Diagrammatic quantum Monte Carlo solution of the two-dimensional cooperon-fermion model

We investigate the two-dimensional cooperon-fermion model in the correlated regime with a continuous-time diagrammatic determinant quantum Monte Carlo algorithm. We estimate the transition temperature $T_c$, examine the effectively reduced band gap and cooperon mass, and find that delocalization of the cooperons enhances the diamagnetism. When applied to diamagnetism of the pseudogap phase in high-$T_c$ cuprates, we obtain results in qualitative agreement with recent torque magnetization measurements.

Figure 1

(a) Elementary interaction events (vertices): two fermions are combined onto a cooperon (left) and vice versa (right). (b) A diagram contributing to the partition function $Z$. (c) A diagram sampled in the DDQMC algorithm: open ends of the fermionic lines imply a sum over all the ways of connecting the vertices by the fermionic lines, which is represented by the corresponding determinant.

Figure 2

Two sets of complementary Monte Carlo updates: (a) adding or removing one pair of vertices and (b) swapping the end points of cooperon lines.

Figure 3

The Feynman diagrams comprising the fermionic (a) and cooperon (b) Green’s function within the random phase approximation.

Figure 4

(a) The chemical potential vs temperature $T/U$. (b) The particle density $n_{\sigma }$ of a single spin component and the cooperon density $n_b$ vs temperature $T/U$. DDQMC results in the thermodynamic limit (blue points) are extrapolated to the infinite system size.

Figure 5

The system size dependence of the on-site double occupancy $n_d$ for various values of $\beta$.

Figure 6

(a) System-size dependence of off-diagonal order $\chi {}_{R_{x,\max },\omega =0}^{\mathrm{od}}$ at distance $R_{x,\max }$. (b) The cooperon Green’s function $G_{R_{x,\max },\omega =0}^b$. $R_{x,\max }=(L-1)/2$ is the maximal distance along the $x$ direction in our system with periodic boundary conditions.

Figure 7

Finite-size scaling of $\chi {}_{k=0,\omega =0}^{\mathrm{od}}$ and $G_{k=0,\omega =0}^b$ according to Eq. (15); the error bars are smaller than the symbol size. The uncertainties of $A=0.55\pm 0.1$ and $T_c=0.03\pm 0.01$ are estimated from a breakdown of the shown data collapse.

Figure 8

(a) Finite-size extrapolation of the renormalized cooperon gap ${\Delta }_{\mathrm{eff}}$. (b) Comparison of the extrapolated ${\Delta }_{\mathrm{eff}}$ at $L\to \infty$ with the RPA results. $T_c=0.03$ obtained in Fig. 7 is shown by the arrow. The error bars are smaller than the symbol sizes.

Figure 9

(a) The suppression of the renormalized cooperon mass with decreasing temperature. The red curve corresponds to the RPA result. The blue (black) dots are for lattice size $L=11,15$. The same fitting process fails for larger lattice sizes $L=21,25$ due to the too small measured $G_{r,\tau }^b$ close to the lattice boundary and very large sampling error bars. (b) Comparison of the renormalized cooperon’s dispersion $\epsilon {}_{k,\text{eff}}^b$ from RPA calculation and our simulation with $L=11$. The fitting process [Eq. (17)] is applied to obtain $\epsilon {}_{k,\text{eff}}^b$ in our simulation. The error bars are smaller than the symbol.

Figure 10

A schematic demonstration of one possible scenario of the anisotropy in momentum space in the pseudogap phase with localized tightly bound cooperons located at the antinodes and a holelike Fermi sea residing on the nodes. The superconductivity may be driven by the interplay between the states at antinodes and nodes.

Figure 11

The temperature dependence of the magnetization at $B=1$ T calculated using Eq. (18).The $x$ axis is normalized by $T_c~0.03$ ($~80$ K with $U=250$ meV). Red dots are from the torque magnetization experiments with $T_c=50$ K.[33] Note that there is no parametric fitting to the experimental data. While strong damping of cooperons at high temperature suppresses the magnetization at $T\gg T_c$, the exponential increase close to $T_c$ will not be affected.