Momentum average approximation for models with boson-modulated hopping: Role of closed loops in the dynamical generation of a finite quasiparticle mass

We generalize the momentum average approximation to study the properties of single polarons in models with boson affected hopping, where the fermion-boson scattering depends explicitly on both the fermion’s and the boson’s momentum. As a specific example, we investigate the Edwards fermion-boson model in both one and two dimensions. In one dimension, this allows us to compare our results with exact diagonalization results, to validate the accuracy of our approximation. The generalization to two-dimensional lattices allows us to calculate the polaron’s quasiparticle weight and dispersion throughout the Brillouin zone and to demonstrate the importance of Trugman loops in generating a finite effective mass even when the free fermion has an infinite mass.

Figure 1

(Color online) Sketch of the three-boson, three-site sequence of processes that give rise to effective second NN fermion hopping. The site occupied by the fermion is shaded and the arrow indicates the direction of the next $t_b$-hopping process. Each $t_b$ hopping either leaves a boson (drawn as a square) at the initial site or absorbs a boson from the arrival site. Both (a) collinear and (b) closed loop processes are allowed in 2D; in 1D only (a) is possible.

Figure 2

(Color online) (a) Sketch of several states included in the ${\text{MA}}^{\left(0\right)}$ approximation discussed here. The polaron cloud is allowed to extend on up to three neighboring sites, located anywhere in the system with any number of bosons (shown as red squares) at each site, and the fermion (shown as a shaded blue circle) at any distance away from the cloud. (b) Example of extra states included in a ${\text{MA}}^{\left(1\right)}$ level approximation, with a boson arbitrarily far away from the polaron cloud.

Figure 3

(Color online) Spectral weights $A\left(k,\omega \right)$ for $k=0$ (top), $k=\frac{\pi }{2}$ (middle), and $k=\pi$ (bottom) from MA (black thin lines) and ED (red thick lines). The MA results are shifted upwards to ease the comparison. Parameters are $\Omega /t_b=2$, $t_f/t_b=0.1$, and $\eta /t_b=0.02$.

Figure 4

(Color online) Polaron energy $E\left(k\right)$ in units of $t_b$ vs $k$, for the parameters indicated. Lines show ED results, symbols show MA results (dashed lines are guide to the eyes).

Figure 5

(Color online) Spectral weights $A\left(k,\omega \right)$ for $k=0$ (top), $k=\frac{\pi }{2}$ (middle), and $k=\pi$ (bottom) from MA and ED. The MA results (black thin lines) are shifted upwards. The parameters are $\Omega /t_b=0.5$, $t_f/t_b=0.04$, and $\eta /t_b=0.002$.

Figure 6

(Color online) Spectral weights $A\left(k,\omega \right)$ for $k=0$ (top), $k=\frac{\pi }{12}$ (middle), and $k=\pi /6$ (bottom) from MA and ED. Again the MA results are shifted upwards. The parameters are $\Omega /t_b=0.5$, $t_f/t_b=4$, and $\eta /t_b=0.006$.

Figure 7

(Color online) Polaron energy $E\left(k\right)$ in units of $t_b$ vs $k$, for the parameters indicated. Dashed lines show MA results, full lines correspond to ED results. The arrows roughly mark a discontinuity in the slope of the dispersion.

Figure 8

(Color online) Sketch of our choice of $\mathbf{\eta },{\mathbf{\eta }}^{\prime }$ indices for the $n$-boson noncollinear three-site boson clouds on the 2D square lattice. In all cases, $m\ge 1$ bosons are at site $i-\mathbf{\eta }$, $p\ge 1$ bosons are at site $i+{\mathbf{\eta }}^{\prime }$ and the remaining $n-m-p$ bosons are at site $i$.

Figure 9

(Color online) Pictorial description of the minimal sets of values $\mathbf{\delta }$ needed for a collinear (left) and noncollinear (right) three-site cloud generalized function $f_{n,m,p}\left(\mathbf{\delta }\right)$. Red solid lines indicate the boson cloud centered at $i$ while the blue dots mark the possible locations of the fermion. The distances from the fermion to the site $i$ define the needed set of $\mathbf{\delta }$ values.

Figure 10

(Color online) Polaron dispersion $E\left(\mathbf{k}\right)$ (upper panel) and $qp$ weight $Z\left(\mathbf{k}\right)$ (lower panel) along the high-symmetry directions in the Brillouin zone of the 2D square lattice, for $t_f=0$, $t_b=\Omega =1$. Lines show MA results, the symbols are fits to Eq. (41).

Figure 11

(Color online) Effective $t_2/t_b$ in (a), $t_3/t_b$ in (b), and $E_P/t_b$ in (c), vs $\Omega /t_b$ for $t_f=0$. The thick lines give fully converged MA results, the dots corresponds to the three-boson solution. The insets show the three-boson results over a larger $\Omega /t$ range and the dashed lines show $1/x^5$ dependence. See text for more details.

Figure 12

Effective hopping integrals $t_2$, $t_3$ and polaron ground-state energy $E_P$ vs the maximum number of allowed bosons $N$. The upper (lower) panels correspond to $\Omega /t_b=0.4$ $\left(\Omega /t_b=1\right)$. In both cases, $t_f=0$.

Figure 13

Polaron dispersion $E\left(\mathbf{k}\right)$ (upper panel) along high-symmetry cuts in the 2D square-lattice Brillouin zone, for $t_f=0.1$, $t_b=\Omega =1$. Lines show MA results, the symbols are fits to Eq. (42). The lower panels show the relative error $\epsilon \left(x\right)=1-x\left(N\right)/x\left(\infty \right)$ in the effective hopping integrals, as well as the polaron ground-state energy, vs the maximum number $N$ of allowed bosons.

Figure 14

(Color online) The effective hopping integrals and the polaron energy, in units of $t_b$, vs $\Omega /t_b$, for $t_f=0.1t_b$ (black solid line) and $t_f=0.2t_b$ (red dashed line). The results are only shown for $\Omega$ values where five or more boson processes become irrelevant.