Bold diagrammatic Monte Carlo technique for frustrated spin systems

Using fermionic representation of spin degrees of freedom within the Popov-Fedotov approach, we develop an algorithm for Monte Carlo sampling of skeleton Feynman diagrams for Heisenberg-type models. Our scheme works without modifications for any dimension of space, lattice geometry, and interaction range, i.e., it is suitable for dealing with frustrated magnetic systems at finite temperature. As a practical application, we compute uniform magnetic susceptibility of the antiferromagnetic Heisenberg model on the triangular lattice and compare our results with the best available high-temperature expansions. We also report results for the momentum dependence of the static magnetic susceptibility throughout the Brillouin zone.

Figure 1

Graphic representation of allowed interaction processes for the Heisenberg model.

Figure 2

Typical low-order diagrams contributing to the particle self-energy and polarization operator within the $G^{2}W$-skeleton scheme (red color denotes spin-exchange coupling). Diagrams for $\Sigma$ and $\Pi$, in their turn, are used to calculate fully dressed $G$ and $W$ functions [see Eq. (7)].

Figure 3

Nonphysical diagram with two special vertices (worms) $\mathcal{S}$ and $\mathcal{T}$ (marked by blue circles). Momentum- and spin-conservation laws would be satisfied at all vertices and interaction lines if one considers $\mathcal{S}/\mathcal{T}$ as a drain/source of momentum $p_w$ and spin projection 1; the same rule is recovered by imagining a line (green dotted) which carries $(p_w,s_z=1)$ from $\mathcal{S}$ to $\mathcal{T}$. If special vertices were not present in the diagram, then this graph would contribute to $\Sigma ({\tau }_1-{\tau }_2)$ after removing the dummy propagator line marked by the cross (if cross marks the dummy $\tilde{W}$ line, then the diagram is contributing to $\Pi$).

Figure 4

Detailed structure of $\mathcal{S}$ and $\mathcal{T}$ vertices. Note that the interaction line containing a worm on one of its ends is also unphysical and the spin-conservation law applies to a pair of vertices.

Figure 5

Two cases for Create and Delete updates which insert/remove a pair of worms at the ends of the particle propagator.

Figure 6

Two cases for Create-H and Delete-H updates which insert/remove a pair of worms and increase the diagram order by adding a Hartree-type bubble.

Figure 7

Upper panel: An illustration of the Move-P update shifting $\mathcal{S}$ along the propagator line. This move changes the momentum of the propagator line and flips its spin as well as the status of interaction lines between physical and unphysical. Lower panel: An illustration of the Move-I update shifting $\mathcal{S}$ along the interaction line. This move changes the auxiliary momentum of the interaction line only.

Figure 8

Upper panel: In the Commute update, the diagram topology is changed by redirecting propagators originating at special vertices $\mathcal{S}$ and $\mathcal{T}$ to have their destination vertices at $\mathcal{B}$ and $\mathcal{A}$, respectively. Lower panel: Typical transformation produced by the Commute update.

Figure 9

Upper panel: Increasing/decreasing the diagram order using a pair of complementary updates Insert and Remove. When the propagators ${\mathcal{V}}_1$-$\mathcal{A}$ and $\mathcal{B}$-${\mathcal{V}}_2$ are linked with the new interaction line $\mathcal{C}$-$\mathcal{D}$ carrying momentum $q$, the closed loop for momentum conservation goes as ${\mathcal{V}}_1$-$\mathcal{C}$-$\mathcal{D}$-${\mathcal{V}}_2$-${\mathcal{V}}_1$. The same loop is used in the Remove update. Lower panel: Diagram transformation when vertices are dressed and undressed with interaction lines.

Figure 10

An illustration of the Move-T update changing the imaginary-time location of a randomly chosen vertex. One is free to change the type of worm from $\mathcal{S}$ to $\mathcal{T}$ or to place it on any of the vertices in both panels.

Figure 11

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) (Ref. 38) is shown by red square and horizontal line.

Figure 12

Uniform susceptibility as a function of the maximum diagram order (black dots) for $T/J=1$. The result of the high-temperature expansion (with Padé approximant extrapolation) (Ref. 38) is shown by red square and horizontal line. Its error bar is based on the difference between various expansion/extrapolation schemes.

Figure 13

Uniform susceptibility as a function of temperature (red dots) for the triangular Heisenberg antiferromagnet calculated within the BDMC approach. NLC expansion results (Ref. 37) based on triangles (labeled as 7T and 8T) and sites (labeled as 12S and 13S) are shown along with two different Padé approximant extrapolations of high-temperature expansions (Ref. 38).

Figure 14

Staggered susceptibility at the wave vector $Q$ as a function of temperature (black dots) plotted for comparison along with the Curie-Weiss law (blue curve) and uniform susceptibility (red dots and line).

Figure 15

Static spin-spin correlation function along the characteristic trajectory in the Brillouin zone.

Figure 16

Modulus of the spin susceptibility in real space at $T/J=0.375$ on the logarithmic scale. Lattice points are enumerated according to their distance from the origin. Spins on red sites (0, 2, 5, 6, 10) are correlated ferromagnetically, while spins on blue and black sites are correlated antiferromagnetically with the spin at the origin. At slightly higher temperature $T/J=0.5$ black points (3, 7) start to correlate ferromagnetically with the spin $S_0$ at the origin, contrary to the classical ground-state pattern depicted to the right.