# Transverse Shifts and Time Delays of Spatiotemporal Vortex Pulses
Reflected and Refracted at a Planar Interface

Maxim Mazanov  V. N. Karazin Kharkiv National University, Kharkiv, 61022,
Ukraine  Danica Sugic  Theoretical Quantum Physics Laboratory, RIKEN Cluster
for Pioneering Research, Wako-shi, Saitama 351-0198, Japan  Miguel A. Alonso
CNRS, Centrale Marseille, Institut Fresnel, Aix Marseille University, UMR
7249, 13397 Marseille CEDEX 20, France  The Institute of Optics, University of
Rochester, Rochester, NY 14627, USA  Franco Nori  RIKEN Center for Quantum
Computing, Wako-shi, Saitama 351-0198, Japan  Theoretical Quantum Physics
Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198,
Japan  Physics Department, University of Michigan, Ann Arbor, Michigan
48109-1040, USA  Konstantin Y. Bliokh  Theoretical Quantum Physics Laboratory,
RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan

###### Abstract

Transverse (Hall-effect) and Goos–Hänchen shifts of light beams
reflected/refracted at planar interfaces are important wave phenomena, which
can be significantly modified and enhanced by the presence of intrinsic
orbital angular momentum (OAM) in the beam. Recently, optical spatiotemporal
vortex pulses (STVPs) carrying a purely transverse intrinsic OAM were
predicted theoretically and generated experimentally. Here we consider the
reflection and refraction of such pulses at a planar isotropic interface. We
find theoretically and confirm numerically novel types of the OAM-dependent
transverse and longitudinal pulse shifts. Remarkably, the longitudinal shifts
can be regarded as time delays, which appear, in contrast to the well-known
Wigner time delay, without temporal dispersion of the reflection/refraction
coefficients. Such time delays allow one to realize OAM-controlled slow
(subluminal) and fast (superluminal) pulse propagation without medium
dispersion. These results can have important implications in various problems
involving scattering of localized vortex states carrying transverse OAM.

##  I Introduction

Small wavepacket shifts and time delays are currently attracting considerable
attention due to their noticeable roles in nanoscience. The first example of
such effects is the Goos–Hänchen shift of the beam reflected/refracted at a
planar interface Goos and Hänchen (1947); Artmann (1948); Merano  _et al._
(2009); Jayaswal  _et al._ (2013); Bliokh and Aiello (2013). This shift is
proportional to the wavevector-gradient of the logarithm of the reflection
coefficient.

The temporal counterpart of this spatial shift is the Wigner time delay of a
wavepacket scattered by a frequency-dependent potential Wigner (1954); Chiao
and Steinberg (1997); de Carvalho and Nussenzveig (2002); Winful (2006); Asano
_et al._ (2016). Correspondingly, this delay is given by the frequency
gradient of the logarithm of the scattering coefficient.

Another example of beam shifts is the transverse Imbert–Fedorov shift
associated with the spin-Hall effect (i.e., a transverse circular-
polarization-induced shift of the reflected/refracted beam) Imbert (1972);
Schilling (1965); Fedoseyev (1988); Onoda  _et al._ (2004); Bliokh and Bliokh
(2006); Hosten and Kwiat (2008); Bliokh and Aiello (2013); Götte and Dennis
(2012); Töppel  _et al._ (2013); Bliokh  _et al._ (2015, 2016); Ling  _et al._
(2017). This shift has a more complicated origin associated with the spin
angular momentum carried by the wave, spin-orbit interaction, and conservation
of the total angular momentum component normal to the interface.

All these shifts and time delays have been studied mostly for Gaussian-like
wavepackets and beams, and all have a typical scale of the wavelength or wave
period, which can be enhanced up to the beam-width or pulse-length scale using
the weak-measurement technique Hosten and Kwiat (2008); Götte and Dennis
(2012); Töppel  _et al._ (2013); Jayaswal  _et al._ (2013); Bliokh  _et al._
(2016); Asano  _et al._ (2016).

It has also been shown that the beam shifts can be modified significantly by
the presence of the intrinsic orbital angular momentum (OAM) in optical vortex
beams Fedoseyev (2001); Dasgupta and Gupta (2006); Fedoseyev (2008); Okuda and
Sasada (2008); Bliokh  _et al._ (2009); Bekshaev (2009); Merano  _et al._
(2010); Dennis and Götte (2012); Bliokh and Nori (2012a); Bliokh and Aiello
(2013). This enhances the Gaussian-beam shifts by the factor of the OAM
quantum number $\ell$ and also produces new types of shifts.

To the best of our knowledge, the role of the intrinsic OAM and vortices on
time delays have not been studied so far. This is because optical vortex beams
are usually monochromatic states unbounded in the longitudinal direction,
while time delays make sense only for finite-length wavepackets.

Recently, a novel type of localized pulses carrying transverse intrinsic OAM —
spatiotemporal vortex pulses (STVPs) — was described theoretically Sukhorukov
and Yangirova (2005); Dror and Malomed (2011); Bliokh and Nori (2012b); Bliokh
(2021, in press) and generated experimentally Jhajj  _et al._ (2016); Hancock
_et al._ (2019); Chong  _et al._ (2020); Hancock  _et al._ (2021); Wan  _et
al._ (2021); Wang  _et al._ (2021) (see also Ref. Dallaire  _et al._ (2009)
for the zeroth-order Bessel STVP without OAM). Such STVPs have geometrical and
OAM properties different from monochromatic vortex beams. (Note that STVPs
should not be confused with principally different space-time wavepackets
considered in Refs. Kondakci and Abouraddy (2017, 2019); Turunena and Friberg
(2010).) Therefore, it is natural to expect that these qualitatively new
objects behave differently in problems involving beam shifts and time delays.

In this work, we consider reflection and refraction of an optical STVP at a
planar isotropic interface. We predict theoretically and confirm numerically a
number of novel spatial shifts and time delays that are controlled by the
value and orientation of the intrinsic OAM of the pulse. Remarkably, time
delays appear in this system without any frequency dependence of the
reflection/refraction coefficients, thereby allowing one to realize slow
(subluminal) and fast (superluminal) pulse propagation without medium
dispersion. This is in sharp contrast to Wigner time delays and is produced by
the coupling of spatial and temporal degrees of freedom in spatiotemporal
vortices. Our results can have important implications in various problems
involving scattering of localized vortex states with transverse OAM, both
classical and quantum.

##  II Laguerre-Gaussian STVPs

We first introduce a STVP propagating along the $z$-axis and carrying
transverse OAM along the $y$-axis. For this, akin to monochromatic Laguerre-
Gaussian (LG) beams Allen  _et al._ (2003); Andrews and Babiker (2012), we
consider a LG-type plane-wave spectrum in the $(z,x)$ plane with the central
wavevector ${\bf k}_{0}=k_{0}\bar{\bf z}$ (the overbar denotes the unit vector
of the corresponding axis) and zero radial quantum number (Fig. 1):

$\tilde{\psi}\left({\bf{k}}\right)\propto{\left[{\gamma\left({{k_{z}}-{k_{0}}}\right)+i\,{\rm
sgn}(\ell){k_{x}}}\right]^{\left|\ell\right|}}e^{-\tfrac{\Delta^{2}}{4}\left[{{\gamma^{2}}{{\left({{k_{z}}-{k_{0}}}\right)}^{2}}+k_{x}^{2}}\right]}.$
(1)

Here, $\ell$ is the integer vortex charge, $\gamma$ is the factor determining
the ellipticity of the STVP profile in the $(z,x)$ plane, and $\Delta$ is the
$x$-width of the pulse ($\gamma\Delta$ being its $z$-length). Note that we do
not include a distribution over $k_{y}$, because for our goals it is
sufficient to consider pulses unbounded along the OAM direction. If needed, an
additional Gaussian distribution over $k_{y}$ can provide localization along
the $y$-axis.

The real-space form of the STVP (1) is given by the Fourier integral
$\psi\left({{\bf{r}},t}\right)\propto\iint{\tilde{\psi}\left({\bf{k}}\right){e^{i{\bf{k}}\cdot{\bf{r}}-i\omega\left({\bf{k}}\right)t}}}d{k_{x}}d{k_{z}}$,
where $\omega\left({\bf{k}}\right)=kc$. For the purpose of this work it is
sufficient to use a paraxial approximation, $k_{0}\Delta\gg 1$, in which only
linear deviations in the transverse wavevector components are considered. This
leads to the following expression for a paraxial LG-type STVP where
diffraction is ignored (Fig. 1):

$\psi\propto{\left[{{\gamma^{-1}}\zeta+i\,{\rm
sgn}(\ell)x}\right]^{\left|\ell\right|}}\\!\exp\\!\left[{-\frac{\left({{\gamma^{-2}}{\zeta^{2}}+{x^{2}}}\right)}{{{\Delta^{2}}}}+i{k_{0}}\zeta}\right]\\!,$
(2)

where $\zeta=z-ct$ is the pulse-accompanying coordinate. Closed-form real-
space expressions that incorporate diffraction both in the paraxial and
nonparaxial regimes will be described in a separate work.

Figure 1:  The phase-intensity distributions of the momentum-space (left) and
real-space (right) wavefunctions of the STVP (1) and (2) with $\ell=1$,
$k_{0}\Delta=0.7$ and $\gamma=1.5$. The brightness is proportional to the
intensity $|\psi|^{2}$, while the color indicates the phase ${\rm Arg}(\psi)$
Thaller (2000).

For our purposes, the key features of such STVPs are: (i) their spatiotemporal
vortex structure near the center:
$\psi\propto{\left[{{\gamma^{-1}}\zeta+i\,{{\rm
sgn}}\\!\left(\ell\right)x}\right]^{\left|\ell\right|}}e^{ik_{0}\zeta}$ , and
(ii) their normalized integral intrinsic OAM Bliokh and Nori (2012b); Bliokh
(2021, in press):

$\left\langle{{\bf L}}\right\rangle=\frac{\iint{{\rm Im}\left[{{\psi^{*}}({\bf
r}\times\\!{\bm{\nabla}})\psi}\right]}\,dxdz}{\iint{{\psi^{*}}\psi}\,dxdz}=\frac{{\gamma+{\gamma^{-1}}}}{2}\,\ell\,\bar{\bf
y}\,.$ (3)

The above equations are written for a scalar wavefunction $\psi$. To consider
polarized optical STVP, one has to multiply each plane wave in the spectrum
(1) by the corresponding electric-field polarization vector ${\bf e}({\bf
k})\bot{\bf k}$. In the paraxial regime this does not affect the shape of the
pulse and its OAM considerably, but polarization plays a crucial role in the
Goos-Hänchen and spin-Hall effects Bliokh and Aiello (2013); Götte and Dennis
(2012); Töppel  _et al._ (2013).

##  III Reflection/refraction of a STVP at an interface

We now consider reflection/refraction of a paraxial STVP at a planar isotropic
(e.g., dielectric) interface. The geometry of the problem is shown in Fig. 2.
The interface is $Z=0$, with the $Z$-axis being directed towards the second
medium. The propagation direction of the incident pulse is determined by the
central wavevector ${\bf k}_{0}=k_{0}(\bar{\bf Z}\cos\theta+\bar{\bf
X}\sin\theta)\equiv k_{0}\bar{\bf z}$. According to Snell’s law, the reflected
and transmitted pulses have the central wavevectors ${\bf
k}^{r}_{0}=k_{0}(-\bar{\bf Z}\cos\theta+\bar{\bf X}\sin\theta)\equiv
k_{0}\bar{\bf z}^{r}$ and ${\bf k}^{t}_{0}=k_{0}^{\prime}(\bar{\bf
Z}\cos\theta^{\prime}+\bar{\bf X}\sin\theta^{\prime})\equiv
k_{0}^{\prime}\bar{\bf z}^{t}$ ($\sin\theta^{\prime}=n^{-1}\sin\theta$,
$k_{0}^{\prime}=nk_{0}$, where $n$ is the relative refractive index of the
second medium), respectively. Here, as usual in beam-shift problems, we use
the accompanying coordinate frames $(x,y,z)$ and $(x^{r,t},y,z^{r,t})$ for the
incident and reflected/transmitted pulses, Fig. 2.

Figure 2:  Schematics of the reflection and refraction of a STVP at a planar
interface. The incident, reflected, and transmitted pulses, together with
their accompanying coordinate frames and intrinsic OAM are shown schematically
for the two geometries (A) and (B) (see details in the text). The longitudinal
shift (time delay) $\langle\zeta\rangle$ and angular shift $\langle
k_{x}\rangle$ are shown for the reflected and transmitted pulses in (a),
whereas the transverse shift $\langle y\rangle$ and angular shift $\langle
k_{y}\rangle$ are shown for the transmitted and reflected pulses in (b).

In contrast to the monochromatic-beam-shift problems, where the orientation of
the OAM is fixed by the beam propagation direction, in our problem the
transverse OAM can have different orientations with respect to the $(x,z)$
plane of incidence. We will consider two basic cases shown in Fig. 2:

(A) The incident STVP is localized in the $(x,z)$ plane, and the intrinsic OAM
$\left\langle{{\bf L}}\right\rangle\parallel\bar{\bf y}$.

(B) The incident STVP is localized in the $(y,z)$ plane and $\left\langle{{\bf
L}}\right\rangle\parallel\bar{\bf x}$.

To describe the main transformations of the reflected and refracted STVPs,
note that the $y$-components of the wavevectors in their spectra are
conserved, $k_{y}^{r,t}=k_{y}$, while the $x$-components in the corresponding
accompanying frames are related as $k_{x}^{r}=-k_{x}$ and
$k_{x}^{t}=(\cos\theta/\cos\theta^{\prime})k_{x}$ Bliokh and Aiello (2013). In
addition, the $z$-components of the wavevectors of the transmitted pulse are
$k_{z}^{t}\simeq nk_{z}$. From this, one can find that the vortex is inverted
in the reflected pulse in the case (A) but not (B), and its intrinsic OAM
becomes (see Fig. 2):

$\displaystyle\left\langle{{{\bf{L}}^{r}}}\right\rangle_{A}=-\left\langle{\bf{L}}\right\rangle=-\frac{{\gamma+{\gamma^{-1}}}}{2}\,\ell\,{\bf{\bar{y}}}\,,$
$\displaystyle\left\langle{{{\bf{L}}^{r}}}\right\rangle_{B}=\frac{{\gamma+{\gamma^{-1}}}}{2}\,\ell\,{\bf{\bar{x}}}^{r}.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (4)

Here and hereafter, the subscripts $A$ and $B$ mark the quantities related to
the cases (A) and (B), respectively.

For the transmitted STVP, the above transformations of the wavevector
components stretch the $x^{t}$-width of the pulse by a factor of
$\cos\theta^{\prime}/\cos\theta$ and squeeze its longitudinal length by a
factor of $1/n$. Therefore, the intrinsic OAM of the refracted pulses becomes

$\displaystyle\left\langle{{{\bf{L}}^{t}}}\right\rangle_{A}=\frac{{\gamma^{\prime}_{A}+{\gamma^{\prime-1}_{A}}}}{2}\,\ell\,{\bf{\bar{y}}},\leavevmode\nobreak\
\leavevmode\nobreak\
\gamma^{\prime}_{A}=\frac{\cos\theta}{n\cos\theta^{\prime}}\gamma,$
$\displaystyle\left\langle{{{\bf{L}}^{t}}}\right\rangle_{B}=\frac{{\gamma^{\prime}_{B}+{\gamma^{\prime-1}_{B}}}}{2}\,\ell\,{\bf{\bar{x}}}^{t},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\gamma^{\prime}_{B}=\frac{\gamma}{n}.\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ (5)

Equations (III) and (III) show that the transformations of the transverse
intrinsic OAM in the case (A) is similar to those of the longitudinal OAM of
monochromatic vortex beams Bliokh  _et al._ (2009); Bliokh and Aiello (2013),
only with the additional $n^{-1}$ factor in $\gamma^{\prime}_{A}$. In turn,
the case (B) differs considerably because the intrinsic OAM and vortex do not
flip in the reflected pulse.

##  IV Transverse shifts and time delays

We are now in the position to calculate small shifts in reflected/refracted
STVPs. Rigorous calculations can be performed by applying the standard
Fresnel-Snell formulas to each plane wave in the incident pulse spectrum; this
is realized numerically in the next section. Here, akin to Ref. Bliokh  _et
al._ (2009), we derive all the OAM-dependent shifts using general
considerations.

First of all, we assume that paraxial polarized optical STVPs experience all
the polarization-dependent shifts known for Gaussian wave beams or packets,
i.e., angular and spatial Goos-Hänchen and spin-Hall shifts $\left\langle
k^{t,r}_{x}\right\rangle_{0}$, $\left\langle x^{t,r}\right\rangle_{0}$,
$\left\langle k^{t,r}_{y}\right\rangle_{0}$, and $\left\langle
y^{t,r}\right\rangle_{0}$ Bliokh and Aiello (2013); Götte and Dennis (2012);
Töppel  _et al._ (2013), where the subscript “0” indicates that the shifts are
calculated for Gaussian states with $\ell=0$. In addition to these shifts, we
will determine the $\ell$-dependent shifts induced by the transverse intrinsic
OAM. There are three types of such shifts.

The first type is related to the conservation of the $Z$-component of the
total angular momentum in the problem and can be associated with the orbital-
Hall effect of light Bliokh and Aiello (2013); Bliokh (2006). In the case (A)
the intrinsic OAM has only the $y$-component, and the conservation law is
satisfied trivially. In the case (B), the incident and reflected pulses have
the same $Z$-components of the normalized intrinsic OAM, $\left\langle
L_{Z}\right\rangle=\left\langle L^{r}_{Z}\right\rangle$, Eqs. (3) and (III),
while the transmitted pulse has a different OAM component: $\left\langle
L_{Z}\right\rangle\neq\left\langle L^{t}_{Z}\right\rangle$, Eqs. (3) and
(III). Similarly to the refraction of monochromatic vortex beams Bliokh and
Aiello (2013); Bliokh  _et al._ (2009); Fedoseyev (2008); Merano  _et al._
(2010), this imbalance between the intrinsic OAM of the incident and
transmitted pulses should be compensated by the transverse $y$-shift of the
refracted pulse producing an extrinsic OAM $\langle L_{Z}^{t}\rangle^{\rm
ext}=\langle y^{t}\rangle\langle k_{X}^{t}\rangle\simeq\langle
y^{t}\rangle\,nk_{0}\sin\theta^{\prime}$. From here, the refracted STVP in the
case (B) should undergo an additional transverse shift (see Fig. 2)

$\left\langle{{y^{t}}}\right\rangle_{B}=\frac{{\left\langle{L_{Z}^{t}}\right\rangle-\left\langle{{L_{Z}}}\right\rangle}}{{n{k_{0}}\sin\theta^{\prime}}}=\frac{{\gamma\ell}}{{2{k_{0}}}}\left({{n^{-2}}-1}\right).$
(6)

In contrast to the analogous shift for refracted monochromatic vortex beams,
the shift (6) is independent of the angle $\theta$ (apart from the small
vicinity of the normal incidence $\theta=0$, which is singular for the
transverse-shift problem). The typical scale of this shift is the wavelength,
although it can be enhanced by high vortex charges $\ell$ or ellipticity
$\gamma\gg 1$ (narrow long pulses).

The second type of $\ell$-dependent shift is related to the angular Goos-
Hänchen and spin-Hall shifts $\langle k_{x,y}\rangle$, see Fig. 2. As has been
shown for monochromatic vortex beams, in the presence of vortex these shifts
acquire an additional factor of $\left(1+|\ell|\right)$ Bliokh  _et al._
(2009); Bliokh and Aiello (2013); Merano  _et al._ (2010), so that the
additional shifts are:

$\left\langle k^{t,r}_{x}\right\rangle_{A}=|\ell|\left\langle
k^{t,r}_{x}\right\rangle_{0},\quad\left\langle
k^{t,r}_{y}\right\rangle_{B}=|\ell|\left\langle
k^{t,r}_{y}\right\rangle_{0}\,.$ (7)

The typical scale of these angular shifts is the inverse Rayleigh range $\sim
1/(k_{0}\Delta^{2})$, and these shifts are independent of the ellipticity
$\gamma$.

Finally, the third type of $\ell$-dependent shifts is related to the cross-
coupling between different Cartesian degrees of freedom in a vortex. Below we
use reasoning similar to that for vortex beams in Refs. Bliokh and Aiello
(2013); Bliokh  _et al._ (2009). In the case (A), the spatiotemporal vortices
in the reflected and transmitted pulses have the forms
$\propto{\left[{-{\gamma^{-1}}{\zeta^{r}}+i\,{\rm
sgn}\\!\left(\ell\right){x^{r}}}\right]^{\left|\ell\right|}}$ and
$\propto{\left[{{\gamma^{\prime-1}_{A}}{\zeta^{t}}+i\,{\rm
sgn}\\!\left(\ell\right){x^{t}}}\right]^{\left|\ell\right|}}$, respectively,
where $\zeta^{r,t}=z^{r,t}-ct$ and $c$ is the speed of light in the
corresponding medium. Among other polarization-dependent shifts, these pulses
experience shifts in momentum space due to the angular Goos-Hänchen effect,
which can be regarded as imaginary shifts in real space Aiello and Woerdman
(2008); Bliokh  _et al._ (2009); Bliokh and Aiello (2013):
${\left\langle{k_{x}^{r}}\right\rangle_{0}}\to\delta{x^{r}}=-i\dfrac{\Delta^{2}}{2}{\left\langle{k_{x}^{r}}\right\rangle_{0}}$
and
${\left\langle{k_{x}^{t}}\right\rangle_{0}}\to\delta{x^{t}}=-i\dfrac{\Delta^{2}}{2}{\left({\dfrac{{\cos\theta^{\prime}}}{{\cos\theta}}}\right)^{2}}{\left\langle{k_{x}^{t}}\right\rangle_{0}}$.
Substituting these shifts to the vortex forms of the reflected and transmitted
pulses, we find that the imaginary $x$-shifts produce real $\ell$-dependent
$\zeta$-shifts as follows (see Fig. 2):

$\left\langle{{\zeta^{r}}}\right\rangle_{A}=-\ell\,\gamma\frac{{{\Delta^{2}}}}{2}{\left\langle{k_{x}^{r}}\right\rangle_{0}},\leavevmode\nobreak\
\leavevmode\nobreak\
\left\langle{{\zeta^{t}}}\right\rangle_{A}=\ell\,\frac{\gamma}{n}\frac{{{\Delta^{2}}}}{2}\frac{{\cos\theta^{\prime}}}{{\cos\theta}}{\left\langle{k_{x}^{t}}\right\rangle_{0}}.$
(8)

Applying analogous considerations to the case (B), with the reflected and
transmitted vortices $\propto{\left[{y+i{\gamma^{-1}}{\rm
sgn}\\!\left(\ell\right){\zeta^{r}}}\right]^{\left|\ell\right|}}$ and
$\propto{\left[{y+i{\gamma_{B}^{\prime-1}}{\rm
sgn}\\!\left(\ell\right){\zeta^{t}}}\right]^{\left|\ell\right|}}$, and angular
Hall-effect shifts ${\left\langle{k_{y}^{r,t}}\right\rangle_{0}}\to\delta
y^{r,t}=-i\dfrac{{{\Delta^{2}}}}{2}{\left\langle{k_{y}^{r,t}}\right\rangle_{0}}$,
where $\Delta$ is the pulse width in the $y$-direction, we obtain

$\left\langle{{\zeta^{r}}}\right\rangle_{B}=-\ell\,\gamma\frac{{{\Delta^{2}}}}{2}{\left\langle{k_{y}^{r}}\right\rangle_{0}},\leavevmode\nobreak\
\leavevmode\nobreak\
\left\langle{{\zeta^{t}}}\right\rangle_{B}=\ell\,\frac{\gamma}{n}\frac{{{\Delta^{2}}}}{2}{\left\langle{k_{y}^{t}}\right\rangle_{0}}.$
(9)

Equations (8) and (9) describe a remarkable qualitatively novel phenomenon:
longitudinal shifts of STVPs reflected/refracted by a planar interface. These
$\zeta$-shifts are equivalent to time delays $\left\langle\delta
t\right\rangle=-\left\langle{{\zeta}}\right\rangle/c$. In contrast to the
Wigner time delays, produced by the temporal dispersion (frequency dependence)
of the scattered potential Wigner (1954); Chiao and Steinberg (1997); de
Carvalho and Nussenzveig (2002); Winful (2006); Asano  _et al._ (2016), here
the time delays appear without any temporal dispersion. The angular Goos-
Hänchen effect ${\left\langle{k_{x}^{r,t}}\right\rangle_{0}}$ originates from
the spatial dispersion (wavevector dependence) of the Fresnel
reflection/transmission coefficients, while the angular spin-Hall shift
${\left\langle{k_{y}^{r,t}}\right\rangle_{0}}$ is a purely geometric
phenomenon which does not require any dispersion Bliokh  _et al._ (2015).

Importantly, such time delays allow one to realize slow (subluminal,
$\left\langle{{\zeta}}\right\rangle<0$) and fast (superluminal,
$\left\langle{{\zeta}}\right\rangle>0$) pulse propagation without any
dispersion in optical media. Unlike previous approaches controlling slow/fast
light via properties of the medium, we can control propagation time via
internal spatiotemporal properties of the pulse. Note, however, that, in
contrast to the wave packets in Ref. Kondakci and Abouraddy (2019), the sub-
or superluminal propagation of the pulses studied here is induced by the
Fresnel-Snell reflection/refraction at an interface rather than by tailoring
the pulse to have a free-space group velocity differing from $c$.

Equations (8) and (9) show that these novel OAM-dependent time delays are a
rather universal phenomenon: they appear in both reflected and transmitted
pulses in both cases (A) and (B). It is natural to expect that such time
delays will appear in a variety of systems, both classical and quantum,
involving scattering of a spatiotemporal vortex with the transverse OAM.

The typical magnitude of the longitudinal shifts (8) and (9) is the
wavelength. However, angular shifts $\left\langle
k^{r}_{x,y}\right\rangle_{0}$ of the reflected pulses, and hence the
corresponding new shifts (7)–(9), are enhanced resonantly for near-$p$
polarization in the vicinity of the Brewster angle of incidence
$\theta_{B}=\tan^{-1}(n)$ Merano  _et al._ (2009); Bliokh and Bliokh (2006);
Dasgupta and Gupta (2006); Qin  _et al._ (2009) (see Fig. 4 below). This is a
general phenomenon of the weak-measurement amplification of shifts for
wavepackets scattered with a near-zero amplitude Asano  _et al._ (2016);
Gorodetski  _et al._ (2012); Götte and Dennis (2013). The maximum weak-
measurement-amplified shift is comparable with the pulse size in the
corresponding dimension, which corresponds to the amplification factor $\sim
k_{0}\Delta\gg 1$.

Figure 3:  Theoretically calculated (curves) and numerically calculated
(symbols) shifts of the reflected ($r$) and transmitted ($t$) STVPs in the
cases (A) and (B) from Fig. 2 as functions of the angle of incidence $\theta$.
The shifts are given by the sums of previously known polarization-induced
contributions at $\ell=0$ (shown by pale dashed curves) and OAM-induced
contributions Eqs. (6)–(9). Parameters are: $\ell=1$, $n=1.5$, $\gamma=0.4$,
$k_{0}\Delta=500$, and ${\bf
e}_{0}\equiv(e_{x},e_{y})=\left(1/\sqrt{3},(1-i)/\sqrt{3}\right)$. The density
plot shows an example of the deformation of the transmitted pulse in the case
(B) with $k_{0}\Delta=1$ and $\gamma=2.5$ (to enhance the shifts with respect
to the pulse size) and its centroid right after the refraction.

##  V Numerical calculations

To verify the above theoretical derivations, we performed numerical
calculations of the reflection/refraction of polarized STVPs at a dielectric
interface by applying exact Fresnel-Snell’s formulas to each plane wave in the
incident pulse spectrum
$\tilde{\bf{E}}({\bf{k}})={\bf{e}}({\bf{k}})\tilde{\psi}({\bf{k}})$. In the
paraxial approximation, this is equivalent to applying an effective
wavevector-dependent Jones matrix ${{\hat{T}}^{r,t}}({\bf{k}})$ to the
polarization of the central plane wave ${\bf e}_{0}={\bf e}({\bf k}_{0})$
Bliokh and Aiello (2013); Götte and Dennis (2012); Töppel  _et al._ (2013), so
that the reflected and transmitted pulse spectra become
$\tilde{{{\bf{E}}}}^{r,t}({\bf{k}})\simeq{{\hat{T}}^{r,t}}({\bf{k}})\,{\bf{e}}_{0}\,\tilde{\psi}({\bf{k}})$.
After that, the spatial and angular shifts are calculated as expectation
values of the corresponding position and momentum operators in the momentum
representation:
$\left\langle{{y^{r,t}}}\right\rangle={{\left\langle{{{\tilde{\bf{E}}}^{r,t}}}\right|i\partial/\partial
k_{y}^{r,t}\left|{{{\tilde{\bf{E}}}^{r,t}}}\right\rangle}\mathord{\left/{\vphantom{{\left\langle{{{\tilde{\bf{E}}}^{r,t}}}\right|i\partial/\partial
k_{y}^{r,t}\left|{{{\tilde{\bf{E}}}^{r,t}}}\right\rangle}{\left\langle{{{\tilde{\bf{E}}}^{r,t}}}\right.\left|{{{\tilde{\bf{E}}}^{r,t}}}\right\rangle}}}\right.\kern-1.2pt}{\left\langle{{{\tilde{\bf{E}}}^{r,t}}}\\!\right.\left|{{{\tilde{\bf{E}}}^{r,t}}}\right\rangle}}$,
$\left\langle{{\zeta^{r,t}}}\right\rangle={{\left\langle{{{\tilde{\bf{E}}}^{r,t}}}\right|i\partial/\partial
k_{z}^{r,t}\left|{{{\tilde{\bf{E}}}^{r,t}}}\right\rangle}\mathord{\left/{\vphantom{{\left\langle{{{{\bf{\tilde{E}}}}^{r,t}}}\right|i\partial/\partial
k_{z}^{r,t}\left|{{{{\bf{\tilde{E}}}}^{r,t}}}\right\rangle}{\left\langle{{{\tilde{\bf{E}}}^{r,t}}}\right.\left|{{{\tilde{\bf{E}}}^{r,t}}}\right\rangle}}}\right.\kern-1.2pt}{\left\langle{{{\tilde{\bf{E}}}^{r,t}}}\\!\right.\left|{{{\tilde{\bf{E}}}^{r,t}}}\right\rangle}}$,
$\left\langle{k_{x,y}^{r,t}}\right\rangle={{\left\langle{{{\tilde{\bf{E}}}^{r,t}}}\right|k_{x,y}^{r,t}\left|{{{\tilde{\bf{E}}}^{r,t}}}\right\rangle}\mathord{\left/{\vphantom{{\left\langle{{{{\bf{\tilde{E}}}}^{r,t}}}\right|k_{x,y}^{r,t}\left|{{{{\bf{\tilde{E}}}}^{r,t}}}\right\rangle}{\left\langle{{{{\bf{\tilde{E}}}}^{r,t}}}\right.\left|{{{{\bf{\tilde{E}}}}^{r,t}}}\right\rangle}}}\right.\kern-1.2pt}{\left\langle{{{\tilde{\bf{E}}}^{r,t}}}\right.\left|{{{\tilde{\bf{E}}}^{r,t}}}\right\rangle}}$,
where the inner product involves integration over the corresponding wavevector
components: $\left(k_{x}^{r,t},k_{z}^{r,t}\right)$ and
$\left(k_{y},k_{z}^{r,t}\right)$ in the cases (A) and (B), respectively.

In doing so, it is sufficient to use the approximation linear in the
transverse wavevector components for all shifts apart from
$\left\langle{{y^{t}}}\right\rangle$, Eq. (6). For this shift it is necessary
to consider the second-order correction from the Snell’s transformation of the
wavevectors, see Eqs. (15) and (19) in Ref. Fedoseyev (2001) for a similar
peculiarity in monochromatic vortex beams. In our case, this second-order
correction is given by ${k_{z}}-{k_{0}}\simeq
k_{z}^{t}-n{k_{0}}-\dfrac{{k_{y}^{2}}}{{2{k_{0}}}}\left({{n^{-2}}-1}\right)$.

Figures 3 and 4 display results of numerical calculations of the shifts
(6)–(9) for an air-glass interface with $n=1.5$, generic incident STVP and
different angles of incidence $\theta$. These calculations demonstrate perfect
agreement with the theoretical predictions. Furthermore, Fig. 3 also shows a
typical real-space intensity profile of the transmitted STVP which exhibits
deformations characteristic for shifts of vortex beams Bliokh and Aiello
(2013); Dennis and Götte (2012). Figure 4 demonstrates weak-meaurement
enhancement Asano  _et al._ (2016); Gorodetski  _et al._ (2012); Götte and
Dennis (2013), by two orders of magnitude, of the longitudinal shifts (time
delays) $\langle\zeta^{r}\rangle$ of reflected pulses for a near-$p$ input
polarization in the vicinity of the Brester angle of incidence,
$\theta\simeq\theta_{B}$.

Figure 4:  Resonant weak-measurement enhancement of the longitudinal shifts
(time delays) of the reflected STVPs for near-$p$ input polarization ${\bf
e}_{0}=\left(1,0.01\right)$ in the vicinity of the Brewster angle of incidence
in the cases (A) and (B). Parameters are: $\ell=1$, $n=1.5$, $\gamma=0.4$,
$k_{0}\Delta=500$.

##  VI Discussion

We have described reflection and refraction of a STVP at a planar isotropic
interface. The problem was considered by adopting previously developed methods
for monochromatic vortex beams. In doing so, spatiotemporal vortices have a
more complicated geometry with a transverse intrinsic OAM, which requires
consideration of two basic cases: (A) the OAM is orthogonal to the plane of
incidence and (B) the OAM lies within this plane. We have described
transformations of the reflected and transmitted pulses in both of these
cases. Notably, reflection in the case (A) can be used to flip the intrinsic
OAM of the pulse, while refraction can be employed for changing the
ellipticity of the pulse.

Most importantly, we have derived analytically and checked numerically all
OAM-dependent spatial and angular shifts of the reflected and transmitted
pulses in the paraxial approximation. These shifts can be divided into three
types: (i) the spatial orbital-Hall-effect shift $\langle y\rangle$ appearing
for the transmitted pulse in the case (B); (ii) the OAM-amplified angular
Goos-Hänchen and Hall-effect shifts $\langle k_{x}\rangle$ and $\langle
k_{y}\rangle$; and (iii) the longitudinal shifts $\langle\zeta\rangle$ which
appear for both reflected and transmitted pulses in both cases (A) and (B).
The latter one is the most remarkable phenomenon, which is equivalent to time
delays $\left\langle\delta
t\right\rangle=-\left\langle{{\zeta}}\right\rangle/c$ of the scattered pulses.
In contrast to the well-known Wigner time delay, this effect occurs without
any temporal dispersion of the scattering coefficients, from the coupling of
spatial and temporal degrees of freedom in spatiotemporal vortices. Such time
delays allow one to realize OAM-controlled slow (subluminal) and fast
(superluminal) pulse propagation without any medium dispersion.

Due to remarkable success in experimental studies of subwavelength shifts of
monochromatic optical beams and Wigner time delays of optical pulses, it is
natural to expect that the new shifts and time delays predicted in this work
could be measured in the near future. Furthermore, our work can stimulate a
number of implications and follow-up studies. In particular, scattering of
quantum spatiotemporal vortices in the geometry (A) can appear in 2D
condensed-matter systems. Furthermore, we have considered only the basic case
of STVP with a purely transverse intrinsic OAM and two basic geometries (A)
and (B) with respect to the interface. In general, one can examine STVPs with
intrinsic OAM arbitrarily oriented with repect to the propagation direction
Bliokh and Nori (2012b); Wan  _et al._ (2021) and interface. One can expect
that in this general case, the pulse shifts could be expressed via suitably
weighted sums of previously considered basic shifts. Finally, including
temporal dispersion of the media and interface into consideration should add
the Wigner time-delay effects, which could be coupled with spatial degrees of
freedom and produce new spatial pulse shifts.

###### Acknowledgements.

We are grateful to V. G. Fedoseyev, A. Y. Bekshaev, and O. Yermakov for
helpful discussions. This work was partially supported by the National
Research Foundation of Ukraine (Project No. 2020.02/0149 “Quantum phenomena in
the interaction of electromagnetic waves with solid-state nanostructures”) and
the Excellence Initiative of Aix Marseille University—A*MIDEX, a French
‘Investissements d’Avenir’ programme. F.N. was supported by Nippon Telegraph
and Telephone Corporation (NTT) Research; the Japan Science and Technology
Agency (JST) via the Quantum Leap Flagship Program (Q-LEAP), the Moonshot R&D
Grant No. JP- MJMS2061, and the Centers of Research Excellence in Science and
Technology (CREST) Grant No. JPMJCR1676; the Japan Society for the Promotion
of Science (JSPS) via the Grants-in-Aid for Scientific Research (KAKENHI)
Grant No. JP20H00134, and the JSPS–RFBR Grant No. JPJSBP120194828; the Army
Research Office (ARO) (Grant No. W911NF-18-1-0358), the Asian Office of
Aerospace Research and Development (AOARD) (Grant No. FA2386-20-1-4069); and
the Foundational Questions Institute Fund (FQXi) (Grant No. FQXi-IAF19-06).

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# Local properties of the $t$-$J$ model in a two-pole approximation within COM

Amir Eskandari-asl1 Adolfo Avella1,2,3 1Dipartimento di Fisica “E.R,
Caianiello”, Universitï¿œ degli Studi di Salerno, I-84084 Fisciano (SA), Italy
2CNR-SPIN, UoS di Salerno, I-84084 Fisciano (SA), Italy  3Unitï¿œ CNISM di
Salerno, Universitï¿œ degli Studi di Salerno, I-84084 Fisciano (SA), Italy

###### Abstract

In this work, we study the $t$-$J$ model using a two-pole approximation within
the composite operator method. We choose a basis of two composite operators –
the constrained electrons and their spin-fluctuation dressing – and
approximate their currents in order to compute the corresponding Green’s
functions. We exploit the algebraic constraints obeyed by the basis operators
to close a set of self-consistent equations that is numerically solved. This
allows to determine the physical parameters of the system such as the spin-
spin correlation function and the kinetic energy. Our results are compared to
those of an exact numerical method on a finite system to asses their
reliability. Indeed, a very good agreement is achieved through a far less
numerically demanding and a more versatile procedure. We show that by
increasing the hole doping, anti-ferromagnetic correlations are replaced by
ferromagnetic ones. The behavior on changing temperature and exchange integral
is also studied and reported.

###### keywords:

$t$-$J$ model , composite operator method (COM) , spin fluctuations

††journal: Journal of Magnetism and Magnetic Materials

##  1 Introduction

Strongly correlated systems have undergone extensive study for several decades
[1, 2, 3]. For lattice systems, the onsite interaction is the most important
one, a feature which is well reflected in the Hubbard model [4, 5, 6]. Despite
its seemingly simple structure, the Hubbard model and its extensions and
derivations have been very successful in studying strongly correlated systems
[7, 8, 9, 10]. In the strong electron-electron repulsion regime, it is
possible to derive an effective model from the Hubbard model in which the
double-occupancy is discarded according to its extreme energy cost: the so
called $t$-$J$ model [11, 12, 13, 14, 15, 16]. The Hubbard and the $t$-$J$
models have been used to theoretically understand and describe so many
interesting phenomena such as Mott-Hubbard metal-insulator transition [17],
non-Fermi-liquid normal phases and high-temperature superconductivity [18, 19,
20, 21, 22, 23, 24, 25, 26, 27], etc. One feature which is very interesting
and still needs to be clarified is the occurrence of a ferro-antiferromagnetic
crossover in the $t$-$J$ model [28, 29]. It is quite well-known that, at half
filling, the system is in the anti-ferromagnetic (AF) Nï¿œel state. Yet,
Nagaoka proved that in the infinite-$U$ regime of the Hubbard model, which
corresponds to vanishing exchange integral in the $t$-$J$ model, if we
introduce one hole into the system the ground state becomes ferromagnetic (FM)
[30, 31, 32]. This idea got generalized in the $t$-$J$ model by so many
successive studies which showed transition to FM phase for finite hole dopings
[33, 34, 35, 36, 37, 38, 39, 40, 41].

In studying strongly correlated systems, quasi-particles play a crucial role.
In the composite operator method (COM), the equation of motion of the
operators corresponding to the most relevant quasi-particles, those associated
with the emergent elementary excitations in the system, are investigated.
Within this method, using the properties of the generalized Green’s functions
(GFs) of the quasi-particle operators and their algebraic constraints, a set
of self-consistent equations are obtained from which the physical properties
can be computed [21, 42, 43, 44, 45, 46]. It’s worth noticing that COM belongs
to the large class of operatorial approaches: the Hubbard approximations [4],
an early high-order GF approach [47], the projection operator method [48, 49],
the works of Mori [50], Rowe [51], and Roth [52], the spectral density
approach [53], the works of Barabanov [54], Val’kov [55], and Plakida [56, 57,
58, 59, 60], and the cluster perturbation theory in the Hubbard-operator
representation [61].

In this work, we consider a two-pole approximation for the $t$-$J$ model on a
two-dimensional lattice and focus on the quasi-particles describing the
constrained electrons and their spin-fluctuation dressing. After computing the
currents of our basis operators, we apply a generalized mean-field
approximation to project them back on the basis. These currents, together with
the integrated spectral weights of the basis, can be exploited to get a set of
self-consistent equations that allow to compute the relevant GFs. The
remaining unknowns can be related to the GFs using the algebraic constraints
obeyed by the composite operators. The solutions of these equations reveal the
physical properties of the system in different parametric regimes. The quality
of the approximation is assessed by comparing our results to those of an exact
numerical study. We find a very good agreement while our method is, on one
hand, numerically less demanding and, on the other hand, more versatile as it
can be generalized to study several different systems and parameter regimes.
We show that the system features AF correlations of the Nï¿œel type near half
filling, while by increasing the hole doping it develops FM ones. At higher
temperatures, both AF and FM correlations are suppressed and the paramagnetic
phase is the favored one. Moreover, as expected, higher and higher values of
the exchange integral favor AF correlations and higher values of doping are
needed for the emergence of FM fluctuations.

The article is organized as follows. In Sec. 2, we introduce the model and the
basis operators we have chosen and describe our method. In Sec. 3, we present
our numerical results, assess them by comparison to exact numerical ones, and
discuss their relevant features. Finally, in Sec. 4, we give our conclusions.

##  2 Model and Method

The $t$-$J$ Hamiltonian is derived in the strongly correlated regime of the
Hubbard model ($t\ll U$) where an exchange integral $J=4t^{2}/U$ emerges [11,
12, 16]. Its explicit form for a two-dimensional lattice is given by

$\displaystyle\mathcal{H}$
$\displaystyle=-4t\sum_{i}\xi^{\dagger}\left(i\right)\cdot\xi^{\alpha}\left(i\right)-\mu\sum_{i}\nu\left(i\right)$
$\displaystyle+\frac{J}{2}\sum_{i}\left[\nu_{k}\left(i\right)\nu_{k}^{\alpha}\left(i\right)-\nu\left(i\right)\nu^{\alpha}\left(i\right)\right],$
(1)

where $t$, $J$, and $\mu$ are the nearest-neighbor hopping integral, the
exchange integral and the chemical potential, respectively. We set $t$ as
energy unit. In this model, double occupancy of sites is prohibited,
accordingly one has to use the operator
$\xi_{\sigma}\left(i\right)=\left(1-n_{\bar{\sigma}}\left(i\right)\right)c_{\sigma}\left(i\right)$,
which describes the transition between empty and singly-occupied sites, with
$c_{\sigma}\left(i\right)$ being the annihilation operator of an electron with
spin $\sigma$ on the site $i$, and
$n_{\sigma}\left(i\right)=c_{\sigma}^{\dagger}\left(i\right)c_{\sigma}\left(i\right)$.
We use the spinorial notation
$\xi^{\dagger}\left(i\right)=\left(\xi_{\uparrow}^{\dagger}\left(i\right),\xi_{\downarrow}^{\dagger}\left(i\right)\right)$
and define (spin) inner product between operators:
$\nu\left(i\right)=\xi^{\dagger}\left(i\right)\cdot\xi\left(i\right)=\sum_{\sigma}\xi_{\sigma}^{\dagger}\left(i\right)\xi_{\sigma}\left(i\right)$
and
$\nu_{k}\left(i\right)=\xi^{\dagger}\left(i\right)\cdot\sigma_{k}\cdot\xi\left(i\right)$
are the charge and spin density operators on site $i$, respectively, with
$\sigma_{k}$ being the Pauli matrices for $k=1,2,3$. Moreover, for every
operator $\phi\left(i\right)$, its projection on the nearest neighbor sites
($\delta_{\left\langle ij\right\rangle}$) is given by
$\phi^{\alpha}\left(i\right)=\sum_{j}\alpha_{ij}\phi\left(j\right)=\frac{1}{4}\left[\phi\left(i_{x}+1,i_{y}\right)+\phi\left(i_{x}-1,i_{y}\right)+\phi\left(i_{x},i_{y}+1\right)+\phi\left(i_{x},i_{y}-1\right)\right]$,
where for a two-dimensional square lattice
$\alpha_{ij}=\frac{1}{4}\delta_{\left\langle ij\right\rangle}$. With some
straightforward calculations, one can rewrite the Hamiltonian as [62]

$\mathcal{H}=\sum_{i}\xi^{\dagger}\left(i\right)\cdot\left[-4t\xi^{\alpha}\left(i\right)+J\left(\widetilde{\xi}_{0}\left(i\right)+\widetilde{\xi}_{s}\left(i\right)\right)-\left(\mu+\frac{J}{2}\right)\xi\left(i\right)\right],$
(2)

where the operators

$\widetilde{\xi}_{0}\left(i\right)=\frac{1}{2}\left(1-\nu^{\alpha}\left(i\right)\right)\xi\left(i\right),$
(3)
$\widetilde{\xi}_{s}\left(i\right)=\frac{1}{2}\nu_{k}^{\alpha}\left(i\right)\sigma_{k}\cdot\xi\left(i\right),$
(4)

describe constrained electronic transitions dressed by nearest-neighbor charge
and spin fluctuations, respectively. As our basis operators we choose the
following set of operators

$\boldsymbol{\psi}\left(i\right)=\left(\begin{array}[]{c}\xi\left(i\right)\\\
\widetilde{\xi}_{s}\left(i\right)\end{array}\right),$ (5)

which reflects the fact that near half filling, the spin fluctuations play the
most important role as the system has a clear tendency towards the AF Nï¿œel
state.

In the Heisenberg picture, the current of the basis operators is

$\boldsymbol{J}\left(i\right)=i\frac{\partial}{\partial
t_{i}}\boldsymbol{\psi}\left(i\right)=\left[\boldsymbol{\psi}\left(i\right),\mathcal{H}\right],$
(6)

where we set $\hbar=1$. One can show that

$\displaystyle J_{\xi}=$
$\displaystyle-2t\left(\xi^{\alpha}\left(i\right)+2\xi_{0}\left(i\right)+2\xi_{s}\left(i\right)\right)$
$\displaystyle+2J\left(\widetilde{\xi}_{0}\left(i\right)+\widetilde{\xi}_{s}\left(i\right)\right)-\left(\mu+J\right)\xi\left(i\right),$
(7) $\displaystyle J_{\widetilde{\xi}_{s}}=$
$\displaystyle-t\nu_{k}^{\alpha}\left(i\right)\sigma_{k}\cdot\left(\xi^{\alpha}\left(i\right)+2\xi_{0}\left(i\right)+2\xi_{s}\left(i\right)\right)$
$\displaystyle+\left(-2t\left[\xi^{\dagger}\left(i\right)\cdot\sigma_{k}\cdot\xi^{\alpha}\left(i\right)\right]^{\alpha}+2t\left[\xi^{\dagger\alpha}\left(i\right)\cdot\sigma_{k}\cdot\xi\left(i\right)\right]^{\alpha}\right)\sigma_{k}\cdot\xi\left(i\right)$
$\displaystyle-\mu\widetilde{\xi}_{s}\left(i\right)-\frac{J}{2}\nu_{k}^{\alpha}\left(i\right)\nu^{\alpha}\left(i\right)\sigma_{k}\cdot\xi\left(i\right)$
$\displaystyle+\frac{J}{2}\nu_{k}^{\alpha}\left(i\right)\nu_{g}^{\alpha}\left(i\right)\sigma_{k}\cdot\sigma_{g}\cdot\xi\left(i\right)+Ji\epsilon_{kgh}\left[\nu_{g}^{\alpha}\left(i\right)\nu_{h}\left(i\right)\right]^{\alpha}\sigma_{k}\cdot\xi\left(i\right),$
(8)

where
$\xi_{0}\left(i\right)=\frac{1}{2}\left(1-\nu\left(i\right)\right)\xi^{\alpha}\left(i\right)$
and
$\xi_{s}\left(i\right)=\frac{1}{2}\nu_{k}\left(i\right)\sigma_{k}\cdot\xi^{\alpha}\left(i\right)$
can be considered as counterparts of $\widetilde{\xi}_{0}\left(i\right)$ and
$\widetilde{\xi}_{s}\left(i\right)$, respectively. Taking into account only
nearest neighbor contributions, these latter higher-order operators can be
approximated as

$\displaystyle\xi_{0}\left(i\right)\simeq$ $\displaystyle
4\widetilde{\xi}_{0}^{\alpha}\left(i\right)-\frac{3}{2}\left(1-\nu\right)\xi^{\alpha}\left(i\right),$
(9) $\displaystyle\xi_{s}\left(i\right)\simeq$ $\displaystyle
4\widetilde{\xi}_{s}^{\alpha}\left(i\right).$ (10)

Moreover, one can approximate $\widetilde{\xi}_{0}\left(i\right)$ by
projecting it on $\xi\left(i\right)$ as [62]

$\widetilde{\xi}_{0}\left(i\right)\approx\frac{2-3\nu+\chi_{c}^{\alpha}}{4\left(1-\frac{\nu}{2}\right)}\xi\left(i\right)-\frac{C_{11}^{\alpha}}{2\left(1-\frac{\nu}{2}\right)}\xi^{\alpha}\left(i\right).$
(11)

Finally, considering a paramagnetic and homogenous phase and using a mean-
field like approximation we can write the currents in the form [42, 43, 44,
21]

$J_{a}\left(i\right)=\sum_{j}\sum_{b}\varepsilon_{ab}\left(i,j\right)\psi_{b}\left(j\right).$
(12)

The Fourier transform of the $\boldsymbol{\varepsilon}$ matrix reads as

$\displaystyle\varepsilon_{11}\left(\boldsymbol{k}\right)=$ $\displaystyle
16t\frac{C_{11}^{\alpha}}{2-\nu}\alpha^{2}\left(\boldsymbol{k}\right)$
$\displaystyle+\left(6t\left(\frac{2}{3}-\nu\right)-8t\frac{2-3\nu+\chi_{c}^{\alpha}}{2-\nu}-2J\frac{C_{11}^{\alpha}}{2-\nu}\right)\alpha\left(\boldsymbol{k}\right)$
$\displaystyle+J\frac{2-3\nu+\chi_{c}^{\alpha}}{2-\nu}-\mu-J,$ (13)
$\displaystyle\varepsilon_{12}\left(\boldsymbol{k}\right)=$
$\displaystyle-16t\alpha\left(\boldsymbol{k}\right)+2J,$ (14)
$\displaystyle\varepsilon_{21}\left(\boldsymbol{k}\right)=$
$\displaystyle-6tC_{11}^{\alpha}\left(1+\frac{1}{2-\nu}\right)\alpha^{2}\left(\boldsymbol{k}\right)$
$\displaystyle+\biggr{(}-\frac{3}{4}t-t\frac{9}{4}\chi_{s}^{\alpha}+6tC_{11}^{\alpha^{2}}-\frac{15}{4}t\left(1-\nu\right)$
$\displaystyle+6t\frac{2-3\nu+\chi_{c}^{\alpha}}{4-2\nu}+3J\frac{C_{11}^{\alpha}}{8-4\nu}\biggr{)}\alpha\left(\boldsymbol{k}\right)$
$\displaystyle+\frac{3J}{8}+\frac{3}{2}tC_{11}^{\alpha}-\frac{3J}{4}\frac{2-3\nu+\chi_{c}^{\alpha}}{4-2\nu},$
(15) $\displaystyle\varepsilon{}_{22}\left(\boldsymbol{k}\right)=$
$\displaystyle
2t\alpha\left(\boldsymbol{k}\right)-\left(\mu+\frac{3J}{4}+\frac{3J}{4}\nu\right),$
(16)

where $\nu=\left\langle\nu\left(i\right)\right\rangle$ is the average electron
number per site which can vary between 0 and 1 (half filling) depending on the
doping.
$C_{ab}^{\alpha^{n}}=\left\langle\psi_{a}^{\alpha^{n}}\left(i\right)\psi_{b}^{\dagger}\left(i\right)\right\rangle$
is the generalized correlation matrix
[$\phi^{\alpha^{n}}\left(i\right)=\sum_{j}\alpha_{ij}^{n}\phi\left(j\right)$]
with $n$ being a non-negative integer: $\alpha_{ij}^{0}=\delta_{ij}$,
$\alpha_{ij}^{1}=\alpha_{ij}$, and for $n>1$,
$\alpha_{ij}^{n}=\sum_{l_{1},..,l_{n-1}}\alpha_{il_{1}}\alpha_{l_{1}l_{2}}...\alpha_{l_{n-1}j}$.
$\chi_{c}^{\alpha}=\left\langle\nu\left(i\right)\nu^{\alpha}\left(i\right)\right\rangle$
and
$\chi_{s}^{\alpha}=\frac{1}{3}\sum_{k=1}^{3}\left\langle\nu_{k}\left(i\right)\nu_{k}^{\alpha}\left(i\right)\right\rangle$
are the charge-charge and spin-spin correlation functions, respectively.

The normalization matrix of the basis operators is defined as

$I_{ab}\left(i,j\right)=\left\langle\left\\{\psi_{a}\left(i\right),\psi_{b}^{\dagger}\left(j\right)\right\\}\right\rangle.$
(17)

Once again, we use mean-field-like approximations and perform Fourier
transformation to obtain

$\displaystyle I_{11}\left(\boldsymbol{k}\right)=$ $\displaystyle
1-\frac{1}{2}\nu,$ (18) $\displaystyle I_{12}\left(\boldsymbol{k}\right)=$
$\displaystyle\frac{3}{4}\chi_{s}^{\alpha}+\frac{3}{2}\alpha\left(\boldsymbol{k}\right)C_{11}^{\alpha},$
(19) $\displaystyle I_{22}\left(\boldsymbol{k}\right)=$
$\displaystyle\frac{3}{16}\left(-\frac{1}{2}\chi_{c}^{\alpha}-\chi_{s}^{\alpha}+\nu\right)-\alpha\left(\boldsymbol{k}\right)C_{12}^{\alpha}$
$\displaystyle+\frac{3\alpha\left(\boldsymbol{k}\right)}{16}C_{11}^{\alpha}+\left(2\alpha^{2}\left(\boldsymbol{k}\right)-\frac{1}{2}\right)\frac{4}{3}C_{12}^{\alpha^{2}},$
(20)

where in the last line we used the so called spherical approximation [63, 42].

In order to obtain the self-consistent set of equations, we define the
retarded GF as follow.

$G_{ab}^{R}\left(i,j\right)=\theta\left(t_{i}-t_{j}\right)\left\langle\left\\{\psi_{a}\left(i\right),\psi_{b}^{\dagger}\left(j\right)\right\\}\right\rangle,$
(21)

where $i$ stands for both time and site indices. For a basis of $n$ operators,
GF is an $n\times n$ matrix (in our case $n=2$). Then, using the current
equations and performing Fourier transformation in space and time one can show

$\boldsymbol{G}^{R}\left(\boldsymbol{k},\omega\right)=\left(\omega-\boldsymbol{\varepsilon}\left(\boldsymbol{k}\right)\right)^{-1}\boldsymbol{I}\left(\boldsymbol{k}\right),$

which results in the following explicit form

$\boldsymbol{G}^{R}\left(\boldsymbol{k},\omega\right)=\sum_{m=1}^{n}\frac{\boldsymbol{\sigma}^{\left(m\right)}\left(\boldsymbol{k}\right)}{\omega-\omega_{m}\left(\boldsymbol{k}\right)+i0^{+}},$
(22)

where $\omega_{m}\left(\boldsymbol{k}\right)$ is the m-th eigenvalue of
$\boldsymbol{\varepsilon}\left(\boldsymbol{k}\right)$, and

$\sigma_{ab}^{\left(m\right)}\left(\boldsymbol{k}\right)=\Omega_{am}\left(\boldsymbol{k}\right)\sum_{c=1}^{2}\Omega_{mc}^{-1}\left(\boldsymbol{k}\right)I_{cb}\left(\boldsymbol{k}\right),$
(23)

in which $\boldsymbol{\Omega}\left(\boldsymbol{k}\right)$ is an $n\times n$
matrix whose columns are the eigenvectors of
$\boldsymbol{\varepsilon}\left(\boldsymbol{k}\right)$. Using Eq, 22, one can
obtain a generalized form of the fluctuation-dissipation theorem as [42]

$\boldsymbol{C}\left(\boldsymbol{k},\omega\right)=2\pi\sum_{m=1}^{n}\left[1-f_{F}\left(\omega_{m}\left(\boldsymbol{k}\right)\right)\right]\boldsymbol{\sigma}^{\left(m\right)}\left(\boldsymbol{k}\right)\delta\left(\omega-\omega_{m}\left(\boldsymbol{k}\right)\right),$
(24)

where $f_{F}$ is the Fermi distribution function. Performing the inverse
Fourier transformation, we obtain

$\displaystyle C_{ab}^{\kappa}=$
$\displaystyle\frac{2\pi}{N}\sum_{\boldsymbol{k}}\kappa\left(-\boldsymbol{k}\right)\sum_{l=1}^{n}\left[1-f_{F}\left(\omega_{l}\left(\boldsymbol{k}\right)\right)\right]\sigma_{ab}^{\left(l\right)}\left(\boldsymbol{k}\right),$
(25)

where $\kappa$ can be any lattice projection operator. This relation shows how
the self-consistent procedure works. For calculating the GFs, we need
$\boldsymbol{I}\left(\boldsymbol{k}\right)$ and
$\boldsymbol{\varepsilon}\left(\boldsymbol{k}\right)$, which are determined by
the correlation functions. On the other hand, the correlation functions are
determined by the GFs through the fluctuation-dissipation theorem, Eq. 25. In
order to close the set of self-consistent equations, we use the following
algebraic constraints obeyed by the basis operators.

$\displaystyle C_{11}^{\delta}=$ $\displaystyle 1-\nu,$ (26)
$C_{12}^{\delta}=0,$ (27) $\displaystyle C_{22}^{\delta}=$
$\displaystyle-\frac{3}{16}\chi_{c}^{\alpha}+\frac{3}{16}\nu.$ (28)

Having a closed set of self-consistent equations, we numerically solve it to
obtain the physical properties of the system.

---
Figure 1:
$C_{11}^{\alpha}=\left\langle\xi^{\alpha}\left(i\right)\xi^{\dagger}\left(i\right)\right\rangle$
, as a function of filling, $\nu$, for $J=0.1$ and temperature T ranging from
0.01 to 1. Circles are ED data extracted from Ref. [64]. $C_{11}^{\alpha}$ is
proportional to the kinetic energy, $K=8tC_{11}^{\alpha}$.

##  3 Results

In this section, we present our numerical results. In Fig. 1, we show
$C_{11}^{\alpha}=\left\langle\xi^{\alpha}\left(i\right)\xi^{\dagger}\left(i\right)\right\rangle$
as a function of electron density per site, $\nu$, for $J=0.1$ and temperature
$T$ ranging from 0.01 to 1. The circles are numerical data extracted from Ref.
[64] and correspond to exact diagonalization (ED) results at zero temperature
for a finite cluster. There is a clear agreement with our results although we
need a small finite temperature to compensate for the finite size effects. At
half filling ($\nu=1$), $C_{11}^{\alpha}$ vanishes, as it is proportional to
the kinetic energy by the relation $K=8tC_{11}^{\alpha}$. Since each site is
occupied exactly by one electron there is no possibility for electrons to
move, and kinetic energy vanishes. Our results show that the kinetic energy
decreases by increasing the temperature, which means the thermally excited
states of the system do not favor hole mobility, as it will be clarified in
the following.

---
Figure 2: $\chi_{s}^{\alpha}$ as a function of $\nu$: (top) same parameters as
Fig. 1; (bottom) $T=0.1$ and $J$ ranging from 0.1 to 0.5.

Although we considered a paramagnetic phase, we can still investigate the
tendency of the system towards other (ordered) phases. In Fig. 2, top panel,
we plot the spin-spin correlation function, $\chi_{s}^{\alpha}$, as a function
of $\nu$, with same parameters as Fig. 1. For low enough temperatures, we have
AF correlations near half filling. This clearly shows that our solution
correctly captures the behavior in this regime, consistently with the well-
established AF Nï¿œel state at half filling. The FM phase in the $t$-$J$ model
has been predicted in the literature [33, 28, 34, 35, 36, 37, 29, 38, 39, 40,
41]: mobile holes can form Nagaoka polarons which results in a FM ordering
[38, 40]. We witness a similar behavior here, i.e., once enough holes are
present in the system, FM correlations clearly emerge and they overcome the AF
ones, whose correlation lengths decrease rapidly with doping [35]. Increasing
the temperature results in weakening of both AF and FM correlations: the
paramagnetic phase becomes the most favorable one. Let us now come back to the
decrease of the kinetic energy on increasing the temperature reported in Fig.
1. This behavior has different explanations in different doping regimes. Near
half filling, the AF correlations get weaker and weaker on increasing the
temperature, inhibiting the virtual hopping processes because of the Pauli
exclusion principle. Accordingly, the kinetic energy decreases. On the other
hand, at intermediate fillings, significant FM correlations result from the
formation of Nagaoka polarons, which requires mobile holes and induce a gain
in kinetic energy. By increasing the temperature, the FM correlations too
become weaker and weaker and, consequently, the kinetic energy decreases also
in this case.

In Fig. 2 bottom panel, we plot the spin-spin correlation function as a
function of $\nu$ for $T=0.1$ and $J$ ranging from 0.1 to 0.5. For larger and
larger values of $J$: $\left(i\right)$ the AF correlations increase, which
shows a stronger tendency towards AF for larger exchange integrals, as
expected; (ii) the emergence of FM correlations requires larger and larger
values of doping in order to overcome the stronger and stronger AF
correlations.

##  4 Summary

In summary, we performed a two-pole study of the $t$-$J$ model within COM. In
our calculations, we considered the constrained electrons and their spin
dressing as fundamental quasi particles. By exploiting mean-field-like
approximations, we projected back the operatorial currents on the basis
operators. We used similar approximations to calculate the normalization
matrix within COM. These relations can be combined with the algebraic
constraints obeyed by the operators to give a closed set of self-consistent
equations which can be numerically solved.

Our results for the kinetic energy are in a very good agreement with those of
ED for finite clusters, while our method is numerically less demanding and
also more versatile. Moreover, we show that the system undergoes a smooth
transition between small and intermediate doping regimes where it features AF
and FM correlations, respectively. By increasing the temperature, both AF and
FM correlations are weakened and, consequently, the kinetic energy decreases
due to the inhibition of exchange virtual processes and polaron formation,
respectively. Increasing the exchange integral strengthens the AF
fluctuations, as expected, and forces the FM fluctuations to emerge at higher
values of doping.

## Acknowledgments

Authors acknowledge support by MIUR under Project No. PRIN 2017RKWTMY.

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