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# Electron pairs bound by the spin-orbit interaction in 2D gated Rashba
materials
with two-band spectrum

Yasha Gindikin  W.I. Fine Theoretical Physics Institute, University of
Minnesota, Minneapolis, MN 55455, USA  Igor Rozhansky  Department of Condensed
Matter Physics, Weizmann Institute of Science, Rehovot, 76100, Israel
Vladimir A. Sablikov  Kotelnikov Institute of Radio Engineering and
Electronics, Russian Academy of Sciences, Fryazino branch, Fryazino, 141190,
Russia

###### Abstract

We show that the bound electron pairs (BEPs) emerge in two-dimensional gated
Rashba materials owing to the interplay of the pair spin-orbit interaction,
produced by the Coulomb fields of interacting electrons, and the peculiarities
of the band structure giving rise to a negative reduced mass of the
interacting electrons. Our consideration is based on the four-band Bernevig-
Hughes-Zhang model with the Rashba spin-orbit interaction created by the
charges on the gate. The binding energy of the BEP varies with the gate
voltage in a wide range across the entire width of the two-particle energy
gap. Although the spin-orbit interaction destroys the spin quantization, the
BEPs have a magnetic moment, which is created mainly by the orbital motion of
the electrons and tuned by the gate voltage.

##  I Introduction

In recent years a great deal of attention is paid to the effects of electron-
electron (e-e) interaction in materials with non-trivial band states [1]. Of
special interest is its interplay with the spin-orbit interaction (SOI) [2, 3,
4, 5, 6], which under certain circumstances can lead to the purely electronic
mechanisms of electron pairing [7]. Studying these mechanisms is important, of
course, not only because of problems of high-temperature superconductivity
[8], but also because exploring the formation of the bound electron pairs
(BEPs) and, generally, few-electron complexes [9, 10] lays the ground for
understanding a more intricate problem of the correlated many-electron state.
The formation of BEPs with the energy lying in the band gap of the many-band
material is also interesting because it gives rise to unconventional transport
of charge and spin even in the absence of band conductivity.

The BEPs have been studied in graphene and bigraphene [11, 12], in flat-band
systems [13, 14], in Dirac semimetals [15], in topological insulators [16], to
name just a few. Electron pairing is often attributed to the peculiarities of
the band structure and, in particular, to the formation of the negative
reduced mass, which governs the relative motion of the electrons in the pair.
In this case even the Coulomb repulsive interaction between the electrons can
bind them into a pair [17].

Another pairing mechanism recently discovered is based on the pair spin-orbit
interaction (PSOI). This is the e-e interaction component that depends on the
spin and momentum of the electrons, which is produced by the Coulomb fields of
interacting electrons in Rashba materials (Ref. [18] is a recent review of the
subject). The PSOI is attractive for the electrons in particular spin
configurations tied to their momentum, and competes with the Coulomb repulsion
of electrons. Under certain conditions, attainable in materials with the giant
Rashba SOI, the PSOI prevails, which leads to the formation of the BEPs [19,
20, 21]. To date, the BEP formation due to the PSOI was explored within the
conduction-band approximation.

This makes it impossible to study the BEPs with high binding energies
comparable to the band gap, which is usually small in materials with a very
strong SOI. However, of greatest interest are the strongly bound pairs
considered as composite particles carrying charge and spin. For this reason,
the theory of the electron pairing due to the PSOI must be generalized to the
multi-band models that properly describe the SOI originating from the
hybridization of the atomic orbitals with different spin and orbital
structures. This is even more so for the effective reduced mass of an electron
pair, the sign and magnitude of which in a multi-band system are determined by
the mixture of atomic orbitals involved in the model, which should be found
self-consistently from the solution of the equation of motion [16].

The goal of the present paper is to explore the energy spectrum and magnetic
properties of the BEPs arising in two-dimensional (2D) systems with the giant
Rashba effect. The particular mechanisms behind the giant SOI can be quite
different in such systems as the heterostructures based on
$\mathrm{LaAlO_{3}/SrTiO_{3}}$ [22, 23], the monolayers of $\mathrm{BiSb}$
[24], the specifically crafted structures based on 2D layers of van der Waals
materials with heavy adatoms [25, 26], and many others [2], but in a single-
band approximation the main effect of the SOI is captured by a widely used SOI
Hamiltonian linear in the wave vector [27]. On the contrary, in a multi-band
situation the Rashba SOI effect is described by a specific model for each
mechanism.

We base our study on the Bernevig–Hughes–Zhang (BHZ) model [28], which is well
established to describe many quasi-2D systems in both topological and trivial
phases. However, in this model, Rashba SOI is due to the $sp^{3}$ band
hybridization mechanism, which as we found results in a PSOI not strong enough
to reliably overcome the Coulomb interaction [21, 29]. Nonetheless, the
situation is significantly improved in the presence of the SOI produced by the
electric field of the gate, which makes it easier to achieve this condition.

Therefore we will study the BEPs in a 2D material described by the BHZ model
with the structure-inversion-asymmetry (SIA) Rashba SOI created by a charged
metallic gate. The gate plays several important roles here. The electrons
induce the image charges on the gate, the normal electric field of which
contributes to the PSOI [30, 31, 21]. The external voltage applied to the gate
creates one-particle Rashba SOI that enhances the electron pairing. And
finally, the gate screens the Coulomb e-e interaction, which affects the
balance between the PSOI and Coulomb interaction by changing the spatial
profile of the effective binding potential.

The SOI is produced by both normal and tangential components of the electric
field in the layer. The in-plane electric field is taken by the BHZ model
exactly within the $sp^{3}$ hybridization scheme, whereas the SOI due to the
normal electric field is accounted for in the generalization of the original
BHZ model [32].

An important point is that the presence of the SOI produced by the electric
field due to the gate totally destroys the spin quantization. As a result no
spin projection is well defined. The only conserving quantity at a given
energy is the normal projection of the total angular moment which is quantized
as $J_{z}=2m\hbar$, $m\in\mathbb{Z}$. If $J_{z}$ is nonzero, one can expect
the magnetic moment of the BEP to be also nonzero. Considering the BEPs as the
composite particles, it is of significant interest to find out the magnitude
of the magnetic moment they carry, and how it depends on the energy of the BEP
and the system parameters. The predicted non-trivial behavior of the orbital
magnetization associated with the BEPs could be probed using the recently
emergent state-of-the-art experimental techniques [33].

We have shown that the interplay of the PSOI and the peculiarities of the
subband structure of the electronic states gives rise to a purely electronic
pairing mechanism. The energy level of the resulting BEPs varies with the gate
voltage in a wide range across the entire width of the two-particle energy
gap. Even though the SOI destroys the spin quantization, BEPs do possess a
magnetic moment, which is created mainly by the orbital motion of the
electrons and is tunable by the gate voltage.

##  II Model and Results

The BHZ model is formulated in the four-band basis
$\mathcal{B}={(\lvert\operatorname{E\\!\uparrow}\rangle,\lvert\operatorname{H\\!\uparrow}\rangle,\lvert\operatorname{E\\!\downarrow}\rangle,\lvert\operatorname{H\\!\downarrow}\rangle)}^{\intercal}$.
The states $\lvert\operatorname{E\\!\uparrow}\rangle$ and
$\lvert\operatorname{E\\!\downarrow}\rangle$ are composed of the electron- and
light-hole band states with the angular momentum projection of $m_{J}=\pm
1/2$, whereas $\lvert\operatorname{H\\!\uparrow}\rangle$ and
$\lvert\operatorname{H\\!\downarrow}\rangle$ are the heavy-hole states with
$m_{J}=\pm 3/2$. The one-body Hamiltonian that governs the envelope wave-
function is [32]:

$\hat{H}=\begin{bmatrix}M+B_{1}k^{2}&Ak_{+}&-i\frac{e\xi}{2}\\{\mathcal{F},k_{-}\\}&0\\\
Ak_{-}&-M-B_{2}k^{2}&0&0\\\
i\frac{e\xi}{2}\\{\mathcal{F},k_{+}\\}&0&M+B_{1}k^{2}&-Ak_{-}\\\
0&0&-Ak_{+}&-M-B_{2}k^{2}\end{bmatrix}\,.$ (1)

Here $M$ is the mass or gap parameter, $B_{1}$ and $B_{2}$ are the dispersion
curvatures in the electron-like and hole-like bands, $A$ defines the band
hybridization, and $\xi$ is a material-dependent Rashba constant. Then, $k$ is
the momentum operator, $k_{\pm}=k_{x}\pm ik_{y}$, and curly brackets stand for
anti-commutator to ensure the hermiticity for the case of the non-uniform
electric field. The normal electric field $\mathcal{F}$ produces the Rashba
SOI that couples only the electron-like states. The formation of the bound
electron pairs proves highly sensitive to the inversion of the electron-like
and hole-like bands. In what follows, we consider only the trivial phase.

In the absence of the SOI, the single-particle spectrum has an energy gap at
$(-M,M)$. With Rashba SOI present, the gap extends from the top of the valence
band at $\varepsilon=-M$ to the bottom of the conduction band, which is
lowered down by the SOI. Things change as we go to the two-body problem [16].

The non-interacting two-body Hamiltonian is constructed as a Kronecker sum
$H_{\mathrm{free}}=H_{1}\oplus H_{2}$ of the one-body terms of Eq. (1). For
electron pairs with zero total momentum, the gap in the two-body spectrum
belongs either to the lower (for $B_{1}>B_{2}$) or to the upper (for
$B_{2}>B_{1}$) half of the energy interval of $(-2M,2M)$. The position and
width of the gap depends on the Rashba SOI magnitude. The continuum boundaries
are plotted by the dashed lines for both cases correspondingly in Fig. 1 and
Fig. 2. The energies of the BEPs lie exactly within the gap.

The Hamiltonian of two interacting electrons is obtained by adding the Coulomb
interaction energy $\mathcal{U}$ to the free Hamiltonian,

$H=H_{1}\oplus H_{2}+\mathcal{U}(\bm{r}_{1}-\bm{r}_{2})I_{16}\,,$ (2)

and by recognizing the fact that the electric field $\bm{\mathcal{E}}(\bm{r})$
of the Coulomb interaction between the electrons depends on their relative in-
plane position $\bm{r}\equiv\bm{r}_{1}-\bm{r}_{2}$, which gives rise to the
PSOI [18].

The model system we consider consists of a 2D layer of Rashba material
proximitized by a charged metallic gate. The gate affects the electron-
electron interaction potential

$\mathcal{U}(r)=\frac{e^{2}}{r}-\frac{e^{2}}{\sqrt{r^{2}+a^{2}}}\,,$ (3)

$a/2$ being the distance to the gate. The contribution of the in-plane
electric field component
$\bm{\mathcal{E}}_{\parallel}(\bm{r})=\frac{1}{e}\grad_{\bm{r}}\mathcal{U}(r)$
to the PSOI is exactly taken into account within the BHZ model via the
hybridization of the electron-like and hole-like bands, similarly to the one-
body SOI produced by an external in-plane electric field [32].

The presence of the gate also gives rise to the normal component of the
electric field $\mathcal{F}(r)\equiv\mathcal{E}_{\perp}(r)+F$ acting on each
of the interacting electrons. It has two terms. The component from the other
electron’s image

$\mathcal{E}_{\perp}(r)=\frac{ea}{{(r^{2}+a^{2})}^{3/2}}\,,$ (4)

which depends on the relative distance between the electrons, contributes to
the PSOI [30, 21]. The uniform component of $e/a^{2}$ from each electron’s own
image sums up with the normal field from the net charge of the gate to give
the total field $F$, which produces the one-body Rashba SOI.

The two-electron basis is the direct product of
$\mathcal{B}_{1}\otimes\mathcal{B}_{2}$. Explicitly, it is written as

$\begin{split}\mathcal{B}_{1}\otimes\mathcal{B}_{2}\equiv&(\lvert\operatorname{E\\!\uparrow}\operatorname{E\\!\uparrow}\rangle,\lvert\operatorname{E\\!\uparrow}\operatorname{H\\!\uparrow}\rangle,\lvert\operatorname{E\\!\uparrow}\operatorname{E\\!\downarrow}\rangle,\lvert\operatorname{E\\!\uparrow}\operatorname{H\\!\downarrow}\rangle,\\\
&\lvert\operatorname{H\\!\uparrow}\operatorname{E\\!\uparrow}\rangle,\lvert\operatorname{H\\!\uparrow}\operatorname{H\\!\uparrow}\rangle,\lvert\operatorname{H\\!\uparrow}\operatorname{E\\!\downarrow}\rangle,\lvert\operatorname{H\\!\uparrow}\operatorname{H\\!\downarrow}\rangle,\\\
&\lvert\operatorname{E\\!\downarrow}\operatorname{E\\!\uparrow}\rangle,\lvert\operatorname{E\\!\downarrow}\operatorname{H\\!\uparrow}\rangle,\lvert\operatorname{E\\!\downarrow}\operatorname{E\\!\downarrow}\rangle,\lvert\operatorname{E\\!\downarrow}\operatorname{H\\!\downarrow}\rangle,\\\
&\lvert\operatorname{H\\!\downarrow}\operatorname{E\\!\uparrow}\rangle,\lvert\operatorname{H\\!\downarrow}\operatorname{H\\!\uparrow}\rangle,\lvert\operatorname{H\\!\downarrow}\operatorname{E\\!\downarrow}\rangle,{\lvert\operatorname{H\\!\downarrow}\operatorname{H\\!\downarrow}\rangle)}^{\intercal}\,.\end{split}$
(5)

###  II.1 BEP wave function

For simplicity, we restrict our consideration to a particular case of the
electron pairs with zero total momentum, although it is known that the center-
of-mass motion of the pair can also lead to electron pairing in Rashba
materials [31, 19]. Such BEPs are formed by two electrons orbiting around
their common barycenter.

In this case, the two-electron wave-function, which represents the spinor of
the 16th rank, depends only on the relative distance
$\mathbf{r}\equiv(r,\varphi)$ between the electrons. The eigenfunctions of the
Hamiltonian of Eq. (2) are also the eigenfunctions of the total angular
momentum along the $z$ direction $\hat{J}_{z}$, which leads to the following
form of the wave-function [16]:

$\begin{split}\Psi_{m}=e^{2im\varphi}&(\psi_{1}(r)e^{-i\varphi},i\psi_{2}(r)e^{-2i\varphi},\psi_{3}(r),i\psi_{4}(r)e^{i\varphi},\\\
&-i\psi_{2}(r)e^{-2i\varphi},\psi_{5}(r)e^{-3i\varphi},i\psi_{6}(r)e^{-i\varphi},\psi_{7}(r),\\\
&-\psi_{3}(r),i\psi_{6}(r)e^{-i\varphi},\psi_{8}(r)e^{i\varphi},-i\psi_{9}(r)e^{2i\varphi},\\\
&i\psi_{4}(r)e^{i\varphi},-\psi_{7}(r),i\psi_{9}(r)e^{2i\varphi},{\psi_{10}(r)e^{3i\varphi})}^{\intercal}\,,\end{split}$
(6)

with $m\in\mathbb{Z}$ being the angular quantum number. Note that the spinor
contains 10 independent components instead of 16 that would be consistent with
the rank of the two-particle Hamiltonian. This is a result of the wave-
function anti-symmetry with respect to the electron permutation. The
eigenvalue of $J_{z}$ is $2m\hbar$. Even values of $2m$ ensure that the wave-
function is anti-symmetric under the electron permutation, which is
accompanied by $\varphi\to\varphi+\pi$.

The spinor components satisfy the system of coupled equations of Eq. (11). The
two-body problem in Rashba materials allows for two seemingly different
mechanisms of electron pairing. First, electron pairs may arise bound by the
Coulomb interaction because of the negative reduced effective mass of two
electrons formed by the hybridization of the electron-like and hole-like bands
[16, 34, 35, 36]. On the other hand, the BEPs can be formed due to the PSOI,
which proves to be attractive for a particular spin configuration of the
electrons. This division is rather arbitrary in the multi-band model, where
the BEPs are formed as a result of the interplay of both mechanisms.

The BEPs energy levels are doubly degenerate with respect to the sign of $m$,
except for the state with $m=0$, which is non-degenerate. The components of
the spinors $\Psi_{m,n_{r}}$ and $\Psi_{-m,n_{r}}$, where $n_{r}\in\mathbb{N}$
stands for the radial quantum number, feature a certain cross-symmetry, which
is a direct consequence of the time-reversal invariance in a bound pair
composed of electrons with opposite momenta:

$\displaystyle\Psi_{m,n_{r}}:$ $\displaystyle\Psi_{-m,n_{r}}:$
$\displaystyle(\lvert\operatorname{E\\!\uparrow}\operatorname{E\\!\uparrow}\rangle)$
$\displaystyle\psi_{1}(r)$ $\displaystyle=\psi_{8}(r)$
$\displaystyle(\lvert\operatorname{E\\!\downarrow}\operatorname{E\\!\downarrow}\rangle)$
$\displaystyle(\lvert\operatorname{E\\!\uparrow}\operatorname{H\\!\uparrow}\rangle)$
$\displaystyle\psi_{2}(r)$ $\displaystyle=\psi_{9}(r)$
$\displaystyle(\lvert\operatorname{H\\!\downarrow}\operatorname{E\\!\downarrow}\rangle)$
(7)
$\displaystyle(\lvert\operatorname{E\\!\uparrow}\operatorname{H\\!\downarrow}\rangle)$
$\displaystyle\psi_{4}(r)$ $\displaystyle=\psi_{6}(r)$
$\displaystyle(\lvert\operatorname{H\\!\uparrow}\operatorname{E\\!\downarrow}\rangle)$
$\displaystyle(\lvert\operatorname{H\\!\uparrow}\operatorname{H\\!\uparrow}\rangle)$
$\displaystyle\psi_{5}(r)$ $\displaystyle=\psi_{10}(r)$
$\displaystyle(\lvert\operatorname{H\\!\downarrow}\operatorname{H\\!\downarrow}\rangle)$

These equations are also valid for $m=0$. In this case they define the
relations between the different components of the spinor $\Psi_{m=0,n_{r}}$.

Figure 1:  The energy level of the BEPs (solid line) as a function of the gate
electric field $F$ within the energy gap of the continuum of unbound states.
The system parameters are
$B_{1}=$120\text{\,}{\mathrm{\text{\r{A}}}^{2}}\mathrm{e}\mathrm{V}$$ and
$B_{2}=$20\text{\,}{\mathrm{\text{\r{A}}}^{2}}\mathrm{e}\mathrm{V}$$. The
continuum boundaries are shown by the dashed lines. Figure 2:  Two energy
levels of the bound electron pairs (green and blue solid lines) as a function
of the gate electric field $F$. The continuum boundaries are shown by the
dashed lines. The system parameters
$B_{1}=$20\text{\,}{\mathrm{\text{\r{A}}}^{2}}\mathrm{e}\mathrm{V}$$ and
$B_{2}=$120\text{\,}{\mathrm{\text{\r{A}}}^{2}}\mathrm{e}\mathrm{V}$$ are
inverted as compared to Fig. 1.

###  II.2 BEP spectrum

The eigenvalue problem of Eq. (11) is treated numerically. The system spectrum
is found by the finite elements method [37] using the Arnoldi eigenvalue
solver [38]. The method of choice to find all the eigenvalues in the energy
gap would be the FEAST solver [39], were it not for the lack of support for
non-hermitian problems in its early version currently used in the Intel Math
Kernel Library.

We choose the system parameters typically found in 2D Rashba materials [40]:
$M=$0.01\text{\,}\mathrm{e}\mathrm{V}$$,
$A=$5\text{\,}\mathrm{\text{\r{A}}}\mathrm{e}\mathrm{V}$$, $\epsilon=20$.
However, we assume larger SOI coupling
$\xi=$5\text{\times}{10}^{3}\text{\,}{\mathrm{\text{\r{A}}}^{2}}$$, as
contrasted to $\xi$ from
$1\text{\times}{10}^{3}\text{\,}{\mathrm{\text{\r{A}}}^{2}}$ to
$2\text{\times}{10}^{3}\text{\,}{\mathrm{\text{\r{A}}}^{2}}$ commonly found in
such system, and quite a short distance of
$a=$20\text{\,}\mathrm{\text{\r{A}}}$$.

Figure 1 shows the position of the energy level of a BEP as a function of the
gate electric field $F$. This is the BEP ground state corresponding to $m=0$
and radial quantum number of $n_{r}=1$. The one-body Rashba SOI produced by
the electric field $F$ is seen to enhance the electron pairing by increasing
the binding energy, in agreement with a conduction-band approximation
treatment [31, 21]. The BEP in the ground state has no definite spin, with
zero spin projection along any direction. There also exist shallow energy
levels close to the lower continuum boundary, not resolvable in the figure.

###  II.3 BEP helical structure and magnetism

The fundamental properties of the BEPs as composite particles that can
manifest themselves in a collective behavior of the many-body system are their
charge and spin. While the charge is well known, it obviously equals $2e$, the
spin is not defined. However, a composite particle certainly has magnetic
properties, which are determined by the helical structure of its quantum
state, which is formed by the relative motion of paired electrons and the
contributions of different atomic orbitals to the spinor of the state.

Before presenting the results, let us dwell in more detail on the essence of
the problem. First we note that the helicity of single-particle states, which
physically means the tight coupling of the momentum to the spin, arises due to
the strong SOI. In the model under study the SOI is inherently presented
through the $sp^{3}$ hybridization. Additional factors producing the SOI, such
as a natural inversion asymmetry caused by the atomic structure of the 2D
layer or the electric fields of external charges, break down the spin
quantization so that no spin projection is defined and therefore the helicity
can not be determined. This, however, does not mean that the helical nature of
the quantum state is destroyed [41]. The point is that the system remains
symmetrical with respect to the time reversal and therefore the eigenstates
are classified by their Kramers index instead of the spin projection. In each
of the Kramers partners the electrons move in opposite direction and their
spin states are described by different four-component spinors related by the
time reversal transformation [42]. The two-particle bound states forming under
this condition have been extremely poorly studied. Since the spin is not a
quantity that characterizes these states, and only the total angular momentum
is a conserved quantity, the magnetic moment of a pair is the measurable
quantity that most adequately characterizes its helical structure.

The total angular momentum $\hat{J}_{z}=\hat{L}_{z}+\hat{S}_{z}+\hat{K}_{z}$
of the BEP has three components: the orbital moment $\hat{L}_{z}$ of the
relative motion of the particles, the electron spin $\hat{S}_{z}$, and the
angular momentum $\hat{K}_{z}$ of the atomic orbitals. All three contribute to
the magnetic moment
$\mu=\mu_{\mathrm{orbital}}+\mu_{\mathrm{spin}}+g\mu_{\mathrm{Bloch}}$ of the
BEP, with material-dependent Landé factor $g$.

The orbital magnetization

$\begin{split}\frac{\mu_{\mathrm{orbital}}}{\mu_{B}}=&{}\frac{m}{\hbar^{2}}\int_{0}^{\infty}\Big{[}-B_{1}\left((2m-1)(\psi_{1}^{2}(r)+\psi_{6}^{2}(r))+(2m-2)\psi_{2}^{2}(r)+4m\psi_{3}^{2}(r)+(2m+1)(\psi_{4}^{2}(r)+\psi_{8}^{2}(r))+(2m+2)\psi_{9}^{2}(r)\right)\\\
&{}+B_{2}\left((2m-2)\psi_{2}^{2}(r)+(2m+1)\psi_{4}^{2}(r)+(2m-3)\psi_{5}^{2}(r)+(2m-1)\psi_{6}^{2}(r)+4m\psi_{7}^{2}(r)+(2m+2)\psi_{9}^{2}(r)+(2m+3)\psi_{10}^{2}(r)\right)\\\
&{}-rA\left(\psi_{2}(r)(\psi_{1}(r)+\psi_{5}(r))+(\psi_{4}(r)-\psi_{6}(r))(\psi_{3}(r)+\psi_{7}(r))-\psi_{9}(r)(\psi_{8}(r)+\psi_{10}(r))\right)\\\
&{}-e\xi
r\mathcal{F}(r)\left(\psi_{1}(r)\psi_{3}(r)-\psi_{2}(r)\psi_{6}(r)-\psi_{3}(r)\psi_{8}(r)+\psi_{4}(r)\psi_{9}(r)\right)\Big{]}4\pi
r\,dr\end{split}$ (8)

is produced by the circular current due to the relative motion of the
electrons within the pair. The magnetic moment is strongly renormalized by the
SOI, which affects the electron velocity.

The spin magnetization

$\begin{split}\frac{\mu_{\mathrm{spin}}}{\mu_{B}}={}&\int\big{(}-\psi_{1}^{2}(r)-2\psi_{2}^{2}(r)-\psi_{5}^{2}(r)\\\
&{}+\psi_{8}^{2}(r)+2\psi_{9}^{2}(r)+\psi_{10}^{2}(r)\big{)}4\pi
r\,dr\end{split}$ (9)

is due to the Zeeman splitting of the spin sector. The last contribution to
the magnetization

$\begin{split}\frac{\mu_{\mathrm{Bloch}}}{\mu_{B}}={}&\int\big{(}-\psi_{2}^{2}(r)+\psi_{4}^{2}(r)-\psi_{5}^{2}(r)\\\
&{}-\psi_{6}^{2}(r)+\psi_{9}^{2}(r)+\psi_{10}^{2}(r)\big{)}4\pi
r\,dr\end{split}$ (10)

is due to the angular moment of the $p$-like basis Bloch states.

The lowest energy level ($n_{r}=1,m=0$) possesses no magnetic moment
associated with any of them, as all three contributions vanish by force of Eq.
(II.1). However, the excited states carry the circular current hence
contributing to the orbital magnetization, possess the non-zero spin
projection on the normal direction, and also feature the finite magnetization
coming from the atomic orbitals.

The formation of a BEP with a nonzero angular momentum due to PSOI encounters
certain limitations caused by the presence of a centrifugal barrier, which
rapidly grows with decreasing the distance from the center and prevents the
attraction due to the PSOI. Calculations show that BEPs with nonzero angular
momentum are formed at a sufficiently small dispersion parameter $B_{1}$ in
the electron subband. This is not difficult to understand, since within the
framework of the used model the SOI acts on the electron subband states,
therefore an increase in the effective mass leads to a decrease in the
centrifugal energy, and the attraction due to the PSOI becomes predominant.
Further study of the magnetic moment is carried out for this situation, which
is quite realizable due to the advances in the band engineering [43].

Figure 3:  The orbital magnetization of both types of the BEPs as a function
of the gate electrical field.

The energy spectrum of the BEPs with $n_{r}=1,m=1$ for this case is shown in
Fig. 2 as a function of the gate voltage. Two different types of the bound
states are seen to exist. At low gate voltage, the BEP is formed with the
energy slightly above the continuum boundary. As the gate electric field
increases, the energy of the state increases too. The second type of the BEP
arises for the electrical field larger the some critical value $F_{c}$, which
is determined by a balance between the Coulomb repulsion of the electrons, the
sign and magnitude of their effective reduced mass, and the effect of the
PSOI. Its energy features the non-monotonous dependence on the gate field,
which strongly affects the hybridization of the electron bands that form the
BEP. When its energy becomes comparable with that of the shallow state, a
visible anti-crossing appears in the spectrum.

The orbital magnetic moment $\mu_{\mathrm{orbital}}$ of two types of the BEPs
is plotted versus $F$ in Fig. 3. Both the magnitude and sign of the
$\mu_{\mathrm{orbital}}$ are tunable by the gate electric field. Increasing
$F$ leads to a strong reconstruction of the orbital and even helical
composition of the BEPs.

The BEP magnetization is dominated by the orbital contribution, as can be seen
from the comparison with Fig. 4, where the spin magnetization and the
magnetization due to the basis atomic orbitals are plotted as a function of
the gate electric field.

Figure 4:  The spin magnetization (solid lines) and magnetization due to the
atomic orbitals (dashed lines) of both types of the BEPs as a function of the
gate electrical field.

##  III Conclusion

We have studied the formation of the BEPs due to the PSOI in a gated 2D
material with a two-band spectrum and a strong Rashba SOI created due to the
$sp^{3}$ band hybridization. The gate plays an important role here as it
creates both the electric field normal to the layer, which is homogeneous
along the plane, and the electric field of the image charges of the electrons,
which contributes to the PSOI along with the in-plane component of the Coulomb
field of interacting electrons. Due to the presence of the gate, the influence
of the PSOI on the BEPs emerging in the system increases significantly, and
can be controlled by the gate field. Thus, it becomes possible to shed light
on the physical mechanism of the formation of the BEPs.

Two mechanisms are discussed that give rise to the effective attraction of the
electrons: the universal mechanism for the formation of a negative effective
reduced mass due to the fact that the quantum state of a pair is formed by
mixing the subband states with opposite masses, and the mechanism of the PSOI,
created by the Coulomb fields of electrons in Rashba materials. Both are
essentially two aspects of the subband hybridization, but reflect different
properties of the subband states. By changing the gate field, one can trace
how the interplay of these two mechanisms manifests itself in the spectrum and
magnetic properties of the BEPs.

We found that an increase in the PSOI leads to the appearance of the BEPs with
a sufficiently high binding energy, the energy level of which, with an
increase in the gate field, moves from the top of the gap in the two-particle
spectrum almost to its bottom. Along with these pairs, there are the states
with an energy slightly above the bottom of the gap, which also exist without
a SOI. Under certain conditions the levels of these two states can converge
and, as a result of the avoided crossing, the states hybridize.

The gate-induced SOI breaks the spin quantization, because of which the spin
projection in any direction is no longer defined. The quantum state of the BEP
is characterized only by the normal projection of the total angular momentum.
Nevertheless, the composite particle formed as a result of the electron
pairing has a magnetic moment directed normally to the layer. Its magnitude is
determined mainly by the orbital relative motion of the electrons, although
the motion of the spin, and the magnetic moment of the atomic orbitals also
contribute to it. In the ground state with zero angular momentum the magnetic
moment is also zero. The magnetic moment exists in the states with nonzero
total angular momentum. The magnitude of the magnetic moment can be fine-tuned
by the gate field, giving rise to a novel mechanism of the electrically
controlled orbital magnetization of the two-electron bound states.

The dependence of the magnetic moment on the gate field is quite different for
the pairs formed mainly due to the negative effective reduced mass or mainly
due to the PSOI. The magnetic moment can even change its sign in the course of
the hybridization of the states, which indicates a rearrangement of the
helical structure of the BEPs. This apparently stems from a change in the
relative motion of electrons during the hybridization of the upper and lower
branches of the spectrum of the BEPs with predominantly different pairing
mechanisms. Such a sign change would be an experimental fingerprint of the
discussed mechanism.

###### Acknowledgements.

Y.G. and I.R. gratefully acknowledge the support and hospitality of the
Weizmann Institute of Science during their stay there. The work of V.A.S. was
carried out in the framework of the state task for the Kotelnikov Institute of
Radio Engineering and Electronics and partially supported by the Russian
Foundation for Basic Research, Project No. 20–02–00126.

##  Appendix A Equation of motion

Hamiltonian of Eq. (2) leads to the following system of coupled equation of
motion for the spinor components of Eq. (6):

$\begin{split}[-2B_{1}(\partial_{r}^{2}+\tfrac{1}{r}\partial_{r}-\tfrac{{(2m-1)}^{2}}{r^{2}})+2M+\mathcal{U}(r)]\psi_{1}(r)-2A(\partial_{r}-\tfrac{2m-2}{r})\psi_{2}(r)+2e\xi\mathcal{F}(r)(\partial_{r}+\tfrac{2m}{r})\psi_{3}(r)+\psi_{3}(r)e\xi\partial_{r}\mathcal{F}(r)&=\varepsilon\psi_{1}(r)\\\
A(\partial_{r}+\tfrac{2m-1}{r})\psi_{1}(r)+[(B_{2}-B_{1})(\partial_{r}^{2}+\tfrac{1}{r}\partial_{r}-\tfrac{{(2m-2)}^{2}}{r^{2}})+\mathcal{U}(r)]\psi_{2}(r)-A(\partial_{r}-\tfrac{2m-3}{r})\psi_{5}(r)&\\\
{}-e\xi\mathcal{F}(r)\partial_{r}\psi_{6}(r)-e\xi(\tfrac{2m-1}{r}\mathcal{F}(r)+\tfrac{1}{2}\partial_{r}\mathcal{F}(r))\psi_{6}(r)&=\varepsilon\psi_{2}(r)\\\
-e\xi\mathcal{F}(r)\partial_{r}\psi_{1}(r)+e\xi(\tfrac{2m-1}{r}\mathcal{F}(r)-\tfrac{1}{2}\partial_{r}\mathcal{F}(r))\psi_{1}(r)+[-2B_{1}(\partial_{r}^{2}+\tfrac{1}{r}\partial_{r}-\tfrac{{(2m)}^{2}}{r^{2}})+2M+\mathcal{U}(r)]\psi_{3}(r)&\\\
+A(\partial_{r}+\tfrac{2m+1}{r})\psi_{4}(r)+A(\partial_{r}-\tfrac{2m-1}{r})\psi_{6}(r)-e\xi\mathcal{F}(r)\partial_{r}\psi_{8}(r)-e\xi(\tfrac{2m+1}{r}\mathcal{F}(r)+\tfrac{1}{2}\partial_{r}\mathcal{F}(r))\psi_{8}(r)&=\varepsilon\psi_{3}(r)\\\
-A(\partial_{r}-\frac{2m}{r})\psi_{3}(r)+[(B_{2}-B_{1})(\partial_{r}^{2}+\tfrac{1}{r}\partial_{r}-\tfrac{{(2m+1)}^{2}}{r^{2}})+\mathcal{U}(r)]\psi_{4}(r)-A(\partial_{r}-\frac{2m}{r})\psi_{7}(r)&\\\
{}+e\xi\mathcal{F}(r)\partial_{r}\psi_{9}(r)+e\xi(\tfrac{2m+2}{r}\mathcal{F}(r)+\tfrac{1}{2}\partial_{r}\mathcal{F}(r))\psi_{9}(r)&=\varepsilon\psi_{4}(r)\\\
2A(\partial_{r}+\tfrac{2m-2}{r})\psi_{2}(r)+[2B_{2}(\partial_{r}^{2}+\tfrac{1}{r}\partial_{r}-\tfrac{{(2m-3)}^{2}}{r^{2}})-2M+\mathcal{U}(r)]\psi_{5}(r)&=\varepsilon\psi_{5}(r)\\\
e\xi\mathcal{F}(r)\partial_{r}\psi_{2}(r)-e\xi(\tfrac{2m-2}{r}\mathcal{F}(r)-\tfrac{1}{2}\partial_{r}\mathcal{F}(r))\psi_{2}(r)-A(\partial_{r}+\frac{2m}{r})\psi_{3}(r)&\\\
+[(B_{2}-B_{1})(\partial_{r}^{2}+\tfrac{1}{r}\partial_{r}-\tfrac{{(2m-1)}^{2}}{r^{2}})+\mathcal{U}(r)]\psi_{6}(r)-A(\partial_{r}+\frac{2m}{r})\psi_{7}(r)&=\varepsilon\psi_{6}(r)\\\
A(\partial_{r}+\tfrac{2m+1}{r})\psi_{4}(r)+A(\partial_{r}-\tfrac{2m-1}{r})\psi_{6}(r)+[2B_{2}(\partial_{r}^{2}+\tfrac{1}{r}\partial_{r}-\tfrac{{(2m)}^{2}}{r^{2}})-2M+\mathcal{U}(r)]\psi_{7}(r)&=\varepsilon\psi_{7}(r)\\\
2e\xi\mathcal{F}(r)(\partial_{r}-\tfrac{2m}{r})\psi_{3}(r)+\psi_{3}(r)e\xi\partial_{r}\mathcal{F}(r)+[-2B_{1}(\partial_{r}^{2}+\tfrac{1}{r}\partial_{r}-\tfrac{{(2m+1)}^{2}}{r^{2}})+2M+\mathcal{U}(r)]\psi_{8}(r)-2A(\partial_{r}+\tfrac{2m+2}{r})\psi_{9}(r)&=\varepsilon\psi_{8}(r)\\\
-e\xi\mathcal{F}(r)\partial_{r}\psi_{4}(r)+e\xi(\tfrac{2m+1}{r}\mathcal{F}(r)-\tfrac{1}{2}\partial_{r}\mathcal{F}(r))\psi_{4}(r)+A(\partial_{r}-\tfrac{2m+1}{r})\psi_{8}(r)&\\\
{}+[(B_{2}-B_{1})(\partial_{r}^{2}+\tfrac{1}{r}\partial_{r}-\tfrac{{(2m+2)}^{2}}{r^{2}})+\mathcal{U}(r)]\psi_{9}(r)-A(\partial_{r}+\tfrac{2m+3}{r})\psi_{10}(r)&=\varepsilon\psi_{9}(r)\\\
2A(\partial_{r}-\tfrac{2m+2}{r})\psi_{9}(r)+[2B_{2}(\partial_{r}^{2}+\tfrac{1}{r}\partial_{r}-\tfrac{{(2m+3)}^{2}}{r^{2}})-2M+\mathcal{U}(r)]\psi_{10}(r)&=\varepsilon\psi_{10}(r)\,.\end{split}$
(11)

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[a,b]Gert Aarts

# Interpreting machine learning functions as physical observables

Dimitrios Bachtis  Biagio Lucini

###### Abstract

We propose to interpret machine learning functions as physical observables,
opening up the possibility to apply “standard” statistical-mechanical methods
to outputs from neural networks. This includes histogram reweighting and
finite-size scaling, to analyse phase transitions quantitatively. In addition
we incorporate predictive functions as conjugate variables coupled to an
external field within the Hamiltonian of a system, allowing to induce order-
disorder phase transitions in a novel manner. A noteworthy feature of this
approach is that no knowledge of the symmetries in the Hamiltonian is
required.

##  1 Introduction

The application of machine learning (ML) in the physical sciences has seen a
dramatic increase in recent years, see e.g. Ref. [1] for a broad overview. In
this contribution, we will discuss some of the work done in Swansea in the
past year [2, 3, 4]; in a parallel talk Ref. [5] was presented, and hopefully
at next year’s Lattice conference Ref. [6] can be discussed in person.

This contribution has three take-home messages, namely

  1. 1.

one can (should) interpret ML outputs as physical observables;

  2. 2.

critical behaviour can be derived from neural network observables only;

  3. 3.

new Hamiltonians can be constructed with the inclusion of ML predictive
functions.

We will discuss these three messages and their implications in the following
sections.

##  2 ML outputs as physical observables

As is well known, ML excels in pattern finding. In the context of statistical
physics and phases of matter, this has been demonstrated in the by-now classic
paper [7], in which it was demonstrated that ML can classify phases and phase
transitions in the two-dimensional Ising model and other lattice systems. The
task is formulated as a supervised learning problem: one generates (or
otherwise obtains) configurations of spins on a two-dimensional lattice for a
variety of temperatures or couplings. Configurations deep in the ordered and
the disordered phases are labelled accordingly and are used to train the ML
algorithm, i.e. adjusting the parameters in the neural network of choice such
that the desired classification is correctly reproduced for the training set.
New, unseen configurations at intermediate temperatures or couplings are then
provided to the network, which returns the probability to be in the
(dis)ordered phase. Repeating this on lattices of various sizes allows one to
study critical scaling and determine properties of the transition in the
infinite-volume limit. This approach has been used in the Ising model,
$q$-state Potts models with varying $q$, scalar $\phi^{4}$ theory, etc. An
example of a convolutional neural network used in our work is shown in Fig. 1.

Figure 1: Example of a convolutional neural network (CNN) used for phase
classification.

Following Ref. [7], the approach has been repeated and extended many times and
is by now a well-established procedure. The obvious question is therefore what
can be added in this direction. As Fig. 1 illustrates, the input in the
network are configurations, which have been generated according to the
Boltzmann weight $e^{-\beta H}/Z$, with $H$ the Hamiltonian of the system,
$\beta$ the inverse temperature or coupling, and $Z$ the partition function.
By considering configurations at fixed $\beta$, the process of computing the
outcome of the neural network is identical to how observables are computed in
numerical simulations in statistical physics, namely by averaging over
configurations. However, the observable in this case is not expressed directly
in terms of the degrees of freedom, such as e.g. the magnetisation is, but is
instead a rather complicated quantity, constructed as described above. Since
the outcome is the probability of being in the (dis)ordered phase, it
nevertheless still has the potential to act similar to an “order parameter” in
the statistical system.

The reasoning above leads to the proposition that the output of a neural
network should be interpreted as an observable in a statistical system [2].
This observation is of interest, since it opens up the possibility to use
“standard” numerical or statistical methods, such as histogram reweighting.
Let $P_{i}$ denote the neural network prediction a configuration with energy
$E_{i}$ is in the broken phase, with $0\leq P_{i}\leq 1$. Averaging over
configurations at fixed $\beta$ then yields the expectation value as provided
by the neural network,

$\langle P\rangle=\frac{1}{Z}\sum_{i}P_{i}e^{-\beta E_{i}},$ (1)

since the configurations labeled by $i$ are Boltzmann distributed. One can now
use standard histogram reweighting [8] to extrapolate from a given $\beta_{0}$
to other $\beta$ values, according to

$\langle
P\rangle(\beta)=\frac{\sum_{i}P_{i}e^{-(\beta-\beta_{0})E_{i}}}{\sum_{i}e^{-(\beta-\beta_{0})E_{i}}},$
(2)

and obtain continuous dependence on the coupling constant.

Figure 2: Probability $\langle P\rangle$ to be in the broken (ordered) phase,
as a function of the coupling $\beta$ in the two-dimensional Ising model on a
$V=128^{2}$ lattice. The open and filled diamonds are predictions from the
neural network, while the line is obtained via histogram reweighting, starting
from the filled diamond at $\beta_{0}=0.438$.

This approach is illustrated in Fig. 2. The CNN shown in Fig. 1 is trained
deeply in the (dis)ordered phases, at $\beta\leq 0.41$ and $\beta\geq 0.47$.
Predictions are made at intermediate values of $\beta$, indicated with the
diamonds. The line is obtained employing histogram reweighting, using only the
datapoint at $\beta_{0}=0.438$ (which is also obtained from the CNN). It is
seen that the line is in agreement with the direct predictions from the CNN
and interpolates smoothly between the disordered and ordered phases. The
insets indicate agreement within statistical uncertainty between the two
approaches. More details can be found in Ref. [2].

Figure 3: Finite-size scaling using histogram reweighing in the Ising model,
for the probability to be in the ordered phase $\langle P\rangle$ (left) and
its susceptibility $\delta P=\beta V\left(\langle P^{2}\rangle-\langle
P\rangle^{2}\right)$ (right).

All results in Fig. 2 are predictions from the CNN, with or without
reweighting. In order to see whether they give a quantitative prediction for
properties of the transition, we carry out a finite-size scaling analysis
using the results obtained with histogram reweighting. The results are shown
in Fig. 3 for the probability $\langle P\rangle$ (left) and its susceptibility
$\delta P=\beta V\left(\langle P^{2}\rangle-\langle P\rangle^{2}\right)$
(right). We will use these data to analyse critical scaling in the next
section.

##  3 Critical behaviour from neural network observables only

Finite-size scaling should give us access to the critical coupling $\beta_{c}$
and critical exponents in the infinite-volume limit. Recall that in the two-
dimensional Ising model, $\beta_{c}=\frac{1}{2}\ln(1+\sqrt{2})$, while the
correlation length and susceptibility diverge as $\xi\sim|t|^{-\nu}$ and
$\chi\sim|t|^{-\gamma}$, where $t$ is the reduced coupling,
$t=(\beta_{c}-\beta)/\beta_{c}$. In a finite system of linear size $L$, this
implies

$|t|\sim\xi^{-1/\nu}\sim L^{-1/\nu},\qquad\qquad\chi\sim|t|^{-\gamma}\sim
L^{-\gamma/\nu}.$ (3)

We use these relations to extract the critical quantities from the observables
determined within the neural network, as shown in Fig. 3. Note in particular
that the susceptibility is not the magnetic susceptibility, but rather the one
linked to the network probability $\langle P\rangle$. The result for the
volume dependence of $\beta_{c}$ is shown in Fig. 4 (left) and the obtained
values for $\beta_{c},\nu$ and $\gamma/\nu$ are given in Table 1. We observe
good agreement with the expected results within the statistical uncertainty.

Figure 4: Determination of the critical parameters $\beta_{c}$ and
$\mu^{2}_{c}$ in the two-dimensional Ising model (left) and the
$\lambda\phi^{4}$ theory (right) respectively using finite-size scaling of CNN
predicted values.

In order to demonstrate that this approach also works in a theory with
continuous degrees of freedom, we repeat the exercise in the two-dimensional
$\lambda\phi^{4}$ scalar field theory [3]. In this case, exact results for the
critical couplings are not available, but it is expected that the model is in
the same universality class as the Ising model. We fixed the coupling
$\lambda=0.7$ and vary the mass parameter $\mu^{2}$ to locate the transition.
Again this is repeated on various volumes. The result for the volume
dependence of $\mu^{2}_{c}$ is shown in Fig. 4 (right), and the obtained
values for $\mu^{2}_{c},\nu$ and $\gamma/\nu$ are given in Table 1. The value
of the critical mass parameter agrees with the one found in the literature
[9], while the value of the critical exponents confirm the universality class.

Ising model | $\beta_{c}$ | $\nu$ | $\gamma/\nu$
---|---|---|---
CNN+Reweighting | 0.440749(68) | 0.95(9) | 1.78(4)
Exact | $\frac{1}{2}\ln(1+\sqrt{2})\approx 0.440687$ | $1$ | $7/4=1.75$
$\phi^{4}$ theory | $\mu_{c}^{2}$ | $\nu$ | $\gamma/\nu$
CNN+Reweighting | -0.95225(54) | 0.99(34) | 1.78(7)
Table 1:  Numerical estimates for the critical parameters $\beta_{c}$ in the
Ising model and $\mu_{c}^{2}$ in the $\lambda\phi^{4}$ theory at fixed
$\lambda=0.7$, as well as for the critical exponents $\nu$ and $\gamma/\nu$,
in the infinite-volume limit.

We conclude that critical behaviour can be derived from observables obtained
within the neural network formulation, with adequate precision.

##  4 Extend Hamiltonians with ML predictive functions

We can now take these insights one step further and prepare new Hamiltonians,
by including the ML predictive function in the Hamiltonian itself [4], i.e. we
couple the ML predictive function to a conjugate source $Y$ which, when varied
to positive and negative values, can drive the system into the
ordered/disordered phases – in both directions – and also allows us to extract
an additional critical exponent.

Let us denote the ML predictive function, i.e the probability to be in the
ordered phase, with $f$, where $0\leq f\leq 1$. Since it is an intensive
quantity, it is multiplied with the volume $V$ and added to the system
Hamiltonian $H$, coupled to the conjugate source $Y$, to arrive at
$H_{Y}=H-VfY$. This formulation can be used to define a generating function,

$\langle f\rangle=\frac{1}{\beta V}\frac{\partial\ln Z_{Y}}{\partial
Y},\qquad\quad Z_{Y}={\mathrm{Tr}\,}e^{-\beta H_{Y}},\qquad\quad H_{Y}=H-VfY,$
(4)

etc, but more importantly, it can induce symmetry breaking without explicit
symmetry breaking.

Figure 5: Response to the external field $Y$ of the probability to be in the
ordered phase $\langle f\rangle$ (left) and its susceptibility $\chi_{f}$
(right) in the two-dimensional Ising model on a $64^{2}$ lattice, for three
values of the coupling $\beta$.

This is demonstrated in Fig. 5, where the response of $\langle f\rangle$ and
its susceptibility $\chi_{f}$ are shown as $Y$ is varied. The three values of
$\beta$ correspond to the symmetric phase ($\beta=0.43$), the known critical
coupling ($\beta_{c}=0.440687$), and the broken-symmetry phase ($\beta=0.45$)
at $Y=0$. We note that the results for nonzero $Y$ are again obtained using
histogram reweighting. We indeed observe that as $Y$ is varied, the system
moves seamlessly from one phase into the other, and in both directions as the
sign of $Y$ is changed. This should be contrasted with the inclusion of an
external magnetic field, in which the symmetry is always broken explicitly,
giving preference to the symmetry-broken phase.

To bring this discussion to an end, we finally note that an external field
introduces a new critical exponent. In the case of an external magnetic field,
the correlation length varies as $\xi\sim|h_{\rm magnetic}|^{-\theta}$, where
in the Ising model $\theta=8/15$. Since it is not guaranteed that the
dependence on $Y$ will be captured by the same exponent, we denote it with
$\theta_{Y}$ and conjecture that $\xi\sim|Y|^{-\theta_{Y}}$. We have
investigated this scaling numerically, using blocking and a renormalisation
group (RG) analysis [4]. This study gives access to $\beta_{c},\nu$ and
$\theta_{Y}$, with the results shown in Table 2. We observe excellent
agreement and confirm that $\theta_{Y}$ can indeed be identified with the
exponent $\theta$.

We reiterate that this analysis only used quantities derived from the neural
network, that there is no need for knowledge of the order parameter, the
pattern of symmetry breaking, the physical meaning of external field, or the
introduction of explicit symmetry breaking.

Ising model | $\beta_{c}$ | $\nu$ | $\theta_{Y},\theta$
---|---|---|---
NN$+$RG | 0.44063(21) | 1.02(2) | $\theta_{Y}=0.534(3)$
Exact | $\frac{1}{2}\ln(1+\sqrt{2})\approx 0.440687$ | $1$ | $\theta=8/15\approx 0.5333$
Table 2: Numerical estimates for $\beta_{c}$ and the critical exponents $\nu$
and $\theta_{Y}$ in the Ising model, using the Hamiltonian extended with the
ML predictive function coupled to a source.

##  5 Summary

In this contribution we reviewed our proposal to identify outputs from neural
networks as observables in statistical physics. The first consequence is that
this allowed us to introduce histogram reweighting to be employed in
supervised machine learning. This yields the possibility to extract critical
properties from a finite-size scaling analysis using quantities derived from
neural networks alone, to relatively high precision. Detailed knowledge of
symmetries and order parameters is not required. By coupling neural network
outputs to an external source, we can drive the system into the
ordered/disordered phases, unlike in the standard way of using an external
magnetic field which always breaks the symmetry explicitly.

We hope these observations can lead to new quantitative studies of phase
transitions based on a synergistic relation between machine learning and
statistical mechanics, with ideas and inspiration which is of interest to both
communities.

Acknowledgments – The authors received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
program under Grant Agreement No. 813942. GA and BL are supported in part by
the UKRI Science and Technology Facilities Council (STFC) Consolidated Grant
No. ST/P00055X/1 and ST/T000813/1. BL is supported in part by the Royal
Society Wolfson Research Merit Award No. WM170010 and by the Leverhulme
Foundation Research Fellowship No. RF-2020-461/9. Numerical simulations have
been performed on the Swansea SUNBIRD system. This system is part of the
Supercomputing Wales project, which is partly funded by the European Regional
Development Fund (ERDF) via the Welsh Government.

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