Monte Carlo simulation of monolayer graphene at nonzero temperature

We present results from lattice simulations of a monolayer graphene model at nonzero temperature. At low temperatures for sufficiently strong coupling the model develops an excitonic condensate of particle-hole pairs corresponding to an insulating phase. The Berezinskii-Kosterlitz-Thouless phase transition temperature is associated with the value of the coupling where the critical exponent $\delta$ governing the response of the order parameter at criticality to an external source has a value close to 15. The critical coupling on a lattice with temporal extent $N_t=32$ [$T=1/\left(N_t{a}_t\right)$ where $a_t$ is the temporal lattice spacing] and spatial extent $N_s=64$ is very close to infinite coupling. The value of the transition temperature normalized with the zero-temperature fermion mass gap ${\Delta }_0$ is given by $\frac{T_{\mathrm{BKT}}}{{\Delta }_0}=0.055\left(2\right)$. This value provides an upper bound on the transition temperature, because simulations closer to the continuum limit where the full U(4) symmetry is restored may result in an even lower value. In addition, we measured the helicity modulus ${\rm Y}$ and the fermion thermal mass ${\Delta }_T\left(T\right)$, the latter providing evidence for a pseudogap phase with ${\Delta }_T>0$ extending to arbitrarily high $T$. Analysis of the dispersion relation suggests that the Fermi velocity is not sensitive to thermal effects.

Figure 1

Exciton condensate $\langle \overline{\chi }\chi \rangle$ versus $m$ from simulations at $g^{-2}=0.375$ on $16\times {32}^2$ and $16\times {48}^2$ lattices.

Figure 2

$\langle \overline{\chi }\chi \rangle$ versus $m$ from a $16\times {32}^2$ lattice.

Figure 3

$\langle \overline{\chi }\chi \rangle$ versus $m$ from a $32\times {64}^2$ lattice.

Figure 4

Fermion correlator for $g^{-2}=0.35,m=0.01$ on a $48\times {24}^2$ lattice.

Figure 5

Fermion mass gap $M_f\left(m\right)$ versus $m$ from simulations with $g^{-2}=0.35,0.375$ on a $48\times {24}^2$ lattice.

Figure 6

${\rm Y}$ versus $j$ from simulations with $g^{-2}=0.45$, $m=0.00125$ on $16\times {32}^2$ and $16\times {48}^2$ lattices.

Figure 7

Chirally extrapolated ${\rm Y}$ versus $j$ for different values of $g^{-2}$ extracted from simulations on a $16\times {32}^2$ lattice.

Figure 8

Fermion correlator with $m=0.00125$ and $g^{-2}=0.45,0.50,0.55$ on a $16\times {32}^2$ lattice. The curves result from fits to data with $t$ odd.

Figure 9

Fermion thermal mass $M_f^T$ versus $m$ for various $g^{-2}$ extracted from simulations on a $16\times {32}^2$ lattice.

Figure 10

Chirally extrapolated thermal mass ${\Delta }_T$ versus $g^{-2}$ extracted from simulations on a $16\times {32}^2$ lattice.

Figure 11

Quasiparticle dispersion relation $E\left(p\right)$ as measured on a $16\times {32}^2$ lattice with $m=0.005$.

Figure 12

The fitted parameter $A$ vs $m$ for various values of $g^{-2}$.