Effect of next-nearest neighbor hopping on the single-particle excitations at finite temperature

Cuprates are archetypal systems of materials, which besides showing uncoventional superconductivity, also exhibit a wide variety of interesting but complex phases as a function of doping and temperature damascelli ; lee . Cu atom in the undoped/parent cuprates has $3d^{9}$ outermost electronic configuration, which gets altered to $3d^{9}\underline{L}$ as a hole is doped Eskes . $\underline{L}$ indicates the fact that the doped hole resides instead on the neighboring oxygen atoms, which forms a bridge between the two neighboring Cu²⁺ as well as a square that surrounds a Cu²⁺ ion. The hole binds to the Cu²⁺ ion leading to the emergence of Zhang-Rice singlet zhang ; Tjeng . The electron doping, whereas, modifies the electronic configuration from $3d^{9}$ to $3d^{10}$. The low-energy physics involving either hole or particle doping have been studied quite extensively in the span of last three decades within the one-orbital Hubbard model, which has provided us with important insight into the understanding of correlation effects Hubbard ; Bulut ; Senechal ; Kung ; Foley ; Rosenberg .

One striking feature of the doping-vs-temperature phase diagram of the high-$T_{c}$ cuprates is the asymmetry with respect to hole or electron doping damascelli ; Pathak . In particular, the long-range antiferromagnetic order (AFM) is found to survive up to only a small hole doping of $\sim$ 1% kurokawa whereas it is robust against a relatively larger electron doping of $\sim$ 15% matsuda . On the other hand, the $d$-wave superconductivity as well as the pseudogap phase exist in a comparatively wide range of hole doping. Recent experiments suggest that the pseudogap phase found upon hole doping may be marked with the presence of variety of symmetry breaking phonomena including the nematic order sato , stripe order Fleck ; huang ; Xu , short- or long-range charge-density wave tranquada ; ghiringhelli ; Borisenko ; Atkinson , pair-density wave daniel etc., while the possibility of coexisting more than one of these is also not ruled out.

The one-orbital Hubbard model with only nearest-neighbor hopping $t$ possesses particle-hole symmetry, therefore it cannot describe the asymmetrical behavior of the phase diagram. Earlier works suggest a crucial role for the next-nearest neighbor hopping parameter $t^{\prime}$ kyung ; kim , which allows the hole/electron to hop within the same sublattice, in describing various spectral features wells , spin-wave excitation spectra singh1 ; tohyama1 , and the asymmetry of the phase diagram. For the hole and electron doping, $t^{\prime}$ is positive and negative, respectively, thus bringing in frustration in the case of former. In presence of next-nearest neighbor hopping, the AFM state is known to be stabilized for a wide electron-doping region, whereas even a single hole doping may prove to be detrimental to it singh2 . Besides, the asymmetry introduced in the density of states by $t^{\prime}$ is also known to enhance the tendency towards ferromagnetic order (FM) upon hole or electron doping  pandey . Furthermore, the maximum value of the transition temperature $T_{c}$ for the high-$T_{c}$ cuprates may exhibit sensitiveness to $t^{\prime}$  pavarini .

Photoemission studies at a temperature well below the Néel temperature $T_{N}$ in the undoped cuprates such as Sr₂CuO₂Cl₂ wells shows characteristics of the quasiparticle excitations, which have several features similar to the one observed in the pseudogap and $d$-wave superconducting phases. First, the quasiparticle peak is sharp near ($\pi$/2, $\pi$/2) and gets broadened on approaching ($\pi$, 0). Secondly, the hole excitation energy is the least near ($\pi$/2, $\pi$/2) while it is the largest close to ($\pi$, 0). The broad peak near ($\pi$, 0) gets sharpened on doping hole. One-orbital Hubbard model with only nearest neighbor hopping or $t$-$J$ model couldn’t reproduce the features in different studies based on different approaches and a crucial role of long-range hopping was emphasized on later, especially, the next-nearest neighbor hopping kyung ; kim ; Nazarenko ; Tanaka .

Recent work at zero doping for $t^{\prime}=0$ indicates that the nature of hole-excitation energy retains its features even beyond $T_{N}$ with the peak height almost independent of momentum along the normal-state Fermi surface dheeraj . The peak-to-peak distance increases on going from $(\pi/2,\pi/2)$ to $(\pi,0)$ along the normal state Fermi surface. However, not much is known about the variation of these spectral features with temperature when $t^{\prime}$ is incorporated into a microscopic model such as Hubbard model. Does the peak-to-peak separation increase on including $t^{\prime}$? If the hole-excitation energy increases on moving from $(\pi/2,\pi/2)$ to $(\pi,0)$ then how does the particle-excitation energy vary? How do these spectral features evolve with change in temperature? Answers to these questions are of significant interest in order to understand the role of $t^{\prime}$ on the pseudogap-like behavior in the half-filled Hubbard model, because an extensive work carried out earlier using Hartree-Fock meanfield Laughlin , dynamical meanfield Sadovskii ; Kuchinskii , quantum Monte Carlo Macridin ; Rosenberg , classical Monte Carlo dheeraj1 ; dheeraj2 , Gutzwiller approximation Sensarma etc., focused mostly on the hole-doped cases.

In this paper, we investigate the role of nearest-neighbor hopping on the single-particle spectral function within one-orbital and half-filled Hubbard model as a function of temperature. We employ exact-diagonalization + Monte-Carlo scheme based on parallelization to extract the characteristics of single-particle spectral function at different temperature. In order to handle a larger system size so that the momentum resolution without any finite-size effect can be achieved, traveling-cluster approximation (TCA) Kumar and twisted-boundary condition Salafranca are used additionally. We arrive at the following major results for the single-particle spectral function: on moving along the normal state Fermi surface from ($\pi$/2, $\pi$/2) to ($\pi$, 0), (i) the hole- and particle-excitation energy increases and decreases, respectively and (ii) the height of the quasiparticle peak for the hole- and particle excitation decreases and increases, respectively. (iii) Below $T_{N}$, the spectral weight is significantly suppressed along (0, 0) $\rightarrow(\pi/2,\pi/2)$ and $(\pi,0)\rightarrow(0,0)$ for the upper band and along $(\pi/2,\pi/2)$ $\rightarrow(\pi,\pi)$ and $(\pi,\pi)\rightarrow(\pi,\pi)$ for the lower band. (iv) For $T\gtrsim T_{N}$, a relatively larger spectral weight near ($\pi$/2, $\pi$/2) is continued to be noticed in comparison to ($\pi$, 0). (v) The hole-excitation energy increases with $t^{\prime}$ and (vi) the dip in the density of states, which persists beyond $T_{N}$ becomes shallower with increasing $t^{\prime}$ indicating that the latter may be unfavorable for the pseudogap-like features, which is also reflected in the behavior of momentum-resolved spectral function.

We start with the following one-orbital Hubbard Hamiltonian

$\displaystyle\mathcal{H}$

$\displaystyle=$

$\displaystyle\sum_{\bf i,{\bf\delta},\sigma}t_{{\bf i},{\bf i+\delta}}d^{\dagger}_{{\bf i}\sigma}d_{{\bf i+\delta}\sigma}-\mu{\sum_{{\bf i},\sigma}}n_{{\bf i}\sigma}+U{\sum_{\bf i}}n_{{\bf i}\uparrow}n_{{\bf i}\downarrow}$

(1)

where the operator $d^{\dagger}_{i\sigma}$($d_{i\sigma}$) creates (annihilates) an electron with spin $\sigma$. ${\bf\delta}$ is a vector which connects a given site to the nearest neighboring and next-neighboring sites. $t_{{i},{i+\delta}}=$ -$t$ and $t^{\prime}$ for the nearest and next-nearest neighbor hopping, respectively. $n_{i\sigma}=d^{\dagger}_{i\sigma}d_{i\sigma}$ is the charge-density operator for spin $\sigma$ electron. $U$ and $\mu$ are the onsite Coulomb repulsion and chemical potential, respectively.

For the simulation, the grand-partition function used corresponding to the original Hamiltonian given by Eq. (1) is

$${\cal Z}=\int{\cal D}\psi{\cal D}\bar{\psi}e^{-{\cal A}[\psi,\bar{\psi}]},$$

(2)

where the action Schulz

	$\displaystyle{\cal A}$	$\displaystyle=$	$\displaystyle\int_{0}^{\beta}d\tau{\sum_{\bf i,{\bf\delta},\sigma}}\bar{\psi}_{i\sigma}(\tau)(({\partial_{\tau}}-\mu)\delta_{{\bf i},{\bf i+\delta}}+t_{{\bf i},{\bf i+\delta}}){\psi}_{{\bf i+\delta}\sigma}(\tau)$		(3)
		$\displaystyle+$	$\displaystyle U{\sum_{\bf i}}\left(\frac{n_{\bf i}^{2}(\tau)}{4}-\left(\vec{S}_{\bf i}(\tau)\cdot{\hat{\bf r}_{i}}\right)^{2}\right).$

${\psi}_{{i}\sigma}(\tau)$ and ${\bar{\psi}}_{{i}\sigma}(\tau)$ are the Grassman variables corresponding to the operators $d_{{i}\sigma}$ and $d^{\dagger}_{{i}\sigma}$, respectively. The form of interaction term used in Eq. (3) follows from

$\displaystyle n_{i\uparrow}n_{i\downarrow}$

$\displaystyle=$

$\displaystyle\frac{n_{i}^{2}}{4}-({S}_{iz})^{2}.$

(4)

${S}_{i\mu}=\frac{1}{2}\sum_{\alpha,\beta}d^{\dagger}_{i\alpha}{\sigma}^{\mu}_{\alpha\beta}d_{i\beta}$ is the $\mu^{th}$ component of local electron-spin operator. ${S}_{iz}=\vec{S}_{i}\cdot\hat{\bf r}$ when the unit vector $\hat{\bf r}$ is oriented along $z$ axis. The Hubbard interaction has the $SU(2)$ rotational symmetry in spin space, therefore

$\displaystyle n_{i\uparrow}n_{i\downarrow}$

$\displaystyle=$

$\displaystyle\frac{n_{i}^{2}}{4}-(\vec{S}_{i}\cdot{\hat{\bf r}_{i}})^{2},$

(5)

where the unit vector $\hat{\bf r}$ may now be oriented along any arbitrary direction.

To make further progress, we use Hubbard-Stratonovich (HS) transformation to decouple the Hubbard interaction by introducing two auxiliary fields, a scalar field $\phi_{i}{(\tau)}$ coupled to the charge density $n_{i}$ and a vector field ${\bf m}_{i}$ coupled to the spin of electron ${\sigma}_{i}$. This modifies the grand-partition function to

$$Z=\int\prod_{i}\frac{d\bar{\psi_{i}}d\psi_{i}d\phi_{i}d{\bf m}_{i}}{4\pi^{2}U}e^{-{\cal A}(\bar{\psi}_{\bf i},\psi_{\bf i},\phi_{i},{\bf m}_{i})}$$

(6)

with contribution to the action due to the on-site interaction being

	$\displaystyle{\cal A}_{int}$	$\displaystyle=$	$\displaystyle\int_{0}^{\beta}d\tau{\sum_{\bf i}}\left\{{\bf i}{\phi_{\bf i}}{\sum_{\sigma}}{\bar{\psi}}_{{\bf i}\sigma}\psi_{{\bf i}\sigma}-{\bf m}_{\bf i}\cdot{\sum_{\sigma\sigma^{\prime}}}{\bar{\psi}}_{{\bf i}\sigma}\vec{\sigma}_{\sigma\sigma^{\prime}}{\psi_{{\bf i}\sigma^{\prime}}}\right\}$		(7)
		$\displaystyle+$	$\displaystyle\frac{1}{U}{\sum_{i}}(\phi_{\bf i}^{2}+{\bf m}_{\bf i}^{2}).$		(8)

In the simulation, the Hubbard-Stratonovich fields are treated as classical fields so that time $(\tau)$ dependence is ignored. Next, we use saddle-point approximation for the scalar field $\phi_{i}$, for which, the spatial fluctuation is ignored, so that $(U/2){\langle}n_{i}\rangle=U/2$ at half filling. Thus, the effective Hamiltonian can be shown to be

	$\displaystyle H_{eff}$	$\displaystyle=$	$\displaystyle\sum_{\bf i,{\bf\delta},\sigma}t_{{\bf i},{\bf i+\delta}}d^{\dagger}_{{\bf i}\sigma}d_{{\bf i+\delta}\sigma}-{\tilde{\mu}}\sum_{i}n_{i}-\frac{U}{2}\sum_{i}{\bf m}_{i}\cdot\vec{\sigma}_{i}$		(9)
		$\displaystyle+$	$\displaystyle\sum_{i}\frac{U}{4}{{\bf m}_{i}^{2}}$
		$\displaystyle=$	$\displaystyle H_{e}+H_{cls},$

where ${\tilde{\mu}}=\mu-U/2$. The field ${\bf m}_{i}$ are scaled by ${\bf m}_{i}\rightarrow\frac{U}{2}{\bf m}_{i}$ so that it can be turned into a dimensionless field. Note the term in the effective Hamiltonian $H_{cls}=\frac{U}{4}{{\bf m}_{i}^{2}}$, which is treated as classical in the simulation process.

The equilibrium configurations for the auxiliary field $\{{\bf m}_{i}\}$ are generated according to the following distribution

$$P\{{\bf m}_{i}\}\propto Tr_{d,d^{\dagger}}e^{-{\beta}H_{e}}e^{-{\beta}H_{cls}}$$

(10)

where the trace over the fermionic degree of freedom cannot be calculated exactly due to the terms in $H_{e}$ coupled to the classical fields. Therefore, the equilibrium field is generated through MC sampling. In each MC update process, $H_{e}$ is diagonalized and then the eigenvalues are used to calculate the change in free energy of the system. ED + MC method allows an access only to a small system size. The observables generated in the simulation suffer from the finite-size effect and therefore limiting the access to a good momentum resolution. However, the limitation can be overcome upon combining three steps in the simulation process. For each update process, instead of considering the full lattice, only a small cluster of sites around the update site is considered. Thus, the process involves the diagonalization of Hamiltonian for the cluster (size $N_{c}{\times}N_{c}$) centered around the update site. The computational cost is reduced by a factor of $\sim N_{c}/N_{l}$, where $N_{l}\times N_{l}$ is the original system size Kumar . We use $N_{l}=40$ and $N_{c}=8$ throughout the current work. The simulation can be further sped up by using parallelized update process, where $N_{p}$, a factor of total number of available processors, sites can be updated simultaneously. The computational cost reduction in this step is achieved up to a factor of $\sim 1/N_{p}$ dheeraj ; anamitra . In order to reduce the finite size effect further, we make use of twisted-boundary condition (TBC) Salafranca , where a superlattice is formed by repeating the original system of size $N_{l}{\times}N_{l}$ and associated field in $x$- and $y$-direction $N_{t}$ times. The spectral function calculated for such a superlattice is equivalent to the spectral function of an effective system size $N_{l}N_{t}{\times}N_{l}N_{t}$. In order to calculate the spectral function, which is to be discussed later, we use $N_{t}=6$ which allows us to access an effective system size of 240 $\times$ 240.

The simulation process is started at a temperature, which is nearly twice of $T_{N}$, and then the system is cooled down in small steps of temperature. At each temperature, first thousand MC sweeps are used to reach equilibrium field configuration $\{{\bf m}_{i}\}$. In the next thousand sweeps, data related to structure factor, spectral function etc. are obtained for different thermal configurations so as to carry out thermal averaging. We set $U$ to be $4t$, which is not far from the screened value as recent works suggest jang . Since the Hubbard model can be mapped to the Heisenberg model with the exchange coupling $4t^{2}/U$, it can be noted that with larger $U$, one expects a larger broadening in the spectral function. This follows from the softening of the AFM state with increasing $U$.

Fig. 1(a) shows the structure factor for the AFM state with ordering wavevector ${\bf Q}=(\pi,\pi)$ given by

$$S({\bf Q})=\frac{1}{N^{2}}{\sum_{{\bf i},{\bf j}}}{\langle}{\bf m}_{\bf i}\cdot{\bf m}_{\bf j}{\rangle}e^{i{\bf Q}\cdot({\bf r}_{\bf i}-{\bf r}_{\bf j})},$$

(11)

where ${\bf r}_{\bf i}$ is the position vector of site ${\bf i}$ and ${\bf m}_{\bf i}$ is the magnetic-vector field at that point. Two features are easily noted. First, the structure factors for different $t^{\prime}$ approach the same value as $T\rightarrow 0$, which agrees with the Hartree-Fock approximation at low temperature. Then, the rise in $S({\bf Q})$, which is indicative of the onset of long-range AFM order, becomes sharper with increasing $t^{\prime}$ because $T_{N}$ gets smaller. It may be recalled that $S({\bf Q})$ remains largely unaffected by the system size except in the vicinity of $T\sim T_{N}$ dheeraj .

Fig. 1(b) shows the onset of short-range magnetic order defined by

$$\phi_{1}=\frac{1}{4N}\sum_{\langle{i,j}\rangle}\langle{\bf m}_{i}\cdot{\bf m}_{j}\rangle,$$

(12)

where ${\langle{i,j}\rangle}$ denotes summation over nearest neighbors. It appears that $\phi_{1}$ is not independent of $t^{\prime}$. In particular, it diminishes with a rise in $t^{\prime}$ when $T\gtrsim T_{N}$. Below $T_{N}$, however, the behavior of short- and long-range magnetic order is similar. Therefore, an important question arises, is non vanishing of the short-range magnetic correlation function linked to the pseudogap-like features in the spectral function? Perhaps, the nature of rise in the structure factor in the vicinity of $T_{N}$ as well as short-range magnetic correlations can be an indicator for the pseudogap-like feature. As we will see below that a sharper rise in the structure factor indicates a smaller temperature window for the pseudogap-like features and vice-versa.

Fig. 2 shows the density of states (DOS) calculated for $T/T_{N}=1$ and $1.5$ using

$$N(\omega)=\sum_{{\bf q},\lambda,{\bf i}}|\psi_{{\bf q},\lambda}({\bf i})|^{2}\delta(\omega-E_{{\bf q},\lambda}).$$

(13)

Here, $E_{{\bf q},\lambda}$ and $\psi_{{\bf q},\lambda}$ are the eigenvalues and eigenvectors for the whole superlattice. It is not difficult to notice the persisting pseudogap-like dip in the DOS at the Néel temperature and beyond. However, most interestingly, the dip becomes shallower with rising $t^{\prime}$ and it can be seen to be almost absent for $t^{\prime}\sim 0.4$ near $T/T_{N}\sim 1.5$ and beyond, whereas it is present at $T/T_{N}\sim 1$. Thus, the next-nearest neighbor hopping appears to be unfavorable for the pseudogap-like features beyond $T_{N}$. This can be noticed also in the $t^{\prime}-T$ phase diagram where the temperature windows for both the long-range AFM order as well as for the pseudogap-like phase shrinks with rising $t^{\prime}$. We have chosen onset temperature of the AFM order to be the temperature where $S({\bf Q})$ starts to rise from zero. Similarly, the onset temperature $T^{*}$ of the pseudogap-like phase , marked by presence of dip in the density of state, is determined by the condition when there is no further change in the dip of the DOS or the dip disappears as temperature rises.

Next, we examine the evolution of quasiparticle dispersion as a function of temperature using the single-particle spectral function

$$A({\bf k},\omega)=\sum_{{\bf q},\lambda}|{\langle}{\bf k}|\psi_{{\bf q},\lambda}\rangle|^{2}\delta(\omega-E_{{\bf q},\lambda}),$$

(14)

where ${\langle}{\bf k}|\psi_{{\bf q},\alpha}{\rangle}={\sum_{l}\sum_{i}}{\langle}{\bf k}|l,i{\rangle}{\langle}l,i|\psi_{{\bf q},\lambda}{\rangle}$, and $l,i$ are superlattice and site indices, respectively. Fig. 4 shows the quasiparticle dispersion for $t^{\prime}/t=0.3$ a value close to the one obtained through various estimates for high-$T_{c}$ cuprates tohyama ; stemmann ; andersen . Well-formed but asymmetrical gap can be seen near $(\pi/2,\pi/2)$ as well as $(\pi,0)$. The hole- and particle-excitation energies are the least near ($\pi/2,\pi/2$) and ($\pi,0$), respectively. The gap does not disappear even at $T/T_{N}=1$ and beyond, which is evident from the suppression of spectral weight at the Fermi level. More specifically, the gap disappears near ($\pi/2,\pi/2$) relatively more quickly with rising temperature and it can be seen to persist near $(\pi,0)$ even at a relatively higher temperature. Another band with a relatively smaller spectral weight is found along $(0,0)\rightarrow(\pi,0)$, $(0,0)\rightarrow(\pi/2,\pi/2)$, and $(\pi/2,\pi/2)\rightarrow(\pi,0)$, which disappears near $T_{N}$ and beyond. In order to look at the comparison of size of quasiparticle peak for the particle and hole excitations, we also plot $A({\bf k},\omega)$, along the high-symmetry directions as shown in the Fig. 5. The spectral weight is the largest near $(0,0)$ and $(\pi,\pi)$, it is the least near $(\pi,0)$ and moderate in the vicinity of $(\pi/2,\pi/2)$.

Further, we look at the evolution of the gap structure along the normal-state Fermi surfaces as a function of temperature for different $t^{\prime}$ (Fig. 6). At lower $t^{\prime}=0.1$, the asymmetry in the particle-hole excitation is weak, however, the gap along the normal-state Fermi surface survives at $T_{N}$ and beyond. On the contrary, the particle-hole asymmetry is much more evident for a relatively larger $t^{\prime}\sim 0.4$ as expected. Near $(\pi/2,\pi/2)$, the hole excitation energy is significantly smaller in comparison to the particle excitation energy. As one moves along $(\pi/2,\pi/2)$ $\rightarrow(\pi,0)$, the hole-excitation energy increases while the particle-excitation energy decreases. Secondly, there is a significant asymmetry in the quasiparticle peak size. It is the largest for the hole excitation near $(\pi/2,\pi/2)$ in comparison to the particle excitation. These features are reversed as one moves towards $(\pi,0)$.

Fig. 7 shows the hole-excitation energy along the normal state Fermi surface as a function of next-nearest neighbor hopping. The excitation energy increases almost linearly on moving from $(\pi/2,\pi/2)$ to $(\pi,0)$ for all $t^{\prime}$. It is also nearly independent of $t^{\prime}$ in the vicinity of $(\pi/2,\pi/2)$, whereas it exhibits an almost linear rise as $t^{\prime}$ approaches $(\pi,0)$. The energy decreases with a rise in temperature as the spectral weight continues to get transferred to the Fermi level. For $T\gtrsim T_{N}$ and higher $t^{\prime}$, the energy nearly vanishes.

One important consequence of inclusion of $t^{\prime}$ is the shift of spectral weight to higher and lower values of the quasiparticle energy depending on whether $t^{\prime}$ induced energy change $4t^{\prime}\cos k_{x}\cos k_{y}$ is positive or negative for a given quasiparticle momentum. This feature can be seen in our results, especially when $T<T_{N}$ and the lower and upper bands are accompanied with significant suppression of spectral weight in parts of the high-symmetry direction. The spectral weight is shifted towards the Fermi level near $(\pi,0)$ and away to a large energy near $(\pi,\pi)$ or $(0,0)$. These features are in agreement with the results obtained via cluster-perturbation theory kohno . More importantly, our calculation establishes that the gap at the Fermi level does not disappear near $T\sim T_{N}$ and beyond though the gain in the spectral weight does take place with rising temperature.

In this work, we have restricted our study to the half-filled Hubbard model, which corresponds to zero doping. However, majority of the theoretical and experimental works have focused on the hole-doped cuprates because that leads to the appearance of unconventional high-$T_{c}$ $d$-wave superconductivity. The doping, however, introduces not only the $d$-wave superconductivity but a variety of other complex phases including the nematic, pair-density wave, striped spin and charge order, charge-density wave etc. The origin of these phases in the hole-doped cuprates are yet to be completely understood. On the other hand, the simulations that we applied to the half-filled Hubbard model provide us with important insight into the role of next-nearest neighbor hopping with regard to the spectral features and therefore fill in the long-standing gap.

The model used in the simulation is not applicable away from half filling because the spatial and thermal fluctuations in the charge degree of freedom is ignored as we have used the auxiliary field associated with the charge-degree of freedom at the saddle-point approximation. An appropriate modification in the model would be necessary to study the role of spin as well as charge fluctuation on the single-particle excitation. Moreover, the absence of the Brinkman-Rice peak of the quasiparticle excitation near the Mott transition for a moderate $U$ is a consequence of the fact that besides treating the charge degree of freedom at the mean-field level, our approach also ignores the temporal fluctuations in the auxiliary fields.

Findings on the role of next-nearest neighbor hopping $t^{\prime}$ shows a very good qualitative agreement of momentum-dependent spectral features with experiments for the undoped cuprates. Interestingly, these features are qualitatively similar to what are observed for the hole-doped cuprates especially in the direction $(\pi/2,\pi/2)\rightarrow(\pi,0)$ along the normal state Fermi surface in the pseudogap and $d$-wave superconducting phase. Within the scheme used in the current work, study of spectral function for the hole-doped case will necessarily involve at least the competition between two tendencies, i.e., AFM ordering and $d$-wave superconducting, which, in turn, will require the inclusion of auxiliary fields associated with $d$-wave superconductivity also. It will be of strong interest to see the consequence of such a competition on the momentum-dependent gap structure in the $d$-wave state as it will help to find the answer to questions such as does the $d$-wave gap get enhanced because of the AFM ordering tendencies? Answer to that question may help in gaining insight into the role of $t^{\prime}$ in increasing the superconducting-transition temperature. Here, it may be recalled that our findings also indicate momentum-dependent gap structure at higher temperature even in the absence of any long-range magnetic order. We also find that the temperature window, where the pseudogap-like features exists, shrinks with an increase in $t^{\prime}$. This raises another pertinent question about the compatibility of the pseudogap phase with $t^{\prime}$ while it may be noted that a larger $t^{\prime}$ is known to enhance $T_{c}$.

To conclude, we have examined in details, role of the next-nearest neighbor hopping on the single-particle excitation near AFM ordering temperature and beyond. Our findings based on an approach free of any finite size effect, while taking into account the thermal and spatial fluctuations, provides important insight into nature of possible hole and particle excitations along the high-symmetry direction in the half-filled Hubbard model. The spectral gap along high-symmetry, which persists even beyond AFM ordering temperature, shows a very good qualitative agreement with the experiments on undoped cuprates whereas the results also indicate that the long-range hopping may not be favorable for the pseudogap phase. On the other hand, the increase in the gap size along the high symmetry direction especially along nodal to anti-nodal point often used in the context of cuprate supercondctors, grows with an increase in the next-nearest neighbor hopping.

D.K.S. was supported through DST/NSM/R&D HPC Applications/2021/14 funded by DST-NSM and start-up research grant SRG/2020/002144 funded by DST-SERB.