IIT-BHU

The standard HVM

Gauhar Abbas111email<EMAIL_ADDRESS>
Department of Physics,
Indian Institute of Technology (BHU),
Varanasi 221005, India

Abstract

> We discuss the standard hierarchical VEVs model, which predicts the leptonic
> mixing angles in terms of the Cabibbo angle, and masses of strange and charm
> quarks as $\sin\theta_{12}^{\ell}\geq 1-2\sin\theta_{12}$,
> $\sin\theta_{23}^{\ell}\geq 1-\sin\theta_{12}$, and
> $\sin\theta_{13}^{\ell}\geq\sin\theta_{12}-\frac{m_{s}}{m_{c}}$ for the
> normal mass ordering. This results in very precise predictions of the
> leptonic mixing angles given by $\sin\theta_{12}^{\ell}=0.55\pm 0.00134$,
> $\sin\theta_{23}^{\ell}=0.775\pm 0.00067$, and
> $\sin\theta_{13}^{\ell}=0.1413-0.1509$. Furthermore, we predict neutrinos to
> be the Dirac kind. The flavour bounds on the model are discussed, and the
> impact of the standard hierarchical VEVs model on the flavonic dark matter
> is highlighted by showing its natural emergence.

> ##  1 Introduction

The origin of the flavour of the standard model (SM) is a fascinating problem
that may lead to the physics beyond the SM. This problem demands an
explanation for the fermionic mass spectrum and mixing of the SM. There exist
different ways to solve this problem. For instance, a technicolour framework
can solve this problem through the sequential multi-fermion chiral condensates
of a dark-technicolor (DTC) symmetry [1, 2]. An Abelian flavour symmetry can
be used as well [3, 4, 5, 6, 7, 8, 9]. We can also have a solution based on
loop-suppressed couplings to the Higgs [10], wave-function localization
scenario [11], the compositeness [12], extra-dimensional framework [13], and
discrete symmetries[14, 15, 16].

Hierarchical VEVs model (HVM) redefines the flavour problem of the SM in terms
of the VEVs hierarchy, and provide an explanation to the quark mixing angles
in terms of the ratios of the hierarchical VEVs [1]. In the ultra-violet
completion (UV) of HVM, the hierarchical VEVs are sequential chiral multi-
fermion condensates of a DTC dynamics depending only on a single parameter
$\Lambda_{\rm DTC}$, i.e., the scale of the underlying non-Ableian DTC gauge
theory[2]. For a recent review of HVM, see reference [17].

In the simplest version of HVM, the SM is subjected to the imposition of the
three Abelian discrete symmetries $\mathcal{Z}_{2}$,
$\mathcal{Z}_{2}^{\prime}$ an $\mathcal{Z}_{2}^{\prime\prime}$, and charged
fermionic mass spectrum is explained in terms of the VEVs of the six gauge
singlet scalar fields $\chi_{i}$ through the dimension-5 operators [1]. In
HVM, the neutrino masses are recovered by adding an additional gauge singlet
scalar field $\chi_{7}$ through the type-i@ seesaw mechanism[18]. However, no
predictions for the neutrino mixing angles were made.

Predicting neutrino mixing angles is one of the most difficult challenges
beyond the SM. The reason is that the observed pattern of the neutrino mixing
angles is entirely different from that of the quark mixing angles. For
instance, a global fit to neutrino oscillation data provides the following
values of the neutrino oscillation parameters for the normal mass ordering
[19],

$\displaystyle\Delta m_{21}^{2}$ $\displaystyle=$
$\displaystyle(7.50^{+0.64}_{-0.56})\times 10^{-5}{\rm eV}^{2},~{}|\Delta
m_{31}^{2}|=(2.55\pm 0.08)\times 10^{-3}\rm{eV}^{2},$ (1)
$\displaystyle\sin\theta_{12}^{\ell}$ $\displaystyle=$
$\displaystyle(0.564_{-0.043}^{+0.044}),~{}\sin\theta_{23}^{\ell}=(0.758_{-0.099}^{+0.023}),~{}\sin\theta_{13}^{\ell}=(0.1483_{-0.0069}^{+0.0067}),$

where the errors are in 3 $\sigma$ range.

We observe that only $\sin\theta_{13}^{\ell}$ is closer to the Cabibbo angle
$\sin\theta_{12}$, and $\sin\theta_{12}-\sin\theta_{13}^{\ell}\approx
0.06923-0.08427$. Furthermore, although the mixing angle
$\sin\theta_{23}^{\ell}$ is of the order one, it is not exactly one. Moreover,
the $\sin\theta_{12}^{\ell}$ is large, however, it cannot be considered order
one.

On the other side, the quark mixing angles are [20],

$\displaystyle\sin\theta_{12}$ $\displaystyle=$ $\displaystyle(0.225\pm
0.00067),~{}\sin\theta_{23}=(0.04182^{+0.00085}_{-0.00074}),~{}\sin\theta_{13}=(0.00369\pm
0.00011).$ (2)

In the case of quark-mixing, $\sin\theta_{23}\approx\sin\theta_{12}^{2}$ and
remarkable observation is $\sin\theta_{13}<\sin\theta_{12}^{3}$.

The really important and challenging question is, if these observations can be
predicted from a fundamental theory. In this work, we shall propose what we
refer to as the standard HVM, and show that the above observations are the
hallmark of the standard HVM.

##  2 The standard HVM

The standard HVM is achieved by enlarging the discrete symmetries used in HVM.
We impose a generic $\mathcal{Z}_{\rm N}\times\mathcal{Z}_{\rm
M}\times\mathcal{Z}_{\rm P}$ flavour symmetry on the SM. This symmetry
naturally emerges through the breaking of three axial $U(1)$ groups in the UV
completion of HVM [2, 17]. It turns out that for the flavour structure of the
standard HVM assuming normal neutrino mass ordering, we must have $\rm N=2$,
$\rm M\geq 3$, and $\rm P\geq 14$. For instance, we use the
$\mathcal{Z}_{2}\times\mathcal{Z}_{4}\times\mathcal{Z}_{14}$ symmetry, and
assign charges to different fermionic and scalar fields under this symmetry,
as shown in table 1.

Fields | $\mathcal{Z}_{2}$ | $\mathcal{Z}_{4}$ | $\mathcal{Z}_{14}$ | Fields | $\mathcal{Z}_{2}$ | $\mathcal{Z}_{4}$ | $\mathcal{Z}_{14}$ | Fields | $\mathcal{Z}_{2}$ | $\mathcal{Z}_{4}$ | $\mathcal{Z}_{14}$ | Fields | $\mathcal{Z}_{2}$ | $\mathcal{Z}_{4}$ | $\mathcal{Z}_{14}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$u_{R}$ | - | $1$ | $\omega^{11}$ |  $d_{R}$, $s_{R}$, $b_{R}$ | + | $1$ | $\omega^{12}$ | $\psi_{L_{3}}^{q}$ | + | $1$ | $\omega^{2}$ | $\tau_{R}$ | + | $\omega^{3}$ | $\omega$
$c_{R}$ | + | $1$ | $\omega^{6}$ | $\chi_{4}$ | + | $1$ | $\omega^{13}$ | $\psi_{L_{1}}^{\ell}$ | + | $\omega^{3}$ | $\omega^{12}$ | $\nu_{e_{R}}$ | + | $\omega$ | $\omega^{8}$
$t_{R}$ | + | $1$ | $\omega^{4}$ | $\chi_{5}$ | + | $1$ | $\omega^{11}$ | $\psi_{L_{2}}^{\ell}$ | + | $\omega^{3}$ | $\omega^{10}$ | $\nu_{\mu_{R}}$ | - | $\omega$ | $\omega^{3}$
$\chi_{1}$ | - | $1$ | $\omega^{2}$ | $\chi_{6}$ | + | $1$ | $\omega^{10}$ | $\psi_{L_{3}}^{\ell}$ | + | $\omega^{3}$ | $\omega^{6}$ | $\nu_{\tau_{R}}$ | - | $\omega$ | $\omega^{3}$
$\chi_{2}$ | + | $1$ | $\omega^{5}$ | $\psi_{L_{1}}^{q}$ | + | $1$ | $\omega^{13}$ | $e_{R}$ | - | $\omega^{3}$ | $\omega^{10}$ | $\chi_{7}$ | - | $\omega^{2}$ | $\omega^{8}$
$\chi_{3}$ | + | $1$ | $\omega^{2}$ | $\psi_{L_{2}}^{q}$ | + | $1$ | $\omega$ | $\mu_{R}$ | + | $\omega^{3}$ | $\omega^{13}$ | $\varphi$ | + | 1 | 1
Table 1: The charges of left- and right-handed fermions and scalar fields
under $\mathcal{Z}_{2}$, $\mathcal{Z}_{4}$ and $\mathcal{Z}_{14}$ symmetries
for the normal mass ordering.

The masses of the charged fermions are produced by the dimension-5 operators
written as,

$\displaystyle{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\Lambda}\Bigl{[}y_{ij}^{u}\bar{\psi}_{L_{i}}^{q}\tilde{\varphi}\psi_{R_{i}}^{u}\chi_{i}+y_{ij}^{d}\bar{\psi}_{L_{i}}^{q}\varphi\psi_{R_{i}}^{d}\chi_{i}+y_{ij}^{\ell}\bar{\psi}_{L_{i}}^{\ell}\varphi\psi_{R_{i}}^{\ell}\chi_{i}\Bigr{]}+{\rm
H.c.},$ (3)

$i$ and $j$ are generation indices, $\psi_{L}^{q},\psi_{L}^{\ell}$ are the
quark and leptonic doublets, $\psi_{R}^{u},\psi_{R}^{d},\psi_{R}^{\ell}$
denote the right-handed up, down-type quarks and leptons, $\varphi$ and
$\tilde{\varphi}=-i\sigma_{2}\varphi^{*}$ represent the SM Higgs field, and
its conjugate and $\sigma_{2}$ is the second Pauli matrix.

The fermionic mass spectrum can be explained in terms of the VEVs pattern
$\langle\chi_{4}\rangle>\langle\chi_{1}\rangle$,
$\langle\chi_{2}\rangle>>\langle\chi_{5}\rangle$,
$\langle\chi_{3}\rangle>>\langle\chi_{6}\rangle$,
$\langle\chi_{3}\rangle>>\langle\chi_{2}\rangle>>\langle\chi_{1}\rangle$, and
$\langle\chi_{6}\rangle>>\langle\chi_{5}\rangle>>\langle\chi_{4}\rangle$. The
mass matrices of up, down quarks and leptons turn out to be,

$\displaystyle{\cal M}_{\cal U}$
$\displaystyle=\dfrac{v}{\sqrt{2}}\begin{pmatrix}y_{11}^{u}\epsilon_{1}&0&y_{13}^{u}\epsilon_{2}\\\
0&y_{22}^{u}\epsilon_{2}&y_{23}^{u}\epsilon_{5}\\\
0&y_{32}^{u}\epsilon_{6}&y_{33}^{u}\epsilon_{3}\end{pmatrix},{\cal M}_{\cal
D}=\dfrac{v}{\sqrt{2}}\begin{pmatrix}y_{11}^{d}\epsilon_{4}&y_{12}^{d}\epsilon_{4}&y_{13}^{d}\epsilon_{4}\\\
y_{21}^{d}\epsilon_{5}&y_{22}^{d}\epsilon_{5}&y_{23}^{d}\epsilon_{5}\\\
y_{31}^{d}\epsilon_{6}&y_{32}^{d}\epsilon_{6}&y_{33}^{d}\epsilon_{6}\\\
\end{pmatrix},{\cal
M}_{\ell}=\dfrac{v}{\sqrt{2}}\begin{pmatrix}y_{11}^{\ell}\epsilon_{1}&y_{12}^{\ell}\epsilon_{4}&y_{13}^{\ell}\epsilon_{5}\\\
0&y_{22}^{\ell}\epsilon_{5}&y_{23}^{\ell}\epsilon_{2}\\\
0&0&y_{33}^{\ell}\epsilon_{2}\\\ \end{pmatrix},$ (4)

where $\epsilon_{i}=\dfrac{\langle\chi_{i}\rangle}{\Lambda}$ and
$\epsilon_{i}<1$.

The masses of charged fermions are,

$\displaystyle m_{t}$ $\displaystyle=$ $\displaystyle\
\left|y^{u}_{33}\right|\epsilon_{3}v/\sqrt{2},~{}m_{c}=\
\left|y^{u}_{22}\epsilon_{2}-\dfrac{y_{23}^{u}y_{32}^{u}\epsilon_{5}\epsilon_{6}}{y_{33}^{u}\epsilon_{3}}\right|v/\sqrt{2},~{}m_{u}=|y_{11}^{u}|\,\epsilon_{1}v/\sqrt{2},$
$\displaystyle m_{b}$ $\displaystyle\approx$ $\displaystyle\
|y^{d}_{33}|\epsilon_{6}v/\sqrt{2},m_{s}\approx\
\left|y^{d}_{22}-\dfrac{y_{23}^{d}y_{32}^{d}}{y_{33}^{d}}\right|\epsilon_{5}v/\sqrt{2},$
$\displaystyle m_{d}$ $\displaystyle\approx$
$\displaystyle\left|y_{11}^{d}-{y_{12}^{d}y_{21}^{d}\over
y^{d}_{22}-\dfrac{y_{23}^{d}y_{32}^{d}}{y_{33}^{d}}}-{{y_{13}^{d}(y_{31}^{d}y_{22}^{d}-y_{21}^{d}y_{32}^{d})-y_{31}^{d}y_{12}^{d}y_{23}^{d}}\over{(y^{d}_{22}-\dfrac{y_{23}^{d}y_{32}^{d}}{y_{33}^{d}})y^{d}_{33}}}\right|\,\epsilon_{4}v/\sqrt{2},$
$\displaystyle m_{\tau}$ $\displaystyle\approx$ $\displaystyle\
|y^{\ell}_{33}|\epsilon_{2}v/\sqrt{2},~{}m_{\mu}\approx\
|y^{\ell}_{22}|\epsilon_{5}v/\sqrt{2},~{}m_{e}=|y_{11}^{\ell}|\,\epsilon_{1}v/\sqrt{2}.$
(5)

The quark mixing angles are [21],

$\displaystyle\sin\theta_{12}$ $\displaystyle\simeq$
$\displaystyle\left|{y_{12}^{d}\epsilon_{4}\over
y_{22}^{d}\epsilon_{5}}\right|={\epsilon_{4}\over\epsilon_{5}},~{}\sin\theta_{23}\simeq\left|{y_{23}^{d}\epsilon_{5}\over
y_{33}^{d}\epsilon_{6}}-{y_{23}^{u}\epsilon_{5}\over
y_{33}^{u}\epsilon_{3}}\right|,~{}\sin\theta_{13}\simeq\left|{y_{13}^{d}\epsilon_{4}\over
y_{33}^{d}\epsilon_{6}}-{y_{13}^{u}\epsilon_{2}\over
y_{33}^{u}\epsilon_{3}}\right|.$ (6)

We can also write,

$\displaystyle\sin\theta_{23}$ $\displaystyle\simeq$
$\displaystyle\left|{y_{23}^{d}\epsilon_{5}\over
y_{33}^{d}\epsilon_{6}}-{y_{23}^{u}\epsilon_{5}\over
y_{33}^{u}\epsilon_{3}}\right|\approx\left|{y_{23}^{d}\over
y_{33}^{d}}\right|\sin\theta_{12}^{2},$ (7) $\displaystyle\sin\theta_{13}$
$\displaystyle\simeq$ $\displaystyle\left|{y_{13}^{d}\epsilon_{4}\over
y_{33}^{d}\epsilon_{6}}-{y_{13}^{u}\epsilon_{2}\over
y_{33}^{u}\epsilon_{3}}\right|\geq\left|{y_{13}^{d}\over
y_{33}^{d}}\right|\sin\theta_{12}^{3}-\left|{y_{13}^{u}\over
y_{33}^{u}}\right|\frac{m_{c}}{m_{t}}\approx\sin\theta_{12}^{3}-2\frac{m_{c}}{m_{t}},$

where we have assumed $\epsilon_{4}/\epsilon_{5}=\sin\theta_{12}=0.225$,
$\epsilon_{5}/\epsilon_{6}=\sin\theta_{12}^{2}$,
$|y_{13}^{d}/y_{33}^{d}|\approx 1$, and $|y_{13}^{u}/y_{33}^{u}|\approx 2$.
Thus, the standard HVM is capable of predicting the quark mixing angle
$\sin\theta_{13}$ to be $\approx\sin\theta_{12}^{3}-2\frac{m_{c}}{m_{t}}$,
which is in agreement with its observed value.

The neutrino masses are recovered by introducing the three right-handed
neutrinos $\nu_{eR}$, $\nu_{\mu R}$, $\nu_{\tau R}$ to the SM, and by writing
the Lagrangian,

$\displaystyle-{\mathcal{L}}_{\rm Yukawa}^{\nu}$ $\displaystyle=$
$\displaystyle
y_{ij}^{\nu}\bar{\psi}_{L_{i}}^{\ell}\tilde{\varphi}\nu_{f_{R}}\left[\dfrac{\chi_{i}\chi_{j}(\text{or}~{}\chi_{i}\chi_{j}^{\dagger})}{\Lambda^{2}}\right]+{\rm
H.c.}.$ (8)

The Dirac mass matrix for neutrinos reads,

${\cal M}_{{\cal
D}}=\dfrac{v}{\sqrt{2}}\begin{pmatrix}y_{11}^{\nu}\epsilon_{1}\epsilon_{7}&y_{12}^{\nu}\epsilon_{4}\epsilon_{7}&y_{13}^{\nu}\epsilon_{4}\epsilon_{7}\\\
0&y_{22}^{\nu}\epsilon_{4}\epsilon_{7}&y_{23}^{\nu}\epsilon_{4}\epsilon_{7}\\\
0&y_{32}^{\nu}\epsilon_{5}\epsilon_{7}&y_{33}^{\nu}\epsilon_{5}\epsilon_{7}\end{pmatrix},$
(9)

and neutrino masses can be written as,

$\displaystyle m_{3}$ $\displaystyle\approx$
$\displaystyle|y^{\nu}_{33}|\epsilon_{5}\epsilon_{7}v/\sqrt{2},m_{2}\approx\left|y^{\nu}_{22}-\dfrac{y_{23}^{\nu}y_{32}^{\nu}}{y_{33}^{\nu}}\right|\epsilon_{4}\epsilon_{7}v/\sqrt{2},m_{1}\approx|y_{11}^{\nu}|\,\epsilon_{1}\epsilon_{7}v/\sqrt{2}.$
(10)

The masses of neutrinos turn out to be $\\{m_{3},m_{2},m_{1}\\}=\\{5.05\times
10^{-2},8.67\times 10^{-3},2.67\times 10^{-4}\\}\,\text{eV}$ for the values of
$y_{ij}^{\nu}$ given in the appendix.

The remarkable predictions are the neutrino mixing angles [21],

$\displaystyle\sin\theta_{12}^{\ell}$ $\displaystyle\simeq$
$\displaystyle\left|{y_{12}^{\ell}\epsilon_{4}\over
y_{22}^{\ell}\epsilon_{5}}-{y_{12}^{\nu}\over
y_{22}^{\nu}}+{y_{23}^{\ell*}y_{13}^{\nu}\epsilon_{4}\over
y_{33}^{\ell}y_{33}^{\nu}\epsilon_{5}}\right|,\sin\theta_{23}^{\ell}\simeq\left|{y_{23}^{\ell}\over
y_{33}^{\ell}}-{y_{23}^{\nu}\epsilon_{4}\over
y_{33}^{\nu}\epsilon_{5}}\right|,~{}\sin\theta_{13}^{\ell}\simeq\left|{y_{13}^{\ell}\epsilon_{5}\over
y_{33}^{\ell}\epsilon_{2}}-{y_{13}^{\nu}\epsilon_{4}\over
y_{33}^{\nu}\epsilon_{5}}\right|.$ (11)

Assuming all the couplings of the order one, we further predict ,

$\displaystyle\sin\theta_{12}^{\ell}$ $\displaystyle\simeq$
$\displaystyle\left|-{y_{12}^{\nu}\over
y_{22}^{\nu}}+{y_{12}^{\ell}\epsilon_{4}\over
y_{22}^{\ell}\epsilon_{5}}+{y_{23}^{\ell*}y_{13}^{\nu}\epsilon_{4}\over
y_{33}^{\ell}y_{33}^{\nu}\epsilon_{5}}\right|\geq\left|-{y_{12}^{\nu}\over
y_{22}^{\nu}}\right|-\left|{y_{12}^{\ell}\over
y_{22}^{\ell}}+{y_{23}^{\ell*}y_{13}^{\nu}\over
y_{33}^{\ell}y_{33}^{\nu}}\right|{\epsilon_{4}\over\epsilon_{5}}\geq
1-2\sin\theta_{12},$ (12) $\displaystyle\sin\theta_{23}^{\ell}$
$\displaystyle\simeq$ $\displaystyle\left|{y_{23}^{\ell}\over
y_{33}^{\ell}}-{y_{23}^{\nu}\epsilon_{4}\over
y_{33}^{\nu}\epsilon_{5}}\right|\geq\left|{y_{23}^{\ell}\over
y_{33}^{\ell}}\right|-\left|{y_{23}^{\nu}\over
y_{33}^{\nu}}\right|{\epsilon_{4}\over\epsilon_{5}}\geq 1-\sin\theta_{12},$
$\displaystyle\sin\theta_{13}^{\ell}$ $\displaystyle\simeq$
$\displaystyle\left|-{y_{13}^{\nu}\epsilon_{4}\over
y_{33}^{\nu}\epsilon_{5}}+{y_{13}^{\ell}\epsilon_{5}\over
y_{33}^{\ell}\epsilon_{2}}\right|\geq\left|-{y_{13}^{\nu}\over
y_{33}^{\nu}}\right|{\epsilon_{4}\over\epsilon_{5}}-\left|{y_{13}^{\ell}\over
y_{33}^{\ell}}\right|{\epsilon_{5}\over\epsilon_{2}}\geq\sin\theta_{12}-\frac{m_{s}}{m_{c}},$

where $m_{s}/m_{c}=\epsilon_{5}/\epsilon_{2}$,. Thus, we conclude that the
leptonic mixing angles are predicted by the standard HVM in terms of the
Cabibbo angle and masses of strange and charm quarks.

We can now predict the leptonic mixing angles precisely using the measurement
of the Cabibbo angle given in equation 2. For the first two mixing angles, we
predict $\sin\theta_{12}^{\ell}=0.55\pm 0.00134$ and
$\sin\theta_{23}^{\ell}=0.775\pm 0.00067$. The theoretical precision of the
$\sin\theta_{12}^{\ell}$ directly depends on the experimental precision of the
Cabibbo angle, and the theoretical precision of the $\sin\theta_{23}^{\ell}$
is identical to that of the Cabibbo angle.

The mixing angle $\sin\theta_{13}^{\ell}$ can be predicted using the values of
running masses of strange and charm quarks given at 1 TeV as
$m_{c}=0.532^{+0.074}_{-0.073}$ GeV and $m_{s}=4.7^{+1.4}_{-1.3}\times
10^{-2}$ GeV[22]. Using these values, we predict $\sin\theta_{13}^{\ell}$ to
be $(0.1169-0.1509)$. However, we observe that the lower end of the
experimental measurement eliminates the range $(0.1169-0.1413)$ of the
theoretical prediction. On the other side, the upper end of the theoretical
prediction rules out the range $(0.1509-0.1550)$ of the experimental
observation. This leads to the final conclusive range of the angle
$\sin\theta_{13}^{\ell}=0.1413-0.1509$.

There is a possibility of the Majorana masses for neutrinos in this work. We
can write the following Weinberg-type operators for the Majorana mass,

$\displaystyle-{\mathcal{L}}_{\rm Weinberg}^{\ell}$ $\displaystyle=$
$\displaystyle
h_{ij}^{\nu}\frac{\bar{\tilde{\psi}}_{L_{i}}^{\ell}\varphi\tilde{\varphi}^{\dagger}\psi_{L_{j}}^{\ell}}{\Lambda}+h_{ij}^{\nu\prime}\frac{\bar{\tilde{\psi}}_{L_{i}}^{\ell}\varphi\tilde{\varphi}^{\dagger}\psi_{L_{j}}^{\ell}}{\Lambda}\left[\dfrac{\chi_{i}(\text{or}~{}\chi_{i}^{\dagger})}{\Lambda}\right]+{\rm
H.c.},$ (13)

where $\tilde{\psi}_{L_{i}}^{\ell}=i\sigma_{2}\psi_{L_{i}}^{c}$. It turns out
that these operators are not allowed by the charge assignments given in table
1. We may also have the right-handed neutrino mass operators $\mathcal{L}_{\rm
M_{R}}$ given by,

$\displaystyle\mathcal{L}_{\rm M_{R}}$ $\displaystyle=$ $\displaystyle
c_{ij}\chi_{i}\bar{\nu_{f_{R}}^{c}}\nu_{f_{R}}+c_{ij}^{\prime}\chi_{i}\bar{\nu_{f_{R}}^{c}}\nu_{f_{R}}\left[\frac{\chi_{i}(\text{or}~{}\chi_{i}^{\dagger})}{\Lambda}\right]+c_{ij}^{\prime\prime}\chi_{i}\bar{\nu_{f_{R}}^{c}}\nu_{f_{R}}\left[\frac{\chi_{i}\chi_{j}(\text{or}~{}\chi_{i}\chi_{j}^{\dagger})}{\Lambda^{2}}\right]+{\rm
H.c.},$ (14)

However, they are forbidden in the standard HVM. Therefore, neutrinos in the
standard HVM, are Dirac type.

Now, we consider the flavour bounds on the standard HVM. In the absence of
interactions among Higgs and singlet scalar fields, there is a global $SU(7)$
symmetry in the scalar sector such that the fields $\chi_{i}$ of the HVM can
be accommodated in its fundamental representation, that is,
$\chi\sim(\chi_{1},\chi_{2},\chi_{3},\chi_{4},\chi_{5},\chi_{6},\chi_{7})^{\rm
T}$. Hence, the scalar potential is written as,

$\displaystyle V$ $\displaystyle=$
$\displaystyle\mu^{2}\varphi^{\dagger}\varphi+\lambda(\varphi^{\dagger}\varphi)^{2}+\mu_{\chi}^{2}\chi^{\dagger}\chi+\lambda_{\chi}(\chi^{\dagger}\chi)^{2}+\lambda_{\varphi\chi}\varphi^{\dagger}\varphi\chi^{\dagger}\chi+\rm
H.c..$ (15)

For phenomenological analysis, we assume $\lambda_{\varphi\chi}=0$ [2]. We can
parametrize the scalar fields as,

$\displaystyle\chi_{i}(x)$
$\displaystyle=\frac{v_{i}+s_{i}(x)+i\,a_{i}(x)}{\sqrt{2}},~{}\varphi(x)=\frac{v+h(x)}{\sqrt{2}}.$
(16)

After the minimization, the masses of the scalars are given by,

$\displaystyle m_{s_{i}}=\sqrt{\lambda_{\chi}}v_{i}.$ (17)

The pseudoscalars can achieve their masses through the potential [23],

$V_{\chi}=-\rho{\chi_{i}^{\tilde{N}}\over\Lambda^{\tilde{N}-4}}+\text{H.c.},$
(18)

where $\tilde{N}$ is the least common multiple of the $\rm N$, $\rm M$ and
$\rm P$ in the $\mathcal{Z}_{\rm N}\times\mathcal{Z}_{\rm
M}\times\mathcal{Z}_{\rm P}$ flavour symmetry.

The masses of pseudo-scalar particles are now given by,

$m_{a_{i}}^{2}={1\over
8}|\rho|\tilde{N}^{2}\epsilon_{i}^{\tilde{N}-4}v_{i}^{2}.$ (19)

The values of the $\epsilon_{i}$ for reproducing masses and mixing parameters
are,

$\epsilon_{1}=3.16\times
10^{-6},~{}\epsilon_{2}=0.0031,~{}\epsilon_{3}=0.87,~{}\epsilon_{4}=0.000061,~{}\epsilon_{5}=0.000270,~{}\epsilon_{6}=0.0054,~{}\epsilon_{7}=7.18\times
10^{-10}.$

From these we can calculate the masses of the pseudoscalars $a_{i}$. For
instance, for $|\rho|=1$, $v_{i}=10^{11}$ GeV and $\tilde{N}=14$111 The
$\tilde{N}=14$ corresponds to the symmetry
$\mathcal{Z}_{2}\times\mathcal{Z}_{\rm M}\times\mathcal{Z}_{14}$ where $\rm
M=7~{}\text{or}~{}14$., we have $m_{a_{4}}\approx 4.1\times 10^{-10}$ GeV,
$m_{a_{5}}\approx 7.1\times 10^{-7}$ GeV, and $m_{a_{6}}\approx 2$ GeV.
Therefore, contribution of these particles to neutral meson mixing
($K,B_{d},B_{s}$) parameters is highly suppressed even for their heavier
masses corresponding to $v_{i}>10^{11}$ GeV [24].

We note that neutrino masses may be inverted in order if the sign of the
$\Delta m_{31}^{2}$ turns out to be negative. This scenario can be realised in
the standard HVM by assigning the charges to the right-handed neutrinos under
the $\mathcal{Z}_{2}\times\mathcal{Z}_{4}\times\mathcal{Z}_{14}$ symmetry as :
$\nu_{e_{R}}:(-,\omega,\omega^{3})$, $\nu_{\mu_{R}}:(-,\omega,\omega)$,
$\nu_{\tau_{R}}:(+,\omega,\omega^{10})$ . The mass matrix of the Dirac
neutrinos becomes,

$\displaystyle{\cal M}_{\rm
Dirac}=\dfrac{v}{\sqrt{2}}\begin{pmatrix}y_{11}^{\nu}\epsilon_{4}\epsilon_{7}&y_{12}^{\nu}\epsilon_{5}\epsilon_{7}&0\\\
y_{21}^{\nu}\epsilon_{4}\epsilon_{7}&y_{22}^{\nu}\epsilon_{5}\epsilon_{7}&0\\\
y_{31}^{\nu}\epsilon_{2}\epsilon_{7}&y_{32}^{\nu}\epsilon_{4}\epsilon_{7}&y_{33}^{\nu}\epsilon_{1}\epsilon_{7}.\end{pmatrix}.$
(20)

The masses of neutrinos are,

$\displaystyle m_{3}$ $\displaystyle\approx$
$\displaystyle|y^{\nu}_{33}|\epsilon_{1}\epsilon_{7}v/\sqrt{2},m_{2}\approx\left|y^{\nu}_{22}\right|\epsilon_{5}\epsilon_{7}v/\sqrt{2},m_{1}\approx\left|y^{\nu}_{11}-\dfrac{y_{12}^{\nu}y_{21}^{\nu}}{y_{33}^{\nu}}\right|\,\epsilon_{4}\epsilon_{7}v/\sqrt{2}.$
(21)

The neutrino mixing angles now become,

$\displaystyle\sin\theta_{12}^{\ell}$ $\displaystyle\simeq$
$\displaystyle\left|{y_{12}^{\ell}\epsilon_{4}\over
y_{22}^{\ell}\epsilon_{5}}-{y_{12}^{\nu}\over
y_{22}^{\nu}}\right|,\sin\theta_{23}^{\ell}\simeq\left|{y_{23}^{\ell}\over
y_{33}^{\ell}}\right|,~{}\sin\theta_{13}^{\ell}\simeq\left|{y_{13}^{\ell}\epsilon_{2}\over
y_{33}^{\ell}\epsilon_{6}}\right|.$ (22)

In this case, our main predictions for the neutrino mixing angle are,

$\displaystyle\sin\theta_{12}^{\ell}$ $\displaystyle\geq$ $\displaystyle
1-\sin\theta_{12},~{}\sin\theta_{13}^{\ell}\geq\frac{m_{c}}{m_{b}},$ (23)

where $m_{c}/m_{b}=\epsilon_{2}/\epsilon_{6}$.

We predict the range $\sin\theta_{13}^{\ell}\leq 0.18519-0.25958$ using the
mass of $b$-quark at 1 TeV $m_{b}=2.43\pm 0.08$ GeV[22]. The current global
fit provides $\sin\theta_{13}^{\ell}=0.14206-0.15569$. Thus we conclude that
the equality sign of this prediction is ruled out by the present data.

##  3 Impact of the standard HVM on the flavonic dark matter

It is possible that all the pseudoscalars, $a_{i}$, become the so-called
flavonic dark matter (FDM)[23]. The pseudoscalar fields $a_{i}$ can become the
FDM for a particular minimum value of $\tilde{N}$. Thus, the standard HVM may
have a multi-component FDM, and masses of the FDM particles are given by [23],

$m_{a_{i}}=0.88\times
10^{16}\left(\epsilon^{\tilde{N}-4}_{i}\tilde{N}^{4}\frac{|\rho|}{a_{0}^{2}}\right)^{2/5}{\rm
eV}.$ (24)

If $a_{i}$ are the FDM, the VEVs $v_{i}$ are bounded by the FCNC process
$K^{+}\to\pi^{+}a_{i}$ to $v_{i}\gtrsim 7\times 10^{11}V^{d}_{21}{\rm GeV}$
where $V^{d}_{21}\approx 0.225$ [25].

In table 2, we show the largest values of the masses of FDMs $a_{i}$ for the
corresponding minimum values of $\tilde{N}$. This is obtained after imposing
the constraint that the decays of the FDMs are forbidden to a pair of electron
and positron, that is, $m_{a_{i}}<2m_{e}$.

FDMs | $a_{1}$ | $a_{2}$ | $a_{3}$ | $a_{4}$ | $a_{5}$ | $a_{6}$ | $a_{7}$
---|---|---|---|---|---|---|---
$\tilde{N}$ | $13$ | $23$ | $765$ | 16 | 18 | $25$ | 10
Mass (eV) | $8.53\times 10^{4}$ | $3.29\times 10^{4}$ | $4.08\times 10^{5}$ | $9.41\times 10^{4}$ | $9.0\times 10^{4}$ | $2.78\times 10^{4}$ | $9.91\times 10^{4}$
Table 2: The minimum values of $\tilde{N}$ for $|\rho|=a_{0}=1$ such that the
pseudoscalars $a_{i}$ are the flavonic dark matter.

For instance, the pseudoscalar $a_{4}$ does not contribute to the neutral
meson-mixing or any flavour observable even for the VEV $v_{4}=10^{19}$ GeV.
Therefore, the pseudoscalar $a_{4}$ is a flavonic dark matter for the slightly
large symmetry $\mathcal{Z}_{2}\times\mathcal{Z}_{4}\times\mathcal{Z}_{16}$.
Thus, we see a natural emergence of the flavonic dark matter in the standard
HVM [24].

##  4 Summary

In this work, we have derived a framework to address, in particular, the
leptonic flavour mixing based on the VEVs hierarchy. We name it the standard
HVM. The standard HVM predicts its unique and defining signatures in the form
of inequalities $\sin\theta_{12}^{\ell}\geq 1-2\sin\theta_{12}$,
$\sin\theta_{23}^{\ell}\geq 1-\sin\theta_{12}$, and
$\sin\theta_{13}^{\ell}\geq\sin\theta_{12}-\frac{m_{s}}{m_{c}}$ for the normal
mass ordering. This results in very precise predictions of the three leptonic
mixing angles given by $\sin\theta_{12}^{\ell}=0.55\pm 0.00134$,
$\sin\theta_{23}^{\ell}=0.775\pm 0.00067$, and
$\sin\theta_{13}^{\ell}=0.1413-0.1509$.

The next generation neutrino experiments such as DUNE, Hyper-Kamiokande, and
JUNO [26] may probe these predictions. Furthermore, we have predicted the
quark mixing angle $\sin\theta_{13}$ to be
$\approx\sin\theta_{12}^{3}-2\frac{m_{c}}{m_{t}}$, which is perfectly in
agreement with the observed value. However, for this prediction, we need to
accept a reasonable assumption that $|y_{13}^{u}/y_{33}^{u}|\approx 2$.

For different $\mathcal{Z}_{\rm N}\times\mathcal{Z}_{\rm
M}\times\mathcal{Z}_{\rm P}$ symmetries, the pseudoscalars $a_{i}$ may become
the FDM provided $v_{i}\gtrsim 7\times 10^{11}V^{d}_{21}{\rm GeV}$. For
instance, pseudoscalar $a_{4}$ emerges as a natural flavonic dark matter in
the standard HVM. In the scenario where all of the pseudoscalars $a_{i}$ are
flavonic dark matter, there will be no signature of the standard HVM in the
quark and charged lepton flavour physics due to a huge scale suppression
caused by the VEVs $v_{i}\gtrsim 7\times 10^{11}V^{d}_{21}{\rm GeV}$. However,
even in this extreme scenario, the predicted leptonic mixing angles will be
the hallmark of the standard HVM for the normal mass ordering.

## Acknowledgement

We thank Neelam Singh for the help in some computation, and for reading the
manuscript. This work is supported by the Council of Science and Technology,
Govt. of Uttar Pradesh, India through the project `` A new paradigm for
flavour problem " no. CST/D-1301, and Science and Engineering Research Board,
Department of Science and Technology, Government of India through the project
`` Higgs Physics within and beyond the Standard Model" no. CRG/2022/003237.

## Appendix

For reproducing the fermion masses and mixing from the standard HVM, the
values of the fermion masses at $1$TeV, given in ref. [22], are used. The
quark mixing parameters are taken from ref. [20]. The neutrino oscillation
data for the normal hierarchy are taken from the global fit results given in
ref. [19]. The coefficients
$y_{ij}^{u,d,\ell,\nu}=|y_{ij}^{u,d,\ell,\nu}|e^{i\phi_{ij}^{q,\ell,\nu}}$ are
scanned in the ranges $|y_{ij}^{u,d,\ell,\nu}|\in[0.9,3]$ and
$\phi_{ij}^{q,\ell,\nu}\in[0,2\pi]$. The fitting results are,

$y^{u}_{ij}=\begin{pmatrix}1.0+1.73i&0&0.58+1.95i\\\
0&-0.89+0.45i&-0.98-0.14i\\\ 0&-0.32+1.60i&1-0.02i\end{pmatrix},$

$y^{d}_{ij}=\begin{pmatrix}-1.77+2.19i&-1.30+2.29i&1.08\,-2.16i\\\
0.76\,-0.95i&1.12\,-2.36i&2.09\,+0.24i\\\
0.31\,+1.98i&-2.25+0.16i&-1.65-2.04i\end{pmatrix},$

The Dirac CP phase in the standard parametrization is $\delta_{\rm
CP}^{q}\approx 1.144$.

$y^{\ell}_{ij}=\begin{pmatrix}1.64\,-0.20i&0.52\,-0.89i&0.36\,-1.04i\\\
0&0.07\,+1.01i&0.42\,+1.02i\\\ 0&0&1.09\end{pmatrix},$

For normal mass ordering the neutrino couplings are,

$y^{\nu}_{ij}=\begin{pmatrix}0.6\,-0.88i&0.87\,-0.41i&1.5\,-0.00004i\\\
0&0.96\,+0.12i&-0.52-1.41i\\\ 0&-1.43+0.28i&1.5\,+0.00004i\end{pmatrix},$

and the leptonic Dirac $CP$ phase is $\delta_{\rm CP}^{\ell}\approx 3.14$.

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††thanks<EMAIL_ADDRESS>
# Magnetic order and strongly-correlated effects in the one-dimensional Ising-
Kondo lattice

Xiaofan Zhou  State Key Laboratory of Quantum Optics and Quantum Optics
Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan 030006,
China  Collaborative Innovation Center of Extreme Optics, Shanxi University,
Taiyuan 030006, China  Jingtao Fan  State Key Laboratory of Quantum Optics and
Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University,
Taiyuan 030006, China  Collaborative Innovation Center of Extreme Optics,
Shanxi University, Taiyuan 030006, China  Suotang Jia  State Key Laboratory of
Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy,
Shanxi University, Taiyuan 030006, China  Collaborative Innovation Center of
Extreme Optics, Shanxi University, Taiyuan 030006, China

###### Abstract

We investigate the magnetic order and related strongly-correlated effects in
an one-dimensional Ising-Kondo lattice with transverse field. This model is
the anisotropic limit of the conventional isotropic Kondo lattice model, in
the sense that the itinerant electrons interact with the localized magnetic
moments via only longitudinal Kondo exchange. Adopting the numerical density-
matrix-renormalization group method, we map out the ground-state phase diagram
in various parameter spaces. Depending on the Kondo coupling and filling
number, three distinct phases, including a metallic paramagnetic, a metallic
ferromagnetic, and a gapped spin-density wave phase, are obtained. The spin-
density wave is characterized by an ordering wave vector which coincides with
the nesting wave vector of the Fermi surface. This makes the corresponding
magnetic transition a spin analog of the Peierls transition occurring in the
one-dimensional metal. Moreover, by analyzing the momentum distribution
function and charge correlation function, the conduction electrons are shown
to behave like free spinless fermions in the ferromagnetic phase. We finally
discuss the effect of the repulsive Hubbard interaction between conduction
electrons. Our work enriches the Kondo physics and deepens the current
understanding of the heavy fermion compounds.

###### pacs:

42.50.Pq

##  I Introduction

As one of the most important canonical models in condensed matter systems, the
Kondo lattice model (KLM) is used to describe the process where a conduction
band, occupied by itinerant fermions, interacting with a lattice of localized
magnetic moments KLM1 . The KLM is relevant to a wide class of real materials
called heavy fermion system KLM2 ; KLM3 ; superconducting ; KLM4 ; KLM5 ; KLM6
; KLM7 ; HF01 ; HF02 ; HF03 , in which huge quasiparticle mass can emerge due
to the formation of Kondo singlets HF1 ; HF2 . The study of such heavy fermion
materials is believed to be critical for understanding diverse phenomena such
as quantum criticality KLM4 ; Critical1 ; Critical01 ; Critical02 ; Critical03
; Critical2 ; Critical04 ; Critical05 ; Critical06  and high-$T_{c}$
superconducting compounds KLM2 ; KLM3 ; superconducting . While the well-
solved single-impurity Kondo problem provides some useful concepts for the
KLM, there are aspects where the single-impurity results have no analog in the
lattice case SKondo1 ; SKondo2 ; SKondo3 . For example, except for a few exact
results in certain limits ExactKLM1 ; ExactKLM2 ; ExactKLM3 , the dynamical
details of the interplay between the local Kondo physics and the nonlocal
Ruderman-Kittel-Kasaya-Yosida (RKKY) interaction remains elusive. As a
consensus that has been reached, under zero temperature, the one-dimensional
(1D) KLM presents three distinct magnetic phases: the anti-ferromagnetic phase
(AFM) with fully opened spin and charge gaps, the metellic paramagnetic phase
(PM) with RKKY correlations, and the metellic ferromagnetic phase (FM) KLM3 ;
KLMP1 ; KLMP2 ; KLMP3 ; KLMP4 ; KLMP5 ; KLMP6 . The AFM is stable down to zero
Kondo coupling when the conduction band is half-filled, whereas the PM-to-FM
transition can occur only away from half-filling HalfKLM1 ; HalfKLM2 ;
HalfKLM3 .

The KLM respects a SU(2) rotational symmetry due to the isotropic Kondo
interaction in the spin space. However, by adding anisotropy between the Kondo
couplings, numerous interesting physics, which is beyond the isotropic model,
can emerge AKLM01 ; AKLM02 ; AKLM03 . For example, the KLM with properly tuned
anisotropy has been utilized to understand the physics of the dissipative two-
state systems AKLM1 ; AKLM2 . Moreover, it has been shown that the
ferromagnetic Kondo couplings with easy-plane anisotropy may yield anomalous
singlet formation AKLM3 . Among various kinds of anisotropy introduced in the
KLM, the Ising-Kondo lattice model (IKL) with only longitudinal Kondo coupling
takes on a special role IKL1 ; IKL2 ; IKL3 ; IKL4 ; IKL5 ; IKL6 ; IKL7 . Its
importance is due both to its simple Ising-type coupling form, amenable to
both analytical and numerical treatments IKL5 , and to its relevance to a
series of real materials IKL1 ; IKL2 ; IKL3 ; IKL4 . The IKL was first
proposed to describe the concurrence of large specific heat jump and weak
antiferromagnetism in URu2Si2 IKL1 . Recently, some thermodynamic properties
of the IKL IKL5 , including the antiferromagnetic topological character IKL6
and the emergent strange metal behaviors IKL7 , were discussed on two-
dimensional lattices. However, a systematic understanding of the magnetic
order in the IKL under various conduction electron concentrations, especially
for the 1D case, is still lacking.

In this work, we investigate the ground-state properties of the 1D IKL with
transverse field by using the numerical density-matrix-renormalizationgroup
(DMRG) method dmrg1 ; dmrg2 . The competition between the nonlocal RKKY
mechanism and the local Kondo physics leads to the emergence of magnetic
correlations and, consequently, various magnetic orders. Apart from the
metallic PM and FM, a gapped spin-density-wave phase (SDW) with long-range
order is unraveled. The ordering wave vector characterizing the SDW is found
to be nothing but the nesting wave vector of the Fermi surface constructed by
the conduction electrons, making the SDW a Peierls-like insulator. We also
discuss the effect of repulsive Hubbard interaction imposed on the magnetic
orders. It is revealed that the Hubbard interaction influences the magnetic
phases in a way much like the Kondo coupling; it can trigger both the PM-to-
SDW and SDW-to-FM transitions, depending on the filling number of the
conduction electrons. Remarkably, we find that, deep inside the FM region, the
conduction electrons behave like free spinless fermions, characterized by a
distinct redistribution of the electron quasi-momentum and the corresponding
signals in the charge correlation function.

##  II Model and method

The IKL in consideration can be described by the following Hamiltonian
($\hbar=1$ throughout)

$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle-t\sum_{\left\langle
i,j\right\rangle,\sigma}\hat{c}_{i,\sigma}^{{\dagger}}\hat{c}_{j,\sigma}+h\sum_{j}\hat{s}_{j}^{z}\hat{S}_{j}^{z}+\Delta\sum_{j}\hat{S}_{j}^{x}$
(1)
$\displaystyle+\frac{U}{2}\sum_{j}\hat{n}_{j,\uparrow}\hat{n}_{j,\downarrow}$

where $\hat{c}_{j,\sigma}^{{\dagger}}$ ($\hat{c}_{j,\sigma}$) is the creation
(annihilation) field operator of the conduction electron with spin $\sigma$
(=$\uparrow,\downarrow$) at lattice site $j$. The conduction electrons can hop
between adjacent sites $\left\langle i,j\right\rangle$ with the hopping rate
$t$. $\hat{S}_{j}^{z}$ denotes the $z$-component of the spin-1/2 operator for
the localized magnetic moment. The spin operator $\hat{s}_{j}^{z}$ for the
conduction electrons is defined by
$\hat{s}_{j}^{z}=\sum_{\tau,\acute{\tau}}\hat{c}_{j,\tau}^{{\dagger}}\sigma_{\tau,\acute{\tau}}^{z}\hat{c}_{j,\acute{\tau}}/2$
where $\sigma^{z}$ is the Pauli-$z$ matrix. With these definitions, the second
term of Hamiltonian (1) denotes the longitudinal Kondo interaction between the
conduction electron and the localized moment at the same site $j$. The
interaction strength $h$ is assumed to be positive, implying an
antiferromagnetic Kondo coupling. Note that for the IKL considered here, the
ferromagnetic coupling with $h<0$ is connected to the antiferromagnetic case
through a simple spin rotation, which is in contrast to the conventional
isotropic KLM. The latter can exhibit dramatic different Kondo physics
depending on the sign of the coupling strength KLM8 ; KLM9 . A transverse
field $\Delta$, introducing additional nonadiabaticity, is applied on the
localized moments IKL1 . The direct repulsive interaction between conduction
electrons is also included through the Hubbard-$U$ term. In this work, our
main focus will be on the ground-state properties of the conduction electrons.
The local moments construct the same magnetism with conduction electrons due
to the longitudinal Kondo coupling term $\sim\hat{s}_{j}^{z}\hat{S}_{j}^{z}$.
In the following discussion, we set the energy scale by taking $t=1$, and we
also take $\Delta=1$ unless otherwise specified.

The Hamiltonian (1), with vanishing transverse Kondo couplings (i.e.,
$\hat{s}_{j}^{-}\hat{S}_{j}^{+}+$ H.c.$\rightarrow 0$), represents essentially
the anisotropic limit of the conventional KLM KLM1 . In the adiabatic limit
with $\Delta=0$, the IKL is exactly sovable since the local moment at each
site is conservative obeying $[\hat{S}_{j}^{z},\hat{H}]=0$. The thermodynamic
properties of the IKL under this condition has been preliminarily touched by
Quantum Monte Carlo simulations IKL5 . A nonzero transverse field $\Delta$,
however, breaks the conservativeness of $\hat{S}_{j}^{z}$ and effectively adds
quantum fluctuations to the localized moments. In this sense, the localized
moments in IKL can be alternatively viewed as lattice degrees of freedom which
impose dynamic potentials on the conduction electrons. The interactions
between fermions and lattice degrees of freedom become specially important in
1D due to the perfect Fermi surface nesting nest1 ; nest2 . One of the most
noted consequences, for example, is the Peierls transition, characterized by
the formation of a charge-density wave ordered at the nesting wave vector
Peierls . In light of the Ising-type interaction between conduction electrons
and localized moments, in the IKL, we expect that an analogous SDW instability
of the conduction electrons can be reached through similar mechanisms. This
point will be further elucidated in the subsequent sections.

The magnetic order of the conduction electrons can be effectively
characterized by the spin structure factor,

$S(k)=\frac{1}{L}\sum_{l,j}\left\langle\hat{s}_{l}^{z}\hat{s}_{j}^{z}\right\rangle
e^{i(l-j)k}$ (2)

where $L$ is the number of lattice sites and $\left\langle...\right\rangle$
denotes the ground-state average. The peak position in $S(k)$, denoted by an
ordering wave vector $k^{\max}$, characterizes the spatial variation of spin
orientations projected into the $z$ direction. Alternatively, we can
characterize the magnetic order using the spin correlation function in real
space

$s(r)=\frac{1}{L}\sum_{l}\left\langle\hat{s}_{l}^{z}\hat{s}_{l+r}^{z}\right\rangle$
(3)

where $r$ is the distance between different sites. The correlation function
$s(r)$ has the advantage of intuitively showing the spatial distribution of
the spin correlations. The equivalence between $S(k)$ and $s(r)$ can be
clearly found by noting that they constitute a Fourier transform pair. In the
spirit of Landau’s paradigm, the magnetic ordering is characterized by
spontaneously broken symmetry at some ordering wave vector $k^{\max}$. This
implies the existence of long-range order, which can be identified by scaling
the spin structure factor Long :

$\lim_{L\rightarrow\infty}\frac{1}{L}S(k^{\max})>0.$ (4)

The long-range nature is also manifested in the correlation function $s(r)$ by
$\lim_{r\rightarrow\infty}s(r)\neq 0$. Therefore, we can identify different
magnetic phases according to Eq. (4) by extrapolating the spin structure
factor to the thermodynamic limit $L\rightarrow\infty$. In this way, we may
obtain the $k^{\max}$-ordered SDW for $k^{\max}\neq 0$ and the FM for
$k^{\max}=0$. If $S(k)/L$ is extrapolated to zero in the thermodynamic limit,
the corresponding phase loses any magnetic long-range order and is thus termed
PM. We emphasize that, due to the space-inversion symmetry of the system, any
peaks in $S(k)$, if exist, should be exactly symmetric about $k=0$.

Here, we perform state-of-the-art DMRG calculations to compute the many-body
ground state of the system, with which various physical observable can be
obtained. In our numerical simulations, the filling number $\rho=N/L$ is a
good quantum number which can be varied from zero to one. Here $N$ is the
total number of conduction electrons. We set lattice size up to $L=60$, and
work with open boundary conditions. For each lattice size, we retain 600
truncated states per DMRG block and perform 20 sweeps with a maximum
truncation error of $\sim 10^{-9}$.

Figure 1: The correlation function $s(r)$ for systems with $\rho=0.75$,
$L=60$, and a varying $h$.

##  III Magnetic correlations induced by Kondo coupling

We first clarify the effect of the Kondo coupling $h$ by setting $U=0$. Before
showing the numerical results, some qualitative insights can be gained by
inspecting different parameter regimes. For the coupling strength comparable
with or less than the Fermi energy of the conduction electrons, the dominant
ordering force turns out to be the RKKY correlation with Ising anisotropy,
which compete with the quantum fluctuations induced by the transverse field
$\Delta$. Increasing $h$ further to the intermediate coupling regime, the
disorder created by $\Delta$ is irrelevant and the interplay between the RKKY-
type interactions and the local Kondo physics becomes important. At extremely
strong-coupling $h\gg 1$, the conduction electrons and the local moments are
tightly bound together to form local spin singlets, giving rise to distinct
local Kondo physics. We thus expect rich long-range magnetic orders may emerge
throughout the whole range of the coupling strength.

Figure 2: (a1)-(c1) The scaled spin structure factor $S(k)/L$ and (a2)-(c2)
the corresponding finite-size scalings at three characteristic wave vectors
for systems with (a) $h=2.0$, (b) $h=10.0$, and (c) $h=22.0$. Different system
sizes are characterized by lines with different colors. In these figures, we
set $\rho=0.75$ and $U=0$.

###  III.1 Phase transitions and long-range order

Let us detailedly discuss the magnetic phase behaviors of the IKL in different
coupling regimes. Figure 1 shows the correlation function $s(r)$ with
$\rho=0.75$ and different $h$. When $h$ is relatively weak, the spin
correlation rapidly decays to zero as the distance $r$ increases. For larger
$h$, we observe a persistent oscillation of $s(r)$ instead. This non-decaying
behavior of spin correlation signals the emergence of a long-range magnetic
order.

The transitions between different magnetic phases becomes clearer if we
investigate the corresponding spin structure factor $S(k)$. The left column of
Fig. 2 [ Figs. 2(a1)-2(c1)] plot $S(k)/L$ for three different values of $h$
with $\rho=0.75$. The results for different system sizes are labeled by lines
with different colors. The values of $L\rightarrow\infty$ are obtained by the
standard finite-size scaling. The right column of Fig. 2 [Figs. 2(a2)- 2(c2)]
demonstrate the scaling details at three characteristic wave vectors. We first
focus on the weak coupling strength with $h=2.0$ [Fig. 2(a1)]. It is shown
that $S(k)/L$ develops weak peaks at $k=k^{\max}=\pm 2k_{F}$ for any finite
system sizes, resulting from a RKKY-type correlation. Here $k_{F}=\pi\rho/2$
is the Fermi momentum of the conduction electron. As the system size
increases, however, the peaks become weaker and weaker and eventually
disappear in the thermodynamic limit $L\rightarrow\infty$. This implies that
the magnetic correlations built up in this regime is essentially short range,
and consequently give rise to the appearance of PM. Something interesting
happens if the coupling strength increases. The spin structure factor with
$h=10.0$ is shown in Fig. 2(b1). It is found that, while the structure with
$k^{\max}=\pm 2k_{F}$ still exists, the peaks become much higher and sharper
compared to $h=2.0$. Accordingly, the scaled structure factor
$S(k=k^{\max})/L$ is extrapolated to a finite value in the thermodynamic limit
[see the red line in Fig. 2(b2)], indicating the existence of a SDW which is
ordered at $k=k^{\max}$. This is in sharp contrast with the isotropic KLM. In
that case, the conduction electrons lose the true SDW order due to additional
fluctuations, although short-range magnetic correlations still exist KLM3 .
Increasing $h$ further, the conduction electrons and localized moments start
to be bound together and the tendency of local Kondo singlet formation
overwhelms that of the RKKY-type correlation. As a result, a sharp peak at
$k=0$ grows up whereas density waves at other wave vectors are greatly
suppressed, as shown in Fig. 2(c1). This structure of $S(k)$ unambiguously
demonstrates the ferromagnetism. We thus find that as $h$ increases, the
system starts from PM, and subsequently traverses the SDW, and eventually
evolves to the FM.

Figure 3: (a) The ordering wave vector $k^{\max}$ and (b) the ordering
strength $S^{\max}$ as functions of $h$ with $\rho=0.75$ and $U=0$.

To provide a more precise picture, we monitor the ordering wave vector
$k^{\max}$ as a function of the coupling strength for $\rho=0.75$. Considering
the inherent inversion symmetry of $S(k)$ about $k=0$, we here focus on the
$k^{\max}\geqslant 0$ part. As shown in Fig. 3(a), $k^{\max}$ is not well
defined for $h<5.8$ due to the general vanishing of $S(k)/L$ in the limit of
$L\rightarrow\infty$, which is the character of the PM. Increasing $h$ above
$5.8$, a well-defined $k^{\max}$ emerges at $2k_{F}$, and keeps at that value
until it drops to zero where the FM is formed. The feature that $k^{\max}$
keeps constant before reaching FM is different from the isotropic KLM, where
the peaks in $S(k)$ can move continuously toward zero as $h$ increases KLM9 .
Figure 3(b) plots $S^{\max}$, which we dub ordering strength, in terms of $h$.
$S^{\max}$ is the maximum of the scaled spin structure factor
$\lim_{L\rightarrow\infty}S(k)/L$ within the Brillion zone. With this
definition, we find the relation
$S^{\max}=\lim_{L\rightarrow\infty}S(k^{\max})/L$ applies in the ordered
phase. It follows that $S^{\max}$, together with the value of $k^{\max}$,
serve as complete order parameters characterizing the ordering process of
different magnetic phases. As shown in Fig. 3(b), while $S^{\max}$ remains
zero in the PM, it gradually increases from zero, reaching the maximum, and
then decreases until vanishes at the critical point $h_{c}=16.4$. Above
$h_{c}$, the system enters the FM with monotonically increased $S^{\max}$.

Figure 4: (a) The correlation function $S_{\text{cf}}(r)$ for different $h$.
(b) The on-site value $S_{\text{cf}}(0)\ $as a function $h$. The other
parameters in these figures are $\rho=0.75$, $U=0$, and $L=60.$ The red dashed
line in (b) denotes the value of the perfect on-site singlets. Figure 5: (a)
The ordering wave vector $k^{\max}$ and (b) the ordering strength $S^{\max}$
as functions of $\rho$ with $h=12.0$ and $U=0$.

The non-monotonically behavior of $S^{\max}$ results from the competition
between the RKKY-type interaction and the Kondo singlet formation. The details
of this competition is partially manifested in the correlation function
between the conduction electron and local moment:

$S_{\text{cf}}(r)=\frac{1}{L}\sum_{l}\left\langle\hat{s}_{l}^{z}S_{l+r}^{z}\right\rangle.$
(5)

As defined in Eq. 5, the correlation function $S_{\text{cf}}(r)$ measures the
non-local character of the Kondo singlet formation KLMP2 . Figure 4(a) plots
$S_{\text{cf}}(r)$ with a varying values of $h$. At weak coupling $h=2$, the
hybridization between the conduction electron and local moment is ineffective,
yielding a rather weak signal in $S_{\text{cf}}(r)$. For larger $h$,
$S_{\text{cf}}(r)$ increases in height and it decays very slowly as $r$
increases, reflecting a sizeable extension of the Kondo singlet. This long-
range behavior of $S_{\text{cf}}(r)$ signals the appearance of a regime where
the RKKY mechanism may work. Increasing the coupling strength further such
that $h\gtrsim 12$, $S_{\text{cf}}(r)$ becomes more and more localized,
implying that the size of the singlet reduces. Consequently, the RKKY
mechanism starts to be suppressed. This point is further strengthen by
examining the on-site singlet correlation $S_{\text{cf}}(0)$ versus $h$. As
shown in Fig. 4(b), as $h$ increases, $S_{\text{cf}}(0)$ approaches
asymptotically that of perfect on-site singlets $-3\rho/4$ (see the red dashed
line in Fig. 4(b)).

Figure 6: The phase diagram in the $h-\rho$ plane with $U=0$. The phase
boundaries have been extrapolated to the thermodynamic limit
$L\rightarrow\infty$.

Up to now, our discussion is limited to a specific band filling of the
conduction electrons, namely $\rho=0.75$. We have observed that the ordering
wave vector of the SDW in this filling is fixed by the Fermi momentum of the
conduction electrons via $k^{\max}=\pm 2k_{F}$. In fact, this relation applies
to any fillings provided the system is within the SDW. To demonstrate this, we
plot $k^{\max}$ and $S^{\max}$ as functions of $\rho$ with $h=12.0$ in Figs.
5(a) and 5(b), respectively. As an immediate finding, while $k^{\max}$ keeps
filling independent in the FM and PM, it follows $\rho$ linearly through
$k^{\max}=\pm 2k_{F}$ in the SDW region [see Fig 5(a)]. Notice that the
ordering wave vector $2k_{F}$ is exactly the nesting wave vector of the Fermi
surface, which characterizes the charge density wave in an Peierls insulator
Peierls . It is thus inferred that the appearance of the SDW in the 1D IKL can
be traced back to the perfect Fermi surface nesting effect. The latter
drastically modifies the band structure of itinerant electrons. From this
point of view, the formation of SDW in the IKL can be regarded as a magnetic
analog of the Peierls transition which occurs in the charge degree of freedom.

###  III.2 Phase diagram

With the understanding above, we map out the phase diagram in the $h-\rho$
plane in Fig. 6. As can be seen, the PM occupies most portion of the phase
diagram for weak coupling strength, due to the dominant quantum fluctuations
of the local moments. As $h$ increases, the ordered phases, including FM and
SDW with various ordering wave vectors, emerge. Whereas the SDW is more
favored when the filling number is closed to $\rho=1$, the FM is formed away
from this half-filling case. The ordering process for $h=12.0$ is illustrated
in Fig. 5(b), where we find $S^{\max}$ increases rapidly as $\rho$ approaches
one. We emphasize that the SDW for $\rho=1$ is an antiferromagnetic insulator
characterized by a Néel configuration with finite spin and charge gaps IKL5 .
Increasing $h$ above $14.0$, the PM generally disappears and the SDW region
starts to shrink. The antiferromagnetic insulator for $\rho=1$, however,
remains stable against the ferromagnetization even in the $h\rightarrow\infty$
limit IKL5 .

Figure 7: Schematic illustration of the magnetic structures when the
conduction electrons are (a) around and (b) far away from half filling. In
these figures, the red and blue balls denote the localized moments and
conduction electrons, respectively.

The mechanism of different magnetic structures closed to and far away from
half filling can be understood as follows. As illustrated in Fig. 7(a), in the
strong coupling regime and around half filling, each neighbor sites of the
conduction electron is likely to be occupied by another one, since double
occupancy of the conduction electrons on the same site would elevate the
energy by $\Delta E\sim h/4$. As a result, the hopping between neighbor sites
is forbidden by the Pauli exclusion principle if the spins of conduction
electrons are aligned. However, if the spins form an antiferromagnetic
configuration, a virtual tunnelling process can occur which lowers the kinetic
energy of the conduction electrons by $\Delta E\sim 4t^{2}/h$, resembling
closely the superexchange process in the Hubbard model Long . Far away from
half filling, on the other hand, the conduction electrons can freely move
along the lattice without costing any additional energy once their spins are
aligned parallel to each other [see Fig. 7(b)].

Figure 8: The entanglement entropy $S_{l}$ in terms of the block size $l$, for
representative points of PM ($h=2.0$, $\rho=0.75$), SDW ($h=12.0$,
$\rho=0.5$), and FM ($h=18.0$, $\rho=0.5$). The scaling behavior for different
phases are plotted by lines with different colors.

The above picture hints that the system is a gapped insulator throughout the
SDW (not only the half-filling case) and becomes metallic when moving into FM
and PM. This conjecture is verified by calculating the scaling of the
entanglement entropy between a block of size $l$ and the rest of the system,
$S_{l}=-$Tr$(\rho_{l}\ln\rho_{l})$, where
$\rho_{l}=$Tr${}_{L-l}\left|\Psi\right\rangle\left\langle\Psi\right|$ is the
reduced density matrix corresponding to the block. For 1D systems, gapped
phases follows an area law, and the entanglement entropy $S_{l}$ would
saturate as $l$ increasesentropy1 . $S_{l}$ in gapless phases, however, does
not saturate but grows logarithmically when increasing $l$ entropy2 . The
scaling of $S_{l}$ for three representative points in the phase diagram are
shown in Fig. 8. While the presence of a gap in the SDW is indicated by the
saturation of $S_{l}$, the FM and PM are both gapless since $S_{l}$
logarithmically diverges with the block’s size.

Figure 9: (a) The momentum distribution function $n(k)$ and (b) the charge
structure factor $C(k)$ for systems with $\rho=0.5$, $U=0$, $L=60$ and a
varying $h$.

###  III.3 Momentum distribution of the conduction electrons

The interplay of the localized and itinerant behaviors in the IKL can be
alternatively characterized by the momentum distribution function of the
conduction electrons,

$n(k)=\frac{1}{L}\sum_{l,j}\left\langle\hat{c}_{l\sigma}^{{\dagger}}\hat{c}_{j\sigma}\right\rangle
e^{i(l-j)k}$ (6)

Figure 9(a) shows the variation of $n(k)$ with increasing $h$ for $\rho=0.5$.
For small $h$, as expected, the Fermi surfaces of the conduction electrons is
clearly formed at $k_{F}=\pm\pi\rho/2$. As $h$ increases, the conduction
electrons start to be hybridized with the local moments, which broadens the
distribution of $n(k)$ and blurs the original Fermi surfaces. For even larger
$h$, two sharp edges in $n(k)$, separating the occupied and unoccupied states,
appear at $k_{F}^{L}=2k_{F}=$ $\pm\pi\rho$. Further increasing $h$ has little
effect on the structure of $n(k)$, indicating that the system has reached a
strong coupling regime with a well-defined large Fermi surface. The variations
in the structure of $n(k)$ is accompanied by corresponding changes in the
charge structure factor, defined by

$C(k)=\frac{1}{L}\sum_{l,j}\left(\left\langle\hat{n}_{l}\hat{n}_{j}\right\rangle-\left\langle\hat{n}_{l}\right\rangle\left\langle\hat{n}_{j}\right\rangle\right)e^{i(l-j)k}.$
(7)

As shown in Fig. 9(b), we plot $C(k)$ for the same values of $h$ as those used
in Fig. 9(a). When $h$ is small, $C(k)$ exhibits two cusps at
$k_{c}=\pm\pi\rho$, manifesting the free particle nature of the conduction
electrons. The cusps, however, is smoothed at intermediate values of $h$,
exhibiting characters of strongly interacting fermions. Moving into the strong
coupling regime with larger $h$, two new cusps are formed at $k_{c}^{L}=\pm
2\pi\rho$. Through a careful analysis of $n(k)$ and $C(k)$ for different
filling number, we find that the appearances of the characteristic wave
vectors $k_{F}^{L}$ and $k_{c}^{L}$ are always accompanied by the formation of
FM. The above signatures of $n(k)$ and $C(k)$ strongly suggest that, in the
strong coupling regime, the conduction electrons behave like free spinless
fermions.

Figure 10: (a) The phase diagram in the $h-U$ plane with $\rho=0.75$ and (b)
the phase diagram in the $\rho-U$ plane with $h=4.0$. The phase boundaries in
(a) and (b) have been extrapolated to the thermodynamic limit
$L\rightarrow\infty$.

To understand this, let us start from a half-filled IKL with $N$ $(=L)$
conduction electrons. In the strong coupling regime, each local moment tend to
catch an electron to form a local Kondo singlet, ending up with a 1D liquid
composed of $L$ singlets. Doping away from half filling amounts to adding
holes to the system. Note that double occupancy of holes on the same lattice
site is naturally excluded by definition. As discussed above, when the spins
of conduction electrons are aligned, the system can lower its energy by
allowing the holes freely moving on the lattice. Otherwise the hoping process
of holes may elevate energy due to the strong Kondo coupling. Suppose now we
have $L-N$ holes ($N<L$) moving in the sea of the singlet background. The
Fermi momentum of these mobile holes is thus $k_{F}^{h}=[1-(L-N_{c})/L]\pi=$
$\rho\pi=k_{F}^{L}$, exactly at the large Fermi surface.

Figure 11: (a1)-(b1) The ordering wave vector $k^{\max}$ and (a2)-(b2) the
ordering strength $S^{\max}$ as functions of $U$ for systems with (a) $h=4.0$
and (b) $h=12.0$. In these figures, we set $\rho=0.75$.

##  IV Effect of the repulsive Hubbard interaction

We proceed to discuss the effect of finite repulsive Hubbard interaction $U$.
Following the same manner used in Sec. III, we present the phase diagram in
the $h-U$ plane for $\rho=0.75$ in Fig. 10(a). As a direct impact of the
Hubbard interaction, the FM becomes more robust, and the phase boundary
between FM and SDW is pushed to lower value of $h$. The stabilization of FM by
$U$ is also predicted in the isotropic KLM KLM7 ; ExactKLM3  and can be
understood on physical grounds. The repulsive interaction enhances the energy
penalty costed by double occupancy of the conduction electrons on the same
lattice site, which promotes the formation of the on-site Kondo singlet. The
latter becomes mobile if the lattice is doped far away from half filling, and
thus favors ferromagnetism through the Kondo mechanism we mentioned at the
last of Sec. III. A more interesting finding which does not exist in the
isotropic KLM is that $U$ can also lower the critical value of $h$ between PM
and SDW. This effect, however, may saturate as $U$ increases to larger value.

To see clearly the role of $U$ in inducing various phase transitions, we plot
$k^{\max}$ and $S^{\max}$ as functions of $U$ for two different values of $h$
in Fig. 11. It is to be seen that, depending on the value of $h$, the
repulsive interaction can drive both PM-to-SDW [Figs. 11(a1) and 11(a2)] and
SDW-to-FM [Figs. 11(b1) and 11(b2)] transitions. We find again that, the
ordering wave vector $k^{\max}$ in the SDW is independent of $U$ and uniquely
specified by the Fermi momentum. As such, the phase transition driven by $U$
is of course filling dependent. The general result is summarized in the
$\rho-U$ phase diagram for $h=4.0$ [see Fig. 10(b)]. We observe that the
filling number influences the phase transitions in a similar fashion as that
in the $h-\rho$ phase diagram (see Fig. 6); the transition from PM to FM turns
out to be easier for low electron filling, and the SDW is stabilized only
closed to half filling.

Figure 12: (a) The momentum distribution function $n(k)$ and (b) the charege
structure factor $C(k)$ for systems with $\rho=0.42$, $h=4.0$, $L=60$, and a
varying $U$.

Along the logic in Sec. III, let us examine the impact of $U$ on the momentum
distribution function and charge structure factor. The variations of $n(k)$
and $C(k)$ with $\rho=0.5$ and $h=4.0$ and a varying values of $U$ are
respectively shown in Figs. 12(a) and 12(b). As depicted in these figures,
when increasing $U$, both $n(k)$ and $C(k)$ exhibit the same behaviors as
those shown in Fig. 9. Especially when $U$ is sufficiently large for which the
FM is reached, the aforementioned large Fermi surface is clearly formed at
$k_{F}^{L}=$ $\pm\pi\rho$, suggesting a ferromagnetic metal. This is
interesting since the Hubbard interaction is commonly believed to render
systems more insulating, whereas its role here is reversed through the
aforementioned Kondo mechanism.

As can be found from above discussion, the role of repulsive Hubbard
interaction presents great similarities with the Kondo coupling $h$, hinting
that it may work through renormalizing $h$. The fully clarification of the
connection between $U$ and $h$ in the IKL may require some more sophisticated
analytical approaches, such as renormalization group and bosonization, which
is out of the scope of the present work and merits a separate study in the
future.

##  V Conclusion

In conclusion, we have studied the ground-state properties of the 1D IKL by
using the numerical density-matrix-renormalizationgroup (DMRG) method. Three
distinct quantum phases, including a metallic PM, a matellic FM, and a gapped
SDW, have been obtained. The SDW is characterized by an ordering wave vector
which coincides with the nesting wave vector of the Fermi surface. This makes
the PM-to-SDW transition a magnetic analog of the Peierls transition which
occurs in the charge degree of freedom of a one-dimensional metal. Moreover,
by analyzing the momentum distribution function and charge correlation
function, the conduction electrons are shown to behave like free spinless
fermions in the FM. The effect of the repulsive Hubbard interaction between
conduction electrons has also been clarified.

## Acknowledgments

This work is supported by the National Key R&D Program of China under Grant
No. 2022YFA1404003, the National Natural Science Foundation of China (NSFC)
under Grant No. 12004230, 12174233 and 12034012, the Research Project
Supported by Shanxi Scholarship Council of China and Shanxi ’1331KSC’.

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