# Evidence for correlated states in a cluster of bosons with Rashba spin-orbit
coupling

Zhihao Xu  Institute of Theoretical Physics, Shanxi University, Taiyuan
030006, China  Zhenhua Yu  Institute for Advanced Study, Tsinghua University,
Beijing 100084, China<EMAIL_ADDRESS>Shizhong Zhang  Department of
Physics and Center of Theoretical and Computational Physics, The University of
Hong Kong, Hong Kong, China<EMAIL_ADDRESS>
###### Abstract

We study the ground state properties of spin-half bosons subjected to the
Rashba spin-orbit coupling in two dimensions. Due to the enhancement of the
low energy density of states, it is expected that the effect of interaction
becomes more important. After reviewing several possible ideal condensed
states, we carry out an exact diagonalization calculation for a cluster of the
bosons in the presence of strong spin-orbit coupling on a two-dimensional disk
and reveal strong correlations in its ground state. We derive a low-energy
effective Hamiltonian to understand how states with strong correlations become
energetically more favorable than the ideal condensed states.

##  I Introduction

A paradigm of traditional bosonic quantum liquid is that of Helium-4. At low
temperature, Helium-4 is an inert element with no relevant internal degrees of
freedom. At zero temperature, about $10$% of the Helium atoms condense into
the lowest energy state with zero momentum Svensson . The effects of the
inter-atomic interaction do not destroy the basic phenomena of Bose
condensation which was first predicted for noninteracting bosons, even though
significant depletion is resulted from inter-particle scattering Yang1962 ;
Leggett . Another general feature of Helium-4 in bulk is that it obeys
Galilean invariance, which plays a crucial role in Landau’s formulation of the
two fluid model for liquid Helium-4 Landau ; Khalatnikov .

In 1995, Bose-Einstein condensation was realized in cold atomic gases. The
observed atomic condensates are very close to an “ideal” condensate for
noninteracting bosons since the inter-atomic interactions are usually very
weak. With the advent of synthetic spin-orbit coupling in cold atomic gases,
the study of bosonic quantum liquid is greatly expanded in several aspects
Dalibard ; Goldman ; Galitski ; Congjun ; HuiZhai1 ; Zhang . First of all,
alkali elements like 87Rb usually have multiple internal hyperfine states, and
in an optical trap all of them can be active and this leads to multitude of
quantum phases in the so-called spinor condensate Ueda ; Ueda1 ; Ho1998 ;
ohmi1998 ; Stenger . Secondly, the inclusion of spin-orbit coupling further
introduces the coupling between the spin and momentum degrees of freedom and
significantly modifies the single particle dispersion relation. In the
experimentally realized case of spin-orbit coupling along one-dimension
Spielman1 ; Spielman2 ; Spielman3 ; JYZhang ; JWPan ; Zwierlein ; JZhang1 ;
JZhang2 ; JZhang3 ; Hamner ; Williams ; Beeler ; LeBlanc ; Olson ; Khamehchi ,
this coupling leads to the so-called stripe and plane wave condensate in
homogeneous systems and in harmonic traps Shizhong ; HuiZhai ; Ozawa ; YPZhang
; YunLi1 ; Santos2011 ; Santos2014  and the associated tricritical point
YunLi2 . Thirdly, for certain symmetric spin-orbit coupling, of Rashba or Weyl
form, the low energy density of states is significantly enhanced such that the
effects of interaction have drastic effects on the existence of Bose
condensate and could in fact destroy its existence XiaolingCui ; QiZhou .
Lastly, the inclusion of spin-orbit coupling breaks the Galilean invariance
which makes the construction of two-fluid model much more involved QZhu ;
WeiZheng ; Valiente ; HPu .

In this paper, we consider spin-half bosons subjected to the Rashba spin-orbit
coupling in two-dimensions. We will concentrate on the interplay between the
enhanced single particle ground state degeneracy and the effects of strong
repulsive interactions. In an infinite system, the Rashba spin-orbit coupling
gives rise to a ring of infinitely degenerate single particle states. We show
that in the truncated Hilbert space spanned by the single particle states on
the ring, trial wave-functions for two and four bosons with strong
correlations built-in can have lower interaction energy than various ideal
condensed states. However, for strong inter-particle interaction, it is also
important to take into account the transverse excitations away from the
degenerate ring. We consider explicitly a disk of radius $R$ and carry out
exact diagonalization calculations for a cluster of bosons in this finite
disk. We find numerical evidences for the correlated nature of the ground
state. We derive a low-energy effective Hamiltonian in the strong spin-orbit
coupling limit and use it to understand why the ground state of the
interacting bosons exhibit strong correlations, analogous to one-dimensional
Mott insulators and quantum Hall states Lee2004 ; Lee2005 .

The paper is organized as follows. In Sec. II, we discuss the general feature
of a spin-half boson subjected to Rashba spin-orbit coupling in two-
dimensional space and obtain its single particle eigen-spectrum and density of
states for later discussions. In Sec. III, we list a host of possible
condensate states and calculate their energies. In Sec. IV.1, we discuss a
trial two-body wave-functions that totally avoid interactions. In Sec. IV.2,
we discuss the possibilities of correlated ground states in the four-particle
case and show that the proposed correlated state is lower in energy than the
best condensate states considered in Sec. III. All the above discussions are
based on trial wave functions constructed from the lowest degenerate ring. We
then highlight the inadequacy of constructing trial states only from the
degenerate ground states on the ring without considering transverse excitation
(finite kinetic energy) when interaction becomes strong. In Sec.V, we take
into account the transverse excitation and perform exact diagonalization
calculation of a cluster of bosons on a disk. To understand the result
obtained, we derive an effective many-body Hamiltonian in the limit of strong
spin-orbit coupling in Sec.V.2. Two appendices (Sec.VII) are given. In Sec.A,
we discuss the exact solution of single particle states on a disk and in
Sec.B, we discuss the detailed construction of the low energy effective
Hamiltonian.

##  II General Setup

In the presence of the Rashba spin-orbit coupling, the single-particle
Hamiltonian in two dimensions (2D) is given by

$H_{\mathrm{kin}}=\frac{\hat{p}_{x}^{2}+\hat{p}_{y}^{2}}{2\mu}+\lambda(\sigma_{x}\hat{p}_{y}-\sigma_{y}\hat{p}_{x}),$
(1)

where $\hat{p}_{x(y)}$ is the momentum along $x$($y$)-direction, $\mu$ is the
mass of the boson, $\lambda$ is the strength of the spin-orbit coupling and
$\sigma_{x(y)}$ is the $x$($y$)-Pauli matrix acting on two internal states of
a boson. There are two branches of single-particle eigenstates

$\chi_{\pm,\mathbf{p}}(\mathbf{r})=\frac{1}{\sqrt{2\Omega}}\left(\begin{array}[]{c}1\\\
\pm iz_{\mathbf{p}}\end{array}\right)e^{i\mathbf{p}\cdot\mathbf{r}},$ (2)

whose eigenvalues are $\epsilon_{\pm,p}=p^{2}/{2\mu}\pm\lambda|\mathbf{p}|$
[see Fig. 1 (A-B)]. Here $\Omega$ is the area of the two-dimensional system,
$z_{\bf p}=e^{i\varphi_{\bf p}}$ with $\varphi_{\bf p}$ being the polar angle
of $\mathbf{p}$, i.e., $p_{x}=p\cos{\varphi_{\mathbf{p}}}$ and
$p_{y}=p\sin{\varphi_{\mathbf{p}}}$. The minimum of the lower Rashba band
occurs at $|\mathbf{p}|=\mu\lambda$. Note that $L_{z}+S_{z}$ is a good quantum
number with $\mathbf{S}=\boldsymbol{\sigma}/2$. Using the fact that
$\exp(i{\bf p}\cdot{\bf r})=\sum_{m}i^{m}J_{m}(pr)\exp(im\theta)$, where
$\theta$ is the angle between ${\bf p}$ and ${\bf r}$, it is straightforward
to show that the alternative spinor wave-functions (not normalized)

$\chi_{\pm,p,m}=\left(\begin{array}[]{c}e^{im\varphi_{\bf r}}J_{m}(pr)\\\ \pm
e^{i(m+1)\varphi_{\bf r}}J_{m+1}(pr)\end{array}\right)$ (3)

satisfy

$\displaystyle H\chi_{\pm,p,m}$ $\displaystyle=$
$\displaystyle\left(\frac{p^{2}}{2\mu}\pm\lambda|\mathbf{p}|\right)\chi_{\pm,p,m},$
(4) $\displaystyle(L_{z}+S_{z})\chi_{\pm,p,m}$ $\displaystyle=$
$\displaystyle(m+1/2)\chi_{\pm,p,m},$ (5)

where $J_{m}(x)$ is the $m$-th order Bessel function, and $\varphi_{r}$ is the
polar angle of $\mathbf{r}$ with respect to ${\bf p}$.

Figure 1: (Color online) (A) Three-dimensional plot of single particle
spectrum with Rashba spin-orbit coupling in two dimensions. The locus of
lowest energy states lies in the circle with $|{\bf p}|=\mu\lambda$ and its
spin direction is indicated in (A). The two branches of the single particle
spectrum touch at $k_{x}=k_{y}=0$ at $E=0$. (B) A cut of the energy spectrum
along one angular direction. It is shown that the lowest energy is given by
$E_{\rm min}=-\mu\lambda^{2}/2$ at $|{\bf p}|=\mu\lambda$. The curvature of
the lower branch at the minimal energy is given by $1/\mu$ and the dispersion
becomes sharper as $\mu\to 0$. (C) The density of states $D(E)$ resembles one-
dimensional system with characteristic $\sim 1/\sqrt{E+\mu\lambda^{2}/2}$
divergence for negative energy, and remains a constant for $E>0$, the ususal
density of states in two-dimensions.

One consequence of the inclusion of the Rashba spin-orbit interaction is that
the low energy density of states is significantly enhanced; see Fig. 1 (C).
The density of states $D(E)$ can be conveniently computed by counting the
number of states with energy below $E$, namely $N(E)=\sum_{\bf
k}[\theta(E-\epsilon_{-,k})+\theta(E-\epsilon_{+,k})]$, where $\theta(x)$ is
the Heaveside step function, and $D(E)=dN(E)/\Omega dE$ gives

$\displaystyle
D(E)=\begin{cases}\frac{\mu^{3/2}\lambda}{\sqrt{2}\pi}\frac{1}{\sqrt{E+\mu\lambda^{2}/2}},&\text{if
$E<0$;}\\\ \frac{\mu}{\pi},&\text{if $E>0$}.\end{cases}$ (6)

Namely, due to the spin-orbit coupling, the low energy
($E\to-\mu\lambda^{2}/2$) density of states diverges as in the one-dimensional
case, while those for $E>0$ retains the usual two-dimensional constant density
of states.

In the following, we assume that the repulsive interactions between the bosons
can be modeled by the contact psuedopotential

$V(\mathbf{r}_{1}-\mathbf{r}_{2})=(U+U_{s}\boldsymbol{\sigma}_{1}\cdot\boldsymbol{\sigma}_{2})\delta(\mathbf{r}_{1}-\mathbf{r}_{2}),$
(7)

where $\boldsymbol{\sigma}_{1,2}$ are the Pauli matrices of the two particles,
and $U$ and $U_{s}$ are positive interaction coupling constants. $U$ describes
the spin-independent density-density interactions, while $U_{s}$ describes the
spin-dependent spin exchange interaction. Due to Bose statistics, the contact
psuedopotential is nonzero only when two bosons are in a spin triplet state;
$V(\mathbf{r})$ in Eq. (7) is equivalent to $\widetilde{U}\delta(\mathbf{r})$
with $\widetilde{U}=U+U_{s}$. We should emphasise here that Eq. (7) is not the
most general interaction between atoms and in fact, only approximate the real
form of the interaction between 87Rb. However, we expect that such a
simplified choice of interaction would not affect the nature of the correlated
ground states which we discuss later, as their nature is rooted in the
degeneracy brought about by the spin-orbit coupling.

##  III Mean field condensate states

Before we discuss the possibility of correlated ground states, let us first
list a few candidate condensate states built from the lowest single particle
states $|\mathbf{p}|=\mu\lambda$. For an ideal condensate of $N$ bosons, the
trial wave-function can be written as

$|\mathrm{Cond}\rangle=\frac{(\phi_{0}^{\dagger})^{N}}{\sqrt{N!}}|0\rangle,$
(8)

with
$\phi_{0}^{\dagger}=\sum_{|\mathbf{p}|=\mu\lambda}c_{\mathbf{p}}a_{\mathbf{p}}^{\dagger}$
and $a_{\mathbf{p}}^{\dagger}$ the lower Rashba branch creation operators. The
expectation value of the interaction energy is given by
($z_{\mathbf{p}}=\exp(i\varphi_{\bf p})$)

$\displaystyle\langle\mathrm{Cond}|H_{\mathrm{int}}|\mathrm{Cond}\rangle$ (9)
$\displaystyle=$
$\displaystyle\frac{1}{2N!}\sum_{\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{k}_{1},\mathbf{k}_{2}}\langle\mathbf{p}_{1},\mathbf{p}_{2}|V|\mathbf{k}_{2},\mathbf{k}_{1}\rangle\langle
0|(\phi_{0})^{N}a_{\mathbf{p}_{1}}^{\dagger}a_{\mathbf{p}_{2}}^{\dagger}a_{\mathbf{k}_{2}}a_{\mathbf{k}_{1}}(\phi_{0}^{\dagger})^{N}|0\rangle$
$\displaystyle=$
$\displaystyle\frac{N(N-1)}{2}\Big{\\{}\frac{1}{4\Omega}\sum_{\mathbf{p}_{1},\mathbf{k}_{1}}\left[(U+U_{s})+2(U-3U_{s})z_{\mathbf{p}_{1}}^{*}z_{\mathbf{k}_{1}}+(U+U_{s})(z_{\mathbf{p}_{1}}^{*}z_{\mathbf{k}_{1}})^{2}\right]c_{\mathbf{p}_{1}}^{*}c_{-\mathbf{p}_{1}}^{*}c_{-\mathbf{k}_{1}}c_{\mathbf{k}_{1}}$
$\displaystyle+\frac{U+U_{s}}{4\Omega}\sum_{\mathbf{p}_{1},\mathbf{p}_{2}}[6+z_{\mathbf{p}_{1}}^{*}z_{\mathbf{p}_{2}}+z_{\mathbf{p}_{1}}z_{\mathbf{p}_{2}}^{*}]|c_{\mathbf{p}_{1}}|^{2}|\mathbf{p}_{2}|^{2}-\frac{U+U_{s}}{\Omega}\sum_{\mathbf{p}_{1}}|c_{\mathbf{p}_{1}}|^{2}|c_{-\mathbf{p}_{1}}|^{2}\Big{\\}}.$

We substitute different forms of $c_{\mathbf{p}}$, and the corresponding
interaction energy expectation values, following the usual scaling
$\sim\widetilde{U}N^{2}/\Omega$ expected for ideal condensates when $N$ is
big, are listed in Table 1. For the case of
$c_{\mathbf{p}}=\frac{1}{\sqrt{2\pi}}[\sin{\alpha}e^{-in\varphi_{\mathbf{p}}}+\cos{\alpha}e^{i\phi}e^{-im\varphi_{\mathbf{p}}}]$
with $m\neq n$, it can be shown that the energy is lowest when
$\alpha=0,\pi/2$ which is shown in Table 1.

Table 1: Interaction energy expectation values of different condensate trial wave-functions. Case | $c_{\mathbf{p}}$ | $\langle\mathrm{Cond}|H_{\mathrm{int}}|\mathrm{Cond}\rangle$
---|---|---
I | $\frac{1}{\sqrt{2\pi}}$ | $\frac{N(N-1)}{2}\frac{\widetilde{U}}{4\Omega}(7-\frac{2}{\pi})$
II | $\delta(\varphi_{\mathbf{p}})$ | $\frac{N(N-1)}{2}\frac{8\widetilde{U}}{4\Omega}$
III | $\frac{1}{\sqrt{2}}[\delta(\varphi_{\mathbf{p}})+e^{i\alpha}\delta(\varphi_{\mathbf{p}}-\pi)]$ | $\frac{N(N-1)}{2}\frac{6\widetilde{U}}{4\Omega}$
IV |  $\frac{1}{\sqrt{2\pi}}e^{in\varphi_{\mathbf{p}}},$ for $n\neq 0,-1$ | $\frac{N(N-1)}{2}\frac{\widetilde{U}}{4\Omega}(6-\frac{2}{\pi})$
V | $\frac{1}{\sqrt{2\pi}}e^{-i\varphi_{\mathbf{p}}}$ | $\frac{N(N-1)}{2}\frac{\widetilde{U}}{4\Omega}(7-\frac{2}{\pi})$

##  IV Few-body trial wave-functions

To demonstrate how the bosons can take advantage of the degeneracy of the
lower Rashba branch at $|\mathbf{p}|=\mu\lambda$ to lower their interaction
energy than an ideal condensate, we construct explicit trial wave-functions
with correlations built in for two and four bosons.

###  IV.1 Two-body

We write down the following two-body wave-function

$\psi_{0}=\sum_{|\mathbf{p}|=\mu\lambda}A_{\mathbf{p}}\left\\{\left[\begin{array}[]{c}1\\\
iz_{\mathbf{p}}\end{array}\right]_{1}\otimes\left[\begin{array}[]{c}1\\\
iz_{-\mathbf{p}}\end{array}\right]_{2}e^{-i\mathbf{p}\cdot(\mathbf{r}_{1}-\mathbf{r}_{2})}+\left[\begin{array}[]{c}1\\\
iz_{-\mathbf{p}}\end{array}\right]_{1}\otimes\left[\begin{array}[]{c}1\\\
iz_{\mathbf{p}}\end{array}\right]_{2}e^{i\mathbf{p}\cdot(\mathbf{r}_{1}-\mathbf{r}_{2})}\right\\},$
(10)

where $A_{\mathbf{p}}$ is the amplitude of the wave-function to be determined
by minimizing the interaction energy. Since we have constrained
$|\mathbf{p}|=\mu\lambda$, the kinetic energy is already at its minimum. It
turns out that by choosing appropriate $A_{\mathbf{p}}\sim
e^{i2n\varphi_{\mathbf{p}}}$, with $n\neq 0,1$, one can show that
$\psi_{0}(\mathbf{r}_{1}=\mathbf{r}_{2})=0$. Due to the contact nature of the
interaction potential, this means that Eq. (10) is an exact eigenstate of two
interacting bosons. The wave-function (10) is the same as the one found in
Ref. QiZhou . Previously Refs. Ozawa  and Sarang2011  have also shown that
two-body scattering states in the lower Rashba branch can have zero
interaction energy once the bare interactions are properly renormalised.

The fact that there exists a great many choice of $n$ such that exact two-body
states without interaction can be constructed raises the interesting
possibility of many-body correlated states, which, while still consists of
states with $|\mathbf{p}|=\mu\lambda$, avoids the interaction energy by
correlating bosons in the manner as embodied in the two-body wave-function,
Eq. (10). Unfortunately, this straightforward generalization from two-body to
many-body could not be consistently carried out, since Eq. (10) correlates two
bosons on the opposite side of the degenerate circle, and for a many-body
system, it is impossible to achieve for any arbitrary pair. This difficulty
can be most easily seen in the four bosons case to be discussed below.

###  IV.2 Four-body

If we try to build explicitly the two-body correlation in the four boson case,
an trial wave-function can be constructed

$|\mathrm{Corr}\rangle=\sum_{|\mathbf{p}|,|\mathbf{k}|=\mu\lambda}A_{\mathbf{p}}a_{\mathbf{p}}^{\dagger}a_{-\mathbf{p}}^{\dagger}B_{\mathbf{k}}a_{\mathbf{k}}^{\dagger}a_{-\mathbf{k}}^{\dagger}|0\rangle,$
(11)

with $A_{\mathbf{p}}=Ae^{i2n\varphi_{\mathbf{p}}}$ and
$B_{\mathbf{k}}=Be^{i2m\varphi_{\mathbf{k}}}$. It is clear that we are simply
building into the four particle wave-function the correlations that we have
identified in the two-body case. If $m=n$, Eq. (11) can be viewed as a
condensate of boson pairs, which is the same as the one used in Ref. Nozieres
to study the condensation of paired bosons. The normalization is
$\langle\mathrm{Corr}|\mathrm{Corr}\rangle=[(1+\delta_{mn})\pi^{2}+2\pi]A^{2}B^{2}$.
Direct evaluation yields

$\displaystyle\langle\mathrm{Corr}|H_{\mathrm{int}}|\mathrm{Corr}\rangle$
$\displaystyle=$
$\displaystyle\frac{12\widetilde{U}}{\Omega}(1+\delta_{mn})\pi^{2}A^{2}B^{2}.$
(12)

Thus the expectation value of the interaction energy for the correlated wave-
function is

$\frac{\langle\mathrm{Corr}|H_{\mathrm{int}}|\mathrm{Corr}\rangle}{\langle\mathrm{Corr}|\mathrm{Corr}\rangle}=\frac{12\widetilde{U}}{\Omega}\frac{1+\delta_{mn}}{1+\delta_{mn}+2/\pi},$
(13)

which is approximately $7.33\times\widetilde{U}/\Omega$ for $m\neq n$ and
$9.1\times\widetilde{U}/\Omega$ for $m=n$, compared with the lowest
interaction energy of an ideal condensate of Eq. (8) given in Tab. 1, Case IV
with $N=4$, $(9-3/\pi)\widetilde{U}/\Omega\approx
8.04\times\widetilde{U}/\Omega$. Thus for four bosons, Eq. (11) with $m\neq n$
has lower interaction energy than the ideal trial condensed states. A super-
fragmented trial state was proposed in Ref. QiZhou , whose interaction energy,
if evaluated with our assumed interaction potential for $N=4$, would be
$3.44\tilde{U}/\Omega$, even lower. This comparison indicates that different
ways to correlate the bosons can suppress the interaction energy to different
degrees.

Despite the effort to build in two-body correlation, the condensate states and
the trial correlated wave functions for four bosons have at least two common
features. Firstly, they are all built from the lowest energy states on the
degenerate ring and secondly their energies are all proportional to $1/\Omega$
and coupling constant $\widetilde{U}$. When $\widetilde{U}=0$, all bosons will
condense in the lowest single particle state. As one increases $\widetilde{U}$
from zero but still keeps it small compared to the single particle excitation
energy, it is expected that the system falls in the mean field regime, as
discussed above. However, as $\widetilde{U}$ increases further, the weakly
dispersing lowest Rashba band becomes very important. One way to see this is
that if one starts with a finite system with area $\Omega$, the single
particle states in the lower Rashba branch with momentum $p-\mu\lambda\lesssim
1/\sqrt{\Omega}$ differ only by energy of order of $1/\Omega$ with the states
in the degenerate ring. On the other hand, the apparent scale of the
interaction energy is also $1/\Omega$ (cf. Eq. (13)). This indicates that, at
least in the strong interaction regime, it is important to consider the
transverse excitations, and their effects have to be taken into account in the
construction of effective Hamiltonian (see Sec. V.2). The above considerations
prompt us to study the bosons in a disk of finite size $\Omega$ in the next
section, where we are able to treat both the kinetic and interaction energies
on the same footing.

##  V Small Clusters in a Disk.

Correlated ground states for a homogeneous interacting Bose gas with the
Rashba spin-orbit coupling XiaolingCui ; QiZhou  have been explored by means
of fermionization Glazman  and by vortex attachement Scarola . Numerical
calculations have also revealed the correlated nature of the system in a
harmonic trap HuiHu2012 .

To understand how strong spin-orbit coupling can help the bosons to suppress
interaction energy, we study a cluster of bosons with the Rashba spin-orbit
coupling in a two-dimensional disk of radius $R$ with a hard wall boundary
condition, while earlier works has focused on two-dimensional harmonic trap
confinement Ramachandhran1 ; Ramachandhran2 ; HuiHu1 ; HuiHu3 ; XiaolingCui .
In this case, the single particle eigen-wave-functions with eigenenergies
$E_{nm}$ have the generic form

$\displaystyle\Phi_{nm}(r,\varphi)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}f_{nm}(r)e^{im\varphi}\\\
\xi_{nm+1}(r)e^{i(m+1)\varphi}\end{array}\right),$ (16)

where $m+1/2$ is the eigenvalue of $S_{z}+L_{z}$, and $n$ is the radial
quantum number determined when $\Phi_{nm}(R,\varphi)=0$ is imposed on the
wave-function in the radial direction. The detailed expressions of $f_{nm}(r)$
and $\xi_{nm+1}(r)$ in terms of Bessel functions are given in Appendix A. In
the following, we use $x\equiv\sqrt{2\mu^{2}\lambda^{2}R^{2}}$ to quantify the
spin-orbit coupling strength.

Figure 2: (Color online) (a) Energy spectrum $E_{nm}$ for single-particle
states at small spin-orbit coupling strength ($x=0.1$) with radial quantum
number $n=0\to 5$ (we take $1/2\mu R^{2}$ as energy units); (b), (c) wave-
function $f_{0m}(r)$ and $\xi_{0m+1}(r)$ of single-particle states in the
$n=0$ manifold at $x=0.1$ change with $r$; (d) Energy spectrum $E_{nm}$ for
single-particle states at large spin-orbit coupling strength ($x=100$) with
radial quantum number $n=0\to 5$; (e), (f) wave-function $f_{0m}(r)$ and
$\xi_{0m+1}(r)$ of single-particle states in the $n=0$ manifold at $x=100$
change with $r$. In (b),(c), (e) and (f) solid line is for $m=0$, dash-dotted
line is for $m=5$, dashed line is for $m=10$ and dotted line is for $m=15$.

Figure 2 shows the energy spectrum $E_{nm}$ and the associated ground state
wave-functions $f_{0m}(r)$ and $\xi_{0m+1}(r)$ ($n=0$) for both small (a-c)
and large (d-f) spin-orbit coupling strengths $x$. We take $1/2\mu R^{2}$ as
energy units throughout. In Figure 2 (a), the energy spectrum for small spin-
orbit coupling ($x=0.1$) is strongly dispersive. The small spin-orbit coupling
can be regarded as a perturbation to the usual quadratic kinetic energy.
However, when the spin-orbit coupling strength is large enough such as $x=100$
as shown in Figure 2 (d), the energy spectrum segregates into distinct energy
bands and the lower energy bands corresponding to smaller $n$ are more flat.
For $x=100$, the band bottom of the lowest energy band is found numerically to
be at energy about $2.47\times 1/2\mu R^{2}-\mu\lambda^{2}/2$. As an example,
the energies for $n=0$ are essentially non-dispersive for the states with
$m\in[-40,40]$. In this limit, the quadratic kinetic energy can be viewed as a
perturbation to the spin-orbit coupling. Figure 2 (b) and (c) show the wave
functions $f_{0m}(r)$ and $\xi_{0m+1}(r)$ versus $r$ for the lowest energy
band $n=0$. For a larger spin-orbit coupling strength ($x=100$), the wave-
functions exhibit increased oscillations compared those with a small spin-
orbit coupling strength. With increasing $|m|$, the weight of the wave-
functions moves towards $r=R$. Here we would like to contrast the behavior of
$f_{0m}(r)$ and $\xi_{0m+1}(r)$ with that in the quantum Hall problem, where
the lowest Landau level wave-functions in the symmetric gauge are localized
within distance $\ell$ at a set of concentric ring with radius
$\sqrt{2m}\ell$, where $\ell$ is the magnetic length. In contrast, the wave-
functions $f_{0m}(r)$ and $\xi_{0m+1}(r)$ are more spread out and with
considerable oscillations for large spin-orbit coupling.

When the spin-orbit coupling strength $x$ is large [Fig. 2 (d)], the $n=0$
energy band is nearly flat and does not overlap with the higher bands for a
large range of $m$ [cf. Figure 2(d)], we can project the Hilbert space into
the lowest band $n=0$. As a result, we shall omit the radial quantum index
$n$. The low energy physics can be described by the Hamiltonian
$H=H_{\mathrm{kin}}+H_{\mathrm{int}}$ with
$H_{\mathrm{kin}}=E_{m}a_{m}^{\dagger}a_{m}$, where $a_{m}^{\dagger}$ is the
creation operator for states with wave-function $\Phi_{0m}({\bf r})$. The
interaction Hamiltonian is given by

$\displaystyle H_{\mathrm{int}}=$
$\displaystyle\sum_{m_{1},m_{2},m_{3},m_{4}}V(m_{1},m_{2},m_{3},m_{4})a_{m_{1}}^{\dagger}a_{m_{2}}^{\dagger}a_{m_{3}}a_{m_{4}}\delta_{m_{1}+m_{2}-m_{3}-m_{4}},$
(17)

where the matrix elements $V(m_{1},m_{2},m_{3},m_{4})$ are given by

$\displaystyle
V(m_{1},m_{2},m_{3},m_{4})=\pi\widetilde{U}\int_{0}^{R}drr[f^{*}_{m_{1}}(r)f_{m_{4}}(r)+\xi^{*}_{m_{1}+1}(r)\xi_{m_{4}+1}(r)][f^{*}_{m_{2}}(r)f_{m_{3}}(r)+\xi^{*}_{m_{2}+1}(r)\xi_{m_{3}+1}(r)].$
(18)

It is worth to mention that the lowest band approximation we take here breaks
down whenever the average energy of a boson (while the interaction energy may
be small, the kinetic energy and thus the total energy can be still quite
large) is comparable to the band gap such that higher bands are substantially
populated.

###  V.1 Exact Diagonalization

We use the above Hamiltonian $H=H_{\text{kin}}+H_{\text{int}}$ projected to
the lowest band to solve the states of $N$ bosons in the strong Rashba spin-
orbit coupling limit by exact diagonalization. In the lowest band, we truncate
the single particle states up to the azimuthal angular momentum $|m|=L$ by
which numerical convergence can be reached. Being specific, we calculate for
$N=2,3,4$ with $x=100$, $\widetilde{U}=4/\mu$ and $L=40$.

Figure 3: (Color online) (a)-(c) Lowest Energies $E_{M}$ (in units of $1/2\mu
R^{2}$) of different total azimuthal angular momentum $M$ for different
$N=2,3,4$, respectively. The inset is the enlarged view around the ground
state. (d)-(f) $V_{M}/N$ (in units of $1/2\mu R^{2}$) for each state of $M$
for $N=2,3,4$ from (d) to (f), respectively. Here, we take $x=100$,
$\widetilde{U}=4/\mu$, and $L=40$. Figure 4: (Color online) (a), (b) The
ground state occupation number $N_{m}$ of the lowest band single particle
states for different $\tilde{U}$. (c), (d) Density distribution $n(r)$ of the
ground state in real space for different $\widetilde{U}$. The left and right
columns are corresponding to $N=3$ and $4$, respectively.

Figure 3 shows the total energy $E_{M}$ and the expectation value of the
interaction Hamiltonian $V_{M}\equiv\left\langle
H_{\mathrm{int}}\right\rangle$ in the eigenstate of a cluster of bosons with
total azimuthal angular momentum is $M$. There are two degenerate ground
states at $M=10$ and $-12$ for $N=2$ (Figure 3(a)), at $M=-1$ and $-2$ for
$N=3$ (Figure 3(b)), and at $M=28$ and $-32$ for $N=4$ (Figure 3(c)). The
double degeneracy of the ground states can be understood in the following way.
Without interaction, if $\Phi_{m}$ of $L_{z}+S_{z}=m+1/2$ is a single particle
eigenfunction with energy $E_{m}$ of Eq. (1), so is
$e^{i\frac{\pi}{2}\sigma_{y}}\Phi_{m}^{*}(\mathbf{r})$. By using
$J_{m}(x)=(-1)^{m}J_{-m}(x)$, we find that
$e^{i\frac{\pi}{2}\sigma_{y}}\Phi_{m}^{*}(\mathbf{r})$ has
$L_{z}+S_{z}=-m-3/2$; both the single particle states of angular momentum
symmetric about $L_{z}+S_{z}=-1/2$ are degenerate. The interaction Hamiltonian
is time-reversal invariant with the time-reversal operator defined as
$\mathcal{T}=e^{-i\frac{\pi}{2}\sigma_{y}}\mathcal{K}$ with the complex
conjugation operator $\mathcal{K}$. Thus in the case of a cluster of $N$
bosons, the energy spectrum should be symmetric with respect to $M=-N/2$,
which agrees with Figure 3.

Figure 3 also shows that there is a large range of $M$ for which the low lying
$E_{M}$ is rather flat. In the flat region, the energy per particle for the
relevant $\widetilde{U}$ we consider is about from $0.1\times 1/2\mu R^{2}$ to
$0.3\times 1/2\mu R^{2}$, measured with respect to the band bottom of the
lower Rashba band. This is much smaller than the energy difference between the
lowest two bands $n=0$ and $1$ at $m=0$, approximately $10/2\mu R^{2}$, and
justifies our lowest band approximation. Whenever the energy per particle
(with respect the lowest band bottom whose value is $2.47\times 1/2\mu
R^{2}-\mu\lambda^{2}/2$) becomes comparable to $10/2\mu R^{2}$, our lowest
band approximation breaks down.

On the other hand, Figure 3 shows that the interaction energy per particle of
the ground state $|s_{0}\rangle$ is orders of magnitude smaller than its
energy per particle. This magnitude difference suggests that the bosons are
taking advantage of the nearly flat lowest band to correlate in a way such
that the interaction energy, usually of order $\ln(\mu\lambda R)/2\mu R^{2}$
(see Eq. (20) below), which for $x=100$ is about $4.3\times 1/2\mu R^{2}$, is
greatly suppressed. Figure 4 (a-b) show that when $\widetilde{U}$ increases,
bosons are prompted to the single particle states of larger $|m|$ in the
lowest band, thus increasing the kinetic energy. Correspondingly, Fig. 4 (c-d)
shows that the density distribution is pushed away from the center due to
occupation of higher angular momentum states $m$. Here $N_{m}$ is the
occupancy of single particle state with angular momentum $m$. With the
spreading of $N_{m}$, it is possible for bosons to explore a larger set of
nearly-degenerate lowest band single particle states to suppress interactions.
This difference also indicates that the kinetic energy of the cluster, which
arises from the weakly dispersing band (whose width is about $0.6\times
1/(2\mu R^{2})$ for $L=40$ and $x=100$), comprises the majority of the energy
of the system. For example, the dispersion of the single particle lowest band
for $x=100$ (cf. Fig. 2(d)) can be well fitted by the formula
$(E_{m}+\mu\lambda^{2}/2)/(1/2\mu R^{2})=5.11\times 10^{-4}(m+1/2)^{2}+2.47$.
Given that the boson occupation number $N_{m}$ is noticeable up to $m=20$ (see
Fig. 4(a) and (b)), the kinetic energy of each boson is estimated to be about
$0.2\times 1/2\mu R^{2}$ when measured from the band bottom, agreeing with our
numerical calculations. Thus it is necessary to start with an effective
Hamiltonian for which the transverse excitations away from the degenerate ring
are included.

Figure 5: (Color online) Correlation function $g_{2}(0,r)$ of the ground state
versus $r/R$ for different $\widetilde{U}$. The probability for two bosons to
stay close to each other is suppressed and the suppression is larger as the
increase of the interaction strength. Here, we show the case of $N=3$. The
asymptotic value is about $2/3\sim 0.67$, differing from $1$ because of the
finite number ($N=3$) of particles considered. Figure 6: (Color online) (a),
(b) Energy $E_{0}$ and the interaction energy $V_{0}=\langle
s_{0}|H_{\mathrm{int}}|s_{0}\rangle$ of the ground state $|s_{0}\rangle$ as
functions of the interaction strength $\widetilde{U}$ in the strongly
correlated regime, respectively ; (c), (d) $E_{0}$ and $V_{0}$ versus
$\tilde{U}$ in the weakly correlated regime, respectively. Here, we take
$N=3$, $L=40$, $x=100$ and use $1/2\mu R^{2}$ as energy units and $1/\mu$ as
units for $\widetilde{U}$.

The correlation of the ground state $|s_{0}\rangle$ is manifested in the
density-density function

$\displaystyle
g_{2}(0,r)=\frac{\left\langle{:n(\mathbf{r})n(0):}\right\rangle}{\left\langle
n(\mathbf{r})\right\rangle\left\langle n(0)\right\rangle},$ (19)

where $n(\mathbf{r})$ is the density operator of the bosons, the symbol $:\,:$
means normal ordering and $\langle\dots\rangle$ is averaging with respect to
$|s_{0}\rangle$. Since $|s_{0}\rangle$ is an eigenstate of total
$L_{z}+S_{z}$, $g_{2}(0,r)$ is a function of $r$. Figure 5 indicates that the
probability for two bosons to stay close to each other is greatly suppressed
at large interaction $\widetilde{U}$, and as $\widetilde{U}$ increases, such
suppression becomes stronger. In Fig. 5, $g_{2}(0,r)$ shows a maximum at about
$r/R=0.25$, which can be understood from the following semi-classical picture.
From Fig. 4 (a) for $N=3$, the most probable relative angular momentum between
two bosons is about $\Delta m=40$. Due to the strong spin-orbit coupling, the
typical linear momentum of a boson is about $\mu\lambda$ and the typical
relative linear momentum between two bosons is about $2\mu\lambda$. Thus the
most probable distance $\Delta r$ between two bosons is about $\Delta
r/R=\Delta m/2\mu\lambda R=\Delta m/\sqrt{2}x\approx 0.28$. Beyond the
maximum, $g_{2}(0,r)$ converges to a constant. The constant is noticeably
smaller than unity due to the boson number $N$ being comparable to unity. Note
that the numerator of $g_{2}(0,r)$ normalises as $\int
d^{2}\mathbf{r}\left\langle{:n(\mathbf{r})n(0):}\right\rangle=(N-1)\langle
n(0)\rangle$, which we have checked numerically, while the denominator of
$g_{2}(0,r)$ normalises as $\int d^{2}\mathbf{r}\left\langle
n(\mathbf{r})\right\rangle\left\langle n(0)\right\rangle=N\langle
n(0)\rangle$. This means that for $N=3$, the asymptotic value of $g_{2}(0,r)$
is about $2/3\sim 0.67$, consistent with numerical findings.

The underlying correlation between the bosons also gives rise to an intriguing
behavior of the interaction energy $V_{0}$ of the ground state $|s_{0}\rangle$
with respect to the interaction strength $\widetilde{U}$. Figure 6 shows the
variation of the ground state energy $E_{0}$ and the interaction energy
$V_{0}$ of the ground state $|s_{0}\rangle$ for $N=3$ with $\widetilde{U}$
both in the strongly and weakly correlated regimes. By the Hellmann-Feynman
theorem, $E_{0}$ must be a strictly increasing function of $\widetilde{U}$,
agreeing with Figure 6 (a) and (c). What is surprising is that, however,
Figure 6 (b) shows that the interaction energy $V_{0}$ is a decreasing
function of $\widetilde{U}$ in the range $\mu\widetilde{U}\in[0.5,0.9]$,
contrary to what one would expect for an ideal condensed state (cf. Tab. 1),
including mean field vortex states. It is also opposite to the
$\widetilde{U}$-dependence of the constructed wave-function
$|\rm{Corr}\rangle$ (Eq. (11)) and the super-fragmented state proposed in Ref.
QiZhou . In the weakly correlated limit where the interaction energy is much
smaller than the energy difference between neighbouring states which is found
to be of order $10^{-3}$ for $x=100$, Figure 6 (d) shows that interaction
energy $V_{0}$ increases linearly with $\widetilde{U}$. So does $E_{0}$ in
Fig. 6 (c). Thus as $\widetilde{U}$ increases, the ground state shall evolve
from a weakly interacting regime where the usual mean field treatment is
applicable to a strongly correlated regime where crucial interaction effects
can not be captured by mean field.

###  V.2 Effective low-energy Hamiltonian

To understand why in the presence of strong spin-orbit coupling it is
favorable for a cluster of bosons to correlate in a way to suppress the
interaction energy, we extract the leading order of the interaction
Hamiltonian Eq. (17). In the limit $x\to\infty$, we can simplify the matrix
elements $V(m_{1},m_{2},m_{3},m_{4})$, and as shown in Appendix B, the final
form of the interaction Hamiltonian takes a particularly simple form, to
leading order in $\ln(\mu\lambda R)$,

$\displaystyle H^{\prime}_{\mathrm{int}}=\frac{4\widetilde{U}}{\pi
R^{2}}\ln(\mu\lambda R)\sum_{l}(A_{l}^{\dagger}A_{l}+B_{l}^{\dagger}B_{l}),$
(20)

where $A_{l}=\sum_{m}\widetilde{N}_{l-m}\widetilde{N}_{m}a_{m}a_{l-m}$ and
$B_{l}=\sum_{m}(-1)^{m}\widetilde{N}_{2l-m}\widetilde{N}_{m}a_{m}a_{2l-m}$.
The dimensionless normalization factor $\widetilde{N}_{m}$ is given in
Appendix B. It is worth to emphasise that Eq. (20) is derived within the
lowest band approximation, which as we argue above holds for our numerical
calculation with $x=100$, cutoff $L=40$ and up to $N=4$. Obviously, for any
states $\left|{\psi}\right\rangle$,
$\left\langle{\psi}\right|H^{\prime}_{\mathrm{int}}\left|{\psi}\right\rangle\geq
0$. The strength of $H^{\prime}_{\text{int}}$ is logarithmically divergent in
the large spin-orbit coupling limit. If we now neglect the weak dispersion of
the lowest band which is of order of $1/(\lambda^{2}R^{4})$ as compared with
$\ln(\mu\lambda R)/R^{2}$, the ground state will be the one which minimises
$H^{\prime}_{\mathrm{int}}$. For a state $\left|{s}\right\rangle$,
$\left\langle{s}\right|H^{\prime}_{\mathrm{int}}\left|{s}\right\rangle=0$ only
if $A_{l}\left|{s}\right\rangle=B_{l}\left|{s}\right\rangle=0$ for any $l$.
The existence of such nontrivial correlated states can be considered in the
following way. We assume that the spin-orbit coupling is so large that the
lowest band can be considered flat for the states of $m\in[-L,L]$. For $N$
bosons, the dimension of the Hilbert space is $(2L+N)!/(2L)!N!$. The
requirement that $A_{l}\left|{s}\right\rangle=B_{l}\left|{s}\right\rangle=0$
for $l\in[2L,-2L]$ imposes $2(4L+1)(2L+N-2)!/(2L)!(N-2)!$ constraints. For
fixed $N$, the constraints can be simultaneously satisfied if $L$ is large
enough; in other words, when the spin-orbit coupling is large enough. On the
other hand, an ideal condensate of the form Eq. (8) can not avoid the
interaction of the form $H^{\prime}_{\text{int}}$ since a condensate wave-
function is determined by assigning $2L+1$ superposition coefficients. Thus in
this limit, the ground state $|s_{0}\rangle$ of a cluster of bosons is
expected to have the correlation such that $\langle
s_{0}|\sum_{l}(A_{l}^{\dagger}A_{l}+B_{l}^{\dagger}B_{l})|s_{0}\rangle\ll 1$,
which is compatible as shown in Fig. 6 (b) where the interaction energy
$V_{0}$ is much smaller than $1/2\mu R^{2}$. It is worth mentioning that
$H^{\prime}_{\text{int}}$ is formally similar to a Hamiltonian whose ground
state can be correlated Mott insulators, corresponding to fractional quantum
hall states Lee2004 ; Lee2005 . The nonlocal nature of $\widetilde{N}_{m}$
(see Appendix B), however, precludes constructing exact ground states of
$H^{\prime}_{\text{int}}$ in our case. It is important to note the difference
between Eq. (20) and other effective interactions derived in Refs. Ozawa  and
QiZhou  for infinite systems; the renormalised effective interaction derived
in Ref. Ozawa  is applicable to the lower branch states while the one in Ref.
QiZhou  is meant for states on the degenerate ring. In all, the above
considerations based on Eq. (20) made it clear why other than an ideal
condensed state, the bosons would prefer to be in a ground state with strong
correlations, whose properties we have revealed in details by the previous
numerical calculation.

##  VI Conclusion

In this work, we describe how the enhanced low-energy density of states
changes the properties of a spin-half boson subjected to the Rashba spin-orbit
coupling. We carry out an exact diagonalisation calculation for a cluster of
bosons with strong spin-orbit coupling and reveal the correlated nature of its
ground state. We derive the corresponding effective Hamiltonian [Eq.(20)],
which is particular simple in form and suggests a correlated ground state that
is analogous to correlated Mott insulator and quantum Hall states, suggested
earlier in the literature.

However, standing alone, the argument based on the effective Hamiltonian (20)
is deficient at least in two aspects. Firstly, the weak dispersion of the
lowest band and sub-leading terms of the interaction Hamiltonian need to be
considered in order to uniquely determine the true ground state. Secondly, the
comparison with ideal condensed states is less relevant if bosons can lower
their energy substantially by strong depleting from condensates. The later
problem requires one to investigate possible variational ground states of a
condensed boson system with strong depletion, which we do not attempt here;
see however, the relevant discussion of Ref. Radic2015 . More extensive
numerical calculations (e.g., variational Monte Carlo) are necessary to
provide more evidences to the correlated nature of the ground state.

##  VII Acknowledgment

We thank W. Cole, N. Trivedi, A. Paramenkanti and J. Yao for their discussions
in the initial stage of this work. Z.X. is supported by NSFC under Grant No.
11547111. Z.Y. is supported by NSFC under Grant No. 11474179 and 11204152, and
the Tsinghua University Initiative Scientific Research Program. S.Z. is
supported by Hong Kong Research Grants Council, GRF HKU-709313P, Collaborative
research fund, HKUST3/CRF/13G, and the Croucher Foundation under the Croucher
Innovation Award.

##  Appendix A Single Particle States on a Disk.

To obtain the explicit form of the function $f_{nm}(r)$ and $\xi_{nm+1}(r)$,
we note that without the hard wall boundary condition, the single particle
wave-function can be written as, since $m=S_{z}+L_{z}$ is a good quantum
number

$\chi_{m}(r,\varphi)=\left(\begin{array}[]{c}\alpha e^{im\varphi}J_{m}(kr)\\\
\beta e^{i(m+1)\varphi}J_{m+1}(kr)\end{array}\right),$ (21)

with $\alpha$ and $\beta$ is the coefficients to be determined later.
Requiring that

$H_{\mathrm{kin}}\chi_{m}(k)=E_{m}\chi_{m}(k),$ (22)

where $E_{m}$ is the eigenvalue with magnetic quantum number $m$. For a
specific value of $E_{m}$, there are two values of $k$ that can be found, by
requiring that

${\rm det}\left[\begin{array}[]{cc}\frac{\hbar^{2}k^{2}}{2\mu}-E_{m}&\lambda
k\\\ \lambda k&\frac{\hbar^{2}k^{2}}{2\mu}-E_{m}\end{array}\right]=0.$ (23)

As a result, we find that
$\epsilon_{m}^{(\pm)}=E_{m}+\mu\lambda^{2}\pm\sqrt{2\mu\lambda^{2}E_{m}+\mu^{2}\lambda^{4}}$
with $\epsilon_{m}^{(\pm)}=\hbar^{2}k_{m}^{(\pm)2}/{2\mu}$. Let us denote the
corresponding wave-functions as $\chi^{\pm}_{m}(r,\varphi)$. The coefficients
are given by

$\displaystyle\alpha_{m}^{(\pm)}$ $\displaystyle=\frac{\lambda
k_{m}^{(\pm)}}{E_{m}-\epsilon_{m}^{(\pm)}}\beta_{m}^{(\pm)}.$ (24)

At the moment, there is no requirement on the overall normalization. Note that
$\alpha_{m}^{(\pm)}$ are functions of $E_{m}$. Now we need to impose the
boundary conditions on the single particle wave-function such that it vanishes
at $r=R$. This can be done by forming a superposition of
$\chi^{\pm}_{m}(r,\varphi)$ (both of energy $E_{m}$), with coefficients
$a_{m},b_{m}$

$\displaystyle\Phi_{m}(r,\varphi)$ $\displaystyle=$ $\displaystyle
a_{m}\left(\begin{array}[]{c}\alpha^{(+)}_{m}e^{im\varphi}J_{m}(k_{m}^{(+)}r)\\\
\beta^{(+)}_{m}e^{i(m+1)\varphi}J_{m+1}(k_{m}^{(+)}r)\end{array}\right)+b_{m}\left(\begin{array}[]{c}\alpha^{(-)}_{m}e^{im\varphi}J_{m}(k_{m}^{(-)}r)\\\
\beta^{(-)}_{m}e^{i(m+1)\varphi}J_{m+1}(k_{m}^{(-)}r)\end{array}\right)$ (29)

Requiring $\Phi_{m}(R,\varphi)=0$ leads to the equation for the coefficients
$a_{m}$ and $b_{m}$ and the condition for them to have non-zero solution is
that the corresponding determinant is zero. Namely, we require

${\rm
det}\left[\begin{array}[]{cc}\alpha^{(+)}_{m}J_{m}(k_{m}^{(+)}R)&\alpha^{(-)}_{m}J_{m}(k_{m}^{(-)}R)\\\
\beta^{(+)}_{m}J_{m+1}(k_{m}^{(+)}R)&\beta^{(-)}_{m}J_{m+1}(k_{m}^{(-)}R)\end{array}\right]=0.$
(30)

From this condition, we determine the allowed set of eigen-energy $E_{nm}$,
which we label by the radial quantum number $n$. Once we know $E_{nm}$, we can
find the corresponding $k^{(\pm)}_{nm}$ and coefficients
$\alpha_{nm}^{(\pm)}$, $\beta_{nm}^{(\pm)}$, $a_{nm}$ and $b_{nm}$ for a
specific radial quantum number $n$. The explicit form of $f_{nm}(r)$ and
$\xi_{nm+1}(r)$ can then be written as

$\displaystyle f_{nm}(r)$
$\displaystyle=a_{nm}\alpha_{nm}^{(+)}J_{m}(k^{(+)}_{nm}r)+b_{nm}\alpha_{nm}^{(-)}J_{m}(k^{(-)}_{nm}r),$
(31) $\displaystyle\xi_{nm+1}(r)$
$\displaystyle=a_{nm}\beta_{nm}^{(+)}J_{m+1}(k^{(+)}_{nm}r)+b_{nm}\beta_{nm}^{(-)}J_{m+1}(k^{(-)}_{nm}r).$
(32)

The full spinor wave-function $\Phi_{nm}(r,\varphi)$ can now be chosen
orthonormal with $\int d\varphi\int
rdr\Phi_{nm}^{\dagger}(r,\varphi)\Phi_{n^{\prime}m^{\prime}}(r,\varphi)=\delta_{n,n^{\prime}}\delta_{m,m^{\prime}}$.
For the numerical calculation shown below, we use $1/(2\mu R^{2})$ as the
units for energy and define $x=\sqrt{2\mu^{2}\lambda^{2}R^{2}}$ to quantify
the strength of spin-orbit coupling.

##  Appendix B Analyzing effective Hamiltonian in limits

Consider insider a two dimensional cylinder of radius $R$. The single particle
wave-function of Eq.(1) in the main text is

$\left(\begin{array}[]{c}e^{im\varphi}J_{m}(kr)\\\
-e^{i(m+1)\varphi}J_{m+1}(kr)\end{array}\right).$ (33)

The wave-function are required to vanish at $r=R$; the single particle wave-
function on the lowest band in the limit $\lambda R\to\infty$ and under the
open boundary condition is described by

$\displaystyle\Phi_{m}(r,\varphi)$ $\displaystyle=$ $\displaystyle
N_{m}\left[J_{m}(k^{-}R)\left(\begin{array}[]{c}J_{m}(k^{+}r)e^{im\varphi}\\\
-J_{m+1}(k^{(+)}r)e^{i(m+1)\varphi}\end{array}\right)-J_{m}(k^{(+)}R)\left(\begin{array}[]{c}J_{m}(k^{(-)}r)e^{im\varphi}\\\
J_{m+1}(k^{(-)}r)e^{i(m+1)\varphi}\end{array}\right)\right]$ (38)
$\displaystyle=$ $\displaystyle
N_{m}\left(\begin{array}[]{c}f_{m}(r)e^{im\varphi}\\\
\xi_{m+1}(r)e^{i(m+1)\varphi}\end{array}\right),$ (41)

where $N_{m}$ is normalization factor. To the lowest order
$k^{\pm}\sim\lambda\pm\sqrt{\Delta}$ where $\Delta=E-E_{\mathrm{min}}\sim
1/R^{2}$ and set the mass of the particle $\mu=1$.

To determine the normalization factor $N_{m}$ and likewise in the following
evaluation for the interaction matrix elements, we expand

$f_{m}(r)=-[J_{m}(k^{+}R)J_{m}(k^{-}r)-J_{m}(k^{-}R)J_{m}(k^{+}r)],$ (42)
$\xi_{m+1}(r)=J_{m}(k^{+}R)J_{m+1}(k^{-}r)-J_{m}(k^{-}R)J_{m+1}(k^{+}r),$ (43)

to the lowest order of $\Delta/\lambda^{2}$ and find

$f_{m}(r)=-2\sqrt{\Delta}[J^{\prime}_{m}(\lambda R)J_{m}(\lambda
r)R-J_{m}(\lambda R)J^{\prime}_{m}(\lambda r)r],$ (44)
$\xi_{m}(r)=2\sqrt{\Delta}[J^{\prime}_{m}(\lambda R)J_{m+1}(\lambda
r)R-J_{m}(\lambda R)J^{\prime}_{m+1}(\lambda r)r].$ (45)

Thus the normalization condition is

$\displaystyle\frac{1}{N_{m}^{2}}$ $\displaystyle=$ $\displaystyle
2\pi\int_{0}^{R}drr[f_{m}^{2}(r)+\xi_{m+1}^{2}(r)]$ (46) $\displaystyle\sim$
$\displaystyle\frac{32}{3\pi\lambda^{2}}R^{2}\Delta[2-(-1)^{m}\sin(2\lambda
R)],$

where in the second line we use the asymptotic approximation of Bessel
function of the first kind $J_{m}(z)\sim\sqrt{\frac{2}{\pi
z}}\cos(z-\frac{m\pi}{2}-\frac{\pi}{4})$.

To calculate the leading order of the interaction, we need to take into
account only

$f_{m}(r)=-2\sqrt{\Delta}J^{\prime}_{m}(\lambda R)J_{m}(\lambda r)R,$ (47)
$\xi_{m+1}(r)=2\sqrt{\Delta}J^{\prime}_{m}(\lambda R)J_{m+1}(\lambda r)R.$
(48)

Let’s define $\widetilde{N}_{m}=2\sqrt{\Delta}N_{m}J^{\prime}_{m}(\lambda
R)\sqrt{R^{3}/\lambda}$ which is of order one when $x\to\infty$. The
interaction matrix element is

$\displaystyle V_{m_{1},m_{2},m_{3},m_{4}}$ $\displaystyle=$
$\displaystyle\frac{2\pi\lambda^{2}\widetilde{U}}{R^{2}}(\prod_{i=1}^{4}\widetilde{N}_{m_{i}})\delta_{m_{1}+m_{2},m_{3}+m_{4}}$
(49) $\displaystyle\int_{0}^{R}drr[J_{m_{1}}(\lambda r)J_{m_{4}}(\lambda
r)+J_{m_{1}+1}(\lambda r)J_{m_{4}+1}(\lambda r)][J_{m_{2}}(\lambda
r)J_{m_{3}}(\lambda r)+J_{m_{2}+1}(\lambda r)J_{m_{3}+1}(\lambda r)]$
$\displaystyle\sim$ $\displaystyle\frac{4\widetilde{U}}{\pi
R^{2}}(\prod_{i=1}^{4}\widetilde{N}_{m_{i}})[1+\cos(\frac{m_{1}+m_{3}-m_{2}-m_{3}}{2}\pi)]\ln(\lambda
R),$

where

$\int_{0}^{R}drrJ_{m_{1}}J_{m_{2}}J_{m_{3}}J_{m_{4}}\sim\frac{1}{2(\pi\lambda)^{2}}[1+\cos(\frac{m_{1}+m_{4}-m_{2}-m_{3}}{2}\pi)+\cos(\frac{m_{1}+m_{3}-m_{2}-m_{4}}{2}\pi)]\ln(\lambda
R).$ (50)

The symmetrized matrix element is

$\widetilde{V}_{m_{1},m_{2},m_{3},m_{4}}=\frac{1}{4}(V_{m_{1},m_{2},m_{3},m_{4}}+V_{m_{2},m_{1},m_{3},m_{4}}+V_{m_{1},m_{2},m_{4},m_{3}}+V_{m_{2},m_{1},m_{4},m_{3}}).$
(51)

Thus

$\widetilde{V}_{l-k,k,k^{\prime},l-k^{\prime}}=\frac{\widetilde{U}}{\pi
R^{2}}(\prod_{i=1}^{4}\widetilde{N}_{m_{i}})[4+2\cos(k^{\prime}-k)\pi+2\cos(l-k-k^{\prime})\pi]\ln(\lambda
R).$ (52)

At this stage, if $l$ is odd

$\widetilde{V}_{l-k,k,k^{\prime},l-k^{\prime}}=\frac{4\widetilde{U}}{\pi
R^{2}}(\prod_{i=1}^{4}\widetilde{N}_{m_{i}})\ln(\lambda R).$ (53)

If $l$ is even,

$\widetilde{V}_{l-k,k,k^{\prime},l-k^{\prime}}=\frac{4\widetilde{U}}{\pi
R^{2}}(\prod_{i=1}^{4}\widetilde{N}_{m_{i}})[1+\cos(k^{\prime}-k)\pi]\ln(\lambda
R).$ (54)

Thus the leading order of the interaction Hamiltonian is

$\displaystyle H^{\prime}_{\mathrm{int}}$ $\displaystyle=$
$\displaystyle\frac{4\widetilde{U}}{\pi R^{2}}\ln(\lambda
R)\left[\sum_{l}\left(\sum_{k}\widetilde{N}_{l-k}\widetilde{N}_{k}a_{l-k}^{\dagger}a_{k}^{\dagger}\right)\left(\sum_{k^{\prime}}\widetilde{N}_{l-k^{\prime}}\widetilde{N}_{k^{\prime}}a_{k^{\prime}}a_{l-k^{\prime}}\right)\right.$
(55)
$\displaystyle\left.+\sum_{l=even}\left(\sum_{k}\widetilde{N}_{l-k}\widetilde{N}_{k}e^{ik\pi}a_{l-k}^{\dagger}a_{k}^{\dagger}\right)\left(\sum_{k^{\prime}}\widetilde{N}_{l-k^{\prime}}\widetilde{N}_{k^{\prime}}e^{-ik^{\prime}\pi}a_{k^{\prime}}a_{l-k^{\prime}}\right)\right].$

The nice form of $H^{\prime}_{\mathrm{int}}$ suggests we define

$A_{l}=\sum_{k^{\prime}}\widetilde{N}_{l-k^{\prime}}\widetilde{N}_{k^{\prime}}a_{k^{\prime}}a_{l-k^{\prime}},$
(56)
$B_{l}=\sum_{k^{\prime}}(-1)^{k^{\prime}}\widetilde{N}_{2l-k^{\prime}}\widetilde{N}_{k^{\prime}}a_{k^{\prime}}a_{2l-k^{\prime}},$
(57)

and have $H^{\prime}_{\mathrm{int}}=\frac{4\widetilde{U}}{\pi
R^{2}}\ln(\lambda R)\sum_{l}(A^{\dagger}_{l}A_{l}+B^{\dagger}_{l}B_{l})$.

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Note on Artin’s Conjecture on Primitive Roots

Sankar Sitaraman

Howard University, Washington, DC

To Gerald, Ralph, Sundar and Susan

Abstract

E. Artin conjectured that any integer $a>1$ which is not a perfect square is a
primitive root modulo $p$ for infinitely many primes $p.$ Let $f_{a}(p)$ be
the multiplicative order of the non-square integer $a$ modulo the prime $p.$
M. R. Murty and S. Srinivasan [10] showed that if
$\displaystyle\sum_{p<x}\frac{1}{f_{a}(p)}=O(x^{1/4})$ then Artin’s conjecture
is true for $a.$ We relate the Murty-Srinivasan condition to sums involving
the cyclotomic periods from the subfields of $\mathbb{Q}(e^{2\pi i/p})$
corresponding to the subgroups $<a>\subseteq\mathbb{F}_{p}^{*}.$

1\. Introduction

Let $a>1$ be an integer which is not a perfect square and let $f_{a}(p)$ be
the multiplicative order of $a$ modulo a given prime $p.$ For such $a$ the
Murty-Srinivasan [10] condition
$\displaystyle\sum_{p<x}\frac{1}{f_{a}(p)}=O(x^{1/4})$ applies, and one could
try to evaluate this sum by looking at the average value of
$\omega(\Phi_{j}(a))$ where $\Phi_{j}(x)$ is the $j-$th cyclotomic polynomial
and $\omega(n)$ is the number of distinct primes dividing $n.$ We were led to
this idea by our thinking of $f_{a}(p)$ as the lengths of the orbits of the
points on the unit circle $S^{1}$ under the “Frobenius” map $\sigma_{a}:x\to
x^{a}.$ This meant looking at the torsion of the group of roots of unity. Note
that for a given prime $p,$ $f_{a}(p)$ is the order of this map acting on the
multiplicative subgroup generated by $\zeta_{p}=e^{\frac{2\pi i}{p}}.$ It is
well known that if $f_{a}(p)=j$ then $p\ |\ \Phi_{j}(a)$ where $\Phi_{j}(x)$
is the $j-$th cyclotomic polynomial. See, for instance, [15], Chapter 2.

Since the $\Phi_{j}(a)$ divide $a^{j}-1$ and we are only concerned with $p<x$
we actually use $\sum\omega_{x}(a^{j}-1)$ where $\omega_{x}(m)$ for integral
$m$ is the number of primes upto $x$ that divide it. The argument to approach
the Murty-Srinivasan condition using the average value of
$\omega(\Phi_{j}(a))$ is also in Murty-Wong [11], Section 5, albeit
implicitly. There an inequality for a sum that contains
$S(x)=\displaystyle\sum_{p<x}\frac{1}{f_{a}(p)}$ is used to give heuristics
for the largest prime factor of $a^{j}-1,j\leq x.$ We show how the sum
$\displaystyle\sum_{j\leq x}{\omega_{x}(a^{j}-1)}$ can be related to
cyclotomic periods after expressing it in terms of finite Fourier series
(exponential sums) and give some bounds for these sums, from which one could
easily get bounds for $\displaystyle\sum_{p<x}\frac{1}{f_{a}(p)}.$

Statement of Main Result: Theorem 1

In what follows $f_{a}(p)$ will be sometimes written simply as $f_{p}$ where
the context is clear, and $\displaystyle e_{p}$ will denote
$\displaystyle\frac{p-1}{f_{p}}.$

$g$ will denote a primitive generator of $\mathbb{F}_{p}^{*}$ and so $a\equiv
g^{e_{p}}\pmod{p}.$ $g_{i}$ will denote the least element of
$\\{1,2,...,p-1\\}$ such that $g_{i}\equiv g^{i}\pmod{p}.$

The cyclotomic periods for $i=1,2,...,e_{p},$ denoted $\eta_{i,p},$ are
defined by

$\eta_{i,p}=\sum_{j=1}^{f_{p}}\zeta_{p}^{g^{i}g^{je_{p}}}=\sum_{j=1}^{f_{p}}\zeta_{p}^{g^{i}a^{j}}=\sum_{j=1}^{f_{p}}\zeta_{p}^{g_{i}a^{j}}.$

They are contained in the subfield of $\mathbb{Q}(\zeta_{p})$ corresponding to
the subgroup $<a>$ generated by $a$ in $\mathbb{F}_{p}^{*}.$ $H_{p}$ will
denote this subgroup.

###### Theorem 1

For $a,$ $f_{a}(p)$ and $\eta_{i,p}$ corresponding to $H_{p}$ defined as
above, we get the following bounds for any $u\leq x$ where $x$ is a fixed
positive integer:

$A:\quad\sum_{j\leq u}{\omega_{x}(a^{j}-1)}\leq O(u\log\log x)+u\sum_{p\leq
x}\frac{\displaystyle\sum_{i=1}^{\frac{p-1}{f_{a}(p)}}|\eta_{i,p}|}{p}.$

$B:\quad\quad\sum_{j\leq u}{\omega_{x}(a^{j}-1)}\leq O(u\log\log x)+2\pi
u\sum_{p\leq
x}\frac{\left(\sum_{k=1}^{p-2}\left|\sum_{m=1}^{k}\eta_{h_{m},p}\right|\right)}{p^{2}}$

where
$\displaystyle\eta_{h_{m},p}=\sum_{j=1}^{f_{a}(p)}\zeta_{p}^{h_{m}a^{j}}$ with
$h_{m}\equiv mf_{a}(p)\pmod{p},m\in[1,p-1].$

Much progress has been made on Artin’s conjecture by Hooley [8], Gupta-Murty
[5], Heath-Brown [7] and others. Heath-Brown proved that Artin’s conjecture is
true for one of 2, 3 and 5.

Our initial motivation was to come up with a Dirichlet series that would help
in obtaining information about the asymptotic behavior of invariants of
cyclotomic fields such as class numbers. We hope to provide such a framework
that would enable us to calculate the asymptotics of $f_{a}(p)$ for a given
$a$ more precisely, as indicated in Sitaraman [12]. For instance, Artin
conjectured that $f_{a}(p)=p-1$ with a certain non-zero density $C$ (equal to
0.374 approximately) known now as the Artin constant.

The author is grateful to Niranjan Ramachandran and Larry Washington for their
hospitality and conversations during his visit to the University of Maryland,
College Park, in 2014-15. He would also like to thank Francois Ramaroson and
Dani Szpruch for helpful conversations, and the anonymous referee for valuable
suggestions to improve the paper.

2\. From the Murty-Srinivasan criterion to Cyclotomic Periods.

2a. Reduction to estimate of $\sum\omega_{x}(a^{j}-1)$

Let $\displaystyle S(x)=\sum_{p<x}\frac{1}{f_{a}(p)}.$ Let $N(j)$ to be the
number of primes $p<x$ with $a$ of order $j$ modulo $p.$ (i.e, $f_{a}(p)=j$).
Then we have $\displaystyle S(x)\leq\sum_{j<x}\frac{N(j)}{j}.$ Since as
mentioned before $f_{a}(p)=j\implies p|\Phi_{j}(a)$ we get

$\displaystyle
S(x)=\sum_{j<x}\frac{N(j)}{j}\leq\sum_{j<x}\frac{\omega_{x}(\Phi_{j}(a))}{j}.$

Since $\Phi_{j}(a)$ is a factor of $a^{j}-1$ we have
$\displaystyle\sum_{j<x}\frac{\omega_{x}(\Phi_{j}(a))}{j}\leq\sum_{j<x}\frac{\omega_{x}(a^{j}-1)}{j}.$
Since it is easier to deal with $a^{j}-1$ we consider the sum
$\displaystyle\sum_{j<x}\frac{\omega_{x}(a^{j}-1)}{j}.$ We will denote this
sum by $W(x).$

Initially we tried to use sieve theoretic techniques to deal with $W(x).$ H.
Halberstam [6] obtained an Erdös-Kac type result for
$\displaystyle\omega(f(m))$ where $f(x)\in\mathbb{Z}[x]$ is an irreducible
polynomial. Namely he showed that $\displaystyle\sum_{m\leq
n}\frac{\omega(f(m))-A(n)}{n\sqrt{A(n)}}$ is asymptotically normally
distributed, where $\displaystyle A(n)=\sum_{p\leq n}\frac{r(p)}{p}$ with
$r(p)$ being the number of solutions to $f(k)\equiv 0\ \text{mod}\ p$ with
$0\leq k<p.$ Halberstam needs $r(p)$ to be uniformly bounded, and that is
possible because the polynomial is fixed. Here we need to average over several
polynomials of varying degree. Even using a later more general version of
Halberstam’s result proved by Granville-Soundararajan [4] we found it
difficult to get a decent estimate for $W(x).$

On the other hand A. T. Felix [3] has shown that on average
$\sum_{p<x}1/f_{a}(p)$ is about $\log x.$ So actual bound could be much
smaller than $O(x^{1/4}).$ This is also the conclusion drawn in Murty-Wong.
Now, using $\omega(a^{j}-1)=O(j)$ one gets the trivial estimate $W(x)=O(x).$
If, as seems possible, the $a^{j}-1$ in general have a random mixture of small
and large primes (rather than a mixture of predominantly small primes), then
it is reasonable to expect that $W(x)=O(x^{\epsilon}),$ thus making $S(x)$
satisfy the Murty-Srinivasan criterion. Indeed, in many cases one seems to
have $\omega_{x}(a^{j}-1)=0.$

The best bound we know for $\displaystyle\sum_{p<x}\frac{1}{f_{a}(p)}$ is
$\sqrt{x}/(\log x)^{1+\delta}$ where $\delta$ is some small positive quantity,
given by P. Erdös and M. R. Murty [2]. The bounds A and B of Theorem 1,
numerically at least, seem to give better estimates.

2b. Proof of bounds A and B of Theorem 1

Using summation by parts, one could reduce the estimate of $W(x)$ to that of
$\displaystyle D(u)$ for $u\leq x$ for positive integers $u,$ where
$\displaystyle D(u)=\sum_{j\leq u}\omega_{x}(a^{j}-1).$

Now it is easy to see that

$D(u)=\sum_{j\leq u}\omega_{x}(a^{j}-1)=\sum_{p\leq x}\frac{1}{p}\sum_{j\leq
u}\sum_{k=1}^{p}\zeta_{p}^{k(a^{j}-1)}.$ $None$

Bound A:

Note that, for a fixed prime $p$ we have, on the one hand,
$\displaystyle\sum_{j\leq
u}\omega_{p}(a^{j}-1)=\left\lfloor\frac{u}{f_{p}}\right\rfloor.$ Here we
abused the notation a little bit to make $\omega_{p}(m)$ equal 1 if $p$
divides $m$ and $0$ otherwise. This will be the case regardless of whether
$f_{p}$ divides $u$ or not. Moreover, if $f_{p}>u,$ then clearly $p$ will not
divide any of the $a^{j}-1,j\leq u,$ and
$\displaystyle\left\lfloor\frac{u}{f_{p}}\right\rfloor=0.$

For the part of the sum $D(u)$ with $k=p$ we have
$\displaystyle\sum_{p<x}\frac{1}{p}\sum_{j\leq
u}\sum_{k=1}^{p}\zeta_{p}^{k(a^{j}-1)}=\sum_{p<x}\sum_{j\leq u}\frac{1}{p}$
and this is $O(u\log\log x).$

In what follows we focus on the $k<p$ part. Also let $u\equiv
u_{f}\pmod{f_{p}}$ with $0\leq u_{f}<f_{p}.$ There will be no confusion owing
to the dependence on $p.$

$D(u)=O(u\log\log x)+\sum_{p\leq
x}\frac{1}{p}\sum_{j=1}^{u}\sum_{k=1}^{p-1}\zeta_{p}^{(a^{j}-1)k}$
$=O(u\log\log x)+\sum_{p\leq
x}\frac{1}{p}\sum_{j=1}^{u_{f}}\sum_{k=1}^{p-1}\zeta_{p}^{(a^{j}-1)k}\quad\quad\qquad\qquad$
$\qquad\quad\quad+\sum_{p\leq
x}\frac{1}{p}\left\lfloor\frac{u}{f_{p}}\right\rfloor\sum_{H_{p}g_{i}\in\mathbb{F}_{p}^{*}/H_{p}}\left(\sum_{k\in
H_{p}g_{i}}\zeta_{p}^{-k}\left(\sum_{j=1}^{f_{p}}\zeta_{p}^{a^{j}k}\right)\right).$
$None$

In the last term of the sum in (2) above we used the facts that the sum over
$j$ is periodic with period $f_{p}$ and also that the inner sum
$\displaystyle\sum_{j=1}^{f_{p}}\zeta_{p}^{a^{j}k}$ is identical for $k$ in
the same coset.

In the second term of the sum above, for a fixed $j$ we have
$\displaystyle\sum_{k=1}^{p-1}\zeta_{p}^{(a^{j}-1)k}=-1$ if $f_{p}\nmid j$
which would be the case for $j\leq u_{f}<f_{p}.$ So we get

$D(u)=O(u\log\log x)+\sum_{p\leq x}\left(-\frac{u_{f}}{p}\right)+\sum_{p\leq
x}\frac{1}{p}\left\lfloor\frac{u}{f_{p}}\right\rfloor\sum_{H_{p}g_{i}\in\mathbb{F}_{p}^{*}/H_{p}}\left(\sum_{k\in
H_{p}g_{i}}\zeta_{p}^{-k}\eta_{i,p}\right)$

The second term of the sum is $O(u\log\log x)$ and for the last term we bound
trivially and then since each coset (i.e, each $\eta_{i,p}$) appears $f_{p}$
times, we get the bound A:

$D(u)=O(u\log\log x)+\sum_{p\leq
x}\frac{1}{p}\left\lfloor\frac{u}{f_{p}}\right\rfloor\sum_{i=1}^{e_{p}}f_{p}|\eta_{i,p}|$
$\implies D(u)\leq O(u\log\log x)+u\sum_{p\leq
x}\frac{\displaystyle\sum_{i=1}^{e_{p}}|\eta_{i,p}|}{p}.$

Note: The inner sum of equation (2), namely
$\displaystyle\sum_{H_{p}g_{i}\in\mathbb{F}_{p}^{*}/H_{p}}\left(\sum_{k\in
H_{p}g_{i}}\zeta_{p}^{-k}\left(\sum_{j=1}^{f_{p}}\zeta_{p}^{a^{j}k}\right)\right),$
can be seen to equal
$\sum_{i}\overline{\eta}_{i,p}\eta_{i,p}=\sum_{i}|\eta_{i,p}|^{2}$ and this is
known to be equal to $p-f_{p}.$ See, for instance, Thaine [14], p. 36. So,
left as it is, this sum does not give anything new. Although this equation
actually gives a formula to compute $f_{p}$ in terms of the cyclotomic
periods, one expects it to be hard to work with, given that $f_{p}$ is hard to
compute even on average.

Even though it might be bigger than original sum, the bound A (as well as
bound B below) at least eliminates $f_{p}$ from the estimate.

Bound B:

To get bound B we use summation by parts. Since, as seen above, the sum for
$j=1$ to $u_{f}$ contributes only $O(u\log\log x)$ as does the sum for $k=p$
for each $p$, we simply focus on the remainder of the expression for $D(u).$
We call this part $D(u)^{\prime}.$

$D(u)^{\prime}=\sum_{p\leq
x}\frac{1}{p}\left\lfloor\frac{u}{f_{p}}\right\rfloor\sum_{H_{p}g_{i}\in\mathbb{F}_{p}^{*}/H_{p}}\left(\sum_{k\in
H_{p}g_{i}}\zeta_{p}^{-k}\eta_{i,p}\right)$

Before summing by parts, first we rearrange the sum so that it goes from
$g_{i}=1$ to $g_{i}=p-1.$

$D(u)^{\prime}=\sum_{p\leq
x}\frac{1}{p}\left\lfloor\frac{u}{f_{p}}\right\rfloor\left(\sum_{g_{i}=1}^{p-1}\zeta_{p}^{-g_{i}}\eta_{i,p}\right)$

Now we rearrange the indices again and try to replace the $g_{i}$ with
$h_{m}\in[1,p-1]$ as $m$ goes from $1$ to $p-1$ such that $h_{m+1}-h_{m}\equiv
f_{p}\pmod{p}.$ This can be done by letting $h_{m}\equiv mf_{p}\pmod{p}.$
Since $p\nmid f_{p}$ the map $m\to h_{m}$ will result in a permutation of
$\\{1,2,...,p-1\\}.$

In particular,
$\displaystyle\sum_{g_{i}}\eta_{i,p}=\sum_{i=1}^{p-1}\eta_{i,p}=\sum_{m=1}^{p-1}\eta_{h_{m},p}.$

Summing by parts,

$D(u)^{\prime}=\sum_{p\leq
x}\frac{1}{p}\left\lfloor\frac{u}{f_{p}}\right\rfloor\left[\left(\sum_{g_{i}}\eta_{i,p}\right)\zeta_{p}^{-h_{p-1}}+\left(\sum_{k=1}^{p-2}\left(\sum_{m=1}^{k}\eta_{h_{m},p}\right)(\zeta_{p}^{-h_{k}}-\zeta_{p}^{-h_{k+1}})\right)\right].$
$None$

Now note that
$\displaystyle\left|\zeta_{p}^{-h_{k}}-\zeta_{p}^{-h_{k+1}}\right|=\left|2\sin\left(\frac{\pi(h_{k+1}-h_{k})}{p}\right)\right|\leq\frac{2\pi
f_{p}}{p}$ because $h_{k+1}-h_{k}=f_{p},$ for $k=1,2,...,p-2.$ Also note that,
since $\displaystyle\sum_{i=1}^{e_{p}}\eta_{i,p}=-1,$ and there are $f_{p}$
full sets of $\eta_{i,p},$ we have $\sum_{g_{i}}\eta_{i,p}=-f_{p}.$ Using
these two computations, we get

$\left|D(u)^{\prime}\right|\leq\sum_{p\leq
x}\frac{1}{p}\left\lfloor\frac{u}{f_{p}}\right\rfloor\left[f_{p}+\left(\sum_{k=1}^{p-2}\left|\sum_{m=1}^{k}\eta_{h_{m},p}\right|\frac{2\pi
f_{p}}{p}\right)\right]\leq u\sum_{p\leq x}\frac{1}{p}+2\pi u\sum_{p\leq
x}\frac{\left(\sum_{k=1}^{p-2}\left|\sum_{m=1}^{k}\eta_{h_{m},p}\right|\right)}{p^{2}}$

Adding the $O(u(\log\log x))$ terms from the sum $D(u)$ that were left out, we
get bound B:

$D(u)\leq O(u\log\log x)+2\pi u\sum_{p\leq
x}\frac{\left(\sum_{k=1}^{p-2}\left|\sum_{m=1}^{k}\eta_{h_{m},p}\right|\right)}{p^{2}}$

Due to the increased possibility of cancellations, we think bound B would be
better than bound A. Numerically that seems to be the case.

2c. Estimates for Bounds A and B

Clearly, the best estimates for the bounds A and B would be obtained using
estimates for the average values of $|\eta_{i,p}|$ and
$\displaystyle\left|\sum_{m=1}^{k}\eta_{h_{m},p}\right|$ over the cyclotomic
periods as well as over the primes. Here we restrict ourselves to seeing what
estimates can be obtained for a fixed prime. As such, in this section we will
drop the subscript $p$ from $\eta_{i,p}$ and $\eta_{h_{m},p}.$

Bound A:

As mentioned in the Note in the previous section, we have
$\sum_{i}\overline{\eta}_{i}\eta_{i}=\sum_{i}|\eta_{i}|^{2}=p-f_{p}.$ From
this we see that all the cyclotomic periods have absolute values bounded by
$\sqrt{p},$ and possibly the absolute values are much less than $\sqrt{p},$ at
least on average. Indeed, we have (looking at the general term from the sum in
bound A):

$\left(\sum_{i}|\eta_{i}|\right)^{2}\leq
e_{p}\sum_{i}|\eta_{i}|^{2}\implies\frac{1}{p}\sum_{i}|\eta_{i}|\leq\frac{\sqrt{e_{p}}}{p}\sqrt{\sum_{i}|\eta_{i}|^{2}}$
$\quad\quad\implies\frac{1}{p}\sum_{i}|\eta_{i}|\leq\sqrt{\frac{p-1}{f_{p}}}\frac{\sqrt{p-f_{p}}}{p}=\sqrt{\frac{1}{f_{p}}-\frac{1}{p}}\times\sqrt{1-\frac{f_{p}}{p}}.$
$None$

(The first inequality above follows from the elementary fact that the average
of a set of positive numbers is smaller than the square root of the average of
their squares).

As expected the estimate from equation (4) does not yield a better estimate
for $D(u)$ (and hence for $\sum_{p<x}1/f_{p}$) than the existing ones.

On the other hand there are several estimates that show that
$\displaystyle\text{max}_{i}\ |\eta_{i}|$ is smaller than $f_{p}$ when $f_{p}$
itself is small. For instance we have

$\text{max}_{i}\ |\eta_{i}|<p^{-3\alpha/8}f_{p}\text{ when
}f_{p}>p^{\frac{1}{3}+\alpha}.$

See Bourgain [1] for a survey of such results. H. Montgomery, R. C. Vaughan
and T. D. Wooley [9] conjecture that $\displaystyle\text{max}_{i}\
|\eta_{i}|<\text{min}\left(\sqrt{p},C\sqrt{f_{p}\times\log p}\right),$ with
the constant $C$ independent of $p.$

Suppose $\displaystyle\text{max}_{i}\ |\eta_{i}|<M_{p}.$ When one uses such an
estimate in bound A one gets

$D(u)=\sum_{j\leq u}\omega_{x}(a^{j}-1)\leq O(u\log\log x)+u\sum_{p\leq
x}\frac{\displaystyle\sum_{i=1}^{e_{p}}|\eta_{i,p}|}{p}$ $\leq O(u\log\log
x)+u\sum_{p\leq x}\frac{1}{p}\left(\frac{p-1}{f_{p}}(M_{p})\right)\leq
O(u\log\log x)+u\sum_{p\leq x}\frac{M_{p}}{f_{p}}.$

Since one key problem is that we don’t know the amount of primes with $f_{p}$
in a given interval $p^{a}\leq f_{p}\leq p^{b}$ it is difficult to average the
bound A over the primes $p.$ Even otherwise, it appears that using existing
estimates for the maximum of the periods it would be difficult to get a useful
estimate for $D(u)$ and hence $\sum_{p<x}1/{f_{p}}.$ One does need a good
estimate for the averages of the $|\eta_{i}|$ as opposed to their maximum.

Bound B:

For bound B we have the following very basic estimate in the form of a
trigonometric sum:

We first estimate $\displaystyle\left|\sum_{m=1}^{k}\eta_{h_{m},p}\right|.$ As
before we omit $p$ in the subscript and denote $h_{m,p}$ by $h_{m}.$ We get:

$\quad\left|\sum_{m=1}^{k}\eta_{h_{m}}\right|=\left|\sum_{m=1}^{k}\sum_{j=1}^{f_{p}}\zeta_{p}^{h_{m}a^{j}}\right|=\left|\sum_{m=1}^{k}\sum_{j=1}^{f_{p}}\zeta_{p}^{h_{m}a^{j}}\right|$
$\implies\left|\sum_{m=1}^{k}\eta_{h_{m}}\right|=\left|\sum_{j=1}^{f_{p}}\sum_{m=1}^{k}\zeta_{p}^{mf_{p}a^{j}}\right|=\left|\sum_{j=1}^{f_{p}}\zeta_{p}^{f_{p}a^{j}}\sum_{m=0}^{k-1}\zeta_{p}^{mf_{p}a^{j}}\right|\leq\sum_{j=1}^{f_{p}}\left|\frac{\sin(\pi
a^{j}f_{p}k/p)}{\sin(\pi a^{j}f_{p}/p)}\right|.$

Putting everything together, we get for Bound B the following estimate in
terms of a trigonometric sum:

$D(u)\leq O(u\log\log x)+2\pi u\sum_{p\leq
x}\frac{1}{p^{2}}\left(\sum_{k=1}^{p-2}\sum_{j=1}^{f_{p}}\left|\frac{\sin(\pi
a^{j}f_{p}k/p)}{\sin(\pi a^{j}f_{p}/p)}\right|\right).$

2d. Implications for Artin’s Conjecture

If $D(u)$ is of order $ux^{1/4}/(\log x)$ or less then after summation by
parts we get that $W(x)$ is of order $x^{1/4}$ and that would verify the
Murty-Srinivasan condition and hence Artin’s conjecture.

Now, when $f_{p}>p^{3/4}$ the $\sum_{p<x}1/f_{p}$ is at most $O(x^{1/4}/(\log
x)).$ So main problem is when $f_{p}<p^{3/4}.$

Erdös and Murty [2] show that for most primes $f_{p}$ is of order bigger than
$\sqrt{p}.$ In the bounds given here most of the contribution would come from
the primes $p$ with small $f_{p}$ and for such $p$ the approximation of
$\sin(\pi f_{p}/p)$ by $\pi f_{p}/p$ would be better.

Finally, the full sums of bounds A and B seem to be at least as big as
$ux^{1/4},$ based on numerical evidence.

Nevertheless, one could try to use the expression for $D(u)$ in terms of
cyclotomic periods (equation (2) and (3)), without taking absolute values as
we did here to get the bounds A and B. If it is possible to sum over the
primes first before summing over the $k$’s à la Voronoi in his estimate for
the Dirichlet divisor problem (cf. for instance, Tenenbaum [13] I.6.4) one
could make progress on the conjecture.

In this direction would the deep estimates of exponential sums over finite
fields developed by Bourgain, Konyagin and others be useful ? See for instance
Bourgain’s survey article [1]. Perhaps those methods would yield good
estimates for average sums for the cyclotomic periods.

## References

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