# Spin Texture in Doped Mott Insulators with Spin-Orbit Coupling

Shuai A. Chen  Institute for Advanced Study, Tsinghua University, Beijing
100084, China  Zheng-Yu Weng  Institute for Advanced Study, Tsinghua
University, Beijing 100084, China  Jan Zaanen  Institute-Lorentz for
Theoretical Physics, Leiden University, P.O. Box 950, The Netherlands

###### Abstract

A hole injected into a Mott insulator will gain an internal structure as
recently identified by exact numerics, which is characterized by a nontrivial
quantum number whose nature is of central importance in understanding the Mott
physics. In this work, we show that a spin texture associated with such an
internal degree of freedom can explicitly manifest after the spin degeneracy
is lifted by a _weak_ Rashba spin-orbit coupling (SOC). It is described by an
emergent angular momentum $J_{z}=\pm 3/2$ as shown by both exact
diagonalization and variational Monte Carlo calculations, which are in good
agreement with each other at a finite size. In particular, as the internal
structure such a spin texture is generally present in the hole composite even
at high excited energies, such that a corresponding texture in momentum space,
extending deep inside the Brillouin zone, can be directly probed by the spin-
polarized angle-resolved photoemission spectroscopy (ARPES). This is in
contrast to a Landau quasiparticle under the SOC, in which the spin texture
induced by SOC will not be protected once the excited energy is larger than
the weak SOC coupling strength, away from the Fermi energy. We point out that
the spin texture due to the SOC should be monotonically enhanced with reducing
spin-spin correlation length in the superconducting/pseudogap phase at finite
doping. A brief discussion of a recent experiment of the spin-polarized ARPES
will be made.

###### Contents

  1. I Introduction
    1. I.1 Background
    2. I.2 Main results
  2. II Models
    1. II.1 The $t$-$J$ and $\sigma\cdot t$-$J$ models
    2. II.2 Spin-orbit coupling
  3. III The Single Hole Study
    1. III.1 Exact diagonalization and variational ansatz in the absence of SOC ($\lambda=0$)
    2. III.2 The single hole state under a weak Rashba spin-orbit coupling ($\lambda\neq 0$)
      1. III.2.1 Exact diagonalization
      2. III.2.2 Variational Monte Carlo calculation
  4. IV Discussion: Finite Doping
  5. V Concluding Remarks
  6. A Sign structures in the $t$-$J$ and $\sigma\cdot$$t$-$J$ models
  7. B One-hole wave function ansatz and emergent symmetry in the $t$-$J$ model
  8. C Rashba interaction
    1. C.1 Total angular momentum $J_{z}$
    2. C.2 ED under periodic boundary condition
  9. D Variational Monte Carlo procedure
    1. D.1 Variational ground state at half-filling
    2. D.2  Variational wave functions for the $t$-$J$-R model
  10. E Spin-polarized spectral function
  11. F Fragility of spin texture in a Fermi Liquid

##  I Introduction

###  I.1 Background

A Landau’s quasiparticle with a definite momentum, charge, and spin-1/2 has
played a fundamental role in a conventional Fermi liquid. Whether a doped
charge (hole) can still propagate like a Bloch wave, is important in
understanding the two-dimensional (2D) Mott insulator at finite doping. The
latter is widely believed to be intrinsically related to the high-$T_{c}$
superconductor in the cuprate [1, 2].

For a single hole injected into a quantum spin antiferromagnetic (AF)
background, although earlier studies [2, 3, 4, 5, 6, 7] have indicated a
Landau-like quasiparticle behavior based on effective approaches, very
recently it has been revealed by the unbiased exact diagonalization (ED) and
density matrix renormalization group (DMRG) numerical calculations [8, 9, 10],
that the doped hole will generate a composite structure by twisting the spin
background via a nontrivial backflow in the $t$-$J$ model. Manifestly a net
spin current will further show up due to the presence of a spin-1/2 partner
around the hole, which gains a nontrivial angular momentum, $L_{z}=\pm 1$, or
a new double degeneracy under an open boundary lattice (OBC) with a $C_{4}$
rotational symmetry [8].

Figure 1: Composite structure of a doped hole in the pure $t$-$J$ model of a
variational ground state description [11], which is schematically illustrated
by (a): the hole (open circle) is associated with a net spin $S_{z}=\pm 1/2$
of a distance $\xi_{sh}$, which forms a circulating spin current around the
hole. Such a hole composite acquires an angular momentum $L_{z}=\pm 1$ with
novel ground state degeneracy in agreement with ED and DMRG [8]; (b) Here
$\xi_{sh}$ is determined by the hole-spin correlator $\langle
n_{i_{0}}^{h}\bar{S}_{i}^{z}\rangle$ in either the $t$-$J$ (red solid) or
$\sigma\cdot t$-$J$ (blue dashed) model, which shows that $\xi_{sh}\simeq$
spin correlation length $\xi$ (the inset). Note that $\xi$ of the half-filling
spin background is tunable, with two limits of long-range ($\xi\gg L$) and
short-range ($\xi\sim 2.7\ll L$) shown here (cf. Appendix D); (c) The spin
current $J^{s}$, defined in Eq. (9), around the hole as measured by the
correlator $\langle n^{h}_{i_{0}}J^{s}_{i,i+\hat{y}}\rangle$ (along the
$\hat{x}$-axis of the lattice) for the $t$-$J$ (red solid) model, which
disappears in the $\sigma\cdot t$-$J$ (blue dashed) model. The VMC calculation
is carried out in a finite-size square lattice of $N=L^{2}=20\times 20$.
Figure 2: A spin texture pattern induced by a weak SOC, which is frozen into
the x-y plane around a doped hole as characterized by a correlator $\langle
n^{h}_{i_{0}}{\mathbf{S}}_{i}^{x,y}\rangle$ with $i_{0}$ located at the center
marked by the open circle. (a) An explicit spin twist in the $t$-$J$ model;
(b) The absence of the spin twist in the $\sigma\cdot t$-$J$ model. Here the
half-filling spin background in the variational ground state study is chosen
the same as in Fig. 1 with $\xi\sim 2.7$ and the SOC strength $\lambda=0.01$
in units of $J=1$ (see text) .

Such a non-Landau-like novel structure of the composite hole revealed by ED
and DMRG calculations can be well reproduced by the variational Monte Carlo
(VMC) study [12, 11] of a variational ansatz state, which is illustrated in
Fig. 1. Generally a net spin-1/2 will be introduced to the ground state by a
doped hole with a characteristic length scale $\xi_{sh}$. Taking the total
$S_{z}=1/2$ without loss of generality, a net spin current around the hole is
generally found as indicated in Fig. 1(a). Namely there is an internal
relative motion between the doped hole and the surrounding spins, which is
characterized by an emergent quantum number, i.e., the angular momentum
$L_{z}$ mentioned above.

The detailed composite structure is further shown in Figs. 1(b) and (c) based
on the correlators, $\langle n^{h}_{i_{0}}\bar{S}^{z}_{i}\rangle$ and $\langle
n^{h}_{i_{0}}J^{s}_{i,i+\hat{y}}\rangle$, which measure the distribution of
the net $S_{z}=1/2$ around the doped hole [(b)] and the corresponding
circulating spin current [(c)], respectively. Here $n^{h}_{i_{0}}$ denotes the
hole number at site $i_{0}$, $\bar{S}^{z}_{i}$ the summation of two nearest-
neighboring spins along the ${x}$-axis, i.e., $S^{z}_{i}+S^{z}_{i+\hat{x}}$,
with $i$ in the opposite sublattice site of $i_{0}$ such that a staggered
oscillation is smoothened, and $J^{s}_{i,i+\hat{y}}$ the transverse spin
current [cf. Eq. (9)]. The variational ground state [11] [cf. Eq. (7)] is
obtained at a lattice of $N=L^{2}=20\times 20$. The average distance of the
$S_{z}=1/2$ from the hole, $\xi_{sh}$, is shown in the inset of Fig. 1(b),
which is essentially proportional to the spin-spin correlation length $\xi$ of
the spin AF background that can be artificially tuned in the VMC calculation
(two cases of $\xi\gg L$ and $\xi\sim 2.7$ in units of the lattice constant
are shown in the main panel with $L$ denoting the sample length). Then, Fig.
1(c) further shows that the spin current around the hole gets enhanced with
reducing $\xi_{sh}\simeq\xi$. It is noted that here we are always focused on
the one-hole case in a finite-size sample. If $L\rightarrow\infty$, a true AF
long-range order $\xi\rightarrow\infty$ will set in, such that spin
excitations become gapless, which may also contribute to the spin currents
induced by the hole in addition to just an $S^{z}=1/2$ shown in Fig. 1(a) at
the finite lattice size. We do not consider this singular limit in the
following.

Note that there are two chiralities of the spin currents in the one-hole
degenerate ground states corresponding to the novel quantum number $L_{z}=\pm
1$ (only $L_{z}=1$ is shown here). By contrast, the chiral spin current can be
turned off with vanishing angular momentum in the so-called
$\sigma\cdot$$t$-$J$ model, in which the doped hole restores the behavior of a
Landau-like quasiparticle [11]. Here the sole distinction between the $t$-$J$
and $\sigma\cdot t$-$J$ models is characterized by the nontrivial backflow of
spin current in the former, which is absent in the latter with the
disappearance of the novelty of the hole composite, although $\xi_{sh}$
remains similar in both cases [cf. Fig. 1(b)].

It is then natural to raise the following question, i.e., how can one detect
such an internal quantum structure of the composite hole experimentally? To
address this question, in this paper, we introduce a _weak_ Rashba spin-orbit
coupling (SOC), which may be present in a double-layer system of the cuprate
materials like Bi$2212$ [13]. Then we show that the hidden transverse spin
current pattern appearing around the hole in Fig. 1 can manifest explicitly an
emergent semiclassical spin texture lying in the $x$-$y$ plane, as illustrated
in Fig. 2(a). Here a real space distribution of the correlator $\langle
n^{h}_{i_{0}}{\mathbf{S}}_{i}^{x,y}\rangle$ is presented with the hole site
$i_{0}$ fixed at the center of the sample (denoted by the open circle in Fig.
2. Namely, the novel internal composite structure of the hole in the pure
(without SOC) $t$-$J$ model can be effectively visualized via a frozen
transverse spin texture, which is induced by a weak SOC lifting the spin
degeneracy and resulting in the locking of $S_{z}$ with $L_{z}$ by a total
angular momentum $J_{z}=L_{z}+S_{z}$. By contrast, such a transverse spin
texture is absent in the $\sigma\cdot t$-$J$ model in the same weak SOC [cf.
Fig.2(b)]. In the latter, the single hole behaves like a Bloch wave without a
surrounding spin current [cf. Fig. 1]. In the following, we further briefly
summarize the main results obtained in the present work and their physical
implications.

###  I.2 Main results

In this work, we show that the hidden spin current pattern of the spin-1/2
partner in a hole composite can manifest itself as an emergent semiclassical
spin texture with the introduction of a weak Rashba SOC. It is explicitly
exhibited in either the spatial space as indicated by Fig. 2(a) above and the
momentum space (see below) as found by ED as well as the VMC calculation.

As shown in Fig. 2(a), such a spin twist as measured by the correlator
$\langle n^{h}_{i_{0}}{\mathbf{S}}_{i}^{x,y}\rangle$ is formed in the $x$-$y$
plane around the hole. Physically it reflects the underlying antiparallel spin
alignments in the $x$-$y$ plane along either the ${x}$ or ${y}$ direction
across the hole to facilitate the hopping. By a sharp contrast, no such a
relative spin twist is shown in Fig. 2(b) for the $\sigma\cdot t$-$J$ model,
which is consistent with the the absence of a spin current structure in Fig.
1(c). Note that $\xi\sim 2.7$ is chosen in the VMC calculation such that the
distribution of the spin-1/2 partner is pulled closer to the hole to enhance
the effect in Fig. 2.

The energy shifts of the ground state characterized by $J_{z}=\pm 3/2$ and the
first excited one at $J_{z}=\pm 1/2$, together with the ground state energy
shift in the $\sigma\cdot t$-$J$ model, are calculated by both ED and VMC
methods as a function of the SOC strength $\lambda$, which are in excellent
agreement as presented in Fig. 3. Here it is interesting to note that the
effect of the SOC in the $\sigma\cdot t$-$J$ model is much weaker than that in
the ground state energy of the $t$-$J$ model, indicating that the spin current
structure substantially enhances the effect of the Rashba SOC.

Furthermore, under the rotational symmetry, a finite angular momentum $J_{z}$
in the $t$-$J$ model also implies a spin twist structure in momentum space,
consistent with the Rashba SOC $\sim{\bf S}_{\bf k}\cdot({\bf
k}\times\hat{{\bf z}})$ where ${\bf S}_{\bf k}$ denotes a single particle’s
spin at momentum ${\bf k}$. Indeed, in Fig. 4, the spin textures of
$\langle{\bf S}_{\bf k}\rangle$ lying in the $x$-$y$ plane, characterized by a
spin helicity in the momentum space, are shown for the ground state of
$J_{z}=\pm 3/2$ and the first excited state of $J_{z}=\pm 1/2$, respectively,
as calculated by ED. Again the spin helicity is much weaker for the
$\sigma\cdot t$-$J$ model [cf. Fig. 4(a)]. In Fig. 5, the ED and VMC results
are further compared in consistency. Both ED and VMC calculations also
indicate that the real-space spin texture in Fig. 2(a) can persist into high
excited states. At a higher excitation energy, the corresponding spin texture
in momentum space will be deeper inside the Brillouin zone, as shown in Fig. 6
obtained by VMC at a larger sample size. Such a hole-like quasiparticle with a
spin texture in momentum space may be thus directly probed by the spin-
polarized angle-resolved photoemission spectroscopy (ARPES) measurement [13].

The persistence of the spin texture in the Brillouin zone, even close to the
$\Gamma$ point in Fig. 6, is a fact that is in sharp contrast to a Landau
Fermi liquid. In the latter, the spin of a quasiparticle may also lock with
its momentum to form a spin helicity in the momentum space by SOC. But it is a
single particle picture without the _spatial_ spin texture [cf. Fig. 2(a)], in
which the SOC-induced spin helicity is expected to be only robust near the
Fermi surface within a very narrow energy interval comparable with the
strength of the weak SOC. Away from the Fermi energy, the spin helicity of a
Landau quasiparticle can be easily washed away by scattering (cf. Fig. 8)
without the protection from the many-body effect in the $t$-$J$ case as
represented by a real-space spin texture associated with $J_{z}\neq 0$.

Finally, we argue that even though this work has mostly focused on a single
hole in a finite size system, where ED and VMC calculations are available to
compare with each other, one may reasonably generalize the main conclusion to
the experimentally relevant regime at finite doping. Firstly, it has been
shown by the VMC approach that the spin texture amplitude gets enhanced with
reducing $\xi$ in the spin background (cf. the inset of Fig. 7), which is due
to the fact that the composite of a hole and its spin-1/2 partner gets smaller
with an enlarged spin current. In order to make quantitative comparison with
the experiment [13], aside from the absolute amplitude of the spin-polarized
spectral function in the inset, a relative polarization ratio of the spectral
function with spin perpendicular to momentum is also shown in the main panel
of Fig. 7 at a given momentum, which indeed indicates a comparable strength
with the experimental observation [13]. Secondly, we point out that the
pairing of the doped holes with the spin texture structure obeying the time-
reversal symmetry is presumably between the composite holes with the opposite
pseudo-spin $J_{z}=\pm 3/2$ in the presence of SOC. Those other quasiparticle
excitations with distinct spin textures, which have higher energies due to the
SOC splitting, are thus effectively excluded out of the pair condensate. In
such a superconducting/pseudogap regime, predominantly the chiral spin texture
associated with the unpaired quasiparticles of $J_{z}=\pm 3/2$ (which exhibit
the _same_ spin texture in momentum space) will be observed by the spin-
polarized ARPES measurement over a finite range of energy, which covers the
extended momentum space away from the Fermi surface region. Such a many-body
picture clearly illustrates why a quasiparticle excitation with spin texture
is “protected”, which can survive even when the thermal energy in
superconducting/pseudogap regime is dominant over the small energy scale of
SOC. By contrast, the emergent Bogoliubove/Landau quasiparticle with Fermi arc
[14] should only contribute to a negligible spin texture under a weak SOC.

The rest of paper will be organized as follows. In Sec. II, the $t$-$J$ and
$\sigma\cdot t$-$J$ models with a Rashba SOC are introduced. In Sec. III, we
focus on a single hole’s motion in a finite-size lattice and discuss the spin
texture structure induced by the SOC based on both ED and VMC approaches. In
Sec. IV, a qualitative discussion at a finite doping is presented. Finally,
Sec. V is devoted to the concluding remarks. Various technical details and
analyses are presented in Appendices.

##  II Models

###  II.1 The $t$-$J$ and $\sigma\cdot t$-$J$ models

Figure 3: The energy shift $\Delta E_{0}^{\mathrm{1h}}$ due to the SOC for the
degenerate ground states ($J_{z}=\pm\frac{3}{2}$) and first excited states
($J_{z}=\pm\frac{1}{2}$) obtained by (a) ED and (b) VMC calculations for the
$t$-$J$ (red dot and red square) and $\sigma\cdot t$-$J$ (blue triangle)
models. Insets: the first derivation of $\Delta E_{0}^{\mathrm{1h}}$ over
$\lambda$. Here $N=4\times 4$ under OBC [cf. Fig. 10(a) for ED under PBC].

In this work we shall examine some novel spin texture structure hidden in the
quasi-hole excitation of the doped Mott insulator described by the $t$-$J$
model [1, 15], whose Hamiltonian is composed of the hopping $H_{t}$ and the
superexchange $H_{J}$ terms as follows,

$\displaystyle H_{t}$ $\displaystyle=-t\sum_{\langle
ij\rangle,\sigma}c_{i\sigma}^{\dagger}c_{j\sigma}+h.c.~{},$ (1) $\displaystyle
H_{J}$ $\displaystyle=J\sum_{\langle
ij\rangle}(\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\frac{1}{4}n_{i}n_{j})~{},$

where $\langle ij\rangle$ denotes two nearest neighbor sites $i$ and $j$, and
$c_{i\sigma}$ creates an electronic hole with the spin index $\sigma=\uparrow$
or $\downarrow$ at site $i$. $\mathbf{S}_{i}$ and $n_{i}$ are the spin and
electron number operators, respectively, and the Hilbert space is always
subject to the no double occupancy constraint

$n_{i}\equiv\sum_{\sigma}c_{i\sigma}^{\dagger}c_{i\sigma}\leq 1~{}.$ (2)

At half-filling $n_{i}=1$, the system becomes the Mott insulator with the spin
degrees of freedom described by the Heisenberg Hamiltonian $H_{J}$. We fix the
model parameters at $t/J=3$ and $J=1$ in the following.

The motion of the holes injected into such a system will strongly interact
with the quantum spin background to acquire a Berry-phase-like emergent sign
structure, i.e., the so-called _phase string_ [16, 17, 18]. This singular
effect is manifested by generating neutral spin current [8, 11] as the
backflow associated with the hopping of the doped holes. An important fact is
that the spin current is conserved, or the phase string created by the hopping
is “irreparable”, which replaces the original fermion signs to become the new
statistical signs in the $t$-$J$ model [18]. One may find a more detailed
discussion on the phase string sign structure in Appendix A.

On the other hand, the phase string sign structure in the doped $t$-$J$ model
can be precisely eliminated by modifying $H_{t}$ into the following form [19,
20]:

$H_{\sigma\cdot t}=-t\sum_{\left\langle ij\right\rangle}\sigma
c_{i\sigma}^{{\dagger}}c_{j\sigma}+h.c.~{},$ (3)

where a spin-dependent sign is inserted to the hopping process, while the
superexchange term $H_{J}$ remains unchanged. It can be shown that the
resulting $\sigma\cdot$$t$-$J$ model with $H_{\sigma\cdot t\text{-}J}\equiv
H_{\sigma\cdot t}+H_{J}$ will no longer generate any spin currents as the
phase string is precisely “switched off” [cf. Appendix A]. Consequently an
injected hole will behave simply like a Landau quasiparticle with no more
internal structure associated with the backflow spin current. Thus, a
comparative study of the $t$-$J$ and $\sigma\cdot$$t$-$J$ models can provide a
unique understanding of the doped Mott physics.

###  II.2 Spin-orbit coupling

In this work, we shall introduce a Rashba SOC term as a perturbation, which is
given as follows [21]

$H_{\text{R}}=\lambda\sum_{i}i(c_{i\uparrow}^{{\dagger}}c_{i+\hat{y}\downarrow}+c_{i\downarrow}^{{\dagger}}c_{i+\hat{y}\uparrow})-(c_{i\uparrow}^{{\dagger}}c_{i+\hat{x}\downarrow}-c_{i\downarrow}^{{\dagger}}c_{i+\hat{x}\uparrow})+h.c.~{}.$
(4)

Here the parameter $\lambda$ characterizes the Rashba SOC strength (valued in
units of $J=1$) and $\hat{x}\left(\hat{y}\right)$ denotes the unit vector
along the $x\left(y\right)$ direction. It is noted that a weak Rashba SOC in
the cuprate may actually arise from the lack of the inversion symmetry in a
double-layer system [13]. In the present approach, we shall neglect the
interlayer $t$-$J$ and SOC coupling between the double layers. Namely one is
still focused on the one layer problem with only retaining an effective weak
intra-layer SOC given in Eq. (4), while the another layer experiences an
opposite SOC [13]. As a perturbation to the $t$-$J$ or $\sigma\cdot$$t$-$J$
model, $H_{\text{R}}$ can further provide a locking between the spin and
orbital angular momenta in the new Hamiltonian
$H_{t\text{-}J\text{-R}}=H_{t}+H_{J}+H_{\text{R}}$ or $H_{\sigma\cdot
t\text{-}J\text{-R}}=H_{\sigma\cdot t}+H_{J}+H_{\text{R}}$.

Then the single hole problem previously explored by both ED and VMC methods
can be easily generalized to the present case, which will reveal interesting
properties of the doped hole as a new twisted quasiparticle in a rather unique
way.

Figure 4:  The spin texture in momentum space. (a) The helicity $h$ defined in
Eq. (10) characterizes the overall strength of the spin texture; (b) and (c)
The corresponding momentum-space spin textures in the ground states
($J_{z}=\pm 3/2$) and the first excited states ($J_{z}=\pm 1/2$) with the same
chirality $\chi=+1$ at a positive $\lambda=0.006$, calculated by ED
($N=4\times 4$ under OBC).  Figure 5: The spin texture in momentum space. (a)
and (b): ED results of the spin polarization
$\langle\mathbf{S}_{\mathbf{k}}\rangle$ lying in the $x$-$y$ plane in the
ground state and the spin-polarized spectral function
$\mathbf{I}_{\mathrm{max}}(\mathbf{k})$, respectively; (c) and (d): The
corresponding VMC results of the variational state ansatz. Here
$\lambda=0.006$ in the $t\text{-}J\text{-R}$ model on a $4\times 4$ lattice
under OBC

##  III The Single Hole Study

As stated in the Introduction, the single-hole ground state of the $t$-$J$
model has exhibited some highly nontrivial phenomenon as shown by ED and DMRG
[8]. Namely, the motion of the hole in the $t$-$J$ model is always accompanied
by a backflow spin current due to the phase string effect, which then leads to
a continuous distribution of momentum for the hole as if there is an
incoherent many-body component in addition to a quasiparticle component in a
finite size sample. In the following, we shall examine the response of such a
two-component single-hole state to a weak Rashba SOC perturbation by ED and
VMC, respectively, which will then reveal the nature of the internal structure
of a doped hole as a unconventional many-body composite in an explicit way.

###  III.1 Exact diagonalization and variational ansatz in the absence of SOC
($\lambda=0$)

To begin with, we note that for one hole injected into a half-filled spin
background in 2D, recent ED and DMRG studies in the pure $t$-$J$ model have
clearly demonstrated [8] that the ground state is characterized by an angular
momentum $L_{z}=\pm 1$ and $S^{z}=\pm 1/2$ under OBC. In other words, the
single hole eigenstates are totally four-fold degenerate, which is not only
true for the ground state but also valid for the excited states as indicated
by ED [8]. It shows that the doped hole acquires an internal structure, which
is associated with a chiral spin currents. This is in contrast to a Landau’s
quasiparticle where the many-body effect only leads to a renormalization of
its inertia by changing the effective mass without affecting the translational
symmetry and $U(1)$ charge nor inducing an internal degree of freedom.

Such a novel structure associated with the doped hole can be captured by a VMC
wave function ansatz [22, 23, 12, 11]. To understand the unique
characterization of this wave function ansatz, one may first recall a
conventional Bloch-wave state given by

$|\Psi_{0}\rangle_{\mathrm{1h}}=\sum_{i}\varphi_{\mathrm{h}}^{0}\left(i\right){c}_{i\alpha}|\phi_{0}\rangle~{},$
(5)

where the single-hole wave function $\varphi_{\mathrm{h}}^{0}$ is a linear
combination of the plane wave $\propto e^{\pm i{\bf k}\cdot{\bf r}_{i}}$ for a
Wannier state ${c}_{i\alpha}|\phi_{0}\rangle$. The latter with a single hole
at site $i$ is created by annihilating an electron from the translationally
invariant half-filling ground state $|\phi_{0}\rangle$, which may be best
approximated by a _bosonic_ resonant-valence bond (RVB) state [24] with the
spin-spin correlation length $\xi\rightarrow\infty$. If the true single-hole
ground state with a total momentum ${\bf k}$ can be adiabatically connected to
Eq. (5) with a finite spectral weight $Z_{{\bf k}}$, we may call the hole
state a Landau-like quasiparticle.

However, instead of Eq. (5), a novel single-hole state wave function has been
found [11] in good agreement with the above ED and DMRG results [8].
Mathematically it is realized by replacing the bare hole creation operator $c$
in Eq. (5) with $\tilde{c}$:

$c_{i\sigma}\rightarrow\tilde{c}_{i\sigma}\equiv
c_{i\sigma}e^{-i\hat{\Omega}_{i}}~{}~{},$ (6)

which leads to a new single-hole ansatz state [12, 11]

$|\Psi\rangle_{\mathrm{1h}}=\sum_{i}\varphi_{\mathrm{h}}\left(i\right){c}_{i\alpha}e^{-i\hat{\Omega}_{i}}|\phi_{0}\rangle~{}.$
(7)

Here the nontrivial phase shift operator takes the form

$\hat{\Omega}_{i}=\sum_{l}\operatorname{Im}\ln(z_{i}-z_{l})n_{l\downarrow}~{},$
(8)

where $n_{l\downarrow}$ denotes the $\downarrow$-spin operator acting on
$|\phi_{0}\rangle$ with $z_{l}$ as the complex coordinate of site $l$. It
encodes the mutual braiding between the hole and the $\downarrow$-spins in the
background spins, as illustrated in Figs. 1, to give rise to a nontrivial
angular momentum $L_{z}=\pm 1$ and a _net_ spin current [11]

$J_{ij}^{s}=i(S_{i}^{+}S_{j}^{-}-S_{i}^{-}S_{j}^{+})~{}.$ (9)

The numerical ED and DMRG (up to $8\times 8$) results [8] for the $t$-$J$
model can be well reproduced by such a variational ground state ansatz in Eq.
(7), with the hole wave function $\varphi_{\mathrm{h}}\left(i\right)$ being
the variational parameter in the VMC calculation [11]. In Eq. (6), a minus
sign is chosen in $\hat{\Omega}_{i}$, which defines one possible chirality of
the ground states. One may also choose a positive sign, i.e.
$e^{+i\hat{\Omega}_{i}}$, to accordingly give rise to a degenerate ground
state with the opposite chirality. A detailed symmetry analysis of the wave
function in Eq. (7) is given in Appendix B.

By contrast, the Landau-like wave function in Eq. (5) is shown [11] to be a
good ground state for the $\sigma\cdot$$t$-$J$ model, in which the phase-
string sign structure is switched off without creating a spin-current backflow
during the hopping process [cf. Fig. 1(c)] even though the spin $S_{z}=\pm
1/2$ has a similar distribution around the hole as in the $t$-$J$ case [cf.
Fig. 1(b)]. Obviously the two wave functions in Eq. (5) and Eq. (7) cannot be
adiabatically connected to each other, indicating the qualitative distinction
and non-perturbative nature of the single-hole state in comparison with the
conventional Landau paradigm. Crucially, the nonlocal phase structure in Eq.
(6) persists through the whole spectrum of the single-hole eigenstates
irrespective of the excited energy. One may also continuously tune the spin-
spin correlation length $\xi$ in $|\phi_{0}\rangle$ variationally (see
Appendix D.1) to simulate the case of an arbitrary finite $\xi$, which should
be self-consistently realized in the ground state at finite doping of holes
[23, 22]. The detailed structures of the single-hole state as variationally
obtained at $\xi\gg L$ and $\xi\sim 2.7$ are illustrated in Figs. 1(b) and
1(c).

After the above brief outline of the single hole state in the pure case, in
the following, one can straightforwardly generalize the above calculations to
$\lambda\neq 0$ and study $H_{t\text{-}J\text{-R}}$ by the VMC approach in
comparison with the ED results. We will see that the exotic non-Landau-like
features of the doped hole will further show up.

###  III.2 The single hole state under a weak Rashba spin-orbit coupling
($\lambda\neq 0$)

####  III.2.1 Exact diagonalization

Let us first start with the ED study under OBC, which is to be compared with a
VMC study based on an analytic ground state ansatz below. In contrast to the
ED under the periodic boundary condition (PBC) given in Appendix C.2, the
ground state degeneracy of the $t$-$J$ model (without SOC) under the OBC (with
$C_{4}$ symmetry) has been shown to be reduced to four-fold characterized by
the orbit angular momentum $L_{z}=\pm 1$ and spin $S_{z}=\pm 1/2$ [11]. Once
$\lambda$ is switched on in $H_{t\text{-}J\text{-R}}$, $S_{z}$ and the angular
momentum $L_{z}$ are no longer conserved separately. Instead, the total
angular momentum $J_{z}=L_{z}+S_{z}$ will remain as a good quantum number (cf.
Appendix C.1), with the ground states characterized by $J_{z}=L_{z}+S_{z}=\pm
3/2$ with two-fold remaining degeneracy by ED. To show the overall strength of
the SOC, the change of the ground state energy as a function of $\lambda$ is
shown in Fig. 3, in which (a) is for ED and (b) is for VMC (see below). The
energy gain induced by SOC as well as the energy splitting between $J_{z}=\pm
3/2$ and $J_{z}=\pm 1/2$ is clearly shown here. By contrast, the SOC effect
for a quasiparticle in the $\sigma\cdot$$t$-$J$ model (blue triangles) is much
reduced. A similar result of the ED calculation under PBC can be found in
Appendix C.2.

To quantitatively characterize the spin texture induced by $\lambda\neq 0$,
one may define a “helicity” as the projection of the averaged spin
$\langle\mathbf{S}_{\mathbf{k}}\rangle$ at the direction perpendicular to the
momentum $\mathbf{k}$ lying in the $x$-$y$ plane:

$h\equiv\sum_{\mathbf{k}}\frac{\langle\mathbf{S}_{\mathbf{k}}\rangle\cdot\mathbf{(}\mathbf{k}\times\hat{\mathbf{z}})}{|\mathbf{k}|}~{},$
(10)

where $\mathbf{S}_{\mathbf{k}}=\frac{1}{2}c_{\mathbf{k}}^{\dagger}\sigma
c_{\mathbf{k}}$ with
$c_{\mathbf{k}}=[c_{\mathbf{k}\uparrow},c_{\mathbf{k}\downarrow}]^{T}$. It
measures the strength of locking between the spin and orbit associated with
the single hole in the momentum space as shown in Fig. 4(a), with the sign
$\chi=\mathrm{sign}(h)$ depicting the chirality of the corresponding spin
texture in Figs. 4(b) and (c). Here the helicity $h$ for $J_{z}=\pm 3/2$ (the
ground states) shows a significantly larger value as compared to that of the
first excited state at $J_{z}=\pm 1/2$ in Figs. 4(b) and (c), which both
possess the same chirality $\chi$ at $\lambda>0$. The sign of $\chi$ does not
depend on the sign of $J_{z}$ but only on the sign of $\lambda$ as shown in
Fig. 4(a). It is easy to understand that the four spin textures in Figs.
4(b)-(c) and their counterparts at $\lambda<0$ just correspond to the eight
degenerate eigenstates of the $t$-$J$ model at $\lambda\rightarrow 0$ in the
thermodynamic limit or under PBC [8, 11], which are split into ground state
and the first excitations in a finite lattice under OBC, and further split at
$\lambda\neq 0$ with distinct chiralities. On the other hand, the helicity $h$
for the $\sigma\cdot$$t$-$J$ model is indeed much reduced as shown in Fig.
4(a).

Figure 6: Spin texture in momentum space as manifested in the spin-polarized
spectral function ${\bf I}_{\mathrm{max}}({\bf k})$, which involves the
excited states as calculated by VMC in a lattice of $8\times 8$ and
$\lambda=0.01$.

####  III.2.2 Variational Monte Carlo calculation

The above ED calculation shows that by introducing a weak Rashba SOC to lift
the separate $S_{z}$ and $L_{z}$ degeneracy, the eigenstate degeneracy
associated with the hole will be reduced to two with the locking of $S_{z}$
and $L_{z}$ in the $t$-$J$ model. Specifically, the total angular momentum
$J_{z}\equiv L_{z}+S_{z}$, which remains to be a good quantum number, will
characterize the composite hole $\tilde{c}$ as a _pseduo-spin_ in replacing
$S_{z}=\pm 1/2$. Namely, the Rashba SOC will select a new quantum number
$J_{z}$, which leads to a chiral spin texture. A detailed symmetry analysis is
given in Appendix C.1.

In the following, we shall employ a variational wave function to understand
the underlying physics involving the many-body response to a weak SOC in the
one-hole-doped Mott insulator. By generalizing the previous four-fold
degenerate wave functions for the pure $t$-$J$ model, one may construct the
following time-reversal invariant new ground states in terms of the linear
combinations of Eq. (7):

$|\Psi_{\mathrm{R}}\rangle_{\mathrm{1h}}=\\!\\!\\!\sum_{i}[\varphi_{h}^{\downarrow}(i)e^{-i\hat{\Omega}_{i}}c_{i\downarrow}+\varphi_{h}^{\uparrow}(i)e^{-i\hat{\Omega}_{i}}c_{i\uparrow}]|\phi_{0}\rangle~{},$
(11)

where $\varphi_{h}^{\uparrow}$ and $\varphi_{h}^{\downarrow}$ are the
variational parameters. One may also construct other types of ansatz by
choosing a different factor $e^{\pm\Omega_{i}}$ in Eq. (11), which
variationally produces excited states. The VMC results are presented as
follows [see Appendix D.2 for the detail].

A spin texture around the hole has been clearly found to lie in the $x$-$y$
plane in Fig. 2(a) with $J_{z}=\pm 3/2$ for the $t$-$J$ model (with
$\lambda=0.01$). Here the spin texture emerges out of the hidden spin current
structure of the non-Landau “twisted” hole in Fig. 1(a) via SOC. Its scale can
be fitted by $\xi_{sh}\simeq\xi$ [cf. the inset of Fig. 1(b)] with $\xi_{sh}$
being the spin-hole correlation length. In Fig. 2, a finite spin-spin
correlation length $\xi\sim 2.7$ in $|\phi_{0}\rangle$ has been used in the
VMC. By comparison, one can examine the spin texture in a Landau-like
quasiparticle in the $\sigma\cdot$$t$-$J$ model, i.e., by turning off the
phase string effect in the $t$-$J$ model. Without creating a spin current
backflow, a doped hole will propagate like a conventional Bloch wave in the
$\sigma\cdot$$t$-$J$ model. At the same strength of $\lambda=0.01$, there is
no more internal spin texture structure around the hole [see the right panel
of Fig. 2(b)]. Here the variational wave function of the $\sigma\cdot$$t$-$J$
model is given by

$|\Psi\rangle_{\sigma\cdot
t\text{-}J\text{-}\mathrm{R}}=\sum_{i}[\varphi_{h}^{\downarrow}(i)c_{i\downarrow}+\varphi_{h}^{\uparrow}(i)c_{i\uparrow}]|\phi_{0}\rangle~{},$
(12)

which is different from those of the $t$-$J$ model by simply switching off the
phase shift operator $\hat{\Omega}_{i}$ in Eq. (11).

The overall agreement of the energies between the ED and VMC calculations are
further illustrated in Fig. 3, including the energy split between $J_{z}=\pm
3/2$ and $J_{z}=\pm 1/2$ in the $t$-$J$ model and the ground state energy
shift in the $\sigma\cdot$$t$-$J$ model. One may see the contributions of the
SOC to the variational energy shifts in Fig. 3(b) as a function of $\lambda$,
which are in excellent agreement with the ED results [Fig. 3(a)]. For the
$\sigma\cdot$$t$-$J$ model, in which a Landau’s quasiparticle picture can be
recovered at $\lambda=0$ [11], one finds a drastically weakened SOC effect at
the same $\lambda$ as compared to the $t$-$J$ model. As pointed out above, the
latter only differs from the former by the phase string sign structure as
represented by the backflow spin current accompanying the hopping. Similar
contrast has been also clearly shown by ED under PBC as presented in Appendix
C.2. Therefore, the phase string effect indeed significantly amplifies the
weak Rashba SOC effect in the $t$-$J$ model.

Furthermore, given the spatial spin texture around the hole in Fig. 2(a) , a
corresponding spin texture described by the average spin
$\langle\mathbf{S}_{\mathbf{k}}\rangle$ in momentum space is expected by the
rotational symmetry, which has been already illustrated in, say, Fig. 4(b) at
$J_{z}=\pm 3/2$. In Fig. 5, the spin-momentum-locking spin textures obtained
by ED and VMC are presented together for comparison, in which
$\langle\mathbf{S}_{\mathbf{k}}\rangle$ in the ground state are shown in (a)
and (c) for a $4\times 4$ lattice.

In Figs. 5(b) and (d), we further present the ED and VMC results for the spin-
polarized spectral function ${\bf I}_{\mathrm{max}}({\bf k})$, which is
defined in Eq. (48) of Appendix E at each given momentum $\mathbf{k}$ with the
maximal magnitude of the spectral function via properly adjusting the excited
energy. In other words, ${\bf I}_{\mathrm{max}}({\bf k})$ can effectively
detect the maximal spin texture in momentum space involving the excited
states, which is experimentally relevant. In Fig. 6, the spin texture in
momentum space is calculated by VMC in a larger ($8\times 8$) lattice such
that more momentum points are available, where the spin vortex structure is
clearly illustrated in the full Brillouin zone, or in other words, persists
over the high energy towards the $\Gamma$ point.

Finally, we emphasize that it is the intrinsic phase string that causes the
hole dressed by a surrounding spin current of the $S=1/2$ partner and the weak
Rashba SOC only lifts the degeneracy to induce a spin texture by breaking the
spin rotational symmetry. If one starts with the large-U Hubbard model, the
high order corrections to the $t$-$J$ model may be at the same order as
$\lambda$, but they do not break the spin rotational symmetry like the SOC
does and it has been shown [25] that the phase string effect still present.
The present conclusion should thus be expected to remain unchanged.

##  IV Discussion: Finite Doping

For the single hole case studied in the previous section, the spin-1/2 partner
forms a spin current vortex surrounding the hole as illustrated in Fig. 1,
which is then “frozen” into a spin texture lying in the $x$-$y$ plane via a
weak SOC as shown in Fig. 2(a). The excellent agreement between the ED and VMC
methods clearly indicates that the single-hole ansatz wave function well
captures the _intrinsic_ structure of the doped hole in a finite size lattice
that cannot be adiabatically connected to a Landau-like quasiparticle, not
only in the absence of SOC but also in the presence of a weak Rashba SOC.

Figure 7: The relative strength $\mathrm{pol}(\omega_{c},\mathbf{k})$ of the
spectral function with spin polarized perpendicular to the momentum [defined
in Eq. (14)] at a given $\mathbf{k}=(\pi/2,\pi/2)$. Here the spin correlation
length $\xi$ is taken in a finite range at different $\lambda$’s (see the
text). Inset: The full magnitude of the spectral function $|{\bf
I}_{\mathrm{max}}({\bf k})|$ as a function of $\xi$. The sample size is
$16\times 16$.

Recently, the VMC approach has been further generalized to the two-hole states
in the absence of SOC [26, 27]. By forming a tightly bound pair, the chiral
spin current gets cancelled and the ground state becomes non-degenerate with
angular momentum $L_{z}=2$ and a d-wave pairing symmetry [cf. Ref. 27], which
is also consistent with the ED and DMRG result [8]. According to the above
discussion, in the absence of the chiral spin current, a weak SOC usually
should not be able to induce a spin texture in a pairing state of holes in
contrast to a single hole case. Therefore, at finite doping, the spin texture
associated with a doped hole is expected to appear only when the unpaired
holes are present in an excited state.

Two questions remain to be answered. One concerns the SOC effect in the
thermodynamic limit, e.g., in the AF long-range ordered state with
$N=L^{2}\rightarrow\infty$ and $\xi\rightarrow\infty$, where ED certainly does
not apply. The VMC calculation indicates that the SOC effect should get
diminished continuously as the spin current associated with the spin-1/2
spreads over in space with the increase of $\xi_{sh}\simeq\xi$. However, in a
true AF long-range ordered state, the gapless spin excitations from the spin
background can be further excited to form an even stronger spin currents
around the hole. In such a limit, the strong spin fluctuations may even cause
the self-localization of the doped hole [11], leading to a very singular
effect in response to an SOC, which is beyond the present VMC approach at a
finite size. On the other hand, the spin texture as the internal structure of
the doped hole discussed in this work is expected to be robust if $\xi$
becomes finite in the spin background with $L\gg\xi$. What we have explored
here is actually about how a hole moving in a spin background of finite $\xi$.
For example, the ED results of a single hole on a $4\times 4$ lattice may be
roughly understood as a finite doping $\sim 1/16$ system. As a matter of fact,
the variational single hole state in Eq. (7) may be created in a spin
background $|\phi_{0}\rangle$ whose spin-spin correlation length $\xi$ can be
artificially tuned (cf. Appendix D.1). For instance, to better illustrate the
spin texture around the hole in Fig. 2(a), a variational $|\phi_{0}\rangle$
with a finite $\xi\sim 2.7$ has been used. Since $\xi_{sh}\simeq\xi\ll L$ in
this case, the spin texture as the integral part of the hole composite is
clearly shown in Fig. 2(a). In other words, if a finite doping mainly changes
the spin background into a short-range AF state (or spin liquid) with the
doped hole still described by $\tilde{c}_{i\sigma}$ in Eq. (6), the above
account of the Rashba SOC effect should be valid in the leading order of
approximation.

So the second question will be whether the present study of the single-hole
behavior under the Rashba SOC can be meaningfully generalized to the
quasiparticle excitation at finite doping, where a short-range AF state with
finite $\xi$ will set in. Namely, a single-particle excitation above the
ground state $|\Psi_{\mathrm{G}}\rangle$ (which may be either a
superconducting or pseudogap state at finite doping, see below) may be
constructed by

$|\psi\rangle_{\mathrm{qp}}=\sum_{i}\varphi_{\mathrm{h}}(i)\tilde{c}_{i\sigma}|\Psi_{\mathrm{G}}\rangle~{},$
(13)

similar to the one-hole state in Eq. (7) with replacing the “vacuum” state
$|\phi_{0}\rangle$ by $|\Psi_{\mathrm{G}}\rangle$. We shall delay the
discussion of such a second question for a moment. Let us first assume that
Eq. (13) is indeed valid, in which $|\Psi_{\mathrm{G}}\rangle$ may be still
treated as a short-ranged AF state $|\phi_{0}\rangle$ with the spin
correlation length $\xi$ tunable. Then one may further make an estimation of
the strength of the SOC effect on the quasiparticle excitation as follows.

The spin-polarized spectral function corresponding to the quasiparticle
excitation in Eq. (13) will be then approximately treated similar to that of
the single hole case under the SOC (cf. Appendix E). The spin texture profile,
determined by VMC in the momentum space based on
${\mathbf{I}}_{\mathrm{max}}(\mathbf{k})$, is similar to Fig. 6 with the
magnitude depending on $\xi$ and $\lambda$. In Fig. 7, a relative strength of
the spin texture at $\mathbf{k}=(\pi/2,\pi/2)$ is shown as a function of $\xi$
at various $\lambda$’s, which is defined by [13]

$\mathrm{pol}(\omega_{c},\mathbf{k})\equiv\frac{\mathcal{A}_{\sigma_{\perp}}(\omega_{c},\mathbf{k})-\mathcal{A}_{\bar{\sigma}_{\perp}}(\omega_{c},\mathbf{k})}{\mathcal{A}_{\sigma_{\perp}}(\omega_{c},\mathbf{k})+\mathcal{A}_{\bar{\sigma}_{\perp}}(\omega_{c},\mathbf{k})}~{},$
(14)

in terms of the spin-polarized ARPES spectral function
$\mathcal{A}_{\sigma}(\omega,\mathbf{k})$ at energy $\omega$ with spin index
$\sigma$ [defined in Eq. (41) in Appendix E)]. Note that here $\sigma_{\perp}$
denotes the spin polarization perpendicular to $\mathbf{k}$
($\bar{\sigma}_{\perp}=-\sigma_{\perp}$) and $\omega_{c}$ the frequency at
which the intensity reaches its maximum. As Fig. 7 shows,
$\mathrm{pol}(\omega_{c},\mathbf{k})$ is roughly flat in a range of finite
$\xi$ with the strength monotonically dependent on $\lambda$, which is
comparable to the experimental measurement in the cuprate [13]. Note that
$\mathrm{pol}$ here only measures the relative spin polarization perpendicular
to $\mathbf{k}$. In the inset of Fig. 7, the total strength of the spectral
function $|{\mathbf{I}}_{\mathrm{max}}|$ (cf. Appendix E) at the same
$\mathbf{k}=(\pi/2,\pi/2)$ is also shown as a function of $\xi$ as
$\xi\rightarrow\infty$, which eventually approaches zero at the AF long-ranged
order limit. Physically it may be understood as the spin current carried by
the spin-$1/2$ partner of the hole composite becomes less and less
concentrated around the hole to result in the reducing SOC effect at large
$\xi$ and $L$.

Now let us come back to address the above second question about the single-
particle excitation at finite doping. We point out that the single-hole ground
state ansatz [Eq. (7)] can be actually regarded as the few hole special cases
[11, 12, 26, 27] of a general ground state ansatz previously proposed to
describe the SC and pseudogap phase at finite doping [23, 22]:

$|\Psi_{\mathrm{G}}\rangle=\hat{{\cal
D}}^{\frac{N^{h}}{2}}|\mathrm{RVB}\rangle~{},$ (15)

in which the AF vacuum state $|\phi_{0}\rangle$ is self-consistently replaced
by $|\mathrm{RVB}\rangle$, which is AF short-range-ordered at a finite doping.
Here the $N^{h}$ holes are paired up and created by

$\hat{{\cal
D}}=\sum_{ij}g_{ij}\tilde{c}_{i\uparrow}\tilde{c}_{j\downarrow}~{},$ (16)

where $\tilde{c}$ defined by Eq. (6) describes the doped holes as twisted
(non-Landau) quasiparticles. Here $|\Psi_{\mathrm{G}}\rangle$ is intrinsically
superconducting at $T=0$ in a finite concentration of holes. The excitations
of free unpaired spinons in $|\mathrm{RVB}\rangle$ can eventually destroy the
SC phase coherence, resulting in the so-called lower pseudogap phase (LPP) or
spontaneous vortex phase, where the pairing in terms of the twisted
quasiparticles, i.e., $\langle\hat{{\cal D}}\rangle\neq 0$ is still present,
but the true Cooper pairing order parameter vanishes [23, 22].

Naturally, an elementary quasiparticle excitation can be constructed, based on
$|\Psi_{\mathrm{G}}\rangle$, as composed of the unpaired twisted quasiparticle
given in Eq. (13). In particular, since the twisted quasiparticles are paired
up via $\hat{{\cal D}}$ in $|\Psi_{\mathrm{G}}\rangle$, the latter may be
reasonably considered to be a “vacuum” as far as $\tilde{c}$ is concerned in
Eq. (13), which thus resembles a $|\phi_{0}\rangle$ just like the single-hole
state in Eq. (7) except for a finite $\xi$. Furthermore, turning on the Rashba
SOC in $H_{t\text{-}J\text{-R}}$ with $\lambda\neq 0$, there will be an
additional important effect occurring in the SC/LPP at finite doping, which
can further enhance the spin texture structure found in the single hole case,
besides the reducing $\xi$ discussed above, as to be outlined below.

Note that in the above SC/LPP ansatz states, the pairing between the twisted
quasi-holes of $\tilde{c}_{i\sigma}$ with $\sigma=\uparrow,\downarrow$ will be
modified by the SOC with the degenerate ground states characterized by a time-
reversal pair of new good quantum numbers: $J_{z}=3/2$ and $J_{z}=-3/2$, i.e.,

$\hat{{\cal
D}}\rightarrow\sum_{ij}{g}(i,j)\tilde{c}_{i,J_{z}=3/2}\tilde{c}_{j,J_{z}=-3/2}~{}.$
(17)

As we have seen, the twisted hole states with both $J_{z}=\pm 3/2$ have the
same chirality of the spin texture. in Eq. (17), the low-lying excitations
will be mainly dominated by those unpaired quasiparticles of $J_{z}=\pm 3/2$
with the same spin texture, where the other excited states with opposite
chirality in the single hole problem should no longer be stable to contribute
to the spin-polarized ${\mathbf{I}}_{\mathrm{max}}$ shown in Fig. 6. Thus,
${\mathbf{I}}_{\mathrm{max}}$ is expected to be further enhanced due to the
fact that the twisted quasiparticles of $J_{z}=\pm 3/2$ are uniquely singled
out via the pairing condensation in $|\Psi_{\mathrm{G}}\rangle$ to exclude all
those excitations of opposite spin textures split by the SOC.

Figure 8: Spin texture configuration due to a weak Rashba SOC as determined by
the spin-polarized spectral function ${\mathbf{I}}_{\mathrm{max}}(\mathbf{k})$
for (a) a free Fermi gas, and (b) a Fermi liquid. The spin textures are
fragile in the Fermi liquid once the scattering between Laudau’s quasiparticle
is taken into consideration (cf. Appendix F for the detail).

##  V Concluding Remarks

As the precursor of a non-Fermi liquid, it has been found recently [11, 27]
that a hole injected into a quantum AF spin background will behave like a non-
Landau quasiparticle. It is a hole composite in which the neighboring spin
background will get “twisted” by the phase string sign structure to facilitate
the hopping of the hole. In the single-hole ansatz state (7), such a many-body
effect is well captured by the nonlocal phase shift factor
$e^{-i\hat{\Omega}_{i}}$. In particular, such a novel structure can become
explicitly manifested if an extra $S_{z}=\pm 1/2$ (introduced by the hole
doped into the spin singlet background) is nearby in a finite system, which
will form a spin current swirling around the hole like in an atom with a
nontrivial angular momentum $L_{z}=\pm 1$.

Then we have shown in this work that by introducing a weak Rashba SOC to lift
the spin-1/2 degeneracy associated with the doped hole, the spin current will
be turned into a semiclassical chiral spin texture lying in the $x$-$y$ plane
in both real and momentum space, which is characterized by a new quantum
number $J_{z}=L_{z}+S_{z}$. It is important to emphasize that a weak SOC does
not change the essential internal structure of the hole composite, but mainly
affects the “soft” spin current contributed by the degenerate $S_{z}=\pm 1/2$,
which are locked with $L_{z}$ due to breaking the spin rotational symmetry by
SOC. In this sense, the present VMC approach is perturbative-like based on the
_good_ (variational) solutions in the pure $t$-$J$ and $\sigma\cdot t$-$J$
models. In particular, the semiclassical spin texture formed by the spin-1/2
around the hole in Fig. 2(a) is merely a “fingerprint” that explicitly reveals
the underlying quantum many-body hole-spin entanglement, which already exists
in the absence of SOC. A recent proposal of the so-called rotational ARPES
spectra [28] may be also used to probe such an internal spin structure
associated with the doped hole.

By using ED and VMC methods, such a spin texture pattern characterized by the
new quantum number $J_{z}$ has been confirmed by both exact numerics and the
variational wave function. The excellent agreement between the ED and VMC
results shows that the hole composite for the $t$-$J$ model has properly
captured the essence of a doped Mott insulator in response to a perturbation
of the SOC. On the other hand, by precisely turning off the phase string in
the $\sigma\cdot t$-$J$ model, both ED and VMC calculations also demonstrate
the disappearance of the novelty of the non-Landau quasiparticle, with a
substantially weakened spin texture entirely induced by the Rashba SOC for a
Landau-like quasiparticle.

Furthermore, we have argued that such a chiral spin texture in the $t$-$J$
model may become directly observable at finite doping in the
superconducting/pseudogap regime, where the majority of the doped holes form a
paired condensate, which effectively selects the hole composites with opposite
quantum numbers $J_{z}=\pm 3/2$ in a weak Rashba interaction with the same
chiral spin texture. In particular, by making the spin AF background short-
ranged, the hole composite becomes “atom-like” as a truly bound entity such
that the spin texture around the hole is both amplified by the Rashba effect
and independent of the lattice size. Consequently, the low-lying
quasiparticles excited from the paired condensate will be predominately
characterized by the same spin texture structure in momentum space, which may
be well observed by the spin-polarized ARPES in momentum space, extending deep
inside the Brillouin zone away from the Fermi energy. Of course, we caution
that the present discussion of the finite doping phase is mostly based on a
generalization of the single hole problem. A further microscopic study at the
same level as dealing with the emergent Bogoliubove/Landau quasiparticle with
Fermi arc [14] will be desirable to describe the spin texture structure under
a weak SOC.

It is important to note that even though a similar spin texture in momentum
space (but not in the real space) commensurate with the Rashba SOC can be
simply found in a noninteracting band structure model as shown in Fig. 8(a)
(cf. Appendix F), the spin texture is actually well protected only within a
very narrow energy regime near the Fermi surface decided by the SOC strength
$\lambda$. As illustrated in Fig. 8(b), the spin texture at momenta away from
the Fermi surface becomes much fragile in a Fermi liquid theory due to a
general scattering effect on the quasiparticles once the excitation energy is
higher than a small scale set by the SOC splitting energy (cf. Appendix F).

We comment on a recent experimental result of the spin-polarized ARPES
experiment on the high-$T_{c}$ cuprate of the Bi-2212 compound [13] in the
pseudogap regime with a $J\sim 120$ meV. The dependence of the in-plane spin
polarization on the quasiparticle momentum observed in the experiment seems to
be compatible with a weak Rashba SOC strength as studied in this work, which
presumably comes from the double interlayer structure [13]. It is an important
observation that the spin polarization extends deep towards the $\Gamma$ point
of the Brillouin zone, which is hard to understand from a conventional Fermi
liquid theory. As discussed above, the spin texture found in the spin
polarization measurement should be restricted by a characteristic energy scale
due to the SOC of a single quasiparticle, which is rather small (e.g.,
$\lambda\sim 0.01J\sim 1.2$ meV) and determines a very narrow regime in the
momentum space near the Fermi energy. By contrast, the spin texture of the new
quasiparticle in a doped Mott insulator is more intrinsically associated with
the internal structure of the doped hole, which is present even for the high
energy excitations. An entirely different mechanism due to quantum many-body
correlations is at work here.

Finally, along with the SOC that breaks the spin rotation, we point out
another important effect involving the Zeeman field, which also breaks the
spin rotational symmetry. Such a Zeeman effect acts on the whole neutral spin
background, in contrast to SOC mainly coupling a hole with its surrounding
spins. Although no spin texture is expected, the Zeeman effect that polarizes
spins can exert strong influence on the charge degree of freedom via an
enhanced spin current due to the enlarged net spin $S_{z}$. Such an important
effect due to phase string is currently under study elsewhere.

###### Acknowledgements.

We acknowledge stimulating discussion with Jingyu Zhao. This work is supported
by MOST of China under Grant No. 2017YFA0302902.

##  Appendix A Sign structures in the $t$-$J$ and $\sigma\cdot$$t$-$J$ models

To be self-contained, in the following we briefly outline the phase-string
sign structure in the $t$-$J$ model and the absence of it in the
$\sigma\cdot$$t$-$J$ model [16, 17, 18].

In the $t$-$J$ model, the Fermi sign structure vanishes at half-filling due to
the no-double-occupancy constraint or the “Mottness”. Upon doping, a reduced
Fermi sign structure starts to recover with the increase of doping
concentration of holes. Specifically, the partition function for the $t$-$J$
model on a square lattice at an arbitrary doping can be expressed by

$Z_{t\text{-}J}=\sum_{c}\tau_{c}\mathcal{Z}[c]~{},$ (18)

where the summation is over $c$, denoting the set of closed loops of the
spatial trajectories of all spins and holes. Accompanying a non-negative
weight $\mathcal{Z}[c]\geq 0$, each path $c$ is associated with a sign factor
$\tau_{c}$ given by

$\tau_{c}\equiv\left(-1\right)^{N_{h}^{\downarrow}\left[c\right]}\times\left(-1\right)^{N_{h}^{h}\left[c\right]}~{}.$
(19)

Here the Berry’s phase-like $\tau_{c}$ replaces the usual Fermion signs in a
partition function of a fermion system due to an explicit restriction of the
Mottness. In $\tau_{c}$, $\left(-1\right)^{N_{h}^{h}\left[c\right]}$ resembles
a conventional Fermi sign structure now only associated with the doped holes,
while $\left(-1\right)^{N_{h}^{\downarrow}[c]}$ depends on the total parity of
exchanges between the holes and $\downarrow$-spins for the closed path set
$c$, which is known as the phase-string sign structure.

By contrast, in the $\sigma\cdot$$t$-$J$ model, the phase-string sign
structure can be simply _switched off_ such that the partition function is
reduced to

$Z_{\sigma\cdot
t\text{-}J}=\sum_{c}\left(-1\right)^{N_{h}^{h}\left[c\right]}\mathcal{Z}\left[c\right]~{},$
(20)

where the non-negative weight $\mathcal{Z}\left[c\right]$ remains the _same_
as in the $t$-$J$ model, with the same Fermi sign structure of the holes:
$\left(-1\right)^{N_{h}^{h}\left[c\right]}$. At half-filling, the $t$-$J$ and
$\sigma\cdot$$t$-$J$ models both reduce to the Heisenberg model.

Therefore, the distinction between the $t$-$J$ and $\sigma\cdot$$t$-$J$ models
only emerges upon doping, which is solely distinguished by the phase-string
sign structure $\left(-1\right)^{N_{h}^{\downarrow}\left[c\right]}$ in Eq.
(19). As it turns out, such a phase string is very singular, which prevents a
perturbative treatment of the $t$-$J$ model, even in the one-hole-doped case.
In the next appendix, we shall discuss a duality transformation and an
emergent symmetry in the one-hole ground state, which are entirely due to the
phase string effect in a non-perturbative way.

##  Appendix B One-hole wave function ansatz and emergent symmetry in the
$t$-$J$ model

The Mott physics in the pure $t$-$J$ model is crucially associated with an
emergent phase string sign structure as outlined above, which will result in a
non-perturbative duality transformation and a new type of ground state ansatz
[12, 11]. In the following, we concentrate on such a one-hole problem from the
perspective of symmetry.

To account for the phase string effect in the one hole case, a many-body phase
shift factor $e^{\pm i\hat{\Omega}_{i}}$ produced by the doped hole has to be
introduced [12, 11] in Eq. (6), with $\hat{\Omega}_{i}$ defined in Eq. (8).
The four-fold degenerate one-hole ground states can be constructed with
$|\Psi_{\mathrm{G}}\rangle_{1,2}$ by removing a $\downarrow$-spin and
$|\Psi_{\mathrm{G}}\rangle_{3,4}$ by removing an $\uparrow$-spin from the
half-filling ground state $|\phi_{0}\rangle$ as follows:

$\displaystyle|\Psi\rangle_{1,2}$
$\displaystyle=\sum_{i}\varphi_{1,2}\left(i\right)e^{\mp
i\hat{\Omega}_{i}}c_{i\downarrow}|\phi_{0}\rangle\text{ },$ (21)
$\displaystyle|\Psi\rangle_{3,4}$
$\displaystyle=\sum_{i}\varphi_{3,4}\left(i\right)e^{\mp
i\hat{\Omega}_{i}}c_{i\uparrow}|\phi_{0}\rangle\text{ }.$ (22)

Here the variational parameters $\varphi_{1,2,3,4}\left(i\right)$ only depend
on the hole’s position. The variational ansatzes in Eqs. (21) and (22) divide
the single-hole Hilbert space into four sectors. The VMC simulation and
analytic analysis have shown [11], consistent with the ED and DMRG results,
that besides $S_{z}=1/2$ for $|\Psi_{\mathrm{G}}\rangle_{1,2}$ and
$S_{z}=-1/2$ for $|\Psi_{\mathrm{G}}\rangle_{3,4}$, each of them acquires an
additional double degeneracies with nontrivial orbital momenta $L_{z}=\pm 1$,
respectively, for a bipartite lattice of $N=2M\times 2M$.

The phase shift factors $e^{\mp i\hat{\Omega}_{i}}$ in Eqs. (21) and (22)
characterize the mutual entanglement between the hole and spins in the AF
background $|\phi_{0}\rangle$ to precisely keep track of the phase-string
structure. Explicitly, we can make a choice
$\hat{\Omega}_{i}=\sum_{l}\mathrm{Im}\ln(z_{i}-z_{l})n_{l}^{\downarrow}$, as
in Eq. (8) for a 2D system. It represents a mutual duality transformation,
which cannot be perturbatively obtained based on a bare hole state created by
$c_{i\sigma}$ on $|\phi_{0}\rangle$. With the nontrivial many-body effect
encoded in $e^{\mp i\hat{\Omega}_{i}}$, the resulting
$\varphi_{1,2,3,4}\left(i\right)$ are consequently treated as the single-hole
wave functions in the VMC calculation, which have turned out to be a very good
approximation [11].

Given the single-particle nature, nevertheless,
$\varphi_{1,2,3,4}\left(i\right)$ are not simply Bloch wave functions. By a
symmetry analysis, we show some intrinsic topological relations among them in
the following.

The anti-unitary time reversal symmetry $\mathcal{T}$ flips spin direction by

$\mathcal{T}c_{\uparrow}\mathcal{T}^{-1}=c_{\downarrow},\mathcal{T}c_{\downarrow}\mathcal{T}^{-1}=-c_{\uparrow},\mathcal{T}i\mathcal{T}^{-1}=-i~{}.$
(23)

Then by noting the time-reversal symmetry in the ground states, one has

$\mathcal{T}|\Psi_{G}\rangle_{1}=|\Psi_{G}\rangle_{3}~{},$ (24)

which requires

$\varphi_{3}\left(i\right)=-\varphi_{1}^{\ast}\left(i\right)e^{i\sum_{l\not=i}\mathrm{Im}\ln\left(z_{i}-z_{l}\right)}~{},$
(25)

with the opposite $S_{z}=\pm 1/2$ and $L_{z}=\pm 1$. Similarly, $\mathcal{T}$
transforms the state $|\Psi_{G}\rangle_{2}$ into $|\Psi_{G}\rangle_{4}$, which
results in

$\varphi_{4}\left(i\right)=-\varphi_{2}^{\ast}\left(i\right)e^{i\sum_{l\not=i}\mathrm{Im}\ln\left(z_{i}-z_{l}\right)}~{}.$
(26)

Furthermore, one may also define a unitary symmetry $\mathbb{Z}_{2}$ generated
by the spin flip operation $\mathcal{R}$, which is defined

$\mathcal{R}c_{\uparrow}\mathcal{R}^{-1}=c_{\downarrow},\quad\mathcal{R}c_{\downarrow}\mathcal{R}^{-1}=-c_{\uparrow}~{},$
(27)

with no extra sign change in the imaginary unit $i$. Then,

$\mathcal{R}|\Psi_{\mathrm{G}}\rangle_{1}=|\Psi_{\mathrm{G}}\rangle_{4}$ (28)

leads to

$\varphi_{4}(i)=-e^{-i\sum_{l\not=i}\mathrm{Im}\ln\left(z_{i}-z_{l}\right)}\varphi_{1}(i)~{},$
(29)

where $|\Psi_{\mathrm{G}}\rangle_{1}$ and $|\Psi_{\mathrm{G}}\rangle_{4}$ have
the same $L_{z}=-1$ and opposite $S_{z}=\pm 1/2$. Similarly, $\mathcal{R}$
transforms the state $|\Psi_{\mathrm{G}}\rangle_{2}$ into
$|\Psi_{\mathrm{G}}\rangle_{3}$. Table 1 lists the corresponding quantum
numbers of the degenerate ground states $|\Psi_{\mathrm{G}}\rangle_{1,2,3,4}$
and the actions of time reversal $\mathcal{T}$ and spin flip symmetries
$\mathcal{R}$.

Table 1: List of the quantum numbers of four degenerate ground states $|\Psi_{\mathrm{G}}\rangle_{1,2,3,4}$ in the pure single-hole doped $t$-$J$.  | Phase string factor | $S_{z}$ | $L_{z}$ | $J_{z}$ | Relation with $|\Psi\rangle_{1}$
---|---|---|---|---|---
$|\Psi_{\mathrm{G}}\rangle_{1}$ | $e^{-i\hat{\Omega}_{i}}$ | $+\frac{1}{2}$ | $+1$ | $+\frac{3}{2}$ |
$|\Psi_{\mathrm{G}}\rangle_{2}$ | $e^{+i\hat{\Omega}_{i}}$ | $+\frac{1}{2}$ | $-1$ | $-\frac{1}{2}$ | $\mathcal{RT}|\Psi\rangle_{1}$
$|\Psi_{\mathrm{G}}\rangle_{3}$ | $e^{-i\hat{\Omega}_{i}}$ | $-\frac{1}{2}$ | $-1$ | $-\frac{3}{2}$ | $\mathcal{T}|\Psi\rangle_{1}$
$|\Psi_{\mathrm{G}}\rangle_{4}$ | $e^{+i\hat{\Omega}_{i}}$ | $-\frac{1}{2}$ | $+1$ | $+\frac{1}{2}$ | $\mathcal{R}|\Psi\rangle_{1}$

##  Appendix C Rashba interaction

###  C.1 Total angular momentum $J_{z}$

The presence of the Rashba interaction breaks conservation of the angular
momentum $L_{z}$ and total spin $S_{z}$. However, the total angular momentum
$J_{z}=L_{z}+S_{z}$ remains as a good quantum number. Here we give a proof. We
aim to prove $UH_{t\text{-}J\text{-R}}U^{{\dagger}}=H_{t\text{-}J\text{-R}}$
for $t$-$J$-R Hamiltonian with $U=\exp(-i\frac{\pi}{2}J_{z})$ generated by
$J_{z}$. For Rashba SOC term
$c_{i\uparrow}^{{\dagger}}c_{i+\hat{x}\downarrow}$ in Eq. (4),

$\displaystyle Uc_{i\uparrow}^{{\dagger}}c_{i+\hat{x}\downarrow}U^{{\dagger}}$
$\displaystyle=e^{-i\frac{\pi}{2}(L_{z}+S_{z})}c_{i\uparrow}^{{\dagger}}c_{i+\hat{x}\downarrow}e^{i\frac{\pi}{2}(L_{z}+S_{z})}$
$\displaystyle=e^{-i\frac{\pi}{2}S_{z}}c_{\hat{R}(i)\uparrow}^{{\dagger}}c_{\hat{R}(i+\hat{x})\downarrow}e^{+i\frac{\pi}{2}S_{z}}$
$\displaystyle=-ic_{\hat{R}(i)\uparrow}^{{\dagger}}c_{\hat{R}(i)+\hat{y}\downarrow},~{}$
(30)

where $\hat{R}(i)$ is the site by rotating site $i$ counterclockwisely 90
degrees. Similarly, for term
$c_{i\uparrow}^{{\dagger}}c_{i+\hat{y}\downarrow}$, we have

$Uc_{i\uparrow}^{{\dagger}}c_{i+\hat{y}\downarrow}U^{{\dagger}}=-ic_{\hat{R}(i)\uparrow}^{{\dagger}}c_{\hat{R}(i)-\hat{x}\downarrow}~{}.$
(31)

Combination of Eqs. (30) and (31) demonstrates the invariance of Rashba
interaction under $U$,

$UH_{\text{R}}U^{{\dagger}}=H_{\text{R}}~{}.$ (32)

The hopping and superexchange terms obviously conserve the total momentum
$J_{z}$. Therefore, we prove that total momentum $J_{z}$ is a good quantum
number for the $t$-$J$-R model.

Figure 9: Two-component structure of the one-hole ground states of the $t$-$J$
model with a weak SOC ($\lambda=0.006J$) obtained by ED under PBC in a
$4\times 4$ lattice: (a) The total spin $S=1/2$ is locked with four total
momenta at $\mathbf{k}_{0}=(\pm\pi/2,\pm\pi/2)$; (b) Corresponding to one of
the $\mathbf{k}_{0}$’s in (a) (the full arrow), a “spin current” pattern of
$\langle J^{s}_{\mathbf{k},\mathbf{k}^{\prime}}\rangle$ in Eq. (33) indicates
a hidden many-body effect (see text).

###  C.2 ED under periodic boundary condition

We present the ED result under the periodic boundary condition (PBC) in the
following. Generally, without SOC, there are eight-fold degenerate ground
states distinguished by momenta
$\mathbf{k}_{0}=\left(\pm\pi/2,\pm\pi/2\right)$ with total $S^{z}=\pm 1/2$ in
the pure one-hole state [8]. The Rashba SOC will lift the spin degeneracy such
that the spin $S=1/2$ will be locked with the four momenta $\mathbf{k}_{0}$ to
give rise to a spin texture of
$\left\langle\mathbf{S}_{\mathbf{k}}\right\rangle$ lying in the $x$-$y$ plane
as illustrated in Fig. 9(a), which is obtained by ED with a very small SOC
strength $\lambda=0.006J$ (see below). Here, corresponding to a ground state
with a given total momentum, say, $\mathbf{k}_{0}=\left(\pi/2,\pi/2\right)$,
$\left\langle\mathbf{S}_{\mathbf{k}}\right\rangle$ is mainly concentrated at
$\mathbf{k}_{0}$ as indicated by the solid arrow in Fig. 9(a) similar to a
_free_ quasiparticle.

However, different from the free quasiparticle picture, in the present ground
state, a nontrivial “spin current” pattern is also found in the momentum space
as shown in Fig. 9(b), which is defined by the correlator

$\langle J_{\mathbf{k},\mathbf{k}^{\prime}}^{s}\rangle\equiv i\left\langle
S_{\mathbf{k}}^{+}S_{\mathbf{k}^{\prime}}^{-}-S_{\mathbf{k}^{\prime}}^{+}S_{\mathbf{k}}^{-}\right\rangle~{}.$
(33)

Here $\langle J_{\mathbf{k},\mathbf{k}^{\prime}}^{s}\rangle$ illustrates
correlations between spins at the neighboring momenta (with
$\mathbf{k}\neq\mathbf{k}^{\prime}$), which is present even in the absence of
the SOC. The nontrivial spin current pattern in Fig. 9(b) only disappears in
the half-filling due to fully frozen charge degree of freedom and the absence
of phase string effect. Hence, $\langle
J_{\mathbf{k},\mathbf{k}^{\prime}}^{s}\rangle\neq 0$ represents an incoherent
component of the one-hole ground state, which manifests as a broaden momentum
distribution of the single hole as a partial momentum can be carried away by
the backflow spin current such that the single-particle momentum $\mathbf{k}$
is no longer conserved [8]. This represents the so-called incoherent component
violating the Landau’s one-to-one correspondence between the momentum
$\mathbf{k}$ of the hole and the total momentum $\mathbf{k}_{0}$. The latter
should be always conserved under PBC due to the translational invariance of
the total system including all spins and the hole. However, due to the spin
current, a partial momentum can be continuously transferred to the spin
background from the hole or _vice versa_ to result in a non-Landau
quasiparticle state, which is essentially a many-body effect [8].

Next we examine the quantitative role of SOC. Figure 10(a) shows the _change_
of the ground-state energy $\Delta E_{0}^{\mathrm{1h}}(\lambda)\equiv
E_{0}^{\mathrm{1h}}(\lambda)-E_{0}^{\mathrm{1h}}(0)$ as a function of
$\lambda$. For the $t$-$J$ model, the linear-$\lambda$ dependence (red line)
and a _finite_ value of its first derivative over $\lambda$ at
$\lambda\rightarrow 0$ [cf. the inset of Fig. 10(a)] strongly suggest that the
spin texture in Fig. 9 at a very small $\lambda=0.006J$ is _not_ completely
new as induced by the SOC, but should rather exist already in the pure $t$-$J$
model to give rise to the first-order perturbative contribution. Namely, a
weak $\lambda$ here mainly selects and picks up a particular linear
recombination state from the pure one-hole ($2\times 4=8$ fold) degenerate
ground states in forming the spin texture in Figs. 9(a) and 9(b).

Indeed, by turning off the phase string effect or the spin current structure
in the $t$-$J$ model to result in a Landau-like quasiparticle in the
$\sigma\cdot$$t$-$J$ model, the change in ground state energy due to SOC
becomes significantly weakened as comparatively shown by Fig. 10(a) and the
inset (blue triangles). There is no more linear-$\lambda$ term. As a matter of
fact, to make the spin texture amplitude comparable, the SOC strength
$\lambda$ has to be enhanced by two orders of magnitude in
$\sigma\cdot$$t$-$J$ model as illustrated in Fig. 10(b). Here the intensity
$|\mathbf{I}_{\max}\left(\mathbf{k}\right)|$ of the spin-polarized spectral
function [defined in Eq. (48) in Appendix E] is shown in Fig. 10(b) at
$\mathbf{k}=\left(\pi/2,\pi/2\right)$.

Figure 10:  (a) Ground state energy shift $\Delta E_{0}^{\mathrm{1h}}$ vs. the
SOC strength $\lambda$ for the $t$-$J$ (red dot) and the $\sigma\cdot$$t$-$J$
(blue triangle) models, respectively. Inset: the corresponding first
derivatives of $\Delta E_{0}^{\mathrm{1h}}$; (b) Intensity of the spin-
polarized single-particle spectral function
$|\mathbf{I}_{\mathrm{max}}(\mathbf{k})|$ at $\mathbf{k}=(\pi/2,\pi/2)$ as a
function of $\lambda$. Here ED results are obtained in a $4\times 4$ lattice
under PBC.

##  Appendix D Variational Monte Carlo procedure

###  D.1 Variational ground state at half-filling

At half filling, both $t$-$J$ and $\sigma\cdot$$t$-$J$ models are reduced to a
pure spin-$1/2$ Heisenberg system. At present, the best variational ground
state is the so-called Liang-Doucot-Anderson bosonic “resonating valence bond”
(RVB) state [24] as

$|\phi_{0}\rangle=\sum_{\upsilon}\omega_{\upsilon}|\upsilon\rangle~{},$ (34)

where each pair of spins in a dimer covering configuration $|\upsilon\rangle$
from different sublattices

$|\upsilon\rangle=\sum_{\\{\sigma\\}}(\prod_{(i,j)\in\upsilon}\epsilon_{\sigma_{i}\sigma_{j}})c_{1\sigma_{1}}^{\dagger}\cdots
c_{N\sigma_{N}}^{\dagger}|0\rangle~{},$ (35)

forms a singlet pairing forced by a Levi-Civita symbol
$\epsilon_{\sigma_{i}\sigma_{j}}$ or, equivalently the Marshall sign [29].
During the calculations, the non-negative amplitude $\omega_{\upsilon}$ can be
factorized as $\omega_{\upsilon}=\prod_{(ij)\in\upsilon}h_{ij}$, where
$h_{ij}$ is a non-negative function depending on sites $i$ and $j$. To obtain
a ground state for AF spin Heisenberg model, we have to optimize all factors
$h_{ij}$. Conversely, with predetermined $h_{ij}$, one can artificially tune
the background spin correlations $\xi$. For example, in Fig. 7, we choose a
form of $h_{ij}$,

$h_{ij}=r_{ij}^{-\alpha}~{},$ (36)

where $r_{ij}$ being the Manhattan distance $|x_{i}-x_{j}|+|y_{i}-y_{j}|$ with
$(x_{i},y_{i})$ being a coordinate of site $i$. The spin correlation length
$\xi$ gradually decreases as the parameter $\alpha$ in Eq. (36) increases. For
example, a state with $\alpha=2$ shows an AF long range order $\xi\gg L$, as
shown in Fig. 1, while a short range ordered state with $\xi\sim 2.7$ can be
obtained by setting $\alpha=4$.

###  D.2  Variational wave functions for the $t$-$J$-R model

Based on the ground states in Eqs. (21) and (22), the wave function
$\varphi_{1,2,3,4}\left(i\right)$ can be determined by VMC. For example,
$\varphi_{1,2}\left(i\right)$ will be obtained by diagonalizing the following
effective Hamiltonian $H_{\mathrm{eff}}$ as the wave function for the
composite quasiparticle $\tilde{c}_{i\downarrow}$ defined in Eq. (6):

$H_{\mathrm{eff}}=-\sum_{\langle
i,j\rangle}\tilde{t}_{ij}^{\downarrow}h_{i}^{\dagger}h_{j}+h.c.~{},$ (37)

where $\tilde{t}_{ij}^{\downarrow}$ represents an effective hopping integral

$\tilde{t}_{ij}^{\downarrow}=t\sum_{\sigma}\langle\phi_{0}|\tilde{c}_{j\downarrow}^{\dagger}c_{j\sigma}c_{i\sigma}^{\dagger}\tilde{c}_{i\downarrow}|\phi_{0}\rangle~{}.$
(38)

Here $\tilde{t}_{ij}^{\downarrow}$ generally takes a complex value due to the
phase shift factor of a twisted particle $\tilde{c}_{i\downarrow}\equiv e^{\mp
i\hat{\Omega}_{i}}{c}_{i\downarrow}$. Thus, it is a Harper-Hofstadter-like
Hamiltonian with a non-uniform flux in Eq. (37) that breaks the translational
symmetry for $\varphi_{1,2}\left(i\right)$. Such an emergent flux can induce
exotic consequences beyond the simple tight binding model. In such an
effective Hamiltonian, $h_{i}^{\dagger}$ creates a ‘twisted’ hole with the
wave function $\varphi_{1,2}\left(i\right)$. Two sectors are available,
denoted as $|\Psi\rangle_{1,2}$ respectively in Eq. (21), which correctly
reproduce two ground states $|\Psi_{\text{G}}\rangle_{1,2}$ with the angular
momentum $L_{z}=\pm 1$. For $S_{z}=-\frac{1}{2}$, similar constructions
$|\Psi\rangle_{3,4}$ in Eq. (22) lead to two ground states
$|\Psi_{\text{G}}\rangle_{3,4}$ with orbital momentum $L_{z}=\mp 1$. Indeed,
excited states can also be determined by Eq. (37) as one may choose excited
wave functions $\varphi_{1,2,3,4}$ determined by Eq. (38) or artificially tune
the spin correlation in $|\phi_{0}\rangle$ [cf. Appendix D.1].

Now let us turn on the SOC with $\lambda\neq 0$. As noted above, $L_{z}=\pm 1$
and $S^{z}=\pm 1/2$ are no longer the good quantum numbers, but the total
angular momentum $J_{z}=L_{z}+S_{z}$ remains. Then one may reconstruct the
one-hole ansatz states in terms of the linear combinations between
$|\Psi_{\mathrm{G}}\rangle_{1,2,3,4}$ and their excited states. From the
symmetry discussed above, Rashba interaction will lift the degeneracy between
$|\Psi_{\mathrm{G}}\rangle_{1,3}$ and $|\Psi_{\mathrm{G}}\rangle_{2,4}$ with
different $J_{z}$’s. A ground state ansatz for the $t$-$J$-R model should come
from a linear combination, i.e.,

$\displaystyle\\!\\!\\!\\!\\!|\Psi_{\mathrm{R}}\rangle_{\mathrm{umdm}}$
$\displaystyle=|\Psi\rangle_{1}+|\Psi\rangle_{3}$
$\displaystyle=\sum_{i}[\varphi_{\downarrow}(i)e^{-i\hat{\Omega}_{i}}c_{i\downarrow}+\varphi_{\uparrow}(i)e^{-i\hat{\Omega}_{i}}c_{i\uparrow}]|\phi_{0}\rangle,$
(39)

or between $|\Psi\rangle_{2}$ and $|\Psi\rangle_{4}$,

$\displaystyle\\!\\!\\!|\Psi_{\mathrm{R}}\rangle_{\mathrm{updp}}$
$\displaystyle=|\Psi\rangle_{2}+|\Psi\rangle_{4}$
$\displaystyle=\sum_{i}[\varphi_{\downarrow}(i)e^{+i\hat{\Omega}_{i}}c_{i\downarrow}+\varphi_{\uparrow}(i)e^{+i\hat{\Omega}_{i}}c_{i\uparrow}]|\phi_{0}\rangle,$
(40)

where hole wave functions $\varphi_{\uparrow}$ and $\varphi_{\downarrow}$ are
variational parameters and the subscripts ‘umdm’ or ‘updp’ represent chirality
of the built-in phase string factor $e^{\pm i\hat{\Omega}_{i}}$ with ‘m’ for
‘$-$’ and ‘p’ for ‘$+$’ in the front of the removed $\uparrow$-spin (‘u’) or
$\downarrow$-spin (‘d’). The true ground states are variationally proved to be
$|\Psi_{\mathrm{R}}\rangle_{\mathrm{umdm}}$ or Eq. (11), and its two-fold
degeneracy is protected by a time reversal symmetry $\mathcal{T}$ in Eq. (23).

Parallel to the ansatz in Eqs. (39) and (40), another two recombinations
$|\Psi_{\mathrm{R}}\rangle_{\text{umdp}}$ and
$|\Psi_{\mathrm{R}}\rangle_{\mathrm{updm}}$ will generate excited states.
Therefore, we have four sectors of variational wave functions
$|\Psi_{\mathrm{R}}\rangle_{\mathrm{umdm}}$,
$|\Psi_{\mathrm{R}}\rangle_{\text{updp}}$,
$|\Psi_{\mathrm{R}}\rangle_{\mathrm{updm}}$ and
$|\Psi_{\mathrm{R}}\rangle_{\text{umdp}}$. Fig. 3(b) shows energies of ground
states with $J_{z}=\pm 3/2$ and the first excited states with $J_{z}=\pm 1/2$
over the SOC strength $\lambda$ via VMC as compared with Fig. 3(a) via ED.

Small Rashba interaction allows perturbative analysis on ground state wave
functions. Conservation of $J_{z}$ requires the ground states for the
$t$-$J$-R model to be composed of one of ground states and one excited state
with $L_{z}=\pm 2$ for the $t$-$J$ model. A sufficiently small SOC strength
$\lambda$ will select one chiral spin pattern with total momentum
$J_{z}=\frac{3}{2}$ or $-\frac{3}{2}$, such that
$|\Psi_{\mathrm{R}}\rangle_{\mathrm{updp}}$ produces ground states. If we
choose $|\Psi\rangle_{2}$ to be $|\Psi_{\mathrm{G}}\rangle_{2}$ with
$J_{z}=\frac{3}{2}$. Then $|\Psi\rangle_{4}$ must have a quantum number
$L_{z}=2$ with $J_{z}=2-\frac{1}{2}=\frac{3}{2}$, which is the first excited
state of the $t$-$J$ model. If we start with $|\Psi_{\text{G}}\rangle_{4}$ as
a choice for $|\Psi\rangle_{4}$, then $|\Psi\rangle_{2}$ is the first excited
state of $t$-$J$ model with $L_{z}=-2$. The two ground states can be
transformed to each other by time-reversal symmetry $\mathcal{T}$ in Eq. (23).
Technically, as proved by previous ED results [8], non-degeneracy between
$|\Psi_{\mathrm{G}}\rangle_{1,2,3,4}$ and first excited states accidentally
originates from the open boundary condition, which leads to dramatically large
energy gain induced by SOC as compared to the $\sigma\cdot$$t$-$J$-R model in
Fig. 3(a) and (b).

##  Appendix E Spin-polarized spectral function

The spin-polarized angle-resolved photoemission spectroscopy (ARPES) can be
used to detect the spin texture of a quasiparticle (hole) excitation. The
normal spectral function for a single quasihole of spin $\sigma$ is defined by

$\displaystyle\mathcal{A}_{\sigma}(\omega,\mathbf{k})$
$\displaystyle=\sum_{n}\langle\Psi_{\mathrm{G}}|c_{\mathbf{k}\sigma}^{{\dagger}}|n\rangle\langle
n|c_{\mathbf{k}\sigma}|\Psi_{\mathrm{G}}\rangle\delta(\omega-
E_{n}+E_{\text{G}})$
$\displaystyle=\frac{1}{\pi}\mathrm{Im}\sum_{n}\frac{\langle\Psi_{\mathrm{G}}|c_{\mathbf{k}\sigma}^{{\dagger}}|n\rangle\langle
n|c_{\mathbf{k}\sigma}|\Psi_{\mathrm{G}}\rangle}{\omega-
E_{n}+E_{\text{G}}-i\eta}~{},$ (41)

where $|\Psi_{\mathrm{G}}\rangle$ is the ground state with energy
$E_{\text{G}}$ and $|n\rangle$ denotes an eigenstate of energy $E_{n}$ with an
extra hole created by $c_{\mathbf{k},\sigma}$ on $|\Psi_{\mathrm{G}}\rangle$.
In Eq. (41), $\eta$ is a broadening introduced to represent energy resolution.
Based on $\mathcal{A}_{\sigma}(\omega,\mathbf{k})$, we may further extract
information on spin polarization, for example, by focusing on the intensity of
the scattered electrons that are parallel or perpendicular to the momentum
$\mathbf{k}$. One may define

$\displaystyle c_{\mathbf{k},\sigma_{\perp}}^{{\dagger}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(c_{\mathbf{k}\uparrow}^{{\dagger}}+ie^{i\theta_{\mathbf{k}}}c_{\mathbf{k}\downarrow}^{{\dagger}}\right)~{},$
(42) $\displaystyle c_{\mathbf{k},\bar{\sigma}_{\perp}}^{{\dagger}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(c_{\mathbf{k}\uparrow}^{{\dagger}}-ie^{i\theta_{\mathbf{k}}}c_{\mathbf{k}\downarrow}^{{\dagger}}\right)~{},$
(43)

to create holes respectively with spin polarization at
$\mathbf{S}_{\sigma_{\perp}}\mathbf{=}\left(\cos\theta_{\mathbf{k}},\sin\theta_{\mathbf{k}},0\right)$
and $\mathbf{S}_{\bar{\sigma}_{\perp}}\mathbf{=}-\mathbf{S}_{\sigma_{\perp}}$
perpendicular to momentum $\mathbf{k}$ lying in the $x$-$y$ plane with
$e^{i\theta_{\mathbf{k}}}=\frac{k_{x}+ik_{y}}{\left|\mathbf{k}\right|}$.
Similarly, the two operators

$\displaystyle c_{\mathbf{k},\sigma_{\parallel}}^{{\dagger}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(c_{\mathbf{k}\uparrow}^{{\dagger}}+e^{i\theta_{\mathbf{k}}}c_{\mathbf{k}\downarrow}^{{\dagger}}\right)~{},$
(44) $\displaystyle c_{\mathbf{k},\bar{\sigma}_{\parallel}}^{{\dagger}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(c_{\mathbf{k}\uparrow}^{{\dagger}}-e^{i\theta_{\mathbf{k}}}c_{\mathbf{k}\downarrow}^{{\dagger}}\right)~{},$
(45)

create holes with spin polarization along
$\mathbf{S}_{\sigma_{\parallel}}\mathbf{=}\left(\sin\theta_{\mathbf{k}},-\cos\theta_{\mathbf{k}},0\right)$
and
$\mathbf{S}_{\bar{\sigma}_{\parallel}}\mathbf{=}-\mathbf{S}_{\sigma_{\parallel}}$,
parallel to momentum $\mathbf{k}$. Then the spin polarization
$I_{\bot}\left(\omega,\mathbf{k}\right)$ at the direction perpendicular to
momentum $\mathbf{k}$ takes the form

$\displaystyle I_{\bot}(\omega,\mathbf{k})$
$\displaystyle=\mathcal{A}_{\sigma_{\bot}}(\omega,\mathbf{k})-\mathcal{A}_{\bar{\sigma}_{\bot}}(\omega,\mathbf{k})~{},$
(46)

and $I_{\parallel}\left(\omega,\mathbf{k}\right)$ at direction parallel to
momentum $\mathbf{k}$:

$\displaystyle I_{\parallel}(\omega,\mathbf{k})$
$\displaystyle=\mathcal{A}_{\sigma_{\parallel}}(\omega,\mathbf{k})-\mathcal{A}_{\bar{\sigma}_{\parallel}}(\omega,\mathbf{k})~{}.$
(47)

Finally one may define an intensity vector
$\mathbf{I}_{\mathrm{max}}(\mathbf{k})$

$\mathbf{I}_{\mathrm{max}}(\mathbf{k})=I_{\parallel}(\omega_{c},\mathbf{k})\mathbf{e}_{\mathbf{k},\parallel}+I_{\perp}(\omega_{c},\mathbf{k})\mathbf{e}_{\mathbf{k},\perp}~{},$
(48)

where $\mathbf{e}_{\mathbf{k},\parallel}$ ($\mathbf{e}_{\mathbf{k},\perp}$) is
the unit vector parallel (perpendicular) to momentum $\mathbf{k}$ and
$\omega_{c}$ denotes the frequency at which
$\sqrt{I_{\perp}(\omega,\mathbf{k})^{2}+I_{\parallel}(\omega,\mathbf{k})^{2}}$
reaches its maximum value $|\mathbf{I}_{\mathrm{max}}(\mathbf{k})|$. The
quantity $\mathbf{I}_{\mathrm{max}}(\mathbf{k})$ as probed by the spin-
polarized ARPES experiment can effectively characterize the spin texture
structure in the momentum space, away from the Fermi energy and deep inside
the Brillouin zone.

##  Appendix F Fragility of spin texture in a Fermi Liquid

In a Fermi liquid, the spin texture induced by Rashba SOC is fragile at
momentum away from the Fermi surface. In this appendix, we give a brief
discussion.

The Landau’s Fermi liquid theory asserts one-to-one correspondence between a
free fermi gas system and an interacting one. The interaction as scattering
between quasiparticles manifests as a self energy $\Sigma(\omega,\mathbf{k})$
in the Green’s function of electrons $c_{\sigma}$

$\displaystyle\mathcal{G}_{\sigma}(\omega,\mathbf{k})$
$\displaystyle=\frac{1}{G_{0}^{-1}(\omega,\mathbf{k})-\Sigma(\omega,\mathbf{k})}$
$\displaystyle=\frac{1}{\omega-[\epsilon_{0}(\mathbf{k})-\mu+\mathrm{Re}\Sigma(\omega,\mathbf{k})]-i\Gamma(\omega,\mathbf{k})}~{},$
(49)

where $G_{0}(\omega,\mathbf{k})=[\omega-(\epsilon_{0}(\mathbf{k})-\mu)]^{-1}$
denotes the propagator for the non-interacting Hamiltonian $H_{0}$, which
generally involves both nearest and next-nearest neighbor hopping, $t$ and
$t^{\prime}$ with a dispersion relation $\epsilon_{0}(\mathbf{k})$,

$\epsilon_{0}(\mathbf{k})=-2t(\cos k_{x}+\cos k_{y})+4t^{\prime}\cos k_{x}\cos
k_{y}~{}.$ (50)

$\mu$ is a chemical potential which determines the doping concentration. The
imaginary part $\Gamma(\omega,\mathbf{k})$ in Eq. (49) of the self energy
$\Sigma(\omega,\mathbf{k})$ dominates the lifetime of a quasiparticle
excitation. A significant consequence is that to the leading order,
$\Gamma(\omega,\mathbf{k})$ depends on the second power of the exciting
energy,

$\Gamma(\omega,\mathbf{k})=c\omega^{2}~{},$ (51)

with a coefficient $c$ determined by quasiparticles’ scatterings.

A sufficiently weak Rashba SOC interaction, on a reasonable assumption, exerts
no influence on the self-energy. In other words, we shall introduce eigenmodes
$d_{\mathbf{k}\pm}^{\dagger}$ to diagonalize the free part $H_{0}+H_{R}$,

$d_{\mathbf{k}\pm}^{\dagger}=\frac{1}{\sqrt{2}}(c_{\mathbf{k}\uparrow}^{\dagger}\pm
e^{i\varphi_{{\bf k}}}c_{\mathbf{k}\downarrow}^{\dagger})~{},$ (52)

which, consequently, modify the Green’s functions

$\displaystyle\mathcal{G}_{+}(\omega,\mathbf{k})$
$\displaystyle=\frac{1}{G_{0}^{-1}(\omega,\mathbf{k})-\Delta_{\mathbf{k}}-i\Gamma(\omega,\mathbf{k})}~{},$
(53) $\displaystyle\mathcal{G}_{-}(\omega,\mathbf{k})$
$\displaystyle=\frac{1}{G_{0}^{-1}(\omega,\mathbf{k})+\Delta_{\mathbf{k}}-i\Gamma(\omega,\mathbf{k})}~{}.$
(54)

Here $\Delta_{\mathbf{k}}=2\lambda\sqrt{\sin^{2}k_{x}+\sin^{2}k_{y}}$ is the
energy splitting induced by SOC in $H_{R}$, and $e^{i\varphi_{{\bf
k}}}=\frac{\sin k_{x}+i\sin k_{y}}{\sqrt{\sin^{2}k_{x}+\sin^{2}k_{y}}}$.

Follow the procedure presented in Sec. E and we can obtain the spin
polarization perpendicular to the momentum $\mathbf{k}$

$\displaystyle I_{\perp}(\omega_{c},\mathbf{k})$
$\displaystyle=\mathcal{A}_{\sigma_{\perp}}(\omega_{c},\mathbf{k})-\mathcal{A}_{\bar{\sigma}_{\perp}}(\omega_{c},\mathbf{k})$
$\displaystyle=2\cos(\theta_{\mathbf{k}}-\varphi_{\mathbf{k}})\frac{2\Delta_{\mathbf{k}}}{c\epsilon_{{\bf
k}}^{2}(c\epsilon_{{\bf k}}^{2}+\Delta_{\mathbf{k}})}~{},$ (55)

where $\omega_{c}$ is defined in Eq. (48) to relate to the maximal spectral
function $\mathbf{I}_{\mathrm{max}}(\mathbf{k})$ and
$e^{i\theta_{\mathbf{k}}}=\frac{k_{x}+ik_{y}}{\left|\mathbf{k}\right|}$. Since
$\Delta_{\mathbf{k}}$ is very small, a quasiparticle with momentum away Fermi
surface towards $\Gamma$ point makes a vanishing contribution, which is
depicted in Fig. 8 with parameters $c=0.005,\lambda=0.01J$ and
$t^{\prime}=0.2t$.

## References

  * Anderson [1987] P. W. Anderson, The resonating valence bond state in La2CuO4 and superconductivity, Science 235, 1196 (1987).
  * Lee  _et al._ [2006] P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a Mott insulator: Physics of high-temperature superconductivity, Rev. Mod. Phys. 78, 17 (2006).
  * Shraiman and Siggia [1988] B. I. Shraiman and E. D. Siggia, Mobile Vacancies in a Quantum Heisenberg Antiferromagnet, Phys. Rev. Lett. 61, 467 (1988).
  * Schmitt-Rink  _et al._ [1988] S. Schmitt-Rink, C. M. Varma, and A. E. Ruckenstein, Spectral Function of Holes in a Quantum Antiferromagnet, Phys. Rev. Lett. 60, 2793 (1988).
  * Kane  _et al._ [1989] C. L. Kane, P. A. Lee, and N. Read, Motion of a single hole in a quantum antiferromagnet, Phys. Rev. B 39, 6880 (1989).
  * Martinez and Horsch [1991] G. Martinez and P. Horsch, Spin polarons in the $t$-$J$ model, Phys. Rev. B 44, 317 (1991).
  * Liu and Manousakis [1991] Z. Liu and E. Manousakis, Spectral function of a hole in the $t$-$J$ model, Phys. Rev. B 44, 2414 (1991).
  * Zheng  _et al._ [2018] W. Zheng, Z. Zhu, D. N. Sheng, and Z.-Y. Weng, Hidden spin current in doped Mott antiferromagnets, Phys. Rev. B 98, 165102 (2018).
  * Zhu  _et al._ [2018] Z. Zhu, D. N. Sheng, and Z.-Y. Weng, Intrinsic translational symmetry breaking in a doped Mott insulator, phys. Rev. B 98, 035129 (2018), arXiv:1707.00068 [cond-mat.str-el] .
  * Zhu and Weng [2015] Z. Zhu and Z.-Y. Weng, Quasiparticle collapsing in an anisotropic $t$-$J$ ladder, phys. Rev. B 92, 235156 (2015), arXiv:1409.3241 [cond-mat.str-el] .
  * Chen  _et al._ [2019] S. Chen, Q.-R. Wang, Y. Qi, D. N. Sheng, and Z.-Y. Weng, Single-hole wave function in two dimensions: A case study of the doped Mott insulator, Phys. Rev. B 99, 205128 (2019).
  * Wang  _et al._ [2015] Q.-R. Wang, Z. Zhu, Y. Qi, and Z.-Y. Weng, Variational wave function for an anisotropic single-hole-doped $t$-$J$ ladder, arXiv e-prints , arXiv:1509.01260 (2015), arXiv:1509.01260 [cond-mat.str-el] .
  * Gotlieb  _et al._ [2018] K. Gotlieb, C.-Y. Lin, M. Serbyn, W. Zhang, C. L. Smallwood, C. Jozwiak, H. Eisaki, Z. Hussain, A. Vishwanath, and A. Lanzara, Revealing hidden spin-momentum locking in a high-temperature cuprate superconductor, Science 362, 1271 (2018).
  * Zhang  _et al._ [2020] J.-H. Zhang, S. Li, Y. Ma, Y. Zhong, H. Ding, and Z.-Y. Weng, Phenomenological single-particle Green’s function for the pseudogap and superconducting phases of high-Tc cuprates, Physical Review Research 2, 023398 (2020), arXiv:1905.04862 [cond-mat.str-el] .
  * Zhang and Rice [1988] F. C. Zhang and T. M. Rice, Effective Hamiltonian for the superconducting Cu oxides, Phys. Rev. B 37, 3759 (1988).
  * Sheng  _et al._ [1996] D. N. Sheng, Y. C. Chen, and Z. Y. Weng, Phase String Effect in a Doped Antiferromagnet, Phys. Rev. Lett. 77, 5102 (1996).
  * Weng  _et al._ [1997] Z. Y. Weng, D. N. Sheng, Y.-C. Chen, and C. S. Ting, Phase string effect in the $t$-$J$ model: General theory, Phys. Rev. B 55, 3894 (1997).
  * Wu  _et al._ [2008] K. Wu, Z. Y. Weng, and J. Zaanen, Sign structure of the $t\text{$-$}J$ model, Phys. Rev. B 77, 155102 (2008).
  * Zhu  _et al._ [2013] Z. Zhu, H.-C. Jiang, Y. Qi, C. Tian, and Z.-Y. Weng, Strong correlation induced charge localization in antiferromagnets, Scientific Reports 3, 2586 (2013), arXiv:1212.6634 [cond-mat.str-el] .
  * Zhu  _et al._ [2014] Z. Zhu, H.-C. Jiang, D. N. Sheng, and Z.-Y. Weng, Nature of strong hole pairing in doped Mott antiferromagnets, Scientific Reports 4, 5419 (2014), arXiv:1312.6893 [cond-mat.str-el] .
  * Rashba [2000] E. I. Rashba, Theory of electrical spin injection: Tunnel contacts as a solution of the conductivity mismatch problem, Phys. Rev. B 62, R16267 (2000).
  * Weng [2011a] Z.-Y. Weng, Mott physics, sign structure, ground state wavefunction, and high-Tc superconductivity, Frontiers of Physics 6, 370 (2011a), arXiv:1110.0546 [cond-mat.supr-con] .
  * Weng [2011b] Z.-Y. Weng, Superconducting ground state of a doped Mott insulator, New Journal of Physics 13, 103039 (2011b), arXiv:1105.3027 [cond-mat.supr-con] .
  * Liang  _et al._ [1988] S. Liang, B. Doucot, and P. W. Anderson, Some New Variational Resonating-Valence-Bond-Type Wave Functions for the Spin-½ Antiferromagnetic Heisenberg Model on a Square Lattice, Phys. Rev. Lett. 61, 365 (1988).
  * Zhang and Weng [2014] L. Zhang and Z.-Y. Weng, Sign structure, electron fractionalization, and emergent gauge description of the Hubbard model, Phys. Rev. B 90, 165120 (2014), arXiv:1406.6867 [cond-mat.str-el] .
  * Chen  _et al._ [2018] S. Chen, Z. Zhu, and Z.-Y. Weng, Two-hole ground state wavefunction: Non-BCS pairing in a $t\text{$-$}J$ two-leg ladder, Phys. Rev. B 98, 245138 (2018).
  * Zhao  _et al._ [2021] J.-Y. Zhao, S. A. Chen, H.-K. Zhang, and Z.-Y. Weng, Two-Hole Ground State: Dichotomy in Pairing Symmetry, arXiv e-prints , arXiv:2106.14898 (2021), arXiv:2106.14898 [cond-mat.str-el] .
  * Bohrdt  _et al._ [2021] A. Bohrdt, E. Demler, and F. Grusdt, Rotational resonances and regge-like trajectories in lightly doped antiferromagnets, Phys. Rev. Lett. 127, 197004 (2021).
  * Auerbach [1994] A. Auerbach,  _Interacting Electrons and Quantum Magnetism_ (Springer-Verlag, 1994).