Electronic quasiparticles in the quantum dimer model: Density matrix renormalization group results

We study a recently proposed quantum dimer model for the pseudogap metal state of the cuprates. The model contains bosonic dimers, representing a spin-singlet valence bond between a pair of electrons, and fermionic dimers, representing a quasiparticle with spin-1/2 and charge $+e$. By density matrix renormalization group calculations on a long but finite cylinder, we obtain the ground-state density distribution of the fermionic dimers for a number of different total densities. From the Friedel oscillations at open boundaries, we deduce that the Fermi surface consists of small hole pockets near $(\pi /2,\pi /2)$, and this feature persists up to a doping density of 1/16. We also compute the entanglement entropy and find that it closely matches the sum of the entanglement entropies of a critical boson and a low density of free fermions. Our results support the existence of a fractionalized Fermi liquid in this model.

Figure 1

A state in the Hilbert state of the dimer model. The blue dimers are bosons representing a spin-singlet pair of electrons. The green dimers are spin-1/2 fermions representing an electron in a bonding orbital between a pair of sites.

Figure 2

The log-scale density of fermionic dimers on a $16\times 4$ lattice. The configuration consists of a single fermionic dimer and 31 bosonic dimers. The dashed line indicates the periodic boundary condition in the $y$ direction; the top dashed line is identified with the bottom solid line. The hopping parameters used are (a) $t_1=-1.05J,t_2=1.95J,t_3=-0.6J$ and (b) $t_1=t_2=t_3=J$. In (a), one can observe the density oscillation with a period of roughly two lattice sites, which corresponds to crystal momenta of $\pi /2$. Note that the Hilbert space is a closely packed dimer configuration, and sites without fermionic dimers are occupied by bosonic dimers.

Figure 3

Fourier transform of the density of fermionic dimers with various total densities. $n$ denotes the number of fermionic dimers in the system. Note that in our $32\times 4$ lattice, $n=8$ corresponds to $1/16$ doping. The hopping parameters used are (a) $t_1=-1.05J,t_2=1.95J,t_3=-0.6J$ and (b) $t_1=t_2=t_3=J$. In (a), there are peaks at $13\pi /16$ and $7\pi /8$, which indicates the fermionic dimers with $k_x=13\pi /32$ and $7\pi /16$ are at the Fermi level, which is a feature missing in (b). The central peak at zero is due to the total density and is normalized to 1.

Figure 4

One-dimensional plot for fermionic dimer density $[\rho \left(x\right)={\sum }_y\rho (x,y)]$ as a function of distance in the $x$ direction when $n=1$. The $x$-axis unit is the lattice constant. Vertical bonds have integer $x$ and are colored yellow; horizontal bonds have half-integer $x$ and are colored blue. Notice the modulation is only present in the horizontal dimers. The system is a $32\times 4$ cylinder with parameters $t_1=t_2=t_3=J$.

Figure 5

Von Neumann entropy of the pure RK model, calculated on a $64\times 2$ cylinder. The red solid line is not the interpolation of the data points but the exact CFT result of Eq. (5) with $c=1$ and $g+c_{\alpha }^{\prime }=1$.

Figure 6

Von Neumann entropy of the FL* phase with different fermionic dimer densities. The system is a $64\times 2$ cylinder with parameters $t_1=t_2=t_3=J$. $L$ is the length of the system, and $l$ is the length of the subsystem. In the case of $n=0$, which is a pure bosonic dimer model near the RK point, we get a nice fit to Eq. (5) with central charge 1.

Figure 7

Fermion contribution to the von Neumann entropy of the dimer model and the Lifshitz transition. $\mathrm{\Delta }{S}_1$ equals $S_1(n=1)-S_1(n=0)$ on the left and $S_1(n=3)-S_1(n=0)$ on the right, where $n$ is the number of fermionic dimers. The system is a $64\times 2$ and $32\times 4$ cylinder with parameters $t_1=t_2=t_3=J$ for the dimer model and free fermions on a 200-site chain for the Lifshitz transition. The $n=1$ and $n=3$ cases for Lifshitz transition correspond to the number of occupied states in the new band.