# Distinct quasiparticle excitations in single-particle spectral function

Jing-Yu Zhao  Institute for Advanced Study, Tsinghua University, Beijing
100084, China  Zheng-Yu Weng  Institute for Advanced Study, Tsinghua
University, Beijing 100084, China

###### Abstract

Recent scanning tunneling microscopy (STM) measurements have observed a multi-
peak energy structure on the positive bias side in dilute-hole-doped cuprates,
where tightly-bound hole pairs are identified as the building blocks that can
continuously persist into the superconducting regime. In this work, we study
the single-particle spectral function based on a two-hole ground state
wavefunction [Phys. Rev. X 12, 011062 (2022)], which can provide a consistent
understanding of the experimental results. Here the wavefunction structure
with a dichotomy of $d$-wave Cooper pairing and $s$-wave “twisted hole”
pairing will lead to two branches in the local spectral function where the
low-lying one corresponds to a conventional nodal quasiparticle, and a higher
energy branch is associated with a “twisted” quasiparticle above the pair
breaking or “pseudogap” energy. These energies can be further extended into
energy spectra in the momentum space, in which the low-energy dispersion
agrees excellently with the Quantum Monte Carlo numerical result. The
implications for the STM spectra in the superconducting state will also be
discussed.

Figure 1: Local single-electron spectral function calculated by VMC, averaged
across the lattice. The red curve on the positive bias ($\omega\geq 0$) side
is measured by injecting an electron into the two-hole paired ground state
given in Eq. (6); The blue curve at $\omega\leq 0$ is calculated by extracting
an electron from the half-filling ground state as given in Eq. (8). The inset
shows the same data with a $V$-shaped background (dashed lines) added for
comparison with the experiment data.

Introduction.— The Cooper pair serves as the fundamental cornerstone in
unraveling the nature of superconducting (SC) mechanism. In doped cuprates, it
is believed that there exist preformed Cooper pairs even before the onset of
the superconducting phase. Recently, the scanning tunneling microscopy (STM)
experiments [1, 2, 3] have refocused on this issue by directly detecting
preformed hole pairs in the localized domains/puddles of lightly doped cuprate
compounds even in the insulating antiferromagnetic (AF) regime, which
continuously persist to the SC regime. The STM spectroscopy [1, 2, 3] further
reveals a multiple peak structure on the positive bias side, with the low-
lying peak becomes sharpened in the SC phase to indicate the coherent peak of
the $d$-wave nodal quasiparticle, while the higher energy peak consistent with
the pseudogap energy scale.

It is noteworthy that a strong pairing of two holes in an AF spin background
[4] has been studied by a ground-state wavefunction utilizing the variational
Monte Carlo (VMC) scheme, which yields an excellent agreement with the exact
diagonalization (ED) and density matrix renormalization group (DMRG) numerical
calculations including the ground-state energy and nontrivial quantum number
of angular momentum $L_{z}=2$. An unconventional pairing mechanism is involved
here as due to the local spin current cancellation, which is much stronger and
short-ranged as compared to either the long-range AF fluctuations or
resonating-valence-bond (RVB) correlations in the spin background. It is due
to the fact that the single hole doped into the AF background will always
induce a vortex of spin current to facilitate its motion, whose ground state
wavefunction describing a “twisted” hole - a bare hole surrounded by a spin
current vortex - has been previously carefully investigated [5, 6, 7, 8] in
the ladder and two-dimensional (2D) cases based on the $t$-$J$ model.
Consequently a strong binding force is found between the two “twisted holes”
with the opposite spin currents, resulting in an $s$-wave pairing of them.
Nevertheless, the overlap of such a ground state with a Cooper pair created by
the bare electron operators is in the $d$-wave-symmetry channel instead [4].

Given the pairing symmetry dichotomy in the two-hole ground state, a local
spectral function can be calculated to provide valuable information on the
internal structure of the wavefunction, which may be directly detected by the
STM measurement. Moreover, since each hole pair is tightly bound, a condensate
of a finite density of the pairs naturally leads to a $d$-wave SC state based
on the two-hole wavefunction [4], such that the calculated STM local spectral
function may also be used to account for the properties of the SC state, to
the zeroth order of approximation, by assuming that each hole pair as a
building block remains robust up to a finite doping as indicated in the
experiments [1, 2, 3, 9, 10]. Furthermore, treating the tightly-bound hole
pairs uniformly and coherently distributing in space, one can similarly
calculate the spectral function in the momentum space such that the single-
hole energy dispersion may be also inferred.

In this paper, building on the two-hole ground state ansatz [4], the single-
particle spectral function is studied, which shows some very unconventional
properties of the single-hole excitation. Specifically a two-component
structure has been revealed. The low-energy branch corresponds to a
conventional quasiparticle excitation (resembling a $d$-wave nodal Bogoliubov
quasiparticle) and the higher-energy component is a fractionalization of such
a quasiparticle into a “twisted” hole loosely bound to a spin antivortex field
in the background. The energy scale is found to be coincide with the binding
energy of two “twisted” holes, resembling to a pseudogap energy scale. We have
also examined the negative bias side by knocking out an electron to create a
hole. The low-energy branch is symmetric to the low-branch of the positive
bias side, with the energy dispersion matching with the quantum Monte Carlo
remarkably well in the whole Brillouin zone. But the high-energy branch
disappears due to an “orthogonality catastrophe” effect brought in by the
aformentioned antivortex field when the overlap with regard to the half-
filling spin background is calculated.

Model and variational wavefunctions.— The $t$-$J$ model is a minimal model
describing the doped Mott insulator physics in the cuprate:

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

where $\langle ij\rangle$ denotes a nearest-neighbor (NN) bond, and the strong
correlation nature originates from the no double occupancy constraint
$\sum_{\sigma}c^{\dagger}_{i\sigma}c_{i\sigma}\leq 1$. Here $\mathbf{S}_{i}$
and $n_{i}$ are spin and electron number operators on site $i$, respectively.
We choose $t/J=3$ throughout the paper.

The $t$-$J$ model reduces to the Heisenberg spin model at half-filling, where
the ground state $|\phi_{0}\rangle$ is a spin-singlet state for a finite-size
sample. Here $|\phi_{0}\rangle$ can be very precisely simulated by a bosonic
resonating-valence-bond (RVB) wavefunction, which produces a long-range
antiferromagnetic (AF) spin correlation in the large sample limit.

Based on $|\phi_{0}\rangle$, a single-hole wavefunction has been recently
constructed. Instead of a bare hole state $c_{i\downarrow}|\phi_{0}\rangle$
with removing a spin-$\downarrow$ electron from the half-filling background,
the ground state ansatz takes the following form [6]

$|\Psi_{\mathrm{1h}}\rangle=\sum_{i}\phi(i)\tilde{c}_{i\downarrow}|\phi_{0}\rangle~{},$
(2)

where a “twisted” hole creation operator is given by

$\tilde{c}_{i\downarrow}=e^{-i\hat{\Omega}_{i}}c_{i\downarrow}~{},$ (3)

and $\phi(i)$ is the wavefunction of such a twisted hole determined by VMC.
Here the specific expression of the phase factor $\hat{\Omega}_{i}$ is defined
by

$\hat{\Omega}_{i}=\sum_{l(\neq i)}\theta_{i}(l)n_{l\downarrow}~{},$ (4)

where $n_{l\downarrow}$ denotes the spin-$\downarrow$ number operator of the
electron at site $l$, and $\theta_{i}(l)$ is the so-called statistical angle
defined by the angle between the arrow pointing from site $l$ to $i$ and the
horizontal line $\mod 2\pi$.

Once employing the “twisted” electron $\tilde{c}_{i\downarrow}$ in Eq. (2),
the VMC calculations can yield very good agreements with ED and DMRG results
on both 2D square lattice [6, 7] and two-leg ladder [5, 11, 8], including an
exotic quantum number, i.e., the orbital angular momentum $L_{z}=\pm 1$ in the
2D ground state. Here the “twisted” quasiparticle $\tilde{c}_{i\downarrow}$
can be identified with a composite structure comprised of a bare hole with a
chiral spin current pattern surrounding around it.

Figure 2: (a) Schematic illustration of the two peaks in the local single-
particle spectral function, in which the higher-energy peak is at the energy
that two holes are first unpaired in the intermediate state and then one of
them gets annihilated by injecting electron as indicated by (b), and the
lower-energy peak corresponds to directly annihilating one of the two holes in
the paired state (c); Here (b) and (c) show the corresponding spin current
patterns (red arrows) associated with both processes with the black circle and
cross denoting the hole and the core of the anti-vortex left by the
annihilated hole, respectively (see text).  Figure 3: Momentum space single-
electron spectrum measured by VMC. (a) and (b) are measured by injecting an
electron into the two-hole paired ground state as Eq. (6). (c) and (d) are
measured by extracting an electron from the half filling ground state as Eq.
(8). The Green’s function Monte Carlo calculation [12] (white cross) is also
shown for comparison with $t=3J$. A cut of the full Brillouin zone is used to
show the result, with the cut lines shown in the corresponding insets. (a) and
(b) share the same color bar, so as (c) and (d).

A strong binding between two “twisted” holes has been then found in the two-
hole ground state, which is described by the wavefunction ansatz [4]:

$|\Psi_{\mathrm{2h}}\rangle=\sum_{i,j}g(i,j)e^{-i(\hat{\Omega}_{i}-\hat{\Omega}_{j})}c_{i\uparrow}c_{j\downarrow}|\phi_{0}\rangle~{},$
(5)

with $g(i,j)$ as the variational parameter. The wavefunction in Eq. (5) has
been extensively studied by the VMC method in Ref. 4 with an excellent
agreement with the exact numerics including the new quantum number $L_{z}=2$
in the ground state. A dichotomy of the pairing symmetry has been identified
[4] here: while $g(i,j)$ is $s$-wave-like for the “twisted” holes,
$|\Psi_{\mathrm{2h}}\rangle$ is shown to be $d$-wave-like as measured by the
Cooper pair order parameter $c_{{\bf k}\uparrow}c_{-{\bf k}\downarrow}$ in
terms of the original electron operators. The strong pairing as evidenced by a
big unpairing energy scale $E_{\mathrm{pair}}\sim 1.97J$ has been found [4]
due to the cancellation between the spin currents of opposite chiralities in
Eq. (5). Namely the “twisted” hole object in Eq. (2) is strongly frustrated,
whose kinetic energy can only get restored by pairing in Eq. (5), which has
been pointed out to be a new pairing mechanism fundamentally distinct from the
usual RVB pairing mechanism.

It has been argued [4] that the tightly-bound two holes in Eq. (5) form a
“building block” of size $\sim 4a_{0}\times 4a_{0}$, which eventually leads to
a superconducting domain or phase at finite-doping. In other words, the Bose
condensate of the two-hole state Eq. (5) may be a good candidate for the
superconducting ground state [13] when a finite density of holes are present.
Since a local two-hole ground state in Eq. (5) may thus be well probed by the
STM experiment, which amounts to the calculation of the local single-particle
spectral function to be studied below.

Single-electron spectral functions.—When an electron (say, with
spin-$\uparrow$ without loss of generality) is injected into the two-hole
paired state, the local spectral function measurable by STM at site $i$ on the
positive bias side ($\omega\geq 0$) can be expressed by

$A_{ii}^{p}(\omega)=-\mathrm{Im}\sum_{n}\frac{\left|\langle\Psi_{1h}(n)|c^{\dagger}_{i\uparrow}|\Psi_{\mathrm{2h}}\rangle\right|^{2}}{\omega-[E_{1h}(n)-E^{G}_{2h}-\mu_{p}]+i\eta}~{},$
(6)

where the chemical potential $\mu_{p}$ is set such that the lowest energy of
$|\Psi_{\mathrm{1h}}(n)\rangle$ shifts to $\omega=0$. Here
$|\Psi_{1h}(n)\rangle$ denotes a single-hole excitation state with energy
$E_{1h}(n)$, whose general form may be constructed based on
$|\Psi_{\mathrm{2h}}\rangle$ in Eq. (5) by annihilating an electron, which is
given as follows

$|\Psi_{\mathrm{1h}}(n)\rangle=\sum_{i,v}\varphi_{n}(i,v)e^{-i(\hat{\Omega}_{i}-\hat{\Omega}_{v})}c_{i\downarrow}|\phi_{0}\rangle~{}.$
(7)

Here $|\Psi_{\mathrm{1h}}(n)\rangle$ differs from the one-hole ground state in
Eq. (7) by an extra antivortex phase factor $e^{i\hat{\Omega}_{v}}$ with $v$
denoting the center of the vortex (usually relaxed from the site to the center
of a plaquette). Note that in the two-hole paired state of Eq. (5), each of
the two holes carries a spin vortex with opposite chiralities. Once an
additional electron is injected into the system to annihilate the hole, the
spin vortex is left by a sudden approximation, which accounts for the
difference between a “twisted” hole and a bare hole according to Eq. (3). The
two basic processes creating such single-hole excitations from the two-hole
ground state is schematically illustrated in Fig. 2 (b) and (c), respectively,
which are realized by either first breaking up the paired “twisted” holes and
then removing one of the bare hole or directly annihilating one of two holes
in tight-bound ground state. The variational wave function $\varphi_{n}(i,v)$
will be determined via the energy optimization.

The calculated $A_{ii}^{p}(\omega)$ is shown in Fig. 1 at $\omega\geq 0$ by
the red curve. A double-peak structure with two characteristic energies,
$\Delta_{d}\sim 0.5J$ and $\Delta_{pg}\sim 2.0J$, are obtained (with $\eta\sim
0.1J$), whose physical meaning will be further discussed later.

On the negative bias side ($\omega\leq 0$), in principle one should calculate
the overlap between the two-hole ground state and the three-hole excited
states by knocking one additional electron out of the system, which would be
beyond the capability of the current method. By assuming the newly created
hole is weakly coupled to the two-hole pair in the background, we shall
calculate the following spectral function with the overlap between the half-
filling AF background $|\phi_{0}\rangle$ and the single-hole state given Eq.
(7) by approximately ignoring the original two-hole pair:

$A_{ii}^{m}(\omega)=-\mathrm{Im}\sum_{n}\frac{\left|\langle\Psi_{1h}(n)|c_{i\downarrow}|\phi_{0}\rangle\right|^{2}}{\omega+[E_{1h}(n)-E^{G}_{0}-\mu_{m}]+i\eta}~{},$
(8)

in which the chemical potential $\mu_{m}$ is chosen such as to move the
excitation edge to zero. The calculated STM spectral function
$A_{ii}^{m}(\omega)$ is shown in Fig. 1 at $\omega<0$ (blue curve). Note that
the relative amplitudes between $A_{ii}^{p}(\omega)$ and $A_{ii}^{m}(\omega)$
are arbitrary here, and in the inset of Fig. 1, a V-shaped background marked
by the dashed lines is added to illustrate the possible contributions from the
background spin-wave excitations which are not considered in the above STM
spectral functions.

Figure 4: Local structures of the single hole state of Eq. (7) at different
energy scales. (a)-(c) shows the local spin current pattern with the hole
projected at the black circle position. The thickness of the red arrow
represents the strength of the spin current. (d)-(f) shows the antivortex
distribution $|\varphi(h_{0},v)|$ with the hole $h_{0}$ fixed at the black
circle position.

The real-space information finds its correspondence in the momentum space
spectrum as well, as displayed in Fig. 3. Calculation of the momentum space
spectrum is essentially the same as the real space one, except that the
$c_{i\sigma}$ operators in Eqs. (6) and (8) should be replaced by $c_{{\bf
k}\sigma}$. All the other parameters are the same. The calculated results for
both the positive bias and negative bias are presented in Fig. 3.

On the negative bias side, there is only one band, which is just a single-hole
excitation created by $c_{{\bf k}\sigma}$ on the antiferromagnetic background
as formulated in Eq. (8). We find good agreements between the quasiparticle
spectrum of the wavefunction Eq. (7) determined by the VMC method and those
calculated by other numerical methods such as quantum Monte Carlo (QMC) [12,
14, 15], ED [16] and other numerical methods [17, 18], as well as other
theoretical approaches such as self-consistent Born approximations (SCBA) [19,
20] and angle-resolved photoemission spectroscopy (ARPES) experiments [21, 22,
23, 24].

On the positive bias side [Figs. 3(a) and (b)], one sees the low-energy peak
in the double-peak structure of the STM spectral function actually corresponds
to a $d$-wave Bogoliubov quasiparticle excitation with a vanishing gap along
the nodal line and a maximum gap near the anti-nodal direction. The whole
dispersion is approximately symmetric to the single-hole spectrum on the
negative bias side [Figs. 3(c) and (d)] except for the disappearance of the
portion along the diagonal scan between $(0,0)$ and $(\pi,\pi)$ due to the
$d$-wave symmetry [4] in the two-hole ground state. On the other hand, the
higher energy branch mimics the behavior of a “twisted” single-hole excitation
with the band bottom at $(\pi/2,\pi/2)$.

The internal structure of the single-hole excitation.—By further analysing
$|\Psi_{\mathrm{1h}}(n)\rangle$ at different energy scales, one may gain more
insights into the underlying physics of the single-hole state. For example,
the local spatial structure surrounding the doped hole in Eq. (7) is
calculated, as shown in Fig. 4, which offers a view of the surrounding neutral
spin current as well as the antivortex amplitude $|\varphi(h_{0},v)|$ around
the doped hole projected at a given position.

Within the low energy band of $\omega\sim\Delta_{d}$ and $\omega\sim 0$ [cf.
Figs. 4(b) and 4(c)], the neutral spin current around the doped hole exhibits
a relatively random distribution without forming a coherent closed vortex
pattern due to the compensation of the antivortex $e^{i\hat{\Omega}_{v}}$,
which is tightly bound to the hole in the nearest plaquettes as illustrated in
Figs. 4(e) and 4(f), consistent with the cartoon illustration in Fig. 2(c). As
a result, the total spin current gets compensated on average, and the single
hole state has a finite overlap with a conventional quasiparticle as measured
in the spectral function.

On the other hand, at $\omega\sim\Delta_{pg}$, the spin current around the
hole adopts the shape as a closed vortex profile shown in Fig. 4(a) with a
broader antivortex distribution [cf. Fig. 4(d)]. Such a pattern is consistent
with the process that the two-hole paired state first undergoes to an
unpairing excitation and then a hole gets annihilated by the injecting
electron to create a final single-hole state of a “twisted” hole loosely bound
to an anitvortex as illustrated in Fig. 2(b). Indeed the energy scale
$\Delta_{pg}\simeq E_{\mathrm{pair}}$ [4] for the two-hole state in Eq. (5).

Finally, it is worth to point out that the distinction between Eq. (2) and
$|\Psi_{\mathrm{1h}}(n)\rangle$ in Eq. (7) represents an important fact that
injecting a bare electron or hole into the two-hole or half-filled ground
state, under a “sudden approximation” in the STM and ARPES experiments, does
not necessarily creates a true single-hole ground state, which marks a
fundamental paradigm shift in understanding the single-particle spectral
function for a Mott insulator. Since a bare electron (hole) involves the
process of
$c_{i\downarrow}\Longleftrightarrow\tilde{c}_{i\downarrow}e^{-i\hat{\Omega}_{i}}$,
a vortex field $e^{i\hat{\Omega}_{v}}$ will be left in the background as a
many-body effect, which effectively prevents Eq. (7) from becoming the
“twisted” single-hole state in Eq. (2) unless a very long-time evolution is
realized. The related issue of “orthogonality catastophe” involving the
relaxation of the antivortex field will need a separate investigation
elsewhere.

Conclusion.—In this work, the concept of preformed Cooper pairs in the doped
Mott insulator has been examined by studying the corresponding single-particle
spectral function. An important finding is the two-component feature of the
single quasiparticle excitation by breaking up a pair of doped holes via
injecting an electron into the sample. Here, besides a $d$-wave nodal
quasiparticle excitation, the spectral function also shows a pseudogap-like
feature at the pair breaking energy, above which a non-Landau “twisted” hole
is identified. On the other hand, on the negative bias side, the low-energy
quasiparticle excitation has been also investigated by knocking out an
electron from the half-filled background, whose energy dispersion obtained by
the present VMC calculation agrees remarkably well with the previous QMC
result. These features provide an insightful understanding of the STM
measurements in the underdoped cuprates, which continuously evolve from the
insulating AF state to the SC phase.

Acknowledgements.— We acknowledge stimulating discussions with Yayu Wang, Hai-
Hu Wen, Jia-Xin Zhang. This work is supported by MOST of China (Grant No.
2021YFA1402101).

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