# Renormalization of the valley Hall conductivity due to interparticle
interaction

D. S. Eliseev  Novosibirsk State Technical University, Novosibirsk 630073,
Russia  A. V. Parafilo  Center for Theoretical Physics of Complex Systems,
Institute for Basic Science (IBS), Daejeon 34126, Korea  V. M. Kovalev
Novosibirsk State Technical University, Novosibirsk 630073, Russia  O. V.
Kibis  Novosibirsk State Technical University, Novosibirsk 630073, Russia  I.
G. Savenko  Department of Physics, Guangdong Technion–Israel Institute of
Technology, 241 Daxue Road, Shantou, Guangdong 15063, China  Technion – Israel
Institute of Technology, 32000 Haifa, Israel  Guangdong Provincial Key
Laboratory of Materials and Technologies for Energy Conversion, Guangdong
Technion–Israel Institute of Technology, Guangdong 515063, China

###### Abstract

We develop a theory of Coulomb interaction-mediated contribution to valley
Hall effect (VHE) in two-dimensional non-centrosymmetric gapped Dirac
materials. We assume that the bare valley Hall current occurs in the system
due to the presence of disorder caused by impurities and is determined by the
valley-selective anisotropic skew scattering. Applying the Boltzmann transport
equation to describe the electron and hole transport in the material, we
calculate the renormalized VHE conductivity due to electron-electron and
electron-hole scattering processes, considering two regimes: (i) an $n$-doped
monolayer hosting a degenerate electron gas, and (ii) an intrinsic
semiconductor with the Boltzmann statistics of electron and hole gases. In
both regimes, the dominant mechanism of interparticle scattering is due to
particles residing in different valleys. Moreover, in case (ii), in addition
to direct scattering, electron-hole annihilation starts to play a role with
the increase in temperature. It might even become the dominant mechanism of
the Coulomb interaction-mediated VHE.

Introduction. The Coulomb scattering of the carriers of charge in solids is
mostly important at sufficiently low temperatures – such temperatures at which
the scattering on optical and acoustic phonons is “frozen”. In semiconductors,
electrons and holes can scatter on each other; thus, electron-electron (e-e),
hole-hole (h-h), and electron-hole (e-h) processes take place. They determine
the temperature behavior of the transport coefficients [1]. The efficiency of
each of these processes depends on the density of particles: In an intrinsic
semiconductor (the insulating phase), e-h scattering can be dominating [2];
instead, in the degenerate electron (hole) gas case, when the Fermi level lies
in either the conduction or the valence band due to doping, the expected
dominant mechanism is e-e or h-h, depending on the position of the Fermi
level. Under optical excitation, Fermi quasi-levels might cross both the
valence and conduction bands simultaneously (it corresponds to non-
equilibrium), and the properties of the semiconductor partially resemble those
of an intrinsic one.

In disordered samples, elastic scattering on impurities usually plays a
considerable role at not-too-high temperatures (before the phonons take over).
Impurities break the Galilean invariance of the system and drastically enhance
the overall particle collision intensity [3]. One should be careful here and
remember that at low temperatures, electron-electron repulsion can be
complemented by their phonons-mediated attraction: Even at very low
temperatures, electrons might start forming Cooper pairs accompanied by the
emission of virtual phonons. Moreover, before the transition to the
superconducting state, the superconducting fluctuations (short-lifetime Cooper
pairs) start to play a role, leading to the paraconductivity [4] as an
additional contribution to normal electron gas conductivity.

The interplay of particle scattering on impurities, Coulomb interaction, and
photoinduced transport in general are especially interesting subjects
regarding the emerging two-dimensional (2D) Dirac materials such as graphene
[5, 6] and transition-metal dichalcogenides (TMDs) [7, 8, 9]. In addition to
the regular translational momentum and spin degrees of freedom, these
materials, possessing a hexagonal lattice, also host the valley degree of
freedom. It represents a quantum number describing the corners $K$ and
$K^{\prime}$ of the Brillouin zone. This nontrivial reciprocal lattice
structure provides new opportunities to control particle transport [6] studied
within the direction of research referred to as valleytronics [9]. The key
property underlying all the valleytronic effects is the valley selection rule,
which literally means that the (low-energy) electrons in each valley interact
with either left or right circular polarization of external light, providing
valley-selective interband transitions in the monolayer.

Figure 1: (a) System schematic: The reciprocal space two-band structure of a
2D material. For a small-bandgap material with its Fermi level is in the
bandgap (intrinsic semiconductor case), the electron and hole densities are
distributed according to the Boltzmann statistics. In the case of a wide-
bandgap semiconductor, electrons and holes are excited due to the sample
illumination by an external electromagnetic field. In the degenerate electron
gas regime, electrons occupy the conduction band up to the Fermi level
(located in the conduction band). (b) Quantum amplitude diagrams describing
the skew electron-impurity scattering, electron-electron and electron-hole
scattering and electron-hole annihilation processes, respectively.

Indeed, many of the Dirac materials represent direct-bandgap semiconductors
(in the vicinity of the valleys $K$ and $K^{\prime}$) and within the optical
range (for MoS2 the gap is $1.66\,\mathrm{eV}$ [8]). It is a platform to study
the valley photogalvanic [10, 11, 12, 13, 14], valley acoustoelectric, and the
valley Hall effects [15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. The valley Hall
effect has attracted particularly high attention in recent years. It directly
relates to the family of anomalous Hall effects: The transport phenomena,
where the transverse current of charged carriers does not directly relate to
the action of the Lorentz force [25, 26, 27]. In the case of VHE, charge
particles propagate in different transverse directions in different valleys.

Three specific mechanisms constitute the basis of VHE in nonmagnetic systems:
(a) the asymmetric (skew) scattering, (b) the side jump, and (c) the anomalous
velocity (or the Berry phase). These mechanisms have recently been studied
theoretically when applied to two-valley Dirac monolayers under different
external drag forces: The static electric field, the phonon, and the photon
drag [22]. However, this theory is based on one-particle approximation,
disregarding possible interparticle scattering processes.

In this Letter, we study the Coulomb interaction-renormalized VHE in gapped
two-valley monolayer materials (such as gapped graphene, TMDs, etc.) in two
regimes: (i) the $n$-doped monolayer (the high-density regime), when the
electron gas is degenerate, and (ii) the low-density regime, when the
monolayer is intrinsic, and both the electrons and holes take part in the
transport phenomena and satisfy the Boltzmann statistics. Case (ii) also
covers the samples exposed to valley-selective light, as discussed below. We
will consider disordered samples at low temperatures when electron (and hole)
impurity relaxation times are much smaller than the interparticle collision
time. In this regime, the interparticle collisions represent a correction to
the bare VHE determined by the particle scattering on impurities [1].
Furthermore, we treat the anisotropic skew scattering of charge carriers on
impurities as the principal mechanism of VHE [22].

It should be noted that in very clean samples, another skew-scattering
mechanism emerges: The interparticle skew scattering (particles skew-scatter
on particles but not impurities). We will disregard this mechanism as we
consider disordered samples, thus focusing on the influence of direct
particle-particle scattering on the disorder-induced VHE. We will show that
inter-particle corrections determine the temperature behavior of VHE at low
temperatures. It should also be noted that at low temperatures, the Cooper
channel of interparticle scattering can contribute to the anomalous Hall
effect [28]; however, we will not consider this regime in this Letter.

General framework of bare impurity-induced VHE. In the calculations, we employ
the two-band spinless model with the parabolic dispersions of electrons and
holes with equal effective masses (Fig. 1). Equilibrium densities of electrons
and holes are determined either by the temperature or by the intra-valley
pumping by means of an external electromagnetic (EM) field. The first case is
realized in small bandgap materials, such as gapped graphene, whereas the
second one takes place in wide-bandgap 2D materials, such as transition metal
dichalcogenide monolayers, where the thermally-activated electron and hole
densities are small due to the large value of the bandgap. We consider the
diffusive regime of the charge carriers’ motion, assuming that the temperature
is low enough such that the particle-impurity collision rate exceeds the
particle-particle collision rate [1]. This approximation allows for using the
Boltzmann equations and treating the particle-particle collision integral via
successive approximations.

Let us first show that this approach provides known formulas for the bare VHE
[22] In the absence of interparticle collisions, the Boltzmann equations for
electrons (momentum $\bf p$) and holes (momentum $\bf k$) can be written as

$\displaystyle{\bf F}\cdot\frac{\partial f_{\bf k}}{\partial{\bf
k}}=Q^{s}_{hi}\\{f_{\bf k}\\}+Q^{a}_{hi}\\{f_{\bf k}\\},$ (1)
$\displaystyle-{\bf F}\cdot\frac{\partial f_{\bf p}}{\partial{\bf
p}}=Q^{s}_{ei}\\{f_{\bf p}\\}+Q^{a}_{ei}\\{f_{\bf p}\\},$ (2)

where ${\bf F}=e{\bf E}$ with $\mathbf{E}$ the electric field, $e>0$, and
$Q^{s}$ and $Q^{a}$ are symmetric and asymmetric parts of electron- and hole-
impurity collision integrals. The former can be treated in the relaxation time
approximation, $Q^{s}_{h(e)i}\\{f_{{\bf k}({\bf p})}\\}=-(f_{{\bf k}({\bf
p})}-n_{{\bf k}({\bf p})})/\tau_{i}$, where $n_{{\bf k}({\bf p})}$ are the
equilibrium hole (electron) distribution functions. The anisotropic collision
integral reads $Q^{a}_{ei}\\{f_{\bf p}\\}=\sum_{{\bf p}^{\prime}}W_{{\bf
p}^{\prime}{\bf p}}f_{{\bf p}^{\prime}}$, where the anisotropic scattering
probability is [22] $W_{{\bf p}^{\prime}{\bf p}}=\eta_{e}W_{0}[{\bf
p}\times{\bf p}^{\prime}]_{z}\delta(\epsilon_{{\bf p}^{\prime}}-\epsilon_{{\bf
p}}),\,W_{0}=-2\pi u_{0}v^{2}/\tau_{i}\Delta^{2}$, with $u_{0}$ the strength
of the short-range impurity potential, and $\Delta$ the monolayer bandgap, $v$
the band parameter (in the framework of the two-band model), $\eta_{e}=\pm 1$
the electron valley index, and finally,
$\epsilon_{\textbf{p}}=\textbf{p}^{2}/2m$ the electron dispersion. For the
holes, the anisotropic scatting integral is determined by a similar
expression, $W_{{\bf k}^{\prime}{\bf k}}=\eta_{h}W_{0}[{\bf k}\times{\bf
k}^{\prime}]_{z}\delta(\epsilon_{{\bf k}^{\prime}}-\epsilon_{{\bf k}})$, with
$\epsilon_{\bf k}=\textbf{k}^{2}/2m$ and $\eta_{h}$ the hole valley index
(note, $\eta_{e}=-\eta_{h}$ for the same valley).

Solving Eqs. (1) and (2) via successive approximations with respect to the
anisotropic impurity scattering (assuming that $\delta
f_{\textbf{p}(\textbf{k})}=f_{\textbf{p}(\textbf{k})}-n_{\textbf{p}(\textbf{k})}$
could be approximated as $\delta f_{\textbf{p}(\textbf{k})}=\delta
f^{(0)}_{\textbf{p}(\textbf{k})}+\delta f^{(1)}_{\textbf{p}(\textbf{k})}$)
gives expressions for the zero-order correction to the distribution functions,

$\displaystyle\delta f^{(0)}_{\bf k}=-\tau_{i}({\bf F}\cdot{\bf v}_{\bf
k})n^{\prime}_{\bf k},\,\,\,\,\delta f^{(0)}_{\bf p}=\tau_{i}({\bf F}\cdot{\bf
v}_{\bf p})n^{\prime}_{\bf p},$ (3)

where $n^{\prime}_{\mathbf{p}}=\partial
n_{\mathbf{p}}/\partial\epsilon_{\mathbf{p}}$, and for the first-order
corrections with respect to anisotropic impurity scattering, $\delta
f^{(1)}_{\bf k}=\tau_{i}Q^{a}_{he}\\{\delta f^{(0)}_{\bf k}\\}$ and $\delta
f^{(1)}_{\bf p}=\tau_{i}Q^{a}_{he}\\{\delta f^{(0)}_{\bf p}\\}$:

$\displaystyle\delta f^{(1)}_{\bf k}=-\tau_{i}^{2}\eta_{h}W_{0}\sum_{{\bf
k}^{\prime}}[{\bf k}\times{\bf k}^{\prime}]_{z}\delta(\epsilon_{{\bf
k}^{\prime}}-\epsilon_{{\bf k}})({\bf F}\cdot{\bf v}_{{\bf
k}^{\prime}})n^{\prime}_{{\bf k}^{\prime}}$ (4) $\displaystyle\delta
f^{(1)}_{\bf p}=\tau_{i}^{2}\eta_{e}W_{0}\sum_{{\bf p}^{\prime}}[{\bf
p}\times{\bf p}^{\prime}]_{z}\delta(\epsilon_{{\bf p}^{\prime}}-\epsilon_{{\bf
p}})({\bf F}\cdot{\bf v}_{{\bf p}^{\prime}})n^{\prime}_{{\bf p}^{\prime}}.$

The total bare VHE electric current density reads ${\bf j}=e\sum_{{\bf k}}{\bf
v}_{\bf k}\delta f^{(1)}_{\bf k}-e\sum_{{\bf p}}{\bf v}_{\bf p}\delta
f^{(1)}_{\bf p}$. For electrons, in the case $\mathbf{F}=eE_{x}\hat{x}$, we
find

$\displaystyle{j}_{y}^{e}$ $\displaystyle=$
$\displaystyle\sigma^{(0)}_{yx}E_{x}=-2\eta_{e}W_{0}e^{2}\tau_{i}^{2}\frac{m^{2}}{(2\pi)^{2}}\langle\epsilon_{\textbf{p}}\rangle
E_{x}\int\limits_{0}^{\infty}d\epsilon_{\textbf{p}}n_{\textbf{p}},$ (5)

where $\sigma^{(0)}_{yx}$ is the transverse conductivity of the bare VHE, and
$\langle\epsilon_{\textbf{p}}\rangle=\sum_{\mathbf{p}}\epsilon_{\mathbf{p}}n_{\mathbf{p}}/\sum_{\mathbf{p}}n_{\mathbf{p}}=\int\limits_{0}^{\infty}d\epsilon_{\textbf{p}}\epsilon_{\textbf{p}}n_{\textbf{p}}/\int\limits_{0}^{\infty}d\epsilon_{\textbf{p}}n_{\textbf{p}}$
is the mean value of the particle kinetic energy. Furthermore, for
$\sigma^{(0)}_{yx}=\sigma^{h}_{yx}+\sigma^{e}_{yx}$ we find (restoring
$\hbar$)

$\displaystyle\sigma^{(0)}_{yx}=-2\sigma_{D}(\eta_{h}+\eta_{e})\left(\frac{W_{0}m^{2}}{2\pi\hbar}\right)\frac{\langle\epsilon_{\mathbf{p}(\mathbf{k})}\rangle\tau_{i}}{\hbar},$
(6)

where $\sigma_{D}=e^{2}n_{i}\tau_{i}/m$ is the Drude conductivity, and $n_{i}$
is the particle density. For an intrinsic monolayer, when electrons and holes
satisfy the Boltzmann statistics,
$\langle\epsilon_{\mathbf{p}}\rangle=\langle\epsilon_{\mathbf{k}}\rangle=T$
with $T$ the temperature, and $n_{i}=\sqrt{n_{e}n_{h}}$ with equal electron
and hole densities, $n_{e}=n_{h}$. Instead, for an n-doped monolayer
(degenerate electron gas case), the contribution from the holes is absent,
whereas $\langle\epsilon_{\mathbf{p}}\rangle=\mu/2$ with $\mu$ the Fermi
energy, and $n_{i}\equiv n_{e}$, where $n_{e}$ is the concentration of
degenerate electrons. Formula (6) coincides with a known result [22].

Electron-electron scattering. The Boltzmann equation, describing (i) the
isotropic and anisotropic electron scattering on impurities and (ii) the
electron-electron scattering reads

$\displaystyle-{\bf F}\cdot\frac{\partial f_{\bf p}}{\partial{\bf
p}}=Q^{s}_{ei}\\{f_{\bf p}\\}+Q^{a}_{ei}\\{f_{\bf p}\\}+Q_{ee}\\{f_{\bf
p}\\}.$ (7)

The first two terms on the right-hand side were defined above. The last term
describes the e-e scattering:

$\displaystyle Q_{ee}\\{f_{\bf p}\\}=2\pi\sum_{{\bf p}^{\prime},{\bf
k}^{\prime},{\bf k}}|U_{{\bf p}^{\prime}-{\bf p}}|^{2}$ (8)
$\displaystyle~{}~{}~{}\times[(1-f_{\bf k})(1-f_{\bf p})f_{{\bf
k}^{\prime}}f_{{\bf p}^{\prime}}-(1-f_{{\bf k}^{\prime}})(1-f_{{\bf
p}^{\prime}})f_{{\bf k}}f_{{\bf p}}]$
$\displaystyle~{}~{}~{}~{}~{}\times\delta(\epsilon_{{\bf
k}^{\prime}}+\epsilon_{{\bf p}^{\prime}}-\epsilon_{{\bf k}}-\epsilon_{{\bf
p}})\delta_{{\bf k}^{\prime}+{\bf p}^{\prime}-{\bf k}-{\bf p}}.$

Here, $U_{{\bf p}^{\prime}-{\bf p}}$ is a Fourier image of the e-e interaction
potential. In this section, both ${\bf p}$ and ${\bf k}$ correspond to
electron momenta (in the subsequent sections, ${\bf k}$ stands for the hole
momentum).

To find the conductivity, the solution of Eq. (7) should be linearized with
respect to the external force ${\bf F}$, thus $f_{\bf p}=n_{\bf p}+\delta
f_{\bf p}$, where $\delta f_{\bf p}$ satisfies the linearized version of Eq.
(7)

$\displaystyle-({\bf F}\cdot{\bf v}_{\bf p})n^{\prime}_{\bf p}=-\frac{\delta
f_{\bf p}}{\tau_{i}}+Q^{a}_{ei}\\{\delta f_{\bf p}\\}+Q_{ee}\\{\delta f_{\bf
p}\\},$ (9)

where the linearized version of e-e collision integrals reads

$\displaystyle Q_{ee}\left\\{\delta f_{\textbf{p}}\right\\}$ $\displaystyle=$
$\displaystyle-2\pi\sum_{\mathbf{p}^{\prime},\mathbf{k},\mathbf{k}^{\prime}}|U_{{\bf
p}^{\prime}-{\bf p}}|^{2}\delta(\epsilon_{{\bf k}^{\prime}}+\epsilon_{{\bf
p}^{\prime}}-\epsilon_{{\bf k}}-\epsilon_{{\bf p}})\delta_{{\bf
k}^{\prime}+{\bf p}^{\prime}-{\bf k}-{\bf p}}$
$\displaystyle\times\Bigl{[}\delta f_{\bf p}[(1-n_{\bf k})n_{{\bf
k}^{\prime}}n_{{\bf p}^{\prime}}+n_{\bf k}(1-n_{{\bf k}^{\prime}})(1-n_{{\bf
p}^{\prime}})]-\delta f_{{\bf p}^{\prime}}[(1-n_{{\bf k}})(1-n_{{\bf
p}})n_{{\bf k}^{\prime}}+n_{\bf k}n_{\bf p}(1-n_{{\bf k}^{\prime}})]$
$\displaystyle~{}~{}~{}+\delta f_{\bf k}[(1-n_{\bf p})n_{{\bf
k}^{\prime}}n_{{\bf p}^{\prime}}+n_{\bf p}(1-n_{{\bf k}^{\prime}})(1-n_{{\bf
p}^{\prime}})]-\delta f_{{\bf k}^{\prime}}[(1-n_{{\bf k}})(1-n_{{\bf
p}})n_{{\bf p}^{\prime}}+n_{\bf k}n_{\bf p}(1-n_{{\bf p}^{\prime}})]\Bigr{]}.$

Since we are calculating the Coulomb corrections to the skew-scattered
electrons, the conductivity must be determined by the product of weak
scattering terms $\sim Q^{a}\cdot Q_{ee}$. Such terms emerge in the second
order corrections of the iteration procedure if we additionally assume $\delta
f_{\textbf{p}}=\delta f^{(0)}_{\textbf{p}}+\delta f^{(1)}_{\textbf{p}}+\delta
f^{(2)}_{\textbf{p}}$. Indeed, the first-order correction reads $\delta
f^{(1)}_{\bf p}=\tau_{i}Q^{a}_{ei}\\{\delta f^{(0)}_{\bf
p}\\}+\tau_{i}Q_{ee}\\{\delta f^{(0)}_{\bf p}\\}$, whereas the second-order
corrections yield $\delta f_{\bf
p}^{(2)}=\tau_{i}(Q_{ei}^{a}\\{\tau_{i}Q_{ee}\\{\delta f_{\bf
p}^{(0)}\\}\\}+Q_{ee}\\{\tau_{i}Q^{a}_{ei}\\{\delta f_{\bf p}^{(0)}\\}\\})$,
where we keep only the cross-terms $Q_{ee}\cdot Q^{a}$, as the terms
proportional to $Q_{ee}^{2},\,(Q^{a}_{ei})^{2}$ do not contain either the
skew-scattering–related terms or the e-e–related ones.

An algebraic analysis (provided in the Supplemental Material
[[SeeSupplementalMaterialforthedetailsofderivationsat\\\\]SMBG]) gives
expressions for the e-e corrections to VHE conductivity. If the electron gas
is degenerate, then (restoring $\hbar$)

$\displaystyle\sigma^{(d)}_{yx}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{3}\sigma_{D}(\eta_{e}-\eta_{e}^{\prime})\left(\frac{W_{0}m^{2}}{2\pi\hbar}\right)\left(\frac{T\tau_{i}}{\hbar}\right)^{2}\left(\frac{e^{2}}{\hbar
v_{F}\varepsilon}\right)^{2}$ (11)
$\displaystyle\times\left[\ln\left(1+\frac{2p_{F}}{\hbar
q_{s}}\right)-\frac{2p_{F}}{\hbar q_{s}+2p_{F}}\right],$

where $\sigma_{D}=e^{2}n_{e}\tau_{i}/m$ is a Drude conductivity. Here, we used
the screened Coulomb interaction $U_{q}=2\pi e^{2}/\varepsilon(q+q_{s})$,
where $q_{s}=me^{2}/\varepsilon\hbar^{2}$ is a screening wave vector and
$\varepsilon$ is a dielectric constant.

In the case of non-degenerate (Boltzmann) statistics of the electron gas, we
can disregard the screening and use pure 2D Coulomb potential to find

$\displaystyle\sigma^{(nd)}_{yx}=\sigma_{D}(\eta_{e}-\eta_{e}^{\prime})5\pi^{2}\left(\frac{W_{0}m^{2}}{2\pi\hbar}\right)\left(\frac{e^{2}\sqrt{n_{e}}}{\varepsilon
T}\right)^{2}\left(\frac{T\tau_{i}}{\hbar}\right)^{2}.$ (12)

The analysis of these two expressions is presented in the Discussion section
below.

Electron-hole scattering. In the case of an intrinsic semiconductor, both
electrons and holes contribute to the Hall conductivity of the system.
Similarly to the e-e case, we start with the Boltzmann equations for electron
and hole nonequilibrium distribution functions,

$\displaystyle-{\bf F}\cdot\frac{\partial f_{\bf p}}{\partial{\bf p}}$
$\displaystyle=$ $\displaystyle-\frac{f_{\bf p}-n_{\bf
p}}{\tau_{i}}+Q^{a}_{ei}\\{f_{\bf p}\\}+Q_{eh}\\{f_{\bf p},f_{\bf k}\\},$
$\displaystyle{\bf F}\cdot\frac{\partial f_{\bf k}}{\partial{\bf p}}$
$\displaystyle=$ $\displaystyle-\frac{f_{\bf k}-n_{\bf
k}}{\tau_{i}}+Q^{a}_{hi}\\{f_{\bf k}\\}+Q_{he}\\{f_{\bf p},f_{\bf k}\\},$

where

$\displaystyle Q_{ei}^{a}\left\\{f_{\bf p}\right\\}$ $\displaystyle=$
$\displaystyle-\eta_{e}W_{0}\frac{2\pi}{\tau_{i}}\sum_{\mathbf{p}^{\prime}}[\mathbf{p}\times\mathbf{p}^{\prime}]_{z}f_{\mathbf{p}^{\prime}}\delta(\epsilon_{\textbf{p}}-\epsilon_{\textbf{p}^{\prime}}),$
$\displaystyle Q_{hi}^{a}\left\\{f_{\bf k}\right\\}$ $\displaystyle=$
$\displaystyle-\eta_{h}W_{0}\frac{2\pi}{\tau_{i}}\sum_{\mathbf{k}^{\prime}}[\mathbf{k}\times\mathbf{k}^{\prime}]_{z}f_{\mathbf{k}^{\prime}}\delta(\epsilon_{\textbf{k}}-\epsilon_{\textbf{k}^{\prime}}).$

In this section, the electron momenta will be denoted by ${\bf p}$ and ${\bf
p}^{\prime}$, and the hole momenta by ${\bf k}$ and ${\bf k}^{\prime}$; and we
will not use the superscripts ‘e’ and ‘h’ in the distribution functions for
brevity.

The linearized e-h collision integral reads

$\displaystyle Q_{eh}\\{\delta f_{\bf p},\delta f_{\bf
k}\\}=-2\pi\sum_{\mathbf{p}^{\prime},\mathbf{k},\mathbf{k}^{\prime}}(|U_{{\bf
p}^{\prime}-{\bf p}}|^{2}+|U_{\mathbf{p}+\mathbf{k}}|^{2})$ (13)
$\displaystyle~{}~{}~{}\times\left(\delta f_{\bf p}n_{\bf k}-\delta f_{\bf
p^{\prime}}n_{\bf k^{\prime}}+\delta f_{\bf k}n_{\bf p}-\delta f_{\bf
k^{\prime}}n_{\bf p^{\prime}}\right)$
$\displaystyle~{}~{}~{}~{}~{}\times\delta(\epsilon_{{\bf
k}^{\prime}}+\epsilon_{{\bf p}^{\prime}}-\epsilon_{{\bf k}}-\epsilon_{{\bf
p}})\delta_{{\bf k}^{\prime}+{\bf p}^{\prime}-{\bf k}-{\bf p}},$

and $Q_{he}\\{\delta f_{\bf p},\delta f_{\bf k}\\}$ acquires the same form.
Here, the first term $U_{\mathbf{p}-\mathbf{p}^{\prime}}$ describes direct
electron-hole scattering, whereas the second term $U_{\mathbf{p}+\mathbf{k}}$
stands for the electron-hole annihilation. The latter can play an essential
role in narrow gap materials such as gapped graphene [30] and, thus, must be
accounted for. The linear corrections satisfy the equations

$\displaystyle\frac{\delta f_{\bf p}}{\tau_{i}}={\bf F}\cdot\frac{\partial
f_{\bf p}}{\partial{\bf p}}+Q_{ei}^{a}\\{\delta f_{\bf p}\\}+Q_{eh}\\{\delta
f_{\bf p},\delta f_{\bf k}\\},$ (14) $\displaystyle\frac{\delta f_{\bf
k}}{\tau_{i}}=-{\bf F}\cdot\frac{\partial f_{\bf k}}{\partial{\bf
k}}+Q_{ei}^{a}\\{\delta f_{\bf k}\\}+Q_{he}\\{\delta f_{\bf p},\delta f_{\bf
k}\\}.$ (15)

We solve these equations by iterations by expanding $\delta
f_{\textbf{p}(\textbf{k})}=\delta f^{(0)}_{\textbf{p}(\textbf{k})}+\delta
f^{(1)}_{\textbf{p}(\textbf{k})}+\delta f^{(2)}_{\textbf{p}(\textbf{k})}+...$
such that $\delta f_{\bf
p}^{(0)}=\tau_{i}(\mathbf{F}\cdot\mathbf{v}_{\mathbf{p}})n_{\mathbf{p}}^{\prime}$,
$\delta f_{\bf
k}^{(0)}=-\tau_{i}(\mathbf{F}\cdot\mathbf{v}_{\mathbf{k}})n_{\mathbf{k}}^{\prime}$,
and $\delta f_{\bf p}^{(1)}=\tau_{i}(Q_{ei}^{a}\\{\delta f_{\bf
p}^{(0)}\\}+Q_{eh}\\{\delta f_{\bf p}^{(0)},\delta f_{\bf k}^{(0)}\\})$,
$\delta f_{\bf k}^{(1)}=\tau_{i}(Q_{ei}^{a}\\{\delta f_{\bf
k}^{(0)}\\}+Q_{he}\\{\delta f_{\bf p}^{(0)},\delta f_{\bf k}^{(0)}\\})$, and
so on.

Performing the algebraic derivations, the Hall conductivity yields [29]

$\displaystyle\sigma^{(eh)}_{yx}$ $\displaystyle=$ $\displaystyle-
W_{0}e^{2}\tau_{i}^{3}\sum_{\mathbf{p},\mathbf{p}^{\prime},\mathbf{k}^{\prime},\mathbf{k}}\left(|U_{\mathbf{p}-\mathbf{p}^{\prime}}|^{2}+|U_{\mathbf{p}+\mathbf{k}}|^{2}\right)$
$\displaystyle\times(k_{y}-p_{y})\left[\eta_{e}(p_{y}\epsilon_{\textbf{p}}-p_{y}^{\prime}\epsilon_{\textbf{p}^{\prime}})-\eta_{h}(k_{y}\epsilon_{\textbf{k}}-k_{y}^{\prime}\epsilon_{\textbf{k}^{\prime}})\right]$
$\displaystyle\times(n_{\bf p}-n_{{\bf p}^{\prime}})(n_{\bf k}-n_{{\bf
k}^{\prime}})\delta(\mathbf{p}+\mathbf{k}-\mathbf{p}^{\prime}-\mathbf{k}^{\prime})$
$\displaystyle\times\int
d\omega\frac{dN_{\omega}}{d\omega}\delta(\epsilon_{\mathbf{k}^{\prime}}-\epsilon_{\mathbf{k}}-\omega)\delta(\epsilon_{\mathbf{p}^{\prime}}-\epsilon_{\mathbf{p}}+\omega).$

Furthermore, it is convenient to split Eq. (Renormalization of the valley Hall
conductivity due to interparticle interaction) into two terms,
$\sigma^{(eh)}_{yx}=\sigma^{(h1)}_{yx}+\sigma^{(h2)}_{xy}$, where
$\sigma^{(h1)}_{yx}$ describes e-h scattering and $\sigma^{(h2)}_{yx}$
corresponds to e-h annihilation. After algebraic calculations [29], we find

$\displaystyle\sigma^{(h1)}_{yx}=\sigma_{D}(\eta_{e}+\eta_{h})5\pi^{2}\left(\frac{W_{0}m^{2}}{2\pi\hbar}\right)\left(\frac{e^{2}\sqrt{n_{i}}}{\varepsilon
T}\right)^{2}\left(\frac{T\tau_{i}}{\hbar}\right)^{2},$ (17)

and

$\displaystyle\sigma^{(h2)}_{yx}=\sigma^{(h1)}_{yx}\frac{2}{5}\int\limits_{0}^{\infty}\frac{x(x^{2}+2)e^{-x^{2}/2}}{(x+\frac{\hbar
q_{T}}{p_{T}})^{2}}dx,$ (18)

where in the last expression, we took into account the static screening for
the non-degenerate statistics of electrons and holes $U_{q}=2\pi
e^{2}/\varepsilon(q+q_{T})$ with $q_{T}=2\pi e^{2}(2n_{i})/T$ the screening
wave vector, and $p_{T}^{2}=2mT$. In Eq.(17) we assume that electron and hole
densities are equal, $n_{i}=n_{e}=n_{h}$.

Results and discussion. Formulas (11), (12), (17) and (18) represent the
central results of this Letter. Let us analyze them separately and compare
them.

For both the cases of degenerate and non-degenerate electron gases (Eqs. (11)
and (12)), the corrections are proportional to $(\eta_{e}-\eta_{e}^{\prime})$,
which means that electrons from valley $K$ scatter with electrons from
$K^{\prime}$, and the scattering of electrons from the same valley is
suppressed. The same argument is valid for the electron-hole scattering: In
Eqs. (17) and (18), the term $(\eta_{e}+\eta_{h})$ is only nonzero if the
particles belong to different valleys (since $\eta_{h}=-\eta_{e}$).

Furthermore, Eqs. (12) and (17) resemble each other up to the valley index-
dependent term, as expected: e-e and e-h-mediated scattering should be similar
for an intrinsic (or nearly intrinsic) semiconductor with equal effective
masses of the carriers of charge.

From Eq. (6), we extract that for a degenerate electron gas, the bare VHE
conductivity reads as

$\displaystyle\sigma^{(0,d)}_{yx}=-2\sigma_{D}\eta_{e}\left(\frac{W_{0}m^{2}}{2\pi\hbar}\right)\frac{\mu\tau_{i}}{2\hbar},$
(19)

and for a nondegenerate gas,

$\displaystyle\sigma^{(0,nd)}_{yx}=-2\sigma_{D}\eta_{e}\left(\frac{W_{0}m^{2}}{2\pi\hbar}\right)\frac{T\tau_{i}}{\hbar}.$
(20)

The Coulomb interaction-induced correction (11) in the case of a degenerate
electron gas depends on the temperature ($\sigma_{yx}^{(d)}\sim T^{2}$) in
contrast to the bare degenerate electron gas VHE if in Eq. (19) we disregard
the dependence of the chemical potential on temperature (which is extremely
weak for a degenerate gas).

Instead, in the case of the non-degenerate electrons, the correction (12)
might not depend on temperature, as compared with the bare VHE (20) (which is
linear in $T$). Indeed, the explicit dependence on temperature disappears in
(12) ($T^{2}$ cancels out in the numerator and denominator there). However,
$n_{e}$ in (12) in the case of intrinsic material at thermal equilibrium might
be temperature dependent. This case is essential for relatively narrow-bandgap
materials, such as a gapped graphene. In the case of wide-bandgap materials,
such as MoS2 in the “intrinsic” regime, the thermally activated electron and
hole densities are very small. In this case, a nonzero VHE current can be
achieved by destroying the time-reversal symmetry by, e.g., the illumination
of the sample by a circularly polarized light with the frequency of the order
of the interband transition energy, thus populating one of the valleys [9] and
keeping the other valley at equilibrium. Under optical excitation, $n_{e}$ and
$n_{h}$ are temperature-independent since these densities are controlled by an
external pump.

Then, for wide-bandgap materials, the correction due to e-h scattering (17) in
the case of the non-degenerate electrons and holes, is also temperature-
independent. However, the e-h annihilation (18) does reveal a peculiar
temperature dependence due to the $q_{T}$-dependent term in the integral. Let
us elaborate on that.

Figure 2: The ratio of e-h exchange (annihilation) and direct e-h scattering-
mediated conductivities in the case of (a) MoS2, where we used the
temperature-independent effective intrinsic density of particles $n_{i}$ (two
values), and (b) gapped graphene with the temperature-dependent $n_{i}$ for
two values for the bandgap $E_{g}$. In (b), the annihilation processes are
dominant over the whole range of reasonable temperatures.

Figure 2(a) shows the ratio $\sigma^{(h2)}_{xy}/\sigma^{(h1)}_{xy}$ for an
intrinsic MoS2 (an example of wide-bandgap material) in the presence of an
additional illumination of one of the valleys. The curve starts from zero and
crosses unity at some temperature. Thus, we conclude that electron-hole
annihilation grows with temperature and can eventually become the predominant
mechanism of the Coulomb-mediated VHE.

The situation is different in gapped graphene (an example of small-bandgap
material). Figure 2(b) shows the ratio
$\sigma^{(h2)}_{xy}/\sigma^{(h1)}_{xy}$, where nonzero $n_{i}$ is due to the
finite temperature, $n_{i}=n_{e}=n_{h}=(mT/2\pi\hbar^{2})\exp(-E_{g}/2T)$, and
$m_{g}=E_{g}/2v_{0}^{2}$ with $v_{0}$ the graphene Fermi velocity. Here, the
processes of e-h annihilation are always dominant (up to very high
temperatures, where phonon-mediated processes become predominant).

Finally, let us compare the Coulomb scattering-mediated effect and the bare
VHE. If we divide the nondegenerate electron gas Hall conductivity Eq. (12) by
the bare conductivity Eq. (20), we acquire the factor $T\tau_{i}\gg 1$ among
other. It represents the ratio of the particles’ kinetic energy and the
impurity-mediated broadening $\hbar/\tau_{i}$. This factor can compensate for
the other small factors in the ratio
$\sigma_{xy}^{(nd)}/\sigma_{xy}^{(0,nd)}$, and it might even result in the
dominance of the Coulomb terms. It should be noted that the considered effects
take place in monolayer materials with the broken inversion symmetry of
crystal lattice. Such a broken symmetry can lead also to many features of
elementary interparticle interactions (electron-electron interaction,
electron-phonon interaction, etc) in various nanostructures (see, e.g., Refs.
[31, 32, 33, 34]).

Conclusions. We developed a theory of a Coulomb interaction-related
contribution to the valley Hall effect and drew the following conclusions.
First, in the case of a degenerate electron gas in a semiconductor, the
scattering of electrons from different valleys plays the major role in Coulomb
interaction-induced valley Hall effect. The Coulomb interaction–mediated
contribution to the Hall conductivity depends on the temperature as $T^{2}$.
Second, in the case of an intrinsic semiconductor embedding non-degenerate
electron and hole gases, the direct Coulomb scattering of particles from
different valleys plays the major role for the effect in question up to some
temperatures. Above a certain temperature, electron-hole annihilation might
become the dominating mechanism, if this temperature is lower than the Debye
temperature, when phonons are expected to take over. The Coulomb interaction-
mediated contribution to the Hall conductivity does not depend on the
temperature in the case of photogenerated non-degenerate electron and hole
gases.

Acknowledgments. This work was supported by the Institute for Basic Science in
Korea (Project No. IBS-R024-D1), Ministry of Science and Higher Education of
the Russian Federation (Project No. FSUN-2023-0006), and the Foundation for
the Advancement of Theoretical Physics and Mathematics “BASIS”.

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# Long-range interactions in a quantum gas mediated by diffracted light

G.R.M. Robb, J.G.M. Walker, G.-L. Oppo & T. A. Ackemann  SUPA and Department
of Physics, University of Strathclyde,Glasgow G4 0NG, Scotland, UK

###### Abstract

A BEC interacting with an optical field via a feedback mirror can be a
realisation of the quantum Hamiltonian Mean Field (HMF) model, a paradigmatic
model of long-range interactions in quantum systems. We demonstrate that the
self-structuring instability displayed by an initially uniform BEC can evolve
as predicted by the quantum HMF model, displaying quasiperiodic ”chevron”
dynamics for strong driving. For weakly driven self-structuring, the BEC and
optical field behave as a two-state quantum system, regularly oscillating
between a spatially uniform state and a spatially periodic state. It also
predicts the width of stable optomechanical droplets and the dependence of
droplet width on optical pump intensity. The results presented suggest that
optical diffraction-mediated interactions between atoms in a BEC may be a
route to experimental realisation of quantum HMF dynamics and a useful
analogue for studying quantum systems involving long-range interactions.

Systems involving long-range interactions, such as those occurring in
gravitational physics or plasma physics, display several unusual behaviours
e.g. extremely slow relaxation and existence of quasi-steady states [1].
Recently, there has been significant interest in quantum systems involving
long range interactions e.g. ion chains, Rydberg gases and cold atomic gases
enclosed in optical cavities [2].

The Hamiltonian Mean Field (HMF) model [1] was introduced as a generic
classical model of long-range interacting systems e.g. self-gravitating
systems [3]. It involves $N$ particles on a ring which experience a pairwise
cosine interaction. It also arises as a model of a system of X-Y rotors
coupled with infinite range. Extension of the HMF model to describe quantum
systems was first carried out by Chavanis [4, 5] and the dynamics of this
quantum HMF model was investigated more recently by Plestid & O’Dell [6, 7]
who demonstrated that the model exhibited violent relaxation of an initially
homogeneous state to a structured state and possessed bright soliton
solutions.

Cold atomic gases enclosed in cavities exhibit phenomena demonstrating
universal behaviours common to many different physical systems e.g. the
behaviour of a cold, thermal gas in a cavity undergoing viscous momentum
damping induced by optical molasses beams is related to the Kuramoto model [8,
9] which describes synchronisation of globally coupled phase oscillators. It
has been shown [10, 11] that in the absence of momentum damping, a thermal gas
in a cavity can exhibit dynamics similar to that of the classical HMF model.
In the case of a quantum degenerate gas, e.g. a Bose-Einstein Condensate
(BEC), its dynamical behaviour in a cavity has been mapped onto the Dicke-
model describing coupled spins and superradiance [12], but to date no
experimental realisation of the quantum HMF model has been described or
proposed.

Here we investigate a system consisting of a BEC interacting with an optical
field via single mirror feedback (SMF) as shown schematically in Fig. 1. In
this BEC-SMF system, coupling between atoms arises due to diffraction,
involves many transverse modes and optical forces directed perpendicular to
the propagation direction of the optical fields. This is significantly
different from cavity systems (such as e.g. [9, 12]), where the dominant
coupling between atoms arises from interference between a pump field and
cavity modes. We show that under certain conditions, the equations describing
the dynamics of the BEC and the optical fields can be mapped onto the quantum
HMF model [4, 5, 6]. Using this connection, we then investigate dynamical
instabilities of initially homogeneous distributions of BEC density and
optical intensity and also the existence of spatially localised states
reminiscent of quantum droplets observed in dipolar BECs [13, 14].

Figure 1: Schematic diagram of the single mirror feedback (SMF) configuration
showing a BEC interacting with a forward propagating optical field ($F$) and a
retroreflected/backward propagating optical field ($B$).

The model we use to describe the BEC-SMF system was originally studied in [15]
as an extension of that used to study self-structuring of a classical, thermal
gas, observed experimentally in [16], with the thermal gas replaced with a
BEC. We consider a BEC with negligible atomic collisions and describe the
evolution of the BEC wavefunction, $\Psi(x,t)$ with the Schrodinger equation :

$i\hbar\frac{\partial\Psi(x,t)}{\partial
t}=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}\Psi(x,t)}{\partial
x^{2}}+\frac{\hbar\delta}{2}s(x,t)\Psi(x,t)$ (1)

where $m$ is the atomic mass, $\delta=\omega-\omega_{a}$ is detuning,
$s(x,t)=|F|^{2}+|B(x,t)|^{2}$ is the atomic saturation parameter due to the
forward and backward optical fields where
$|F|^{2}=\frac{I_{F}}{I_{sat}\Delta^{2}}$ ,
$|B|^{2}=\frac{I_{B}}{I_{sat}\Delta^{2}}$ and $I_{F}$, $I_{B}$ are the
intensities of the forward (F) and backward (B) fields respectively. $I_{sat}$
is the saturation intensity on resonance, $\Delta=\frac{2\delta}{\Gamma}$ and
$\Gamma$ is the decay rate of the atomic transition. It has been assumed that
$|\Delta|\gg 1$ and that consequently $s\ll 1$ so that the atoms remain in
their ground state. In addition, longitudinal grating effects due to
interference between the counterpropagating optical fields are neglected.

In order to describe the optical field in the gas we assume that the gas is
sufficiently thin that diffraction can be neglected, so that the forward field
transmitted through the cloud is

$F_{tr}=\sqrt{p_{0}}\exp\left(-i\chi_{0}n(x,t)\right)$ (2)

where $p_{0}=|F(z=0)|^{2}$ is the scaled pump intensity,
$\chi_{0}=\frac{b_{0}}{2\Delta}$ is the susceptibility of the BEC, $b_{0}$ is
the optical thickness of the BEC at resonance and $n(x,t)=|\Psi(x,t)|^{2}$ is
the local BEC density, which for a BEC of uniform density is $n(x,t)=1$.

The backward field, $B$, at the BEC completes the feedback loop. As the field
propagates a distance $2d$ from the BEC to the mirror and back, optical
diffraction plays a critical role by converting phase modulations to amplitude
modulations and consequently optical dipole forces. The relation between the
Fourier components of the forward and backward fields at the BEC is

$B(q)=\sqrt{R}F_{tr}(q)e^{i\frac{q^{2}d}{k_{0}}}$ (3)

where $R$ is the mirror reflectivity, $q$ is the transverse wavenumber and
$k_{0}=\frac{2\pi}{\lambda_{0}}$. It was shown in [15] that this system
exhibits a self-structuring instability where the optical fields and BEC
density develop modulations with a spatial period of
$\Lambda_{c}=\frac{2\pi}{q_{c}}$, where the critcal wavenumber, $q_{c}$, is

$q_{c}=\sqrt{\frac{\pi}{2}\frac{k_{0}}{d}}.$ (4)

The reason for this instability is that BEC density modulations (which produce
refractive index modulations) with spatial frequency $q_{c}$, produce phase
modulations in $F_{tr}$ which are in turn converted into intensity modulations
of $B$ (see Eq. (3)). These intensity modulations produce dipole forces which
reinforce the density modulations, resulting in positive feedback and
instability of the initial, homogeneous state. A condition of this instability
is that the pump intensity exceeds a threshold value, $p_{th}$ [15], which for
$q=q_{c}$ can be written as

$p_{th}=\frac{2\omega_{r}}{b_{0}R\Gamma},$ (5)

where $\omega_{r}=\frac{\hbar q_{c}^{2}}{2m}$.

The optomechanical self-structuring exhibited by the BEC-SMF model of Eq.
(1)-(3) derived in [15] can be reduced to that of the quantum HMF model,
originally proposed in [4, 5] and revisited in [6, 7]. We express the optical
intensity, $s(x,t)$, in terms of $n$ (density) using Eq. (2). Assuming
$\chi_{0}n\ll 1$ as in [17], then
$F_{tr}\approx\sqrt{p_{0}}(1+i\chi_{0}n(x,t)).$ It is assumed that the BEC
density and (backward) optical field consist of a spatially uniform component
and a spatial modulation with spatial frequency, $q_{c}$, so

$B(q_{c})=\sqrt{R}F_{tr}(q_{c})e^{i\frac{q_{c}^{2}d}{k_{0}}}=i\sqrt{R}F_{tr}(q_{c})$
(6)

i.e. phase modulation of $F_{tr}$ becomes amplitude modulation of $B$.
Expressing

$\displaystyle
F_{tr}(x,t)=F_{tr}^{(0)}+F_{tr}^{(q_{c})}e^{iq_{c}x}+F_{tr}^{(-q_{c})}e^{-iq_{c}x}$
$\displaystyle n(x,t)=1+n^{(q_{c})}e^{iq_{c}x}+{n^{(q_{c})}}^{*}e^{-iq_{c}x}$

then substitution of the above into Eq. (2) shows that

$\left.\begin{array}[]{c}F_{tr}^{(0)}=\sqrt{p_{0}}(1+i\chi_{0})\approx\sqrt{p_{0}}\\\
F_{tr}^{(q_{c})}=i\sqrt{p_{0}}\chi_{0}n^{(q_{c})}\\\
F_{tr}^{(-q_{c})}=i\sqrt{p_{0}}\chi_{0}{n^{(q_{c})}}^{*}\end{array}\right\\}.$
(7)

Using a similar expansion of $B(x,t)$ and then Eq. (6),(7) produces

$B=\sqrt{Rp_{0}}-\sqrt{Rp_{0}}\chi_{0}n^{(q_{c})}e^{iq_{c}x}-\sqrt{Rp_{0}}\chi_{0}{n^{(q_{c})}}^{*}e^{-iq_{c}x}.$

Writing $n^{(q_{c})}=|n^{(q_{c})}|e^{-i\phi}$, then

$B=\sqrt{Rp_{0}}-2\sqrt{Rp_{0}}\chi_{0}|n^{(q_{c})}|\cos(q_{c}x-\phi),$ (8)

which allows us the optical field intensities in Eq. (1) to be written in
terms of the BEC density :

$s(x,t)\approx p_{0}+Rp_{0}-4Rp_{0}\chi_{0}|n^{(q_{c})}|\cos(q_{c}x-\phi).$
(9)

Note that if the assumption $\chi_{0}n\ll 1$ was relaxed, additional terms
with spatial frequency $2q_{c}$ would also be present. As $n^{(q_{c})}$ is
described by
$n^{(q_{c})}=\frac{1}{L}\int_{0}^{L}|\Psi(x,t)|^{2}e^{-iq_{c}x}\;dx$ , where
$L$ is the BEC length then

$|n^{(q_{c})}|\cos(q_{c}x-\phi)=\frac{1}{2\pi}\int_{0}^{2\pi}|\Psi(\theta^{\prime},t)|^{2}\cos(\theta-\theta^{\prime}))\;d\theta^{\prime}$

where $\theta=q_{c}x$ and it has been assumed that $\Psi$ is spatially
periodic with period $\Lambda_{c}$. Consequently, Eq. (9) can be written as

$s(x,t)=p_{0}+Rp_{0}-4Rp_{0}\chi_{0}\Phi(\theta,t)$ (10)

where the non-local potential , $\Phi(\theta,t)$, is
$\Phi(\theta,t)=\frac{1}{2\pi}\int_{0}^{2\pi}|\Psi(\theta^{\prime},t)|^{2}\cos(\theta-\theta^{\prime}))\;d\theta^{\prime}.$
The constant term in Eq. (10) results in a constant potential energy
contribution to Eq.(1), which can be eliminated by the transformation, $\Psi$
via $\Psi=\Psi^{\prime}\exp(-i\frac{(1+R)p_{0}\delta}{2}t)$, so that the
Schrodinger equation from Eq. (1) becomes a Gross-Pitaevskii Equation (GPE)
analogue

$i\frac{\partial\Psi^{\prime}}{\partial
t}=-\omega_{r}\frac{\partial^{2}\Psi^{\prime}}{\partial\theta^{2}}-\epsilon\Phi(\theta,t)\Psi^{\prime}$
(11)

where $\epsilon=2\delta
Rp_{0}\chi_{0}=\frac{Rp_{0}b_{0}\Gamma}{2}=\frac{p_{0}}{p_{th}}\omega_{r}.$
Eq.(11) has the same effective GPE-like form as that of the quantum HMF model
[4, 6]. Note that $\epsilon>0$ always, which corresponds to the case of the
ferromagnetic quantum HMF model. The order parameter or magnetization, $M$, is
essentially the Fourier component of the BEC density with spatial frequency
$q_{c}$ [5, 6],

$M=\left|\frac{1}{2\pi}\int_{0}^{2\pi}|\Psi|^{2}e^{i\theta}\;d\theta\right|.$
(12)

In order to demonstrate that Eq. (1)-(3) can exhibit dynamical behaviour
associated with the quantum HMF model, we consider two example cases : strong
driving, far above threshold i.e. $p_{0}\gg p_{th}$, and weak driving, just
above threshold i.e. $p_{0}$ only marginally exceeds $p_{th}$. These cases of
strong and weak driving can be interpreted physically as that where the
structuring nature of the instability completely dominates delocalising
quantum pressure in the BEC, and that where the effects of quantum pressure
are significant, respectively. In both cases we restrict the values of
$b_{0}$, $\Delta$ etc. such that $\chi_{0}n\ll 1$, for consistency with the
assumption made when deriving Eq. (11) from Eq. (13). Fig. 2 shows an example
of self-structuring displayed by the BEC-SMF model, Eq. (1)-(3), in the case
where the system is driven strongly, far above the instability threshold i.e.
$p_{0}\gg p_{th}$. The system spontaneously develops a modulated optical
intensity and modulated density with a spatial period of $\Lambda_{c}$. The
spatio-temporal distribution of the BEC density and optical intensity develop
intricate ”chevron” structures similar to those observed in [6] produced by a
”quantum Jeans instability” [5].

Figure 2: Evolution of BEC density and optical intensity for strong driving,
calculated from Eq. (1)-(3). Parameters used : $b_{0}=100$, $\Delta=500$,
$p_{0}=10p_{th}=2\times 10^{-9}$, $R=1$, $\frac{\omega_{r}}{\Gamma}=10^{-8}$.

Fig. 3 shows an example of self-structuring when the system is driven weakly,
marginally above the instability threshold. Again, both the BEC and optical
field spontaneously develop a modulation with a spatial period of
$\Lambda_{c}$, but the evolution of the system is qualitatively different from
the strongly driven case shown in Fig. 2. In the weakly-driven case, the BEC
density distribution consists of what were termed ”monoclusters” in [6] and
the chevrons are absent. The temporal behaviour is also different in the two
cases. For weak driving, after development of the optical and BEC structures
they disperse and reform regularly whereas in the strongly driven case the
temporal behaviour is more complex, with a quasiperiodic sequence of dispersal
and revival.

Figure 3: Evolution of BEC density and optical intensity for weak driving,
calculated from Eq. (1)-(3). Parameters used : $b_{0}=100$, $\Delta=500$,
$p_{0}=1.1p_{th}=2.2\times 10^{-10}$, $R=1$,
$\frac{\omega_{r}}{\Gamma}=10^{-8}$.

This mapping between the BEC-SMF model of Eq. (1)-(3) and the quantum HMF
model when $\chi_{0}n\ll 1$ allows us to gain some insight into the behaviour
of the BEC-SMF system. It explains the similarity in the evolution of the BEC
density shown in Fig. 2 with that displayed by the quantum HMF model in [6]
i.e. the chevron structures. In the weakly driven regime, it allows additional
insight if we assume a wavefunction of the form

$\Psi(\theta,t)=c_{0}(t)+c_{1}(t)\cos(\theta)$ (13)

i.e. representing two states, one of which, $|0\rangle$, is spatially uniform,
and the other, $|1\rangle$, which is spatially periodic with spatial period
$\Lambda_{c}$. Using this two-state ansatz, the effective GPE equation of the
quantum HMF model in Eq. (11) can be rewritten as an equation for the order
parameter/ or ”magnetization” , $M$ (see supplementary material):

$\left(\frac{dM}{dt}\right)^{2}+\frac{\epsilon^{2}}{2}M^{4}-\omega_{r}^{2}\left(\frac{\epsilon}{\omega_{r}}-1\right)M^{2}=0$
(14)

which has the solution

$M(t)=\sqrt{2}\frac{\omega_{r}}{\epsilon}\sqrt{\frac{\epsilon}{\omega_{r}}-1}\;\mbox{sech}\left[\omega_{r}\sqrt{\frac{\epsilon}{\omega_{r}}-1}(t-t_{0})\right]$
(15)

where
$t_{0}=\frac{\cosh^{-1}\left(\frac{\sqrt{2}\frac{\omega_{r}}{\epsilon}\sqrt{\frac{\epsilon}{\omega_{r}}-1}}{M_{0}}\right)}{\omega_{r}\sqrt{\frac{\epsilon}{\omega_{r}}-1}.}$
and $M_{0}=M(t=0)$.

Fig. 4 (inset) shows the evolution of $M$ as calculated from Eq. (15) and from
the BEC-SMF model (Eq. (1)-(3)), when the system is driven weakly. The
analytical expression for $M$ in Eq. (15) and the numerical calculation agree
well for the first period of the evolution, which in the numerical simulation
then repeats periodically as in fig. 3. The behaviour of the system in the
weakly driven regime is therefore similar to that of a two-state quantum
system where the BEC density (and consequently the optical intensity)
oscillates spontaneously in time between a spatially uniform state and a
spatially structured state. Eq. (15) predicts that the maximum value of the
order parameter, $M$ scales with distance from threshold
$\propto(p_{0}-p_{th})^{1/2}$, similar to the mean-field Ising model. Fig. 4
shows that this scaling behaviour is produced by the BEC self-structuring
model (Eq. (1)-(3)).

Figure 4: Maximum value of $M$ as a function of $p_{0}$, calculated from Eq.
(1)-(3). All other parameters used are as for Fig. 3. Inset shows evolution of
$M$ calculated from Eq. (15) (dashed line), and from a numerical solution of
Eq. (1)-(3)) (full line), for one period of oscillation when the system is
driven weakly ($p_{0}=1.05p_{th}=2.098\times 10^{-10}$). All other parameters
used are as for Fig. 3.

In addition to formation of global structures i.e spatially periodic patterns,
it has been shown that spatially localised structures can also arise in the
BEC-SMF system [17]. These structures were termed ”droplets” in [17] due to
the similarity with quantum droplets in dipolar BECs [13, 14]. An example of a
stable droplet in the BEC-SMF system is shown in Fig. 5 as calculated from
Eq.(1..3). It can be seen that a BEC of width smaller than $\Lambda_{c}$
maintains its shape due to its interaction with the optical field which it
generates. The existence of soliton solutions for the quantum HMF model was
discovered in [7], which showed that they are similar to strongly localised
gap solitons which can exist for BECs in optical lattices [18], with the
difference that in the quantum HMF model the lattice is not externally
imposed, but self-generated by the BEC. Here we show that the mapping of the
BEC-SMF system as described by Eq. (1)-(3) to the quantum HMF model as
described by Eq. (11) allows determination of the width of the droplet and its
dependence on the parameters of the system e.g. pump intensity, $p_{0}$.

Figure 5: Evolution of (a) BEC density and (b) optical intensity distribution
calculated from Eq. (1)-(3) showing a stable, localised droplet. The
parameters used were : $p_{0}=6.3\times 10^{-8}$, $b_{0}=20$, $\Delta=1600$,
$\omega_{r}/\Gamma=1.00\times 10^{-7}$ and $R=0.99$. The initial BEC
wavefunction was Gaussian with width $\sigma_{x}/\Lambda_{c}=0.07$. (c) Plot
(on log-log scale) of stable droplet width, $\sigma_{x}$ vs pump intensity,
$p_{0}$ calculated from Eq.(1)-(3). The parameters used were as for (a) and
(b).

Assuming that the profile of the BEC density is Gaussian with width
$\sigma_{x}$ i.e. $\Psi(x)\propto\exp(-\frac{x^{2}}{2\sigma_{x}^{2}})$, then
the value of $\sigma_{x}$ which minimises the energy functional,
$E(\sigma_{x})$, defined as

$E(\sigma_{x})=\frac{1}{2\pi}\int_{0}^{2\pi}\Psi^{*}\left[-\omega_{r}\frac{\partial^{2}\Psi^{\prime}}{\partial\theta^{2}}-\epsilon\Phi(x,t)\right]\Psi\;d\theta$
(16)

can be shown to be (see supplementary material)

$\frac{\sigma_{x}}{\Lambda_{c}}=\frac{1}{2\pi}\left(\frac{\omega_{r}}{\epsilon}\right)^{1/4}\propto(p_{0}b_{0}R)^{-1/4}.$
(17)

This is consistent with a more rigorous derivation of soliton solutions for
the quantum HMF model [7], with density profiles described by parabolic
cylinder functions of characteristic width $\propto\epsilon^{-1/4}$. The
predicted dependence of droplet width, $\sigma_{x}$, on pump intensity,
$p_{0}$, is confirmed in Fig. 5, where the stable droplet width is calculated
from Eq. (1)-(3) for different pump intensities and is plotted against
$p_{0}$. The power-law scaling $\sigma_{x}\propto p_{0}^{-1/4}$ predicted by
energy minimisation of the quantum HMF model agrees well with the results of
the simulations so long as $\Delta$ is sufficiently large that condition
$\chi_{0}n\ll 1$ is well satisfied. This scaling behaviour shows that the
profile and characteristic width of these optomechanical droplets are more
closely related to those of localised gap solitons [7, 18] in a self-generated
lattice than to other types of localised structures e.g. non-linear
Schrodinger equation solitons or quantum droplets observed in dipolar BECs
[13, 14].

In conclusion, we have shown that a BEC interacting with an optical field via
a feedback mirror can be a realisation of the quantum HMF model. We
demonstrated that the self-structuring of an initially uniform BEC displays
features observed previously in the quantum HMF model: for strong driving,
chevrons appear in the BEC density; for weak driving, the BEC behaves as a
two-state quantum system, with the order parameter or magnetisation evolving
as a series of sech pulses. The mapping to the quantum HMF model also allowed
prediction the dependence of BEC droplet width on pump intensity, which agreed
well with simulations of the BEC-SMF model. These results suggest that optical
diffraction-mediated interaction between atoms in a BEC may be a promising
candidate for experimental realisation of quantum HMF dynamics and
consequently be a versatile testing ground for models of quantum systems
involving long-range interactions.

###### Acknowledgements.

We acknowledge useful discussions with G. Morigi.

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