Large-$N$ expansion based on the Hubbard operator path integral representation and its application to the $t\text{−}J$ model. II. The case for finite $J$

We have introduced a new perturbative approach for $t\text{−}J\text{−}V$ model where Hubbard operators are treated as fundamental objects. Using our vertices and propagators we have developed a controllable large-$N$ expansion to calculate different correlation functions. We have investigated charge density-density response and the phase diagram of the model. The charge correlations functions are not very sensitive to the value of $J$ and they show collective peaks (or zero sound) which are more pronounced when they are well separated (in energy) from the particle-hole continuum. For a given $J$ a Fermi liquid state is found to be stable for doping $\delta$ larger than a critical doping ${\delta }_c$. ${\delta }_c$ decreases with decreasing $J$. For the physical region of the parameters and, for $\delta <{\delta }_c$, the system enters in an incommensurate flux or DDW phase. The inclusion of the nearest-neighbor Coulomb repulsion $V$ leads to a charge density wave phase when $V$ is larger than a critical value $V_c$. The dependence of $V_c$ with $\delta$ and $J$ is shown. We have compared the results with other ones in the literature.

Figure 1

Summary of the Feynman rules. Solid line represents the propagator $G_{\left(0\right)}$ (Eq. (7)) for the correlated fermion $f_p$. Dashed line represents the $6\times 6$ boson propagator $D_{\left(0\right)}$ (Eq. (8)) for the $6$-component field $\delta X^a$. Note that the component $(1,1)$ of this propagator is directly associated with the $X^{00}$ charge operator. Dotted line is the propagator $B$ for the boson ghost field ${\mathcal{Z}}_p$. ${\Lambda }_a^{p{p}^{\prime }}$ (Eq. (9)) and ${\Lambda }_{ab}^{p{p}^{\prime }}$ represent the interaction between two fermions $f_p$ and one and two bosons $\delta X^a$ respectively. ${\Gamma }_a^{p{p}^{\prime }}$ and ${\Gamma }_{ab}^{p{p}^{\prime }}$ represent the interaction between two ghost fields ${\mathcal{Z}}_p$ and one and two bosons $\delta X^a$ respectively.

Figure 2

The four different contributions ${\Pi }_{ab}^{\left(i\right)}\left(i=1,2,3,4\right)$ to the irreducible boson self-energy ${\Pi }_{ab}$.

Figure 3

Density-density correlation functions for $J=0.3$, $V=0.0$, and $\delta =0.20$ for different $\mathbf{q}$’s in the BZ. The density-density correlation functions contain collective peaks due to infinite diagram summation seen in Fig. 2. The collective peaks are well pronounced when they are above the particle-hole continuum [see, for example, $\mathbf{q}=(\pi ,0)$]. They are less pronounced when they are superimposed to the particle-hole continuum [see, for example, $\mathbf{q}=(2\pi ∕5,\pi ∕5)$]. $\omega$ is in units of $2t$ and $t$ is considered to be 1.

Figure 4

Phase diagram in the ${\delta }_c\text{−}J$ plane for $V=0.0$. The solid line defines the stability border for the homogeneous Fermi liquid (HFL). For a given $J$ the HFL is stable for $\delta >{\delta }_c$. The thick dashed line around $J\sim 0.5$ separates the flux phase (FP) from the bond order phase (BOP).

Figure 5

Incommensuration $x$ vs ${\delta }_c$ for $V=0.0$. $x$ is defined as ${\mathbf{q}}_{\mathit{c}}=(1,x)\pi$. ${\mathbf{q}}_{\mathit{c}}$ is the $\mathbf{q}$ vector, where the instability takes place. The instability is incommensurate for all $J$ except for $J\to 0$ where $x\to 0$ $\left[{\mathbf{q}}_{\mathit{c}}\to (\pi ,\pi )\right]$. At the separating border of the FP from the BOP there is a jump in the value of ${\mathbf{q}}_{\mathit{c}}$.

Figure 6

Static charge susceptibility $\mathrm{Re}\left[\tilde{D}(\mathbf{q},i{\omega }_n=0)\right]$ vs $\mathbf{q}=(\pi ,q)$ for $J=0.3$, $V=0.0$ and for $\delta =0.15$ and $\delta =0.14$. For $\delta =0.14$ (solid line) the curve looks similar to the case for $\delta =0.15$ (dashed line). For $\delta =0.14$ the HFL is not stable and there is no strong sign of the instability in the static charge susceptibility. In the inset we plot the $\det \left(D_{ab}^{-1}\right)$ (see text for discussions) vs $\mathbf{q}=(\pi ,q)$. For $\delta =0.14$, the $\det \left(D_{ab}^{-1}\right)$ clearly changes sign in a small region around ${\mathbf{q}}_{\mathit{c}}\sim (\pi ,2.7)$.

Figure 7

Charge density wave (CDW) stability line $V_c$ vs $\delta$ for $J=0.0$ (dashed line) and $J=0.3$ (solid line). For a given $\delta$, the system is in a CDW state for $V>V_c$. The value of $V_c$ increases with increasing $J$. For $J=0.3$ the first presented doping is $\delta =0.15$. (For $J=0.3$ the HFL is not stable for $\delta <0.14$ at $V=0.0$.)

Figure 8

Static charge susceptibility for $J=0.3$ and $\delta =0.25$ vs $\mathbf{q}=(q,q)$ for two different values of $V$, $V=1$ and $V=0.5$. $V=1$ is near (from below) the value of $V_c=1.1$. In contrast to Fig. 6, the CDW instability has a clear signal in the static charge susceptibility. For $V=1$ there is a clear indication of a divergence at $(\pi ,\pi )$. For $V=0.5\left(V⪡V_c\right)$, the static charge susceptibility is flat.