# Detecting, distinguishing, and spatiotemporally tracking photogenerated
charge and heat at the nanoscale

Hannah L. Weaver  [  Cora M. Went  [  Joeson Wong  [  Dipti Jasrasaria  [
Eran Rabani  [  Harry A. Atwater  [  Naomi S. Ginsberg
<EMAIL_ADDRESS>[

###### Abstract

Since dissipative processes are ubiquitous in semiconductors, characterizing
how electronic and thermal energy transduce and transport at the nanoscale is
vital for understanding and leveraging their fundamental properties. For
example, in low-dimensional transition metal dichalcogenides (TMDCs), excess
heat generation upon photoexcitation is difficult to avoid since even with
modest injected exciton densities, exciton-exciton annihilation still occurs.
Both heat and photoexcited electronic species imprint transient changes in the
optical response of a semiconductor, yet the unique signatures of each are
difficult to disentangle in typical spectra due to overlapping resonances. In
response, we employ stroboscopic optical scattering microscopy (stroboSCAT) to
simultaneously map both heat and exciton populations in few-layer MoS2 on
relevant nanometer and picosecond length- and time scales and with 100-mK
temperature sensitivity. We discern excitonic contributions to the signal from
heat by combining observations close to and far from exciton resonances,
characterizing photoinduced dynamics for each. Our approach is general and can
be applied to any electronic material, including thermoelectrics, where heat
and electronic observables spatially interplay, and lays the groundwork for
direct and quantitative discernment of different types of coexisting energy
without recourse to complex models or underlying assumptions.

University of California, Berkeley]Department of Physics, University of
California, Berkeley, CA 94720, United States California Institute of
Technology]Department of Physics, California Institute of Technology,
Pasadena, CA 91125, United States California Institute of
Technology]Department of Applied Physics and Materials Science, California
Institute of Technology, Pasadena, CA 91125, United States Current address:
Department of Chemistry, University of Chicago, Chicago, IL 60637, United
States University of California, Berkeley]Department of Chemistry, University
of California, Berkeley, CA 94720, United States Current address: Department
of Chemistry, Columbia University, New York, NY 10027, United States
University of California, Berkeley]Department of Chemistry, University of
California, Berkeley, CA 94720, United States Materials Sciences Division,
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, United States The
Sackler Center for Computational Molecular and Materials Science, Tel Aviv
University, Tel Aviv, Israel 69978 California Institute of
Technology]Department of Applied Physics and Materials Science, California
Institute of Technology, Pasadena, CA 91125, United States University of
California, Berkeley]Department of Chemistry, University of California,
Berkeley, CA 94720, United States Department of Physics, University of
California, Berkeley, CA 94720, United States Molecular Biophysics and
Integrated Bioimaging Division, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, United States Materials Sciences Division, Lawrence
Berkeley National Laboratory, Berkeley, CA 94720, United States Kavli Energy
NanoScience Institute, Berkeley, CA 94720, United States STROBE, NSF Science &
Technology Center, Berkeley, California 94720, United States

##  1 Introduction

Heat and charge coexist in many semiconducting materials following
photoexcitation or electrical injection. Here “heat” refers to heating induced
lattice fluctuations, and “charge” refers to excited state electrons, holes
and their correlated combinations, e.g., excitons. In low-dimensional
materials, Auger-Meitner (A-M) processes, density-dependent heat-generating
exciton annihilation events, are prevalent even at modest exciton densities
due to enhanced Coulomb interactions, 1 and nonradiative pathways often
dominate due to defects and natural background doping. 2, 3 The combination of
these effects can lead to a significant fraction of absorbed light energy
undergoing transduction to lattice heating, which coexists with other charge
excitations like excitons. Similarly, in nanoscale electronic devices, charge
carrier scattering with phonons leads to Joule heating and elevated device
temperatures that impair efficient electronic dynamics, including transport,
due to increased scattering with the lattice 4. As device dimensions and
volume for heat dissipation continue to decrease, material interfaces
increase, and carrier-boundary scattering also plays a key role in self-
heating, limiting thermal and electrical conductivity 5. Each of these
dissipative effects in optoelectronic devices is important to discern so that
metrics such as photoluminescent quantum yield (PLQY) and carrier diffusion
length can be optimized. Additionally, thermal management strategies often
leverage inherent material anisotropies in energy flow which give rise to
thermoelectric capabilities, the ability to reversibly convert an electric
potential, $V$, to a temperature gradient, $\nabla T$, or vice versa. In
either case, distinguishing heat from charge when they coexist in a material
and discerning their unique photoinduced dynamics is vital not only for
informing design principles for directing heat and charge in emerging
materials but also for drawing well-informed conclusions about intrinsic
material properties.

Distinguishing between heat and charge with optical measurements is, however,
challenging. For example, both ground state and transient excited state
optical spectroscopy manifest complex perturbations to the location,
amplitude, and width of electronic resonances. 6, 7, 8 Even if signatures of
unique excitations are quantitatively observed, because their signatures can
overlap spectrally and temporally, they remain difficult to quantitate despite
judicious choices in excitation and probing wavelengths, time-dependent
signatures, measuring as a function of voltage bias, and measuring a pump
fluence- or temperature dependence. In particular, local heating could
influence or masquerade as electronic excitations in semiconducting materials,
and because it is spectrally ubiquitous, isolating the electronic dynamics is
especially challenging. Physically, heating leads to increased lattice
fluctuations, which change the mass density, an effect measured by the thermo-
optic coefficient and which may be observed as a frequency-independent change
in the dielectric function. 9, 10 Near an electronic resonance, however,
heating broadens and shifts the resonance feature, an effect that adds to or
cancels any transient photoinduced changes from the electronic carriers
themselves. 8, 7 This behavior is especially complicated in transition metal
dichalcogenides (TMDCs) where heat dissipation following photoexcitation is
prevalent and different exciton resonance features may spectrally overlap. 11

The growing collection of spatiotemporally resolved optical microscopies are
excellent candidates for characterizing photogenerated energy and its
transport in semiconducting materials because they provide spatial dynamics in
addition to the information provided by more traditional time-resolved
spectroscopy. These techniques, e.g., micro-time-resolved photoluminescence
(microTRPL), 12, 13, 14, 15 transient absorption microscopy (TAM), 16, 17, 18,
variations of transient reflectance microscopy, 19, 20, 21 including
stroboscopic scattering microscopy (stroboSCAT), 22 directly measure the
progressive expansion of initially localized populations of impulsively
photogenerated energy carriers with nanoscale sensitivity and down to
ultrafast time resolution. TRPL is, however, restricted to detecting electron-
hole radiative recombination, and, to our knowledge, there are no reports of
TAM separately resolving heat directly in addition to charge. Fortunately, due
to its high sensitivity to changes in the real part of the dielectric
function, even beyond transient reflectance alone, stroboSCAT has investigated
heat flow in metallic composite films 23, 24 and has distinguished heat and
charge flow in p-doped silicon based on different signs of imaging contrast, a
substantial separation in time scales, and corroboration with commonly cited
respective diffusivities. 22 To address the additional challenges associated
with characterizing electronic dynamics and transport in ultrathin
semiconductors, especially when heat and electronic diffusivities are not as
dissimilar as in silicon and where photogenerated heat is prominent,
additional strategies must be developed.

Here, we directly co-measure both heat and excitons in few-layer molybdenum
disulfide (MoS2) on relevant nanometer and picosecond length- and time scales
using stroboSCAT. Through a combination of near- and far-from resonant
stroboSCAT probing conditions, one of which isolates heat alone, and
calibration with steady state temperature-dependent reflectance contrast
spectroscopy, we observe and quantitatively discern transient thermal and
electronic contributions to the photogenerated dynamics. This new strategy
enables isolation and characterization of the excitonic dynamics without
concern for mistaking thermal contributions for electronic ones. This
capability with few-layer MoS2 complements the capabilities of microTRPL,
which is largely restricted to monolayer TMDC measurements 13, 14, 15 because
additional layers lead to very low PLQY due to the emergence of an indirect
band gap. Furthermore, we corroborate our results with a spatiotemporal model
for heat and exciton populations. With the ability to detect temperature
elevations as low as 100 mK, our study suggests that even a modest temperature
elevation has a substantial effect on the optical response in few-layer MoS2.
More broadly, this work establishes a strategy for isolating electronic
dynamics and transport in a wide range of conventional and emerging
semiconductors that offers great potential for more incisively investigating
thermal management and thermoelectric energy conversion.

Figure 1: (a) Calculated single particle band structure for 4L MoS2 from
Reference 25 showing the above-band gap pump excitation (green arrow) at the
$K$-point of the Brillouin zone, 26 followed by fast thermalization (orange)
and intervalley scattering (gray) to the lowest energy indirect exciton (blue
arrow). (b) Optical reflectance image of the entire MoS2 flake with the
measured 4L hBN-encapsulated region outlined in yellow. (c) Steady state
photoluminescence (PL) spectrum (black) showing strong emission from the
direct A exciton near 1.77 eV with a blue shoulder corresponding to emission
from the direct B exciton. Emission from the indirect exciton (1.4 eV) is
comparatively weaker. The reflectance contrast spectrum (blue) exhibits
dispersive resonance features near the PL peaks.

To demonstrate the capability of identifying and simultaneously tracking
electronic and thermal energy evolution, we focus on a hBN-encapsulated four-
layer (4L) MoS2 flake fabricated using the hot pick-up technique and supported
by a glass substrate. 27 Before proceeding with spatiotemporally resolved
transport measurements, we pre-characterize the sample. The relevant portion
of the electronic structure is shown schematically in Figure 1a. 25 Optical
reflectance microscopy readily identifies the encapsulated 4L region outlined
in yellow in Figure 1b. Additional sample thickness characterization is found
in Figure S1. The steady state photoluminescence spectrum in Figure 1c shows
characteristic strong emission from the spin-orbit split A ($\sim$1.8 eV) and
B ($\sim$2 eV) direct excitons, with a strong B:A photoluminescence intensity
and dispersive resonance amplitude ratio in the blue reflectance contrast
spectrum indicative of good sample quality. 28 Photoluminescence due to
recombination of the lowest-energy indirect exciton (IDE) in the near-infrared
($\sim$1.4 eV), indicated by the blue arrow in Figure 1a, is comparatively
weak.

We use stroboSCAT to directly visualize heat and exciton transport in the 4L
MoS2. stroboSCAT is a differential pump-probe technique that leverages the
sensitivity of optical scattering to image energy migration. This sensitivity
arises from homodyne detecting even the high angle scattered field via
interference with a reference, in this case specularly reflected probe light.
The physical process is similar to transient reflectance spectroscopy, except
for the important difference that it takes place in a microscope with a wide-
field – and often highly off-resonant – probe pulse capable of amplifying the
signal that scatters off of a more localized pump-induced excitation
distribution. 22 These particular experiments have nanometer spatial precision
and $\sim$200 picosecond time resolution. Two pulsed diode laser sources are
directed into the sample with a high numerical aperture microscope objective
that also collects the signal and directs it to a CMOS camera (Methods).
First, a near-diffraction-limited focused pump pulse of 2.8 eV photons (green
arrow in Figure 1a) generates a Gaussian spatial distribution of
photoexcitations in the sample which are then probed after a controllable time
delay by a single-wavelength widefield laser pulse of either 2.4 or 1.8 eV.
Specifically in 4L MoS2, the pump pulse generates excitations at the $K$-point
of the Brillouin zone. 26 Efficient sub-ps intervalley scattering (horizontal
gray lines in Figure 1a) and $\sim$fs phonon-assisted relaxation to the band
edge (orange in Figure 1a) occur within the $\sim$200 ps experimental
instrument response function (IRF), and we therefore expect the IDE to be the
dominant electronic excitation on the time scale of our measurements. 16 A
differential signal image is generated through the difference of the image at
a time delay after the pump and an image taken without a pump, normalized to
the latter. The contrast in the image, $(R_{\text{pump on}}-R_{\text{pump
off}})/R_{\text{pump off}}$, is commonly referred to as $\Delta R/R$. Despite
the fact that the 4L MoS2 is not luminescent, the presence of photoexcitations
modifies the material’s local dielectric function, generating transient
contrast that evolves as a function of space and time according to the
quantity and location of decaying and diffusing photoexcitations. Energy
population dynamics are described by integrated population decays as a
function of time. More importantly, transport is characterized via the mean
squared expansion (MSE) of the population in space as a function of time,
$\text{MSE}=\sigma^{2}(t)-\sigma^{2}(0)=2Dt$ where $D$ is the diffusivity and
$\sigma(t)$ is the width of the Gaussian population at time delay, $t$,
assuming a cylindrically symmetric distribution. In principle, any
photoexcitation, e.g., charge carriers, excitons, phonons, may be detected in
this way since the measurement observable, elastically backscattered light,
does not rely on the material absorbing or emitting light at a particular
frequency. Furthermore, near an absorption resonance, the sign of the
differential contrast can be tuned above (bright) or below (dark) the baseline
(gray) background. For a dispersive optical resonance, the transient response
leads to oppositely signed $\Delta R/R$ on either side of the resonance. 29
Heat, which also modulates electronic resonances, may also modify $\Delta
R/R$.

To isolate the effect of heating on the optical response, steady state
reflectance contrast (RC) spectra 30 are measured at a range of temperatures
from room temperature to 90∘C, as described below and in the Methods. RC
spectra are obtained by measuring reflected spectra from the sample atop of
the substrate ($R$) and separately under the bare glass substrate ($R_{0}$)
and calculating $RC=(R-R_{0})/R_{0}$. RC spectra over the measured temperature
range are presented in Figure 2a, showing the characteristic A and B exciton
resonance peaks redshifting and broadening with increasing temperature. From
these spectra, we identify two spectral regimes, near- and far-from resonance,
which tune the relative contribution to the stroboSCAT signal from heat and
excitons, enabling distinction between the two, as described below. We select
probe energies for spatiotemporal imaging from available discrete laser diode
sources indicated by the red (near resonant, 700 nm or 1.77 eV) and green
(far-from resonant, 515 nm or 2.41 eV) vertical lines in Figure 2a.

##  2 Results and analysis

Figure 2: (a) Temperature-dependent reflectance contrast spectra over a range
of temperature set points near the A and B exciton resonances. The near- and
far-from-resonant probe energies are indicated with the red and green lines,
respectively. (b) Expected differential contrast due to heating,
$\left(\frac{\Delta
R}{R}\right)_{\text{heating}}=\frac{RC_{\text{hot}}-RC_{\text{room
temp}}}{RC_{\text{room temp}}}$, at 1.77 eV. The vertical axis is multiplied
by -1 for direct comparison to widefield stroboSCAT measurements. The linear
fit (blue line) has a fixed intercept through the origin. (c) stroboSCAT time
series captured with a near-resonant (top) and far-from-resonant (bottom)
probe. The focused 2.8 eV pump generates a peak initial exciton density of
$3.5\times 10^{13}$ cm-2.

In the same sample region, we collect two complementary stroboSCAT
measurements over a 7 ns time window by probing at the near- and far-from
resonant energies (Figure 2b). In both measurements, the same applied pump
pulse fluence of 35 µJ/cm2 generates an estimated peak exciton density of
$3.5\times 10^{13}$ cm-2 (see Supporting Information), falling in an
intermediate density regime where A-M interactions play a significant role but
still being an order of magnitude below the Mott transition at which excitons
would dissociate. 31 Hot excitons thermalize and scatter to the indirect band
edge within the IRF, transferring their excess energy to the lattice via
efficient phonon emission. We therefore expect to observe both long-lived
(several ns) IDEs and lattice heating simultaneously on our measurement time
scales. In the far-from resonant stroboSCAT measurement, we observe negative
(dark) contrast alone that decays and expands over several nanoseconds (Figure
2b, bottom), whereas in the near-resonant stroboSCAT measurement at the top of
Figure 2b, we observe bright positive contrast beyond a dark, negative-
contrast center that similarly decays over several nanoseconds, with positive
contrast dominating after 1 ns. We assume that each sign of contrast, positive
or negative, in each measurement is generated by either heat or excitons, the
two dominant forms of energy in the material following photoexcitation.
Temperature-dependent RC spectra predict negative differential contrast due
strictly to heating in the near-resonant measurement (Figure 2c and details
below), therefore we deduce that positive contrast in the same measurement
must be due to the presence of excitons. Using these assignments, we observe
that excitons diffuse faster than heat, giving rise to positive amplitude
extending beyond the heat-dominated negative contrast. This assignment is
consistent with previous reports of exciton and heat diffusivity in MoS2, in
which exciton diffusivities are up to a few cm2/s 32, 33 while reported heat
diffusivities are slower at $\sim$0.2 cm2/s. 34

To most readily compare the datasets at the two imaging wavelengths, we apply
a pixel size correction to account for their different point spread functions
(PSFs) (Figure S2). With this correction, the spatial extent of the positive
signal measured with the near-resonant probe is obviously larger than the
corrected negative signal measured with the far-from resonant probe (Figure
S3). This observation suggests that unique dynamics give rise to the differing
spatial extent of the positive and negative contrast signals respectively
probed near and far from resonance, furthermore confirming that they represent
distinct photoexcited species. As the positive contrast has already been
assigned to an exciton population, we deduce that the negative contrast far-
from resonance must be generated by heat. A single Gaussian function fits all
far-from resonant data well, therefore we deduce that the measurement is
dominated by the optical response due to heating, and any potential
contribution from excitons (positive or negative) is below our detection
limit. It is therefore possible to isolate the positive exciton contribution
to the near-resonant dataset despite its spatial overlap with the negative
heat contribution by subtracting the far-from resonant heat-dominated
measurement using “image arithmetic” (Figure 3a). This strategy requires
quantitative knowledge of the difference in strength of the optical response
due to heating at the two probe energies, the magnitude of which depends on
the proximity to exciton resonances. We introduce a scaling factor, $\eta$,
into this image subtraction to account for this difference:
$\eta=\frac{\text{strength of optical response due to heat at 1.8
eV}}{\text{strength of optical response due to heat at 2.4 eV}}$. The far-from
resonant stroboSCAT measurement characterizes the optical response due to
heating at 2.4 eV. To quantitate the optical response due only to heating near
resonance at 1.8 eV, we refer to the temperature-dependent reflectance
contrast spectra in Figure 2a. To reframe the RC in units of stroboSCAT
contrast, we calculate the difference in reflectance contrast at 1.8 eV found
at each given elevated temperature and the value measured at room temperature
and normalize it to the room-temperature response, $\Delta
R/R_{\text{heating}}$ (Figures S4 and S5). The result is plotted in Figure 2c
for each heater set point up to 90∘C ($\Delta T=70$ K) with a linear fit, the
slope of which, -0.00042(5)/K, quantifies the predicted differential
stroboSCAT contrast associated with a given temperature increase. To estimate
the temperature at which to compare to $\eta$, we use a spatiotemporal kinetic
model, described in more detail below, to iteratively fit exciton and heat
experimental profiles until the predicted maximum achieved temperature in the
MoS2 matches the value of $\eta$ used in the fit. With this method we estimate
that the sample reaches a maximum temperature of 18 K above room temperature
after accounting for fast interfacial heat transfer to surrounding hBN, a
reasonable estimate that is close to the predicted temperature increase from
thermalization to the band edge (see Supporting Information). The scaling
factor $\eta$ is therefore given by the ratio between the predicted maximum
stroboSCAT contrast due to heat near resonance divided by the maximum
stroboSCAT contrast at time zero in the far-from resonant measurement:
$\eta=\frac{-0.00042(5)/\text{K}\times 18\text{ K}}{-0.00124(2)}=6.1(4)$.

Figure 3: (a) Image arithmetic to isolate exciton distributions (purple) from
thermal ones (green) when they coexist, with a scaling constant accounting for
the wavelength-dependent sensitivity to optical perturbations depending on
proximity to electronic resonances. (b) Azimuthal averages for the data shown
in the bottom row of Figure 2b representing thermal distributions. (c,d)
Integrated population decay fit to a single exponential plus power law (c) and
mean squared expansion fit to a line (d) for the thermal distributions in (b).
(e) Azimuthally averaged isolated exciton profiles after scaled image
subtraction of the near- and far-from-resonant datasets in Figure 2b. (f,g)
Integrated population decay fit to a biexponential (f) and mean squared
expansion fit to a line starting at 0.5 ns (g) for the exciton distributions
in (e). (h,i) Predicted thermal (h) and exciton (i) distributions from a
spatiotemporal kinetic model best fit of the experimental data in (b,e).

With the interpretation and strategy developed above, we obtain the population
dynamics and transport parameters for both heat and excitons in the
encapsulated 4L MoS2. We determine the thermal dynamics by fitting each
azimuthally averaged spatial distribution of each time point in the far-
detuned stroboSCAT measurement to a Gaussian function (Figure 3b). We find
that the integrated temperature profile has an initial fast $\sim$300 ps decay
due to interfacial transfer to the encapsulating hBN, and then heat transfers
more slowly, limited by the rate of heat diffusion in the hBN (Figure 3c).
Although hBN is a good thermal conductor, its capacity to sink heat generated
in MoS2 is limited by the evolving temperature gradient between the two
materials and the finite volume of hBN, and since the amplitude of the
Gaussian temperature distribution in the hBN drops due to lateral heat
diffusion as $t^{-1/2}$, this scaling determines the rate-limiting thermal
transfer for the MoS2 in the $\sim$ns time frame (see Supporting Information).
We obtain an in-plane heat diffusivity in the MoS2 of 0.17 cm2/s, consistent
with the reported lateral thermal conductivity for MoS2 (Figure 3d). 34

In order to quantitatively discern exciton dynamics from heat dynamics, we
developed a careful strategy of frame-by-frame azimuthal profile subtraction
of the near- and far-from resonant stroboSCAT datasets (Figure 3a). First,
each Gaussian fit profile representing the heat population (temperature
profile) in the far-detuned measurement is width- and amplitude-adjusted by a
time-dependent PSF correction factor. This operation generates the shape of
the isolated heat distribution that would have been measured with the near-
resonant probe. Next, we multiply the PSF-corrected heat profiles by the
above-deduced scaling factor, $\eta$, to quantitatively represent the
differential contrast profile due to heating near resonance (Figure S3).
Finally, we azimuthally average the total near-resonant stroboSCAT signal and
subtract the PSF-corrected and scaled heat-only component obtained from the
far-from resonant dataset. The isolated radial exciton profiles are near-
Gaussian, as shown in Figure 3e. The spatially integrated exciton population
dynamics fit best to a biexponential exciton decay, which we attribute to
density-dependent A-M interactions that dominate at early time delays when
exciton densities are higher ($\tau_{1}=310$ ps), followed by slow
nonradiative recombination of the IDE over $\tau_{2}\sim 8$ nanoseconds
(Figure 3f). The exciton profiles at time delays earlier than 500 ps
(grayscale) do not appear to expand, presumably due to the pump-induced
profile and its early time changes being beneath the spatiotemporal resolution
of these measurements. The extracted exciton diffusivity for $\geq$500 ps,
once it is possible to see the profiles expanding, is 0.5 cm2/s, in agreement
with other measurements in few-layer TMDCs (Figure 3g). 35, 36 Repeating this
dynamical analysis over a range of values of $\eta$ from 1.4 to 7 enables an
estimate of the uncertainties in the extracted diffusivities, based on
uncertainty in the initial maximum sample temperature elevation that we can
safely bound between 4 and 20 K.

We support these findings with a spatiotemporal kinetic model that describes
the coupled dynamics of excitons and heat. Additional details are included in
the Supporting Information. We developed a simple set of coupled equations to
capture the expansion and decay of excitons whose energy is overwhelmingly
(due to very low PLQY) converted to heat via either nonradiative hot carrier
relaxation following optical excitation or A-M decay: 13, 37

$\dot{N}(r,t)=D_{\text{X}}(r,t)\nabla^{2}N(r,t)-\frac{1}{\tau_{\text{X}}}N(r,t)-R_{\text{A-M}}N^{2}(r,t)+G(r,t)$
(1) $\dot{T}(r,t)=\alpha
N^{2}(r,t)-\frac{1}{\tau_{\text{T}}}[T(r,t)-T_{0}]+D_{\text{T}}\nabla^{2}T(r,t)+\beta
N(r,t)+\gamma G(r,t)$ (2)

Equation 1 describes the evolution of the exciton population, $N(r,t)$. The
first term describes exciton diffusion due to the pump-induced exciton
population gradient, with an exciton diffusivity of $D_{\text{X}}$. The second
term describes single exciton recombination, where $\tau_{\text{X}}$ is the
recombination lifetime. The third term describes biexciton recombination due
to A-M interactions, where $R_{\text{A-M}}$ is the A-M coefficient. Finally,
$G(r,t)$ describes the generation of excitons from the pump pulse, which has a
temporal pulse width of 72 ps and a spatial width, $\sigma$, of 168 nm.
Equation 1 describes the temperature or heat population, $T(r,t)$. The first
term accounts for the temperature increase due to nonradiative relaxation of
hot excitons that are created via A-M recombination, where
$\alpha=\frac{R_{\text{A-M}}E_{\text{G}}}{c}$, $E_{\text{G}}$ is the indirect
band gap energy, and $c$ is the specific heat. The second term describes the
decay of the temperature profile back to its initial room temperature value,
$T_{0}$, with a lifetime $\tau_{\text{T}}$, and the third term describes heat
diffusion with diffusivity $D_{\text{T}}$. The fourth term describes the
temperature increase due to nonradiative single-exciton recombination, where
$\beta=\frac{(1-\text{PLQY})E_{\text{G}}}{\tau_{\text{X}}c}$. The last term
describes the heat generated when excitons relax to the band edge after pump
excitation, where $\gamma=\frac{E_{\text{P}}-E_{\text{G}}}{c}$ and
$E_{\text{P}}$ is the pump pulse energy. In addition, when propagating these
equations in time, the heat decay phenomenologically transitions to a
$t^{-1/2}$ scaling after 700 ps to represent thermal transfer to hBN discussed
in the Supporting Information. To fit the model to the time series of exciton
profiles extracted above, along with the corresponding heat profiles obtained
far-from resonance, we allow $\tau_{\text{X}}$ and $D_{\text{X}}$ to vary only
within the experimentally determined uncertainty and $R_{\text{A-M}}$ to vary
within the literature estimates.38 We approximate the indirect band gap by the
peak position of the IDE PL (1.4 eV). All other parameters are fixed by our
experimentally obtained values or from literature values.39, 26

Figure 3h,i and Figure S6 show excellent qualitative agreement between the
modeled and experimentally obtained exciton and thermal profiles. The model
predicts an initial maximum exciton density of $\sim 8\times 10^{12}$ cm-2,
lower than our experimental estimate based on the pump fluence and sample
absorbance. We expect this discrepancy arises because the model accounts for
the finite pump duration and subsequent exciton decay occurring within the
experimental IRF. (If we suppress A-M interactions, exciton decay, and exciton
expansion in the model, we recover the initial condition for $N(0,0)=3.5\times
10^{13}$ cm-2.) The predicted maximum temperature at experimental time-zero,
(predominantly from exciton thermalization after above band gap excitation),
is 18 K, which is the value that is consistent with $\eta$ of 6.1 (see
Supporting Information). The best fit $R_{\text{A-M}}$ coefficient, a
parameter we cannot constrain with our experiments due to the experimental
IRF, is $2.6\times 10^{-3}$ cm-2, which is lower than for monolayer TMDCs, but
still within the expected range for multilayer TMDCs. 40, 14, 38 Overall, the
model supports our experimental finding that excitons diffuse slightly faster
than heat, importantly enabling spatial differentiation between the two, which
we described as instrumental in the differential contrast assignment.

##  3 Discussion

Having described our observations and analysis strategies and corroborated our
results with simulations, we turn to a discussion of our findings and of what
they reveal and implicate. Below, we first establish the consistency of our
physical model with other transient microscopy results that indirectly reveal
a role for heat to impact exciton populations in 2D TMDCs. Second, we explore
stroboSCAT’s favorable temperature sensitivity, which, under the conditions
employed in this work, is as good as $\sim$100 mK. Third, we explore the value
in employing our spatio-spectro-temporal approach and discuss complementary,
related strategies that could be used as a general toolkit to draw from,
depending on the particular photophysical details of any given material, given
the ubiquity of heat generation. Finally, we suggest strategies to treat
increasingly complex combinations of energy carriers, and we point toward the
utility of our approach in elucidating and leveraging new mechanisms of
electron-phonon coupling in thermal management and thermoelectrics.

To test our model-related findings, we note that Perea-Causín et al. 37 also
employed transient optical microscopy to study the interplay between heat and
excitons in 2D TMDCs, albeit with different experimental parameters. When
registering the exciton photoluminescence of monolayer WS2, following
ultrafast pump excitation leading to a far greater nonequilibrium-exciton
effective temperature elevation, they observed remarkable “halo-like” spatial
photoluminescence profiles, which they attributed to radially outward
excitonic transport driven by strong temperature gradients. To relate our
model to these results, we added the Seebeck term $\frac{\sigma S}{q}\Delta
T$, as in Perea-Causín et al., 37 to Equation 1 to reflect that exciton
diffusivity is driven not only by exciton density gradients but also by
thermal ones. Here, $\sigma(r,t)=N(r,t)q\mu$ is the electric conductivity for
exciton density $N$, elementary charge $q$, and exciton mobility, $\mu$. $S$
is the Seebeck coefficient, an intrinsic material parameter. By using the
Perea-Causín parameters into the model, we indeed generate halo-shaped
excitonic profiles (Figure S7). If, however, we retain the Seebeck term to
attempt to fit our experimental results that have a less impulsive pump pulse
and furthermore allow $\eta$ to vary, we find that an exceedingly large
Seebeck coefficient ($S>10,000$ V/K) would be required to generate halo-like
exciton profiles. For this reason, we rule out this Seebeck regime to model
our observations in 4L MoS2. This finding is also consistent with results from
Zipfel et al. 14 who also do not observe exciton halos in encapsulated samples
with $R_{\text{A-M}}\lesssim 10^{-3}$ cm-2.

Figure 4: (a) Gaussian amplitude fit decay for the far-from resonant probe
dataset. The vertical axis is calibrated using the maximum predicted sample
temperature elevation from the spatiotemporal model (18 K) and the measured
maximum differential contrast value. The inset shows the azimuthal averages in
Figure 3b with a rescaled vertical temperature axis. Error bars represent the
standard error of the mean of the azimuthal averaging. (b) Far-from resonant
probe images labeled with the maximum temperature achieved in each frame. All
images are shown on the same contrast scale. White traces are Gaussian fits to
the raw azimuthally averaged data.

We also wish to highlight a valuable byproduct of developing thermal
sensitivity capabilities in a transient optical microscopy – namely that
stroboSCAT can serve as a highly sensitive non-contact thermometry with
excellent spatial resolution compared to infrared analogs. For example, based
on the 18 K temperature elevation calculation in the 4L MoS2, we reframe in
Figure 4 the relaxation of the temperature in Figure 2b and Figure 3f to
establish the sensitivity of this thermometry. Figure 4a relabels the axis of
Figure 3f with the time-dependent peak temperature at the center of the heat
distribution. The radial profiles of corresponding time delays, also in Figure
3b, are shown as an inset, and some example $\Delta R/R$ images are included
in Figure 4b. At the longer time delays, we thus establish the ability to
resolve $\sim$0.1 K temperature elevation. Although it is difficult to
directly compare with the metrics of more conventional thermal imaging, 41
this value seems on par with it and enjoys substantially higher spatial
resolution. While the present data were collected by only averaging for 7
minutes per time point, we estimate, based on the 12-bit CMOS camera well
depth and the shot noise limit, that our sensitivity should indeed be in the
range of 100 mK. With increased averaging, microscope stabilization, and
detector sensitivity, this limit could be pushed into the tens of mK regime,
making stroboSCAT an exquisite thermometry approach with added high spatial
resolution. Not only is this newfound spatially-resolved sensitivity to
temperature very powerful for discerning heat and charge in photoexcited
materials, but it should find great utility in thermal management
characterization in the semiconductor device sector.

Regarding the toolkit that we have developed and its more general
applicability, our spatio-spectro-temporal approach is able to characterize
the complex, overlapping electronic and thermal system dynamics even though
their contributions would be difficult to disentangle from spectroscopic data
alone; discerning these dynamics in 4L MoS2 was possible despite an absence of
strongly distinct time scales in the excitonic and thermal population
dynamics. Another strategy to achieve the same result could be to use the
temperature-dependence of the reflectance contrast to calibrate the response
to heat at 2.4 eV, not only at 1.8 eV, where the proximity to a zero-crossing
precluded this process in the current work. Careful selection of another far-
from resonance probe wavelength could enable this alternative strategy for
future work. As a general strategy, for each specific sample, one must
carefully consider the best approach to isolate the electronic contribution to
the photoexcited signal based on spectral information, as discussed here, and
the extent to which the thermal and electronic diffusivities create a
separation of time scales, which is minimal here but was, for example,
sufficient to investigate silicon.22 Furthermore, there are generally multiple
different processes by which heat is generated that can occur on different
time scales relative to the generation and evolution of electronic excitation.
Regardless, the impact of heat on transient measurements is substantial in
many materials, not only for few-layer TMDCs. 42, 43, 7 Any type of non-
radiative process, from above gap excitation, to nonlinear processes like
annihilation, including Auger-Meitner effects, to ‘standard’ non-radiative
decay, will generate heat. By measuring the photoinduced response in a
reflectance geometry, the thermal response is revealed most clearly due to its
superior sensitivity to the real part of the dielectric function, relative to
transmission-based (or photoluminescence) measurements. Again, while it has
been painstakingly investigated in semiconductor transient spectroscopy, 8, 7
which is largely performed in reflectance geometry due to the opacity of
semiconductors in the visible and near-infrared parts of the spectrum, the
higher sensitivity and the addition of the spatial variable that stroboSCAT
affords provide additional helpful constraints to discern heat and charge. The
spatial coordinate is helpful not only because of access to instantaneous
spatial distributions but also to the time rate of change in the
spatiotemporal evolution, which yields transport parameters such as
diffusivity. 44 We therefore anticipate this newfound ability to characterize
the coexistence, transport, and interplay of heat and charge in materials to
be highly general and to enable a more detailed and reliable mechanistic
understanding of a material’s physical properties and phenomena far more
broadly than in TMDCs alone.

Regarding strategies for the future, the possibility exists to interrogate
materials with more than two types of energy carriers, for example, free
carriers, trions and excitons, coexist together with one another and also with
heat. In 4L MoS2, we probed sufficiently far from a zero crossing in the
transient reflectance spectrum in all cases to avoid sign-changes in the
differential contrast associated with each type of energy. In materials with
additional, distinct electronic species, the combination of measuring at
additional pump energies and fluences to tune the relative densities of
distinct photoinduced energy carriers could enable them to be distinguished.
Furthermore, a continuously tunable probe source could be leveraged to
increase the signal-to-noise ratio of photoexcitations that may not
appreciably modify the local dielectric function. 45, 46

We also envision a range of additional utilities of stroboSCAT in elucidating
mechanisms of electron-phonon coupling and deepening our understanding of
intrinsic thermal–electronic energy conversion and transport. Spatiotemporally
monitoring charged photoexcitations and phonons simultaneously opens new doors
for discovering mechanisms of electron-phonon scattering. In particular,
understanding nonradiative decay pathways facilitated by traps, interfaces,
defects, A-M interactions and natural background doping will elucidate design
principles for engineering higher PLQY materials and directed or enhanced
diffusion lengths. 47, 48, 49, 50 Characterizing the potential interplay
between heating and electronic energy flow could inform thermal management
strategies by revealing the dominant factors and mechanisms that tune
electron-phonon coupling. Furthermore, while thermal management aims to
mitigate the impact of unwanted and deleterious heat dissipation, for example
in semiconductor electronics, stroboSCAT also has the capability to directly
measure transport anisotropies and thermoelectric effects in which heat is
harnessed to do useful electronic work. For example, directly measuring the
intrinsic Seebeck coefficient, an important factor in the intrinsic figure of
merit for thermoelectrics, across different device configurations or material
thicknesses may address challenges in efficiently upcycling heat loss through
conversion to electricity or otherwise controlling heat flow in operating
devices that suffer from poor performance due to self heating. 51

##  4 Conclusion

In conclusion, we use a combination of optical scattering microscopy and
temperature-dependent reflectance contrast spectroscopy to co-measure and
discern the photoinduced dynamics of heat and excitons in 4L MoS2. This
capability is a generalizable consequence of stroboSCAT’s unique spatially-,
spectrally-, and temporally-resolved contrast mechanism that is sensitive to
any perturbation that modifies a material’s local dielectric function. The
spatiotemporal energy maps play a key role in identifying overlapping
energetic populations with distinct contributions to the differential
contrast. Our results agree with previous characterizations of few-layer MoS2,
are robust to experimental uncertainties in the estimated sample temperature
elevation, and demonstrate a temperature sensitivity as good as 100 mK,
ushering in a new era for spatiotemporally-resolved optical microscopy to
discern charge and heat and their potential interplay. With quantitative
energy-carrier-specific tracking down to few-ps time scales, directly
characterizing and explaining the factors that give rise to the optoelectronic
properties of a wide range of emerging semiconducting materials, including
intrinsic thermoelectrics and low-dimensional devices, is now possible without
having to rely on complex models or assumptions.

We thank W. Tisdale and M. Delor for valuable discussions and input in the
earlier stages of this work. stroboSCAT method development for this work has
been supported by STROBE, A National Science Foundation Science & Technology
Center under Grant No. DMR 1548924. Sample preparation, stroboSCAT and all
reflectance measurements, and all data analysis were supported by the
“Photonics at Thermodynamic Limits” Energy Frontiers Research Center of the
U.S. Department of Energy, Office of Basic Energy Sciences under award no. DE-
SC0019140. Modelling was supported by the Center for Computational Study of
Excited State Phenomena in Energy Materials (C2SEPEM) under the U.S.
Department of Energy, Office of Science, Basic Energy Sciences, Materials
Sciences and Engineering Division under contract no. DE-AC02-05CH11231, as
part of the Computational Materials Sciences Program. H. L. Weaver
acknowledges a National Science Foundation Graduate Research Fellowship (DGE
1106400). D.J. acknowledges the support of the Computational Science Graduate
Fellowship from the U.S. Department of Energy under Grant No. DE-SC0019323.
N.S.G. acknowledges an Alfred P. Sloan Research Fellowship, a David and Lucile
Packard Foundation Fellowship for Science and Engineering, and a Camille and
Henry Dreyfus Teacher-Scholar Award.

##  5 Materials and Methods

###  5.1 MoS2 preparation and characterization

We use the hot pick-up technique to fabricate hBN-encapsulated few-layer MoS2
heterostructures on coverglass. 27 Briefly, hBN and MoS2 are exfoliated onto
285 nm of thermally oxidized SiO2 on Si. We make stamps consisting of PDMS
covered with a thin film of the thermoplastic polymer polycarbonate (PC).
Using these stamps, we first pick up the top hBN, then the desired MoS2 flake,
and then the bottom hBN, all at 50∘C. We deposit the stack onto #1.5
coverglass, which serves as the substrate for all stroboSCAT measurements, at
180∘C, and then dissolve the PC in chloroform.

We characterize the hBN-encapsulated MoS2 sample using optical microscopy,
Raman spectroscopy, and atomic force microscopy (AFM). The 4L MoS2 flake
measured in the main text is outlined in blue in Figure S1a,b. The AFM image
in Figure S1b shows that our sample has large ($>7\times 7$ µm), homogeneous,
bubble-free areas, which are ideal for measuring in stroboSCAT. The separation
between the two MoS2 Raman peaks demonstrates that these samples are 4 layers
thick Figure S1c,d. 52. The hBN layer between the MoS2 flake and glass
substrate is thin enough ($\sim$5 nm) to allow easy optical access to the MoS2
layer within the 1.4 NA objective’s depth of field.

###  5.2 stroboSCAT measurements

stroboSCAT measurements were performed at room temperature using three
Picoquant pulsed laser diode sources, one to excite the sample at 440 nm (LDH-
D-C-440) and two probe sources at 515 nm (LDH-D-C-520) and 700 nm (LDH-
D-C-705). The base laser repetition rate was set to 10 MHz with the pump
modulated at 660 Hz. Pump-probe time delays were electronically controlled by
the laser driver with $<$20 ps precision. The pump and probe beams were
spatially filtered through 25 and 50 µm pinholes, respectively, before being
combined with a dichroic mirror (DMLP505, Thorlabs) and directed into the
objective of an inverted microscope stage with a 50/50 beamsplitter. The pump
was focused onto the sample with this high numerical aperture oil-immersion
objective (Leica HC PL APO 63x/1.40NA) to a FWHM of $\sim$300 nm. The probe
was focused into the back focal plane of the same objective to enable
widefield illumination in the sample plane. The backscattered light was
collected through the same objective, filtered (ThorLabs FB520-10 or Chroma
ET720/60m) to reject the pump light, and focused onto a CMOS detector
(PixeLINK PL-D752) using a 500 mm focal length imaging lens, resulting in an
overall image magnification of 157.5$\times$. The instrument response function
was measured to be $\sim$240 ps, primarily limited by the pulse duration of
the diode sources (60-110 ps). Extensive details of the stroboSCAT technique
are described in previous work. 22

Differential $\Delta R/R$ stroboSCAT images at each probe time delay are
constructed from the difference between a widefield image collected after a
pump pulse excitation and a widefield image collected without pump pulse
excitation, normalized to the latter: $\Delta R/R=(R_{\text{pump
on}}-R_{\text{pump off}})/R_{\text{pump off}}$. Each of these sequentially-
taken individual images is obtained within the 1.3 ms camera exposure. A total
of 3500 pairs of pump-on + pump-off image exposures at each time delay were
collected in each of 10 scans over the full set of time delays. The results of
each time delay from these 10 scans were averaged together for a total
averaging time of 7 minutes per time-delayed differential image. Images are
contrast-adjusted so that the grayscale baseline is in the center of the scale
with the maximum signal magnitude defining both positive and negative scale
bounds. The -5 ns time delayed image is subtracted from each subsequent time
delayed image as a baseline correction.

###  5.3 Temperature-dependent reflectance contrast spectroscopy

Broadband emission from a stabilized tungsten-halogen lamp source (ThorLabs
SLS201L) is spatially filtered through an optical fiber then focused into a
0.9 NA air objective in an Leica DMi8 inverted microscope, illuminating the
sample within a $\sim$1 $\mu$m spot size. The reflected light output is fiber-
coupled to a calibrated Princeton Instruments Spectrometer (HRS300) with
$\sim$1.5 nm spectral resolution. Spectra were collected by averaging 500
frames (10 ms exposure per frame) together after heating and equilibrating the
sample using a PID-controlled metal ceramic heater (ThorLabs HT19R).

###  5.4 Spatiotemporal model

A set of coupled equations representing exciton and heat dynamics and
transport and their interchange was recast in natural units and solved using
the pdepe function in MATLAB. Multistart optimization was performed with 500
starting points over a constrained 3-parameter space with least squares
minimization to the measured heat and isolated exciton profiles. More details
are provided in the Supporting Information.

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# Spin-charge Separation in Nodal Antiferromagnetic Insulator

Su-Peng Kou  Department of Physics, Beijing Normal University, Beijing, 100875
P. R. China

###### Abstract

In this paper, by using two dimensional (2D) Hubbard models with $\pi$-flux
phase and that on a hexagonal lattice as examples, we explore spin-charge-
separated solitons in nodal antiferromagnetic (AF) insulator - an AF order
with massive Dirac fermionic excitations (see detail in the paper). We
calculate fermion zero modes and induced quantum numbers on solitons (half
skyrmions) in the continuum limit, which are similar to that in the quasi one-
dimensional conductor polyacetylene (CH)x and that in topological band
insulator. In particular, we find some novel phenomena : thanks to an induced
staggered spin moment, a mobile half skyrmion becomes a fermionic particle;
when a hole or an electron is added, the half skyrmion turns into a bosonic
particle with charge degree of freedom only. Our results imply that nontrivial
induced quantum number on solitons may be a universal feature of spin-charge
separation in different systems.

PACS numbers : 71.10.Fd, 75.10.Jm, 75.40.Gb

The Fermi liquid based view of the electronic properties has been very
successful as a basis for understanding the physics of conventional solids.
The quasi-particles of Fermi liquid carry both spin and charge quantum
numbers. However, in some cases, spin-charge separation occurs, providing a
new framework for thinking about the given systems. It indicates that the
systems have two independent elementary excitations, neutral spinon and
spinless holon, respectively, as opposed to single quasi-particle excitation
in conventional solids.

The first example is electronic systems in one spatial dimensionle . The idea
of solitons with induced quantum numbers starts with a beautiful result
obtained in the context of relativistic quantum field theories by Jackiw and
Rebbijackiw . Based on this idea, spin-charge separated solitons had a lasting
impact on condensed matter physics. In the long molecule-chain of trans-
Polyacetylene, spin-charge separation can occur in term of soliton
statespolrev . Due to induced fermion quantum numbers, the soliton may be
neutral particles with spin 1/2, or spinless with charge $\pm e$. In two
dimensional electronic systems, spin-charge separation has been a basic
concept in understanding doped Mott-Hubbard insulator related to high-$T_{c}$
cuprates pwa ; KRS . It is supposed that the particles can be liberated at low
energies, with spin-charge separation being an upshot in the “resonating
valence bond” (RVB) spin liquid state which is proposed by Anderson as a new
state of matterpwa . Recently topological band insulator (TBI) has attracted
considerable attention because of their relevance to the quantum spin Hall
effectkm ; sh . It is pointed out that there may exist spin-charge separated
solitons in the presence of $\pi$ flux with induced quantum numbersqi ; ran ;
ran1 .

In this paper we focus on a special class of antiferromagnetic (AF) ordered
state - nodal AF insulator, and we will show how spin-charge separation
occurs. Nodal AF insulator is an AF order (long range or short range) with
massive Dirac fermionic excitations. When there is no AF order, fermionic
excitations become nodal quasi-particles. There are two examples of nodal AF
insulator in condensed matter physics - one is an AF order on a honeycomb
lattice, the other is $\pi$-flux phase together with a nonzero Neel order
parameter. Based on these examples, our results confirm that induced quantum
number on solitons is an important feature of the spin-charge separation in
different systems.

_Formulation_ $\mathcal{-}$ To develop a systematical formulation, we start
with the extended Hubbard models,

$H=-\sum\limits_{\langle
i,j\rangle,\sigma}t_{ij}\hat{c}_{i,\sigma}^{\dagger}\hat{c}_{j,\sigma}+U\sum_{j}\hat{n}_{j\uparrow}\hat{n}_{j\downarrow}-\mu\sum\limits_{i,\sigma}\hat{c}_{i,\sigma}^{\dagger}\hat{c}_{i,\sigma}+h.c..$
(1)

Here $\hat{c}_{i,\sigma}^{\dagger}$ and $\hat{c}_{j,\sigma}$ are electronic
creation and annihilation operators. $U$ is the on-site Coulomb repulsion.
$\sigma$ are the spin-indices for electrons. $\mu$ is the chemical potential.
$\langle i,j\rangle$ denotes two sites on a nearest-neighbor link.
$\hat{n}_{j\uparrow}$ and $\hat{n}_{j\downarrow}$ are the number operators of
electrons with up-spin and down spin. On a honeycomb lattice, the nearest
neighbor hopping is a constant, $t_{i,j}=t;$ on a square lattice with
$\pi$-flux phase, it can be chosen as $t_{i,i+\hat{x}}=\chi,$
$t_{i,i+\hat{y}}=i\chi$ st ; st0 ; pi ; wen . The partition function of the
extended Hubbard models is written as
$\mathcal{Z}=\int\mathcal{D}\overline{c}\mathcal{D}ce^{-\int_{0}^{\beta}d\tau
L},$ where

$\displaystyle L$ $\displaystyle=$
$\displaystyle\sum_{j,\sigma}\bar{c}_{j,\sigma}\left(\partial_{\tau}-\mu\right)c_{j,\sigma}+\sum_{\langle
i,j\rangle,\sigma}t_{ij}\bar{c}_{i,\sigma}c_{j,\sigma}$
$\displaystyle-U\sum_{j}n_{j\uparrow}n_{j\downarrow}.$

$\bar{c}_{i,\sigma}$ and $c_{j,\sigma}$ are Grassmann variables describing the
electronic fields.

Firstly let us derive long wave-length effective Lagrangian of the hopping
term in the extended Hubbard models. Although $\pi$-flux phase does not break
translational symmetry, we may still divide the square lattice into two
sublattices, $A$ and $B$. After transforming the hopping term into momentum
space, we obtain $E_{f}=2\chi\sqrt{\cos^{2}k_{x}+\cos^{2}k_{y}}$. So there
exist two nodal fermi-points at
$\mathbf{k}_{1}=(\frac{\pi}{2},\frac{\pi}{2}),$
$\mathbf{k}_{2}=(\frac{\pi}{2},-\frac{\pi}{2})$ and the spectrum of fermions
becomes linear in the vicinity of the two nodal points. On a honeycomb
lattice, after dividing the lattice into two sublattices, $A$ and $B$, the
dispersion is obtained in Ref.her ; her1 ; pei . There also exist two nodal
points, $\mathbf{k}_{1}=\frac{2\pi}{\sqrt{3}}(1,\frac{1}{\sqrt{3}})$ and
$\mathbf{k}_{2}=\frac{2\pi}{\sqrt{3}}(-1,-\frac{1}{\sqrt{3}})$ and the
spectrum of fermions becomes linear near $\mathbf{k}_{1,2}$. In the continuum
limit, the Dirac-like effective Lagrangian describes the low energy fermionic
modes for both cases

$\mathcal{L}_{f}=i\bar{\psi}_{1}\gamma_{\mu}\partial_{\mu}\psi_{1}+i\bar{\psi}_{2}\gamma_{\mu}\partial_{\mu}\psi_{2}$
(3)

where
$\bar{\psi}_{1}=\psi_{1}^{\dagger}\gamma_{0}=(\begin{array}[]{llll}\bar{\psi}_{\uparrow
1A},&\bar{\psi}_{\uparrow 1B},&\bar{\psi}_{\downarrow
1A},&\bar{\psi}_{\downarrow 1B}\end{array})$ and
$\bar{\psi}_{2}=\psi_{2}^{\dagger}\gamma_{0}=(\begin{array}[]{llll}\bar{\psi}_{\uparrow
2B},&\bar{\psi}_{\uparrow 2A},&\bar{\psi}_{\downarrow
2B},&\bar{\psi}_{\downarrow 2A}\end{array})$her ; her1 ; pei . $\gamma_{\mu}$
is defined as $\gamma_{0}=\sigma_{0}\otimes\tau_{z},$
$\gamma_{1}=\sigma_{0}\otimes\tau_{y},$
$\gamma_{2}=\sigma_{0}\otimes\tau_{x},$
$\sigma_{0}=\left(\begin{array}[]{ll}1&0\\\ 0&1\end{array}\right)$.
$\tau^{x},$ $\tau^{y},$ $\tau^{z}$ are Pauli matrices. We have set the Fermi
velocity to be unit $v_{F}=1$.

In the strongly coupling limit, $U>>t$, there always exists an AF order in the
extended Hubbard models. Introducing Stratonovich-Hubbard fields for the spin
degrees of freedom wen , we obtain the partition function as
$Z=\int\mathcal{D}\overline{c}\mathcal{D}c\mathcal{D}\mathbf{B}e^{-\int_{0}^{\beta}d\tau
L}$, where the Lagrangian is given by

$\displaystyle L$ $\displaystyle=$
$\displaystyle\sum_{j,\sigma}\bar{c}_{j,\sigma}\left(\partial_{\tau}-\mu\right)c_{j,\sigma}+\sum_{\langle
i,j\rangle,\sigma}t_{ij}\bar{c}_{i,\sigma}c_{j,\sigma}$
$\displaystyle-\frac{3}{2U}\sum_{j}\mathbf{B}_{j}^{2}+U\sum_{j}(-1)^{j}\mathbf{B}_{j}\cdot\bar{c}_{j}\mathbf{\sigma}c_{j}$

with Pauli matrices
$\mathbf{\sigma=}(\sigma^{x},\sigma^{y},\sigma^{z})\mathbf{.}$ Here
$\mathbf{B}_{j}$ is a vector denoting spin configurations,
$\mathbf{B}_{j}=\left|B_{j}\right|\mathbf{n}_{j}$ where
$\left|B_{j}\right|=\phi_{0}$ represents the value of localized spin moments
and $\mathbf{n}_{j}$ is a unit vector describing the Néel order parameter. In
AF ordered state, the mass gap of the electrons is given as $m=\phi_{0}U$.

Then starting from Eq.(Spin-charge Separation in Nodal Antiferromagnetic
Insulator), we get the same long wave-length effective model of nodal AF
insulator at ; ki ; senthil

$\mathcal{L}_{\mathrm{eff}}=\sum\limits_{\alpha}i\bar{\psi}_{\alpha}\gamma_{\mu}\partial_{\mu}\psi_{\alpha}+m(\bar{\psi}_{1}\mathbf{n\cdot\sigma}\psi_{1}-\bar{\psi}_{2}\mathbf{n\cdot\sigma}\psi_{2}).$
(5)

$\alpha=1,$ $2$ labels the two Fermi points

_Zero modes on half skyrmions_ \- In this section we will study the properties
of topological solitons. Instead of considering topological solitons with
integer topological charge (skyrmions), we focus on solitons with a half
topological charge,
$\int\frac{1}{2\pi}\mathbf{n}\cdot\partial_{x}\mathbf{n}\times\partial_{y}\mathbf{n}$
$d^{2}\mathbf{r}=\pm\frac{1}{2}$ with $\mathbf{r=(}x,$ $y\mathbf{).}$ Such
soliton is called a half skyrmion (meron or anti-meron). A meron with a narrow
core size (the lattice size $a$) is characterized by
$\mathbf{n=r}/\mid\mathbf{r}\mid,$ $\mathbf{r}^{2}=x^{2}+y^{2}$ pol ; meron ;
half ; ba ; ng ; ots ; weng1 ; kou . To stabilize such a solitons, one may add
a small easy-plane anisotropy of the Néel order.

Around a meron configuration, the fermionic operators are expanded as

$\displaystyle\hat{\psi}_{\mathbf{\alpha}}(\mathbf{r},t)$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{k}\neq 0}\hat{b}_{\mathbf{\alpha
k}}e^{-iE_{\mathbf{k}}t}\psi_{{\mathbf{\alpha}}\mathbf{k}}(\mathbf{r})$
$\displaystyle+\sum_{\mathbf{k}\neq 0}\hat{d}_{\mathbf{\alpha
k}}^{\dagger}e^{iE_{\mathbf{k}}t}\psi_{\mathbf{\alpha
k}}^{\dagger}(\mathbf{r})+\hat{a}_{\mathbf{\alpha}}^{0}\psi_{\mathbf{\alpha}}^{0}(\mathbf{r}),\text{
}$

where $\hat{b}_{\alpha\mathbf{k}}$ and $\hat{d}_{\alpha\mathbf{k}}^{\dagger}$
are operators of $\mathbf{k}\neq 0$ modes that are irrelevant to the soliton
states discussed below. $\psi_{\mathbf{\alpha
k}}^{\dagger}(\mathbf{r})=(\begin{array}[]{llll}\psi_{\uparrow\mathbf{\alpha
Ak}}^{0*},&\psi_{\uparrow\mathbf{\alpha}B\mathbf{k}}^{0*},&\psi_{\downarrow\mathbf{\alpha
Ak}}^{0*},&\psi_{\downarrow\mathbf{\alpha}B\mathbf{k}}^{0*}\end{array})$ are
the functions of zero modes. $\hat{a}_{\alpha}^{0}$ are annihilation operators
of zero modes.

To obtain the zero modes, we write down two Dirac equations from Eq.(5)

$i\partial_{x}\gamma_{1}\psi_{1}^{0}+i\partial_{y}\gamma_{2}\psi_{1}^{0}+m\mathbf{n\cdot\sigma}\psi_{1}^{0}=0$
(7)

and

$i\partial_{x}\gamma_{1}\psi_{2}^{0}+i\partial_{y}\gamma_{2}\psi_{2}^{0}-m\mathbf{n\cdot\sigma}\psi_{2}^{0}=0.$
(8)

Firstly we solve the Dirac equation for $\psi_{1}^{0}$. With the ansatz
$\psi_{1}^{0}=\left(\begin{array}[]{l}\xi_{1}(\tilde{x})e^{-i\theta}\\\
\xi_{2}(\tilde{x})\\\ \xi_{3}(\tilde{x})\\\
\xi_{4}(\tilde{x})e^{i\theta}\end{array}\right),$ we have

$\displaystyle\partial_{\tilde{x}}\xi_{2}$ $\displaystyle=$
$\displaystyle\xi_{3},\text{ }\partial_{\tilde{x}}\xi_{3}=\xi_{2},$ (9)
$\displaystyle\partial_{\tilde{x}}\xi_{1}$ $\displaystyle=$
$\displaystyle-\frac{\xi_{1}}{\tilde{x}}+\xi_{4},\text{
}\partial_{\tilde{x}}\xi_{4}=-\frac{\xi_{4}}{\tilde{x}}+\xi_{1}$

where $\mathbf{r=}\mid\mathbf{r}\mid(\cos\theta,\sin\theta)$ and
$\tilde{x}=\frac{\mid\mathbf{r}\mid}{m}$. The solution has been obtained in
Ref.zeromode  as

$\xi_{1}(\tilde{x})=\xi_{4}(\tilde{x})=0,\text{
}\xi_{2}(\tilde{x})=\xi_{3}(\tilde{x})=\tilde{x}^{\frac{1}{2}}K_{\frac{1}{2}}(\tilde{x})$

where $K_{\frac{1}{2}}(\tilde{x})$ is the modified Bessel function. So the
solution of $\psi_{1}^{0}$ becomes $\left(\begin{array}[]{l}0\\\
\tilde{x}^{\frac{1}{2}}K_{\frac{1}{2}}(\tilde{x})\\\
\tilde{x}^{\frac{1}{2}}K_{\frac{1}{2}}(\tilde{x})\\\ 0\end{array}\right).$

To solve $\psi_{2}^{0},$ we transform the equation
$i\partial_{i}\gamma_{i}\psi_{2}^{0}-m\mathbf{n\cdot\sigma}\psi_{2}^{0}=0$
into

$Ui\partial_{i}\gamma_{i}\tilde{\psi}_{2}^{0}U^{-1}+mU\mathbf{n\cdot\sigma}\tilde{\psi}_{2}^{0}U^{-1}=0,$
(10)

where $U=e^{i\pi\gamma_{0}/2},$ $U\gamma_{i}U^{-1}=-\gamma_{i}$ and
$U^{-1}\psi_{2}^{0}U=\tilde{\psi}_{2}^{0}.$ Then the solution of
$\psi_{2}^{0}\ $is obtained as $\left(\begin{array}[]{l}0\\\
-\tilde{x}^{\frac{1}{2}}K_{\frac{1}{2}}(\tilde{x})\\\
\tilde{x}^{\frac{1}{2}}K_{\frac{1}{2}}(\tilde{x})\\\ 0\end{array}\right).$

It is noticable that from above solutions of zero modes, the components
$\psi_{\uparrow 1A}^{0},$ $\psi_{\downarrow 1B}^{0},$ $\psi_{\downarrow
2A}^{0}$ and $\psi_{\uparrow 2B}^{0}$ are all zero.

_Topological mechanism of spin-charge separation -_ For the solutions of zero
modes, there are four zero-energy soliton states $\mid\mathrm{sol}\rangle$
around a half skyrmion which are denoted by $\mid 1_{+}\rangle\otimes\mid
2_{+}\rangle,$ $\mid 1_{-}\rangle\otimes\mid 2_{-}\rangle,$ $\mid
1_{-}\rangle\otimes\mid 2_{+}\rangle$ and $\mid 1_{+}\rangle\otimes\mid
2_{-}\rangle.$ $\mid 1_{-}\rangle$ and $\mid 2_{-}\rangle$ are empty states of
the zero modes $\psi_{1}^{0}(\mathbf{r})$ and $\psi_{2}^{0}(\mathbf{r});$
$\mid 1_{+}\rangle$ and $\mid 2_{+}\rangle$ are occupied states of them. Thus
we have the relationship between $\hat{a}_{\mathbf{\alpha}}^{0}$ and
$\mid\mathrm{sol}\rangle$ as

$\hat{a}_{1}^{0}\mid 1_{+}\rangle=\mid 1_{-}\rangle,\text{
}\hat{a}_{1}^{0}\mid 1_{-}\rangle=0,\text{ }\hat{a}_{2}^{0}\mid
2_{+}\rangle=\mid 2_{-}\rangle,\text{ }\hat{a}_{2}^{0}\mid 2_{-}\rangle=0,$
(11)

Firstly we define total induced fermion number operators of the soliton
states, $\hat{N}_{F}=\sum\limits_{\alpha}\hat{N}_{\alpha,F}$ with

$\displaystyle\hat{N}_{\alpha,F}$ $\displaystyle\equiv$
$\displaystyle\int:\hat{\psi}_{\alpha}^{\dagger}\hat{\psi}_{\alpha}:d^{2}\mathbf{r}$
$\displaystyle=$
$\displaystyle(\hat{a}_{\alpha}^{0})^{\dagger}\hat{a}_{\alpha}^{0}+\sum\limits_{\mathbf{k}\neq
0}(\hat{b}_{\alpha\mathbf{k}}^{\dagger}\hat{b}_{\alpha\mathbf{k}}-\hat{d}_{\alpha\mathbf{k}}^{\dagger}\hat{d}_{\alpha\mathbf{k}})-\frac{1}{2}.$

$:\hat{\psi}_{\alpha}^{\dagger}\hat{\psi}_{\alpha}:$ means normal product of
$\hat{\psi}_{\alpha}^{\dagger}\hat{\psi}_{\alpha}$. From the relation between
$\hat{a}_{\mathbf{\alpha}}^{0}$ and $\mid\mathrm{sol}\rangle$ in Eq.(11), we
find that $\mid 1_{\pm}\rangle$ or $\mid 2_{\pm}\rangle$ have eigenvalues of
$\pm\frac{1}{2}$ of the total induced fermion number operator $\hat{N}_{F}$,

$\displaystyle\hat{N}_{1,F}|1_{\pm}\rangle$ $\displaystyle=$
$\displaystyle\pm{\frac{1}{2}}|1_{\pm}\rangle,\text{
}\hat{N}_{1,F}|2_{\pm}\rangle=0,\text{ }$ (13)
$\displaystyle\hat{N}_{2,F}|2_{\pm}\rangle$ $\displaystyle=$
$\displaystyle\pm{\frac{1}{2}}|2_{\pm}\rangle,\text{
}\hat{N}_{2,F}|1_{\pm}\rangle=0.$

Another important induced quantum number operator is staggered spin operator,
$\hat{S}_{(\pi,\pi)}^{z}=\frac{1}{2}\sum\limits_{i\in
A}\hat{c}_{i}^{\dagger}\sigma_{z}\hat{c}_{i}-\frac{1}{2}\sum\limits_{i\in
B}\hat{c}_{i}^{\dagger}\sigma_{z}\hat{c}_{i}=\frac{1}{2}\int:[(\hat{\psi}_{\uparrow
1A}^{\dagger}\hat{\psi}_{\uparrow 1A}+\hat{\psi}_{\downarrow
1B}^{\dagger}\hat{\psi}_{\downarrow 1B}-\hat{\psi}_{\downarrow
1A}^{\dagger}\hat{\psi}_{\downarrow 1A}$ $-\hat{\psi}_{\uparrow
1B}^{\dagger}\hat{\psi}_{\uparrow 1B})+(\hat{\psi}_{\uparrow
2A}^{\dagger}\hat{\psi}_{\uparrow 2A}+\hat{\psi}_{\downarrow
2B}^{\dagger}\hat{\psi}_{\downarrow 2B}-\hat{\psi}_{\downarrow
2A}^{\dagger}\hat{\psi}_{\downarrow 2A}-\hat{\psi}_{\uparrow
2B}^{\dagger}\hat{\psi}_{\uparrow 2B})]:d^{2}\mathbf{r}.$ For the four
degenerate zero modes, it can be simplified into
$\hat{S}_{(\pi,\pi)}^{z}\mid\mathrm{sol}\rangle=\frac{1}{2}(\hat{N}_{2,F}-\hat{N}_{1,F})\mid\mathrm{sol}\rangle.$
Let us show the detailed calculations. From the zero solutions of
$\psi_{\uparrow 1A}^{0},$ $\psi_{\downarrow 1B}^{0},$ $\psi_{\downarrow
2A}^{0}$ and $\psi_{\uparrow 2B}^{0}$, we obtain four equations

$\displaystyle(\int$ $\displaystyle:$ $\displaystyle\hat{\psi}_{\uparrow
1A}^{\dagger}\hat{\psi}_{\uparrow
1A}:d^{2}\mathbf{r})\mid\mathrm{sol}\rangle\equiv 0,$ (14)
$\displaystyle(\int$ $\displaystyle:$ $\displaystyle\hat{\psi}_{\downarrow
1B}^{\dagger}\hat{\psi}_{\downarrow
1B}:d^{2}\mathbf{r})\mid\mathrm{sol}\rangle\equiv 0,$ $\displaystyle(\int$
$\displaystyle:$ $\displaystyle\hat{\psi}_{\downarrow
2A}^{\dagger}\hat{\psi}_{\downarrow
2A}:d^{2}\mathbf{r})\mid\mathrm{sol}\rangle\equiv 0,$ $\displaystyle(\int$
$\displaystyle:$ $\displaystyle\hat{\psi}_{\uparrow
2B}^{\dagger}\hat{\psi}_{\uparrow
2B}:d^{2}\mathbf{r})\mid\mathrm{sol}\rangle\equiv 0.$

Using above four equations, we obtain

$\hat{S}_{(\pi,\pi)}^{z}\mid\mathrm{sol}\rangle=\frac{1}{2}\int
d^{2}\mathbf{r}:(-\hat{\psi}_{\uparrow 1A}^{\dagger}\hat{\psi}_{\uparrow
1A}-\hat{\psi}_{\downarrow 1B}^{\dagger}\hat{\psi}_{\downarrow
1B}-\hat{\psi}_{\downarrow 1A}^{\dagger}\hat{\psi}_{\downarrow
1A}-\hat{\psi}_{\uparrow 1B}^{\dagger}\hat{\psi}_{\uparrow 1B}$
$+\hat{\psi}_{\uparrow 2A}^{\dagger}\hat{\psi}_{\uparrow
2A}+\hat{\psi}_{\downarrow 2B}^{\dagger}\hat{\psi}_{\downarrow
2B}+\hat{\psi}_{\downarrow 2A}^{\dagger}\hat{\psi}_{\downarrow
2A}+\hat{\psi}_{\uparrow 2B}^{\dagger}\hat{\psi}_{\uparrow
2B}):\mid\mathrm{sol}\rangle=-\frac{1}{2}(\hat{N}_{1,F}-\hat{N}_{2,F})\mid\mathrm{sol}\rangle.$

Then we calculate two induced quantum numbers defined above. Without doping,
the soliton states of a half skyrmion are denoted by $\mid
1_{-}\rangle\otimes\mid 2_{+}\rangle$ and $\mid 1_{+}\rangle\otimes\mid
2_{-}\rangle$. One can easily check that the total induced fermion number on
the solitons is zero from the cancelation effect between two nodals
$\hat{N}_{F}\mid 1_{-}\rangle\otimes\mid 2_{+}\rangle=\hat{N}_{F}\mid
1_{+}\rangle\otimes\mid 2_{-}\rangle=0.$ It is consistent to the earlier
results that forbid a Hopf term for the low energy theory of two dimensional
Heisenberg modelwenhopf . On the other hand, there exists an induced staggered
spin moment on the soliton states $\mid 1_{-}\rangle\otimes\mid 2_{+}\rangle$
and $\mid 1_{+}\rangle\otimes\mid 2_{-}\rangle,$

$\displaystyle\hat{S}_{(\pi,\pi)}^{z}$ $\displaystyle\mid$ $\displaystyle
1_{-}\rangle\otimes\mid 2_{+}\rangle=\frac{1}{2}\mid 1_{-}\rangle\otimes\mid
2_{+}\rangle,$ (15) $\displaystyle\hat{S}_{(\pi,\pi)}^{z}$ $\displaystyle\mid$
$\displaystyle 1_{+}\rangle\otimes\mid 2_{-}\rangle=-\frac{1}{2}\mid
1_{+}\rangle\otimes\mid 2_{-}\rangle.$

The induced staggered spin moment may be straightforwardly obtained by
combining the definition of $\hat{S}_{(\pi,\pi)}^{z}$ and Eq.(13) together.

When half skyrmions become mobile, their quantum statistics becomes important.
Let us examine the statistics of a half skyrmion with an induced staggered
spin moment. In CP(1) representation of $\mathbf{n,}$ a ”bosonic spinon” is
introduced by $\mathbf{n}=\mathbf{\bar{z}\sigma z}$ with
$\mathbf{z}=\left(\begin{array}[]{l}z_{\uparrow}\\\
z_{\downarrow}\end{array}\right)$ and $\mathbf{\bar{z}z=1}$. Since each
”bosonic spinon” $\mathbf{z}$ carries $\pm\frac{1}{2}$ staggered spin moment,
an induced staggered spin moment corresponds to a trapped ”bosonic spinon”
$\mathbf{z}$. On the other hand, a half skyrmion can be regarded as a
$\pi-$flux of the ”bosonic spinon”,
$\int\frac{1}{2\pi}\mathbf{n}\cdot\partial_{x}\mathbf{n}\times\partial_{y}\mathbf{n}$
$d^{2}\mathbf{r}=\frac{1}{2\pi}\int\epsilon_{\mu\nu}\partial_{\mu}a_{\nu}d^{2}\mathbf{r}=\pm\frac{1}{2}$
with
$a_{\mu}\equiv\frac{i}{2}(\mathbf{\bar{z}}\partial_{\mu}\mathbf{z}-\partial_{\mu}\mathbf{\bar{z}z}).$
To be more explicit, moving a ”bosonic spinon” $z$ around a half skyrmion
generates a Berry phase $\phi$ to
$\mathbf{z}\rightarrow\mathbf{z}^{\prime}=\left(\begin{array}[]{l}z_{\uparrow}e^{i\phi}\\\
z_{\downarrow}e^{i\phi}\end{array}\right)$ where
$\phi=\int\epsilon_{\mu\nu}\partial_{\mu}a_{\nu}d^{2}\mathbf{r}=\pm\pi$. As a
result, a ”bosonic spinon” $\mathbf{z}$ and a half skyrmion (meron or
antimeron ) share mutual semion statistics. Binding the trapped ”bosonic
spinon”, a mobile half skyrmion becomes a fermionic particle. We may use the
operator $\hat{f}_{\sigma}$ to describe such neutral fermionic particle with
half spin. The relation between the zero energy states and the fermionic
states is given as $\mid 1_{+}\rangle\otimes\mid
2_{-}\rangle=\hat{f}_{\downarrow}^{\dagger}\mid 0\rangle_{f}\text{ and }\mid
1_{-}\rangle\otimes\mid 2_{+}\rangle=\hat{f}_{\uparrow}^{\dagger}\mid
0\rangle_{f}$ (The state $|0\rangle_{f}$ is defined through
$\hat{f}_{\uparrow}|0\rangle_{f}=\hat{f}_{\downarrow}|0\rangle_{f}=0$). We
call such neutral object (fermion with $\pm\frac{1}{2}$ spin degree freedom) a
(fermionic) ”spinon”.

Now we go away from half filling. It is known that when a hole (electron) is
doped, it is equivalence to removing (adding) an electron. Without considering
the existence of half skyrmions, the hole (electron) will be doped into the
lower (upper) Hubbard band. The existence of zero modes on half skyrmions
leads to the appearance of bound levels in the middle of the Mott-Hubbard gap
meron . The hole (electron) will be doped onto the bound states on the half
skyrmion and then one of the zero modes is occupied. When one hole is doped,
the soliton state is denoted by $|1_{-}\rangle\otimes|2_{-}\rangle$. One can
easily check the result by calculating its induce quantum numbers. On the one
hand, there is no induced staggered spin moment, $\hat{S}_{(\pi,\pi)}^{z}\mid
1_{-}\rangle\otimes\mid 2_{-}\rangle=0.$ On the other hand, the total fermion
number is not zero,
$\hat{N}_{F}|1_{-}\rangle\otimes|2_{-}\rangle=-|1_{-}\rangle\otimes|2_{-}\rangle.$
These results mean that such soliton state is a spinless ”holon” with positive
charge degrees of freedom. After binding a fermionic hole, the soliton state
(holon) does not have an induced staggered spin moment. Thus the holon obeys
bosonic statistics and becomes a charged bosonic particles. When one electron
is doped, the soliton state is denoted by $|1_{+}\rangle\otimes|2_{+}\rangle$.
The induced quantum numbers of it are
$\hat{N}_{F}|1_{+}\rangle\otimes|2_{+}\rangle=+|1_{+}\rangle\otimes|2_{+}\rangle$
and $\hat{S}_{(\pi,\pi)}^{z}\mid 1_{+}\rangle\otimes\mid 2_{+}\rangle=0.$ Such
soliton state is also a bosonic particle with a negative charge but without
spin degrees of freedom. We call such a soliton state an ”electon” to mark
difference with the word ”electron”.

| $\mid 1_{+}\rangle\otimes\mid 2_{+}\rangle$ | $\mid 1_{-}\rangle\otimes\mid 2_{+}\rangle$ | $\mid 1_{+}\rangle\otimes\mid 2_{-}\rangle$ | $\mid 1_{-}\rangle\otimes\mid 2_{-}\rangle$
---|---|---|---|---
$N_{F}$ | $1$ | 0 | 0 | -1
$S_{(\pi,\pi)}^{z}$ | $0$ | $1/2$ | $-1/2$ | 0
Table 1: Quantum numbers of the degenerate soliton states.

Finally we get a __ topological mechanism of spin-charge separation in nodal
AF insulators. There exist two types of topological objects - one is the
fermionic spinon, the other is the bosonic holon ( or the bosonic electon ).

In 1D system, real spin-charge separation may occur. As far as the low energy
physics is concerned, the spin and charge dynamics are completely decoupled
from each other. In 2D, real spin-charge separation in a nodal AF insulators
can not occur in long range AF order. In the future we will study the
deconfinement condition of spin-charge separated solitons and explore the
properties of deconfined phases with real spin-charge separation.

_Summary_ \- By using 2D $\pi$-flux phase Hubbard model and the Hubbard model
on a honeycomb lattice as examples, we explore spin-charge separation in nodal
AF insulator. The crus crux of the matter in this paper is the discovery of
induced staggered spin moment $S_{(\pi,\pi)}^{z}$ on half skyrmions in nodal
AF insulators. Based on such nontrivial induced quantum number, we classify
four degenerate soliton states with zero energy - two of them ($\mid
1_{-}\rangle\otimes\mid 2_{+}\rangle$ and $\mid 1_{+}\rangle\otimes\mid
2_{-}\rangle$) represent the up-spin and down-spin states for a fermionic
”spinon”, another state ($|1_{-}\rangle\otimes|2_{-}\rangle$) represents a
”holon” and the last one ($|1_{+}\rangle\otimes|2_{+}\rangle$) denotes an
”electon”.

This research is supported by NFSC Grant no. 10574014.

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