Quantum phases of the planar antiferromagnetic $J_1-J_2-J_3$ Heisenberg model

We present results for a complementary analysis of the frustrated planar $J_1-J_2-J_3$ spin-1/2 quantum antiferromagnet (AFM). Using dynamical functional renormalization group, high-order-coupled cluster calculations, and series expansion based on the flow equation method, we have calculated generalized momentum-resolved susceptibilities, the ground-state energy, the magnetic-order parameter, and the elementary excitation gap. From these, we determine a quantum phase diagram that shows a large window of a quantum paramagnetic (QP) phase situated among the Néel, spiral, and collinear states, which are present already in the classical $J_1-J_2-J_3$ AFM. Our findings are consistent with substantial plaquette correlations in the QP phase. The extent of the QP region is found to be in satisfying agreement between the three different approaches we have employed.

Figure 1

(a) $J_1-J_2-J_3$ model. Solid dots refer to lattice sites. Only representative third-nearest-neighbor exchange paths are depicted. (b) The classical phase diagram of the $J_1-J_2-J_3$ model.

Figure 2

The FRG scheme in diagrammatic form. Lines without a slash represent the Green’s functions, and lines with a slash represent the single-scale propagators. The different vertices are given by circles. Equations (a) and (b) show the FRG flow equations for the self-energy and the two-particle vertex, respectively. Note that Eq. (b) does not distinguish between the particle-particle channel and the different particle-hole channels. For a full presentation, see, e.g., Ref. 25. The Katanin truncation scheme is given by the replacement (c). In (d), the relation between the spin-correlation function $\chi$ and the two-particle vertex is displayed.

Figure 3

GS energy at $J_2=0$: CCM, SE, and ED from this paper, NNVB basis from Ref. 19, PEPS from Ref. 21, and QMC from Ref. 52. CCM extrapolation see text. All energies are given in units of $J_1$.

Figure 4

Combined quantum phase diagram of the $J_1$-$J_2$-$J_3$ model. (Solid lines) Onset of magnetic flow from FRG on $11\times 11$ sites with 46 frequency points. Error bars of size 0.1 (larger than 0.1) are due to the finite $J_{2,3}$ mesh (uncertainties in the flow of the susceptibility). See text. (Dashed lines) Lines of vanishing-order parameter from the LSUB$n$ CCM with fixed- (variable)-exponent extrapolations from $n=4,6,8,10$ for diamonds (circles). See text. (Dashed-dotted lines) Triplet-gap closure from fifth-order CUT SE. Small- (Large)-$J_3$ lines are [1, 3]([2,2])-DlogPadé approximants. Shaded region refers to the difference between bare series and DlogPadés. N, C, and S denote Néel, collinear, and spiral states. QP refers to a generic QP for CCM and FRG and to a plaquette VBC for SE.

Figure 5

Magnetic-order parameter $m$ obtained within the CCM-LSUB$n$ approximation as well as extrapolated data using the extrapolation scheme $m(n)=b_0+b_1(1/n){}^{1/2}+b_2(1/n){}^{3/2}$ and LSUB$n$ data for $n=4,6,8,10$, see Sec. 2b. (a) In the Néel phase and (b) in the collinear phase.

Figure 6

(Lower part) The brackets present the wave vectors $(k_x,k_y)$ in units of $\pi$ at the maximum of the static susceptibility from FRG. Solid lines indicate the classical critical lines. The blue (lower left), green (lower right), and red (upper) regions correspond to the Néel, collinear, and spiral states, respectively. Thin frames around the brackets mark regions of uncertain flow behavior. (Upper part) Results for the pitch $q$ within the CCM-LSUB$8$ calculation.

Figure 7

Static susceptibility for wave vectors $k_x,k_y\in [0,\pi ]$ for various values $J_2$ and constant $J_3=0.4$. The black dots mark the positions of the maxima.