beta^{(c)})\underbrace{\varepsilon(m)}_{=\tfrac{1}{2}Jm^{2}}\big{)}\;.$
(39)

For large $N$ and $\beta^{(c)}-\kappa<1$ (see Appendix (D)) the sum may be
replaced by an integral in $m$ over a Gaussian distribution,i.e.

$Z(\beta^{(c)})=\frac{2^{N}}{\sqrt{N}}\;\frac{1}{\sqrt{J(\beta^{(c)}-\kappa)}}\;.$
(40)

Straightforward the cumulant generating function,

$\Phi(\beta^{(c)})=-\ln(Z(\beta^{(c)}))\;,$ (41)

determines with Eqs. (14,28) the mean energy

$\langle\varepsilon\rangle_{GFS}=\partial_{\beta^{(C)}}\Phi(\beta^{(c)})=\tfrac{1}{2}\;\frac{1}{\beta^{(c)}-\kappa}=\bar{\varepsilon}\;.$
(42)

This self consistent expression is definitely not trivial. The mean energy
$\langle\varepsilon\rangle_{GFS}=J/2\langle m^{2}\rangle$ of a GFS within its
ensemble results purely from energy fluctuations away from its most probable
state, the degenerated ground state at $\varepsilon^{*}=0$. In contrast, in
extensive systems with short ranged interaction, i.e. additive systems, mean
energy and energy of the mode are almost identical Swendsen and Wang (2016),
even, if fluctuations play a role Falcioni  _et al._ (2011).

Though the potential $\Phi$ reveals the mean energy and other moments from its
cumulants, it has not the property of a Helmholtz free energy in the sense of
a Legendre transformation, the inversion of which would reveals the mode of
energy. As noted above this transformation requires that the mode of the
energy is a local maximum, where temperature of the system and bath become
equivalent. Then, by saddle point approximation, in the thermodynamic limit
free energy and cumulant generating function would be directly related by
$F(\beta^{(c)})\approx-1/\beta^{(c)}\Phi(\beta^{(c)})$, and mode and mean of
energy would become equivalent, which is not the case here.

##  IV Challenge for the 2nd law of thermodynamics

Figure 1: Relaxation of the conditional probability distribution
$p(m_{2}|m_{1},t)$ (Eq. (57)), with initial magnetic moment $m_{1}=50$ (green)
and three further subsequent points in time, $\alpha\;t$
($\alpha=DJ(\beta^{(c)}-\kappa)$), color coded in red, blue and black. The GFS
consists of $N=1000$ spins, i.e. temperature is $\kappa=-1/1000\;J^{-1}$. The
canonical temperature $\beta^{(c)}$ was adjusted to a mean energy in the bath
of
$\langle\varepsilon\rangle_{GFS}=\tfrac{1}{2}\;(\beta^{(c)}-\kappa)^{-1}=\tfrac{J}{2}\langle
m_{2}^{2}\rangle=\tfrac{J}{2}\;30^{2}$. At $\alpha\;t=10$ the equilibrium
distribution $p_{GFS}^{(m)}(m_{2})=\lim_{t\to\infty}p(m_{2}|m_{1},t)$ is
almost reached. The harmonic potential $V$ affecting the transitions between
magnetic moments in orange (see text). In grey the evolution of the
probability of the mode $p(m_{\mu})$. Note: probabilities are given for
discretized magnetic moments.

Referring to Boltzmann temperature a GFS is hotter than its bath assembled
from many weakly coupled GFSs (Eqs. (28,29)). This provokes the thought
experiment: a single GFS with energy $\varepsilon_{1}$ taken out of its bath 1
and brought in contact for a time $t$ with another bath 2 at same temperature,
should deliver heat according to the gradient of respective Boltzmann
temperatures, i.e.
$Q(t)=-\Delta\varepsilon(t)=-(\varepsilon_{2}(t)-\varepsilon_{1})>0$, with
$\varepsilon_{2}(t)$ as the GFS’s energy in bath 2 after time $t$ . This is in
particular intriguing, because prediction of heat flow from hot to cold is one
of the main merits one normally assigns to Boltzmann’s temperature Cerino  _et
al._ (2015); Swendsen and Wang (2015). Obviously such a behavior would be in
conflict with the 2nd law of thermodynamics as it would imply GFS mediated
heat flow between two baths at the same temperature. Hence, the concept of
heat flow from hot to cold defined by Boltzmann temperatures must be
reconsidered.

Indeed such heat transfer increases the combined system’s entropy, $\Delta
S_{GFS}(t)+\Delta S_{bath}(t)=-(\kappa-\beta^{(c)})Q(t)>0$ (Eqs. (22, 28,
34)), i.e. the combined system is more likely to drift towards lower energy of
the GFS. Apparently repeating this procedure would imply unlimited heat
transfer. However, as we will show, fluctuations inherent to this procedure
just compensate the effect of the the stochastic drift.

Since elementary stochastic spin flip processes directly translate into
changes of the magnetic moment, $|\delta m|=2$, it is more convenient to
switch to this state variable. The (equilibrium) probability to find a GFS in
bath 1 with magnetization $m_{1}$ before taking it out is (see Eq. (34) and
spin dynamics simulations in appendix C)

$\displaystyle p_{GFS}^{(m)}(m_{1})$ $\displaystyle=$ $\displaystyle
2\sqrt{\tfrac{J(\beta^{c}-\kappa)}{2\pi}}e^{-\tfrac{J}{2}(\beta^{c}-\kappa)m_{1}^{2}}\;.$
(43)

With $p(m_{2}|m_{1},t)$ as conditional probability for the transition
$m_{1}\to m_{2}$ after contact time $t$ in bath 2, its sequential probability
derives as $p(m_{2}|m_{1},t)p_{GFS}^{(m)}(m_{1})$. Exploiting conservation of
probability, and detailed balance makes the energy balance for many
repetitions of the above protocol vanish, since

$\displaystyle\langle\Delta\varepsilon(t)\rangle_{m_{2},m_{1}}$
$\displaystyle=$
$\displaystyle\langle\varepsilon_{2}(t)-\varepsilon_{1}\rangle_{m_{2},m_{1}}=-\langle
Q(t)\rangle$ (44) $\displaystyle=$
$\displaystyle\sum_{m_{2},m_{1}}p(m_{2}|m_{1},t)p_{GFS}^{(m)}(m_{1})\tfrac{1}{2}J(m_{2}^{2}-m_{1}^{2})$
(46) $\displaystyle=$
$\displaystyle\sum_{m_{2}}\tfrac{1}{2}Jm_{2}^{2}\underbrace{\sum_{m_{1}}p(m_{2}|m_{1},t)p_{GFS}^{(m)}(m_{1})}_{=p_{GFS}^{(m)}(m_{2}),\;{\text{detailed
balance}}}$ (48) $\displaystyle-$
$\displaystyle\sum_{m_{1}}\underbrace{\sum_{m_{2}}p(m_{2}|m_{1},t)}_{\begin{subarray}{c}=1\\\
{\text{conservation of
probability}}\end{subarray}}p_{GFS}^{(m)}(m_{1})\tfrac{1}{2}Jm_{1}^{2}$ (51)
$\displaystyle=$ $\displaystyle 0\;.$ (53)

Hence, though the single GFS of bath 1 is, by Boltzmann temperature, hotter
than the assembly in bath 2 ($\kappa<\beta^{(c)}=\beta_{\text{bath 2}}$),
fluctuations emerging from the distribution $p_{GFS}^{(m)}(m_{1})$ and the
transition dynamics $p(m_{2}|m_{1},t)$ impede net energy transfer between the
baths, independent from the contact time $t$. Therefore one can state, that
equal canonical temperatures predict absence of heat flow, despite a non-
vanishing gradient of Boltzmann temperatures.

To illustrate this formal derivation we assume that the stochastic transitions
between magnetic moments of a GFS in a bath result from Markovian spin flip
processes. Then, in the continuum limit, $\delta m\to 0$, Equation (43)
predicts dynamics of the magnetic moments as that of an Ornstein-Uhlenbeck
process, i.e. diffusion within the harmonic potential
$V(m)=\frac{1}{2}J(\beta^{(c)}-\kappa)m^{2}$. The probability density
555probability distributions on discrete magnetic moments $p(m)\;(\delta m=2)$
and density functions $\rho(m)\;(\delta m\to 0)$ in the continuum limit are
related by $p(m)=\int_{m-1}^{m+1}dm^{\prime}\rho(m^{\prime})$ evolves
according to the Smoluchowski Equation (see ref. Gardiner (1995) and appendix
C)

$\partial_{t}\rho(m,t)=D\;\partial_{m}\bigg{(}\partial_{m}-\underbrace{J(\kappa-\beta^{(c)})\;m}_{=-\partial_{m}V(m)}\bigg{)}\;\rho(m,t)\;,$
(54)

with diffusion coefficient $D$. The conditional probability density
$\rho(m_{2}|m_{1},t)$, i.e. Greens function, becomes with the dynamic constant
$\alpha=D\;J(\beta^{(c)}-\kappa)$ the Gaussian

$\displaystyle\rho(m_{2}|m_{1},t)$ $\displaystyle=$
$\displaystyle\tfrac{1}{\sqrt{2\pi}\sigma(t)}\exp\left(-\tfrac{(m_{2}-m_{\mu}(t))^{2}}{2\sigma^{2}(t)}\right),\;\text{with}$
(55) $\displaystyle m_{\mu}(t)$ $\displaystyle=$ $\displaystyle
m_{1}e^{-\alpha t},\;\text{and}\;\sigma^{2}(t)=\frac{1-e^{-2\alpha
t}}{J(\beta^{(c)}-\kappa)}$ (57)

as mode and variance, respectively. Hence, the most frequent magnetic moment
drifts from its initial position towards the ground states,
$m_{\mu}(0)=m_{1}\to 0$, (Fig. (1) and comparison of analytical results with
spin-dynamics simulations in appendix C), i.e. parallel to the gradient of
Boltzmann temperatures $\kappa<\beta_{bath}$ from the hot GFS ($\kappa$) to
the cooler bath ($\beta_{bath}=\beta^{(c)}$, see Eq. (28)).

Figure 2: Fluctuation mediated energy dissipation of a GFS in bath 2,
visualized by relaxation of the conditional probability $p(m_{2}|m_{1},t)$
(Eq. (57)) and corresponding energy balance
$\Delta\varepsilon_{m_{2}|m_{1}}(t)$ (Eq. (58)) for an initial moment
$m_{1}=50$ (a,b), and $m_{1}=20$ (c,d). Other parameters are as in Fig. (1)).
Gray/black and red/lightred in (a,c) imply energy uptake/release of the GFS.
According to Eq. (61) heat $Q$ is released to bath 2 for $m_{1}=50$ (b)
(negative areas under the curves and time course in insert) and taken up from
bath 2 for $m_{1}=20$ in (d) (positive area under the curve and insert).

The fraction of GFSs having undergone the transition $m_{1}\to m_{2}$ in bath
2, contribute to the energy balance with

$\Delta\varepsilon(t)_{m_{2}|m_{1}}=\rho(m_{2}|m_{1},t)\left(\tfrac{1}{2}Jm_{2}^{2}-\tfrac{1}{2}Jm_{1}^{2}\right)\;.$
(58)

The impact of fluctuations from transition dynamics on energy dissipation,
i.e. heat transfer from the GFS to bath 2, derives from Eq. (57) as
conditional average

$\displaystyle Q(t)$ $\displaystyle=$ $\displaystyle-\int
dm_{2}\;\Delta\varepsilon_{m_{2}|m_{1}}(t)=-\tfrac{J}{2}\left(\sigma^{2}(t)+m^{2}_{\mu}(t)-m_{1}^{2}\right)$
(59) $\displaystyle=$
$\displaystyle\underbrace{-\tfrac{J}{2}\left(\tfrac{1}{J(\beta^{C}-\kappa)}-m_{1}^{2}\right)}_{=\varepsilon(m_{1})-\langle\varepsilon\rangle_{GFS}}\left(1-e^{-2\alpha
t}\right)\;.$ (61)

Therefore the gradient between initial energy of the particular GFS taken out
from bath 1, $\varepsilon(m_{1})$, and the mean energy of GFSs in bath 2,
$\langle\varepsilon\rangle_{GFS}$, (Fig. (2)) is parallel to the direction of
heat flow. Depending on the initial energy $\varepsilon(m_{1})$, this flow may
be parallel or anti-parallel to the gradient of Boltzmann temperatures.

Additional effects of fluctuations originating from the equilibrium
distribution in bath 1 become evident by randomly picking GFSs out of it,
followed by insertion in bath 2. Then average energy balance vanishes,

$\langle\Delta\varepsilon(t)\rangle_{m_{2},m_{1}}=\int
dm_{2}dm_{1}\;\Delta\varepsilon_{m_{2}|m_{1}}(t)=0\;,$ (62)

i.e. no heat flows between baths at equal canonical temperatures. For
different canonical temperatures $\beta^{(c)}_{1}\neq\beta^{(c)}_{2}$ the
energy balance becomes

$\displaystyle\langle\Delta\varepsilon(t)\rangle_{m_{2},m_{1}}$
$\displaystyle=$
$\displaystyle\left(1-e^{-2\alpha_{2}t}\right)\tfrac{J}{2}\left(\tfrac{1}{J\beta^{(c)}_{2}-\kappa}-\tfrac{1}{J\beta^{(c)}_{1}-\kappa}\right)$
(63) $\displaystyle=$
$\displaystyle\left(\beta^{(c)}_{1}-\beta^{(c)}_{2}\right)\tfrac{1-e^{-2\alpha_{2}t}}{2(\beta^{(c)}_{2}-\kappa)(\beta^{(c)}_{1}-\kappa)}\;,$
(65)

i.e. heat $Q(t)=-\langle\Delta\varepsilon(t)\rangle_{m_{2},m_{1}}$ flows
parallel to the gradient of the canonical temperatures.

##  V Summary and conclusion

GFSs exhibit constant negative temperature, which makes them an attractive
paradigm to challenge our notion of thermodynamics. In contrast to
conventional paradigms for negative temperature, e.g. paramagnetic spin gases,
they are not subject for discussions referring to instability Ramsey (1956);
Romero-Rochín (2013); Struchtrup (2018). Neither do GFSs develop different
phases, as systems in Ref. Ramírez-Hernández  _et al._ (2008), which made them
instable when brought in weak thermal contact. A set of weakly coupled GFSs
is, with reference to Boltzmann temperature, cooler than the single GFS
itself, and may even take positive temperatures. If this set consists of many
GFSs, i.e. when this combined system may be considered as a heat bath, its
Boltzmann temperature becomes equivalent with the canonical temperature of the
GFS constituents, derived from an information technical approach. Hence,
Boltzmann - and canonical temperature of a GFS differ. This in-equivalence of
Boltzmann and canonical temperature of a GFS also implies that a GFS cannot be
assigned a Helmholtz free energy within its bath by a Legendre transform, as
can be done for normal, i.e. additive systems. Nevertheless, the associated
partition function reveals correctly the mean energy of a GFS.

The GFSs also differ from systems with long ranging interactions with ensemble
in-equivalence Baldovin  _et al._ (2021); Ramírez-Hernández  _et al._ (2008);
Baldovin (2018), including also those showing this in the negative temperature
domain Caglioti  _et al._ (1992); Smith and O’Neil (1990); Miceli  _et al._
(2019). This implies that fixing the energy (microcanonical ensemble) reveals
for an observable a different result than fixing at the corresponding
temperature (canonical ensemble). Often, a convex entropy function, i.e.
negative specific heat, implies thermodynamic instability for a combined
system and potentially coexistence of different phases. In contrast the
constant, i.e. energy independent, temperature of the GFS imply an indifferent
rather than unstable state of an assembly of GFSs, and different phases play
no role.

The difference of Boltzmann and canonical temperature of a GFS, i.e. the
difference of Boltzmann temperatures of a single GFS and that of a bath
consisting of many copies of this GFS, leads apparently to a conflict with the
2nd law of thermodynamics, because, at a first glance, this difference would
imply heat transport. However, the Boltzmann temperature gradient predicts
solely in which direction of energy uptake or release the most probables
state, i.e. the mode, of the combined system drifts. But fluctuations related
to the stochastic dynamics of energy exchange and the equilibrium distribution
of a GFS within its bath make heat flow parallel to the canonical temperature
gradient, which may be opposed to that of Boltzmann temperatures.

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##  Appendix A Number of states

The number of microstates, which an assembly of $M$ GFSs with magnetic moments
$\boldsymbol{m}=(m^{(1)}),\ldots,m^{(M)})^{T}$ comprises, is according to Eq.
(3) in the main text

$\displaystyle|\Sigma_{\boldsymbol{m}}|$ $\displaystyle=$
$\displaystyle\prod_{i=1}^{M}\binom{N}{\tfrac{N+m^{(i)}}{2}}\;.$ (A.1)

The energy of this assembly is given by the Hamiltonian

$H(\boldsymbol{m})=\frac{J}{2}\;\sum_{i=1}^{M}(m^{(i)})^{2}=\frac{J}{2}\;\|\boldsymbol{m}\|^{2}\;.$
(A.2)

with the Euclidean norm of the magnetic moments
$\|\boldsymbol{m}\|=\sqrt{m^{T}m}$. The number of microstates for some energy
$E$ is then the sum over the number of states of magnetic moments
$\boldsymbol{m}$, which fulfill $H(\boldsymbol{m})=E$, i.e.

$|\Sigma_{E}|=\sum_{\boldsymbol{m}}|\Sigma_{\boldsymbol{m}}|\;\text{rect}(2(\|\boldsymbol{m}\|-\sqrt{2E/J}))\;.$
(A.3)

Here $\text{rect}(x)$ denotes the boxcar function with $\text{rect}(x)=1$ if
$|x|\leq 1/2$, and otherwise zero, which takes into account the partitioning
of the space of magnetic moments.666This counting of states within an interval
of $\Delta m=2$ around some moment $\|m\|=\sqrt{2E/J}\pm 1$ is slightly
different from the density of energy states, which is often taken as the base
for determining Boltzmann entropy. This density would be obtained by counting
states within some fixed energy interval. Our counting algorithm, which aims
at the number of microstates for some energy $E=J/2\;\|m\|^{2}$, exactly
reveals the correct number of states for $M=1$, i.e. the binomials of Ea.
(A.1), and is rigorously extended to $M>1$. Of note is that both approaches
lead to the same result for larger $M$, as these relative difference of these
differently defined entropies scales with $1/M$.

The binomials determining the number of microstates in Eq. (A.1) may be
approximated by Gaussians,

$|\Sigma_{\boldsymbol{m}}|\approx\left(\tfrac{1}{\sqrt{2\pi
N}}\;2^{(N+1)}\right)^{M}\;\underbrace{e^{-\frac{1}{N}\;\frac{1}{2}\|\boldsymbol{m}\|^{2}}}_{=e^{\kappa
H(\boldsymbol{m})}}\;,$ (A.4)

which makes Eq. (A.3)

$|\Sigma_{E}|\approx e^{\kappa
E}\;\sum_{\boldsymbol{m}}\text{rect}(2(\|\boldsymbol{m}\|-\sqrt{2E/J}))\;.$
(A.5)

The sum is proportional to the surface ${\cal{A}}_{M}$ of the $M-$dimensional
hypersphere with radius $m_{M}=\sqrt{2E/J}$,

${\cal{A}}_{M}={\cal{S}}_{M}\;(2E/J)^{\frac{M-1}{2}}\;$ (A.6)

with

${\cal{S}}_{M}=2\pi^{M/2}/\Gamma(M/2)\;,$ (A.7)

as the surface of the unit hypersphere. As magnetic moments are partitioned in
units of 2 (one spin flip process), the $M-$dimensional space of magnetic
moments is partitioned in cubic voxels of volume $2^{M}$. The magnetic moments
with energy $E$ are then the voxels found within a spherical shell, the width
of which is $2$ (see boxcar function in Eq. (A.3) and surface given by Eq.
(A.6). This surface itself is partitioned by the facet of the $M-$dimensional
voxel, i.e. the $M-1$ dimensional squares with sidelength $2$. Therefore the
number of realizations of magnetic moments is

$\displaystyle\sum_{\boldsymbol{m}}\text{rect}(\|\boldsymbol{m}\|-\sqrt{2E/J})=\frac{{\cal{A}}_{M}}{2^{M-1}}\;,$
(A.8) (A.9)
$\displaystyle={\cal{S}}_{M}\;(2E/J)^{\frac{M-1}{2}}\;2^{-(M-1)}\;.$ (A.10)

Hence, with this Equation the number of microstates of the assembly of GFSs
with energy $E$ in Eq. (A.3) becomes in the Gaussian approximation (Eq. (A.5))

$\displaystyle|\Sigma_{E}|$ $\displaystyle\approx$
$\displaystyle{\cal{S}}_{M}\;(2E/J)^{\frac{M-1}{2}}2^{-(M-1)}\left(\tfrac{1}{\sqrt{2\pi
N}}\;2^{(N+1)}\right)^{M}\;e^{\kappa E}$ (A.11) $\displaystyle=$
$\displaystyle{\cal{S}}_{M}\;(E/J)^{\frac{M-1}{2}}\tfrac{2^{\frac{M-1}{2}}}{2^{(M-1)}}\left(\tfrac{1}{\sqrt{\pi
N}}\;2^{N}\sqrt{2})\right)^{M}\;e^{\kappa E}$ (A.13) $\displaystyle=$
$\displaystyle{\cal{S}}_{M}\sqrt{2}\;\left(\tfrac{2^{N}}{\sqrt{\pi
N}}\right)^{M}(E/J)^{\frac{M-1}{2}}e^{\kappa E}\;.$ (A.15)

The quality of this Gaussian based approximation compared with direct counting
of microstates in Eq. (A.3) is shown in Fig. (A.1).

Figure A.1: Entropy $S=\ln(|\Sigma_{E}|$ vs. energy $E$, expressed by the norm
of the magnetic moment $m_{M}=\sqrt{2E/J}$, of an assembly of
$M=1,\;2,\;{\text{and}}\;4$ GFSs, each consisting of $N=1000$ spins. Compared
are the entropy obtained by counting the number of states from Eq. (A.3) (blue
points) with the Gaussian based approximation in Eq. (A.15) (red line). The
energy where the inverse temperature $\beta_{M}=\partial_{E}S$ vanishes (see
Eq. (15) in main text), i.e. at $m_{M,\beta_{M}=0}=\sqrt{(M-1)N}\approx
0\;(M=1),\;45\;(M=2),\;54\;(M=4)$ is shown by the red disk.

##  Appendix B Canonical distribution derived from information theory

The information theory related ansatz focuses on how microstates $\sigma$ are
distributed within a large assembly of GFSs. The microstate of the latter
derives from the individual microstates of its $M$ constituents,
$\sigma^{(i)},\;i=1,\ldots,M$, as
$\boldsymbol{\sigma}=(\sigma^{(1)},\ldots,\sigma^{(M)})^{t}$, under the
constraint of energy conservation, $\sum_{i=1}^{M}H^{(i)}(\sigma^{(i)})=E$.
The relative distribution frequency of the microstates is

$\displaystyle\varpi(\sigma)$ $\displaystyle=$ $\displaystyle
n(\sigma)/M\;,\text{with}$ (B.1) $\displaystyle n(\sigma)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{M}\delta_{\sigma,\sigma^{(i)}}\;,$ (B.2)

as the absolute frequencies.

The degeneracy of a given distribution $\varpi(\sigma)$, i,e, the number of
microstates $\boldsymbol{\sigma}$ which distribute accordingly, derives from
combinatorics as

${\cal{D}}[\varpi]=M!/\prod_{\sigma}n(\sigma)!\;.$ (B.3)

The logarithm of this degeneracy is proportional to the information one gains
from knowing the distribution to knowing its micro state variable,
$\varpi(\sigma)\to\boldsymbol{\sigma}$, i.e. knowing the distribution is less
informative than the knowledge of the microstate. As equipartition predicts
equal probability of all microstates $\boldsymbol{\sigma}$, which fulfill
energy conservation, the degeneracy of a distribution is proportional to the
probability of its occurence, $w[\varpi]$. Applying Stirling formula to Eq.
(B.3), yields the probability to find the distribution $\varpi(\sigma)$ as

$\displaystyle w[\varpi]$ $\displaystyle\sim$
$\displaystyle{\cal{D}}[\varpi]=\exp(MS_{Sh})\;,\;\text{with}$ (B.4)
$\displaystyle S_{Sh}[\varpi]$ $\displaystyle=$
$\displaystyle-\sum_{\sigma}\varpi(\sigma)\ln(\varpi(\sigma))\;$ (B.5)

as Shannon entropy.

##  Appendix C Simulation of spin dynamics of a single GFS

###  C.1 General aspects and determination of equilibrium probability of a GFS
within a heat bath

Figure C.1: Simulated stochastic trajectory in the space of magnetic moments
$m$ for $t/\Delta t=10^{6}$ time steps. Starting point was at $m(0)=0$. Like
in the example in the main text the GFS consists of $N=1000$ spins, i.e. its
inverse temperature is $\kappa=-1/1000\;J^{-1}$, and the inverse temperature
of the bath $\beta_{bath}$ was adjusted to a value that the mean energy of the
GFS within is
$\langle\varepsilon\rangle_{GFS}=\tfrac{1}{2}\;(\beta_{bath}-\kappa)^{-1}=\tfrac{J}{2}\langle
m_{2}^{2}\rangle=\tfrac{J}{2}\;30^{2}$. Figure C.2: Comparison between
equilibrium distribution $p^{(m)}_{GFS}$ of the magnetic moment obtained from
the simulated stochastic trajectory in Fig. (C.1) (relative frequency of
visits in brown histogram) and the canonical distribution (dotted line)
obtained from $\exp(-1/2\;J(\beta^{(c)}-\kappa)m^{2})$ by discretization of
magnetic moments in intervals of $|\delta m|=2$ (see main text). Note that
canonical and bath temperature are equivalent, $\beta_{bath}=\beta^{(c)}$ (see
main text).

We assume that the microstate of a GFS $\sigma=(s_{1},\ldots,s_{N})^{T}$ with
spin orientations $s_{i}=\pm 1$ is subjected to stochastic forces originating
from the heat bath with inverse temperature $\beta_{bath}$. These forces
induce with some probability $p$ after some time $\Delta t$ the flipping of a
single arbitrary spin $s_{i}\to-s_{i}$. The dipole moment $m=\sum_{i}s_{i}$
changes by $\delta m=-2s_{i}$, and energy

$\displaystyle\Delta E$ $\displaystyle=$ $\displaystyle\tfrac{1}{2}J\;\delta
m^{2}+J\;m\;\delta m$ (C.1) $\displaystyle=$ $\displaystyle 2J(1-s_{i}m)\;.$
(C.2)

Note that the in- or decrement of the magnetic moment is $|\delta m|=2$.
According to the Glauber algorithm Glauber (1963) the probability that spin
$i$ flips is

$\displaystyle p_{s_{i}\to-s_{i}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\;\frac{\exp(-\beta_{bath}\Delta
E/2)}{\cosh(\;\beta_{bath}\Delta E/2\;)}$ (C.3) $\displaystyle=$
$\displaystyle\frac{1}{1+\exp(\beta_{bath}\Delta E)}\;,$ (C.5)

i.e. vice vera with probability $1-p_{s_{i}\to-s_{i}}$ the GFS remains in its
state $\sigma$. Translation of this Markov process from the space of
microstates $\sigma$ into that of magnetic moments $m$ requires that the
random selection after $\Delta t$ of some arbitrary spin $i$ as a flip
candidate is replaced by random selection between a spin down or up
polarization. To be consistent with the arbitrary selection of a spin this
random choice is given by the fraction of the respective sorts, i.e. $(N\mp
m)/(2N)$ to select a down or up spin, respectively.

For simulation of the stochastic trajectory of the magnetic moment of the GFS,
first the polarization was randomly assessed, which determined whether
magnetization changed potentially by $\delta m=2$ if down -, or $\delta m=-2$
if up polarization was chosen, respectively. The Glauber’s transition
probability in Eq. (C.5) then determines, whether the spin flip process, i.e.
change in magnetization, was realized or not. Figure (C.1) shows a stochastic
trajectory over a long period of time ($10^{6}\Delta t$), which visits
magnetic moments with equilibrium probability derived from Eq. (43) in the
main text (Fig. (C.2)) .

###  C.2 Relaxation of a GFS within a heat bath

The above considerations are easily transferred to simulate the relaxation of
a GFS with a given initial energy/magnetic moment within a heat bath. From
some starting magnetic moment stochastic trajectories are followed and
sampling at points in time reveals the assigned probability distribution. To
evaluate the dynamics of relaxation it is useful to derive the Smoluchowski
diffusion equation applied in the main text from the simulation approach.

The probability that the magnetic moment of the GFS undergoes a stochastic
transition to its neighbor value $m\to m\pm|\delta m|$ after $\Delta t$ is
given by Eq. (C.6) and considerations above about selection of spin
polarization,

$w_{m\to m\pm|\delta m|}=\underbrace{\frac{N\mp
m}{2N}}_{\begin{subarray}{c}\text{fraction of}\\\ \text{down/up spins
}\end{subarray}}\;\underbrace{\frac{1}{1+\exp\left(\beta_{bath}\Delta E_{m\to
m\pm|\delta m|}\right)}}_{\text{Glauber factor}}\;,$ (C.6)

with $\Delta E_{m\to m\pm|\delta m|}=E_{m\pm|\delta m|}-E_{m}=J(1/2\delta
m^{2}\pm m|\delta m|)$ as the energy difference between final and inital
state. Similarly the transition probability from the neighboring moment values
to $m$, i.e. $m\pm|\delta m|\to m$, are obtained. The corresponding hopping
rates between nearest neighbors are given by $w/\Delta t$. The evolution of
the probability distribution $p(m,t)$ may be written in form of a master
equation with the rates given above,

$\displaystyle\frac{\Delta p(m,t)}{\Delta t}=$ $\displaystyle\frac{1}{\Delta
t}\bigg{(}w_{m+|\delta m|\to m}\;\;p(m+|\delta m|,t)+w_{m-|\delta m|\to
m}\;\;p(m-|\delta m|,t)$ (C.7) $\displaystyle-$ $\displaystyle(w_{m\to
m+|\delta m|}+w_{m\to m-|\delta m|})\;\;p(m,t)\bigg{)}.$ (C.9)

To approximate the above Equation in the continuum limit, i.e. $|\delta m|\to
0$, $\Delta t\to 0$ and replacement of the discrete probability $p$ by the
density $\rho$, one must consider, that variations of the magnetic moment are
partitioned in units of 2. Hence, the appropriate differentials are $\delta
m\to 2\;dm$ and $\Delta t\to dt$. Inserting into Eq. (C.9) and expanding the
terms on the right to the second order $(dm)^{2}$ makes the Master Equation a
Smoluchowski diffusion equation where harmonic forces
$F(m)=-\partial_{m}\left(\tfrac{1}{2}J\;(\beta_{bath}-\kappa)m^{2}\right)$
determine the drift term, i.e. with $\kappa=-1/(JN)$ as the negative
temperature of the GFS, and diffusion coefficient $D$

$\partial_{t}\rho(m,t)=\underbrace{\frac{(dm)^{2}}{dt}}_{=D}\bigg{(}\partial_{m}^{2}\rho(m,t)-\partial_{m}\bigg{(}\underbrace{-J\;\left(\beta_{bath}-\kappa\right)m}_{=F(m)}\;\rho(m,t)\bigg{)}\bigg{)}\;.$
(C.10)

The Greens function, i.e. evolution of the conditional probability density of
this Equation, is a shifting Gaussian, starting as a delta distribution
($t=0$) and asymptotically ($t\to\infty$) ending as the Gaussian equilibrium
density (see main text). The discrete probability distribution is regained
from density by $p(m)=\int_{m-1}^{m+1}d\tilde{m}\;\rho(\tilde{m})$. Figure
(C.3) shows the congruence of relaxation of probability distributions obtained
from spin dynamic simulations and the analytical solution (see main text).

Figure C.3: Comparison of spin dynamic simulations with analytical results
from Eq. (C.10) for relaxation of the distribution of the magnetic moment
within a heat bath. Consistently with Glauber dynamics the diffusion constant
$D$ was chosen unity as well as interaction energy $J$. Parameters of the GFS
as spin number $N=1000$, and, hence, its inverse temperature $\kappa=1/(JN)$
and that of the heat bath $\beta_{bath}=1/30^{2}+\kappa=0.\bar{1}\times
10^{-3}$ are that of Figs. (C.1,,C.2) and the example in the main text. 8000
trajectories were simulated starting at $m_{1}=50$ and followed over 4500 time
steps. Four points in time are shown: $t=0,\;300,\;900,\text{and}\;4500$,
referring to the green, red, blue and grey histograms/curves. With the dynamic
constant $\alpha=DJ(\beta_{bath}-\kappa)=1/900$, which determines speed of
relaxation, this refers to $\alpha t=0,\;1/3,\;1,\text{and}\;5$ (see main
text).

##  Appendix D Binomial versus Gaussian approximation for determination of the
cumulant generating function

Figure D.1: Cumulant generating functions,
$\Phi(\beta^{(c)}=-\ln(Z(\beta^{(c)}))$, with partition functions $Z$ (a) and
mean energy $\langle\varepsilon\rangle_{GFS}$ (b) of a GFS vs. inverse
canonical temperature $\beta^{(c)}$ of the bath. The right chart in each row
is a magnified section of the left for small inverse temperatures. The
partition function is obtained from Eqs. (38) either, - with the exact
binomial expression of entropy,
$\exp(S(\varepsilon(m))=|\Sigma_{m}|=\binom{N}{\tfrac{N+m}{2}}$ of Eq. (3)
(green), - its Gaussian approximation in Eq. (39) (red dotted), - and the
integral approximation of the latter sum in Eq. (40) (black). Note that the
integral approximation implies $\sum_{m}f(m)\to 1/2\int dmf(m)$ as the
magnetic moment is partitioned in units of $\delta m=2$. As in the previous
examples the number of spins of a GFS is $N=1000$, and hence its inverse
temperature $\kappa=-1/1000J^{-1}$. Mean energies were either obtained by
numerical differentiation
$\langle\varepsilon\rangle_{GFS}=\partial_{\beta^{(c)}}\Phi=-\partial_{\beta^{(c)}}\ln(Z(\beta^{(c)}))$
for the binomial and Gaussian partition sum, or by Eq. (42) for the integral
approximation of the Gaussian sum.

The validity of the Gaussian approximation of the exact, binomial distributed
number of microstates is tested for determination of the partition sum (Eq.
(38)) and thereof derived cumulant generating function and and mean energy. In
addition the integral approximation of the Gaussian partition sum in Eq. (40)
is investigated. Figure (D.1) shows congruent curves of the exact (green)
function, its Gaussian approximation (red dotted) and the integral
approximation of the latter (black).