Bold Diagrammatic Monte Carlo Method Applied to Fermionized Frustrated Spins

We demonstrate, by considering the triangular lattice spin-$1/2$ Heisenberg model, that Monte Carlo sampling of skeleton Feynman diagrams within the fermionization framework offers a universal first-principles tool for strongly correlated lattice quantum systems. We observe the fermionic sign blessing—cancellation of higher order diagrams leading to a finite convergence radius of the series. We calculate the magnetic susceptibility of the triangular-lattice quantum antiferromagnet in the correlated paramagnet regime and reveal a surprisingly accurate microscopic correspondence with its classical counterpart at all accessible temperatures. The extrapolation of the observed relation to zero temperature suggests the absence of the magnetic order in the ground state. We critically examine the implications of this unusual scenario.

Figure 1

Uniform susceptibility calculated within the $G^{2}W$-skeleton expansion as a function of the maximum diagram order retained in the BDMC simulation (black dots) for $T/J=2$. The result of the high-temperature expansion (with Padé approximant extrapolation) [8, 9] is shown by the red square and horizontal line.

Figure 2

Uniform susceptibility as a function of temperature (red dots) for the triangular Heisenberg antiferromagnet calculated within the BDMC approach. NLC expansion results based on triangles (labeled as 7T and 8T) and sites (labeled as 12S and 13S) [9] are shown along with two different Padé approximant extrapolations [8].

Figure 3

Left and middle panels: Perfect match of the normalized quantum and classical responses at the corresponding temperatures, $T$ and $T_{\mathrm{cl}}(T)$. Points are ordered according to their distance from the origin, $r$, as is illustrated in the right top corner. The sign of the correlator is indicated explicitly next to the point. Right Panel: Mapping between the classical and quantum temperatures. The solid (continued as dashed) line is the fitting function, $y=(4x^2+Ax+B)/(3x+C)$, with $A=0.462$, $B=1.065$, $C=3.825$, which satisfies the asymptotic law $y=4x/3$ expected in the high-temperature limit shown by the dash-dotted line. The red arrow indicates the position of the chiral transition advocated in Refs. 27, 30, 31. For comparison, the square lattice data (stars) are shown in the inset.

Figure 4

Blue (square) symbols: local static susceptibility $\chi (r=0,{\omega }_m=0)$, multiplied by $4T$, as a function of $T/J$. Black circles show $T$ dependence of the (4 times) local spin correlation function $\chi (r=0,\tau =\beta /2)$.