Quantum paramagnetism and helimagnetic orders in the Heisenberg model on the body centered cubic lattice

We investigate the spin $S=1/2$ Heisenberg model on the body centered cubic lattice in the presence of ferromagnetic and antiferromagnetic nearest-neighbor $J_1$, second-neighbor $J_2$, and third-neighbor $J_3$ exchange interactions. The classical ground state phase diagram obtained by a Luttinger-Tisza analysis is shown to host six different (noncollinear) helimagnetic orders in addition to ferromagnetic, Néel, stripe, and planar antiferromagnetic orders. Employing the pseudofermion functional renormalization group (PFFRG) method for quantum spins ($S=1/2$) we find an extended nonmagnetic region, and significant shifts to the classical phase boundaries and helimagnetic pitch vectors caused by quantum fluctuations, while no new long-range dipolar magnetic orders are stabilized. The nonmagnetic phase is found to disappear for $S=1$. We calculate the magnetic ordering temperatures from PFFRG and quantum Monte Carlo methods, and make comparisons to available data.

Figure 1

Cubic unit cell of the bcc lattice together with its lattice vectors ${\mathbf{a}}_1=(\frac{1}{2},\frac{1}{2},\frac{1}{2}),{\mathbf{a}}_2=(-\frac{1}{2},-\frac{1}{2},\frac{1}{2})$, and ${\mathbf{a}}_3=(\frac{1}{2},-\frac{1}{2},-\frac{1}{2})$, assuming the lattice constant $a=1$. The Heisenberg couplings at nearest neighbor $J_1$, second neighbor $J_2$, and third neighbor $J_3$ defining the Hamiltonian [Eq. (1)] are also shown. The bcc lattice is a bi-bipartite lattice; i.e., it is a bipartite lattice that itself consists of two interpenetrating bipartite simple cubic lattices (labeled by black and yellow spheres) displaced by ${\mathbf{a}}_1$. The coordination numbers for different neighbors are nearest neighbor, $z_1=8$; second neighbor, $z_2=6$; and third neighbor, $z_3=12$.

Figure 2

Illustration of the classical spin configurations found in the $J_1-J_2-J_3$ Heisenberg model on the bcc lattice. The global orientation of the spins and, where applicable, the chirality of the spin spirals are not determined by the model [Eq. (1)]. The simplest ground states of this model are the (a) ferromagnet, and (b) Néel antiferromagnet. For the latter, the bcc lattice can be divided into two simple cubic lattices, one being the body centers of the other, which are ordered antiferromagnetically. (c) The $(\pi ,\pi ,\pi )$ state can be decomposed into two interpenetrating Néel-ordered simple cubic lattices, and in this illustration we choose to show them rotated by an angle $\pi /2$ relative to each other. (d) The $(\pi ,\pi ,0)$ antiferromagnet consists of ferromagnetic 110-type planes of spins aligned antiparallel to neighboring planes. The remaining six spin configurations come in pairs of corresponding spirals for ferromagnetic and antiferromagnetic nearest-neighbor coupling $J_1$. (e) In the $(q,0,0)$ state, spins are spiraling when moving along one spatial direction and are ordered ferromagnetically in the 100 planes perpendicular to this direction. The corresponding $(2\pi -q,0,0)$ state (h) for antiferromagnetic $J_1$ is the same state, but with flipped spins at the body center positions. (f) The $(q,q,0)$ state similarly features spins spiraling in two spatial directions with the same pitch, leading to ferromagnetically ordered 110 planes parallel to the third direction. The corresponding $(2\pi -q,q,0)$ state (i) features the same spiraling behavior in two directions, but these spirals now have different chiralities, albeit the same pitch. On top of this, the spins on the body centers are flipped. Note that the relative chirality of the spirals is fixed within the Heisenberg model in contrast to the absolute chirality. The $(q,q,q)$ state (g) features spiraling with the same pitch and chirality in all three spatial directions, which means spins are ferromagnetically aligned in 111 planes. For antiferromagnetic $J_2$, the corresponding state (j) has a wave vector of $(2\pi -q,q,q)$. This implies that the spins are spiraling backward when moving along one particular spatial direction and, in addition, the spins on the body centers are again flipped. Spins are now aligned parallel on $-111$ planes.

Figure 3

Diagrammatic representation of the FRG equations for (a) the self-energy $\mathrm{\Sigma }$ and (b) the two-particle vertex $\mathrm{\Gamma }$. Fermionic propagators are drawn as lines with an arrow, where gray slashes indicate that a $\mathrm{\Lambda }$ derivative acts on the cutoff function (resulting in so-called single-scale propagators). Gray slashes crossing two propagators indicate that the derivative acts on the product of both propagators. (c) Within the Katanin truncation the contracted three-particle vertex [last term on the right hand side in (b)] is approximated by the depicted diagram representing a self-energy correction to the two-particle vertex flow. Note that the self-energy correction is only depicted for the particle-particle channel while further diagrammatic contributions can be constructed from the other channels. (d) Different interaction channels may be inserted into each other to yield additional two-loop diagrammatic corrections. Note that only one particular nested two-loop diagram is shown corresponding to one possible choice of inserting two interaction channels into each other.

Figure 4

The classical phase diagrams of the $J_1-J_2-J_3$ Heisenberg model on the bcc lattice for (a) ferromagnetic $J_1$ and (c) antiferromagnetic $J_1$. Brown solid (dashed) lines mark phase boundaries with first (second) order phase transitions. The corresponding $S=1/2$ quantum phase diagrams are shown in (b) for ferromagnetic $J_1$ and (d) for antiferromagnetic $J_1$. Note the change in the phase boundaries and the appearance of a paramagnetic phase in both quantum phase diagrams. The classical phase boundaries overlaid in (b) and (d) are meant as an aid to visualize the modifications in the quantum phase diagram with respect to the corresponding classical one.

Figure 5

Representative RG flows ($\mathrm{\Lambda }$ is in units of $|J_1|$) of the maximum of the magnetic susceptibilities (in units of $1/|J_1|$) in momentum space for the ordered regimes and the PM regimes of Fig. 4 and Fig. 4. The susceptibilities are evaluated for FM $J_1$ (left panel) at the following $(J_2/|J_1|,J_3/|J_1\left|\right)$ points: FM order at $(0,0),(\pi ,\pi ,0)$ order at $(1.5,1),(\pi ,\pi ,\pi )$ order at (1.5, 0), 1D spiral $(q,0,0)$ at $(-0.5,1)$, 2D spiral $(q,q,0)$ at (0.5, 1), 3D spiral $(q,q,q)$ at (0.26, 0.2), and PM at (0.8, 0.3). For AF $J_1$ (right panel), the magnetic susceptibilities are plotted at the same parameter points as for FM $J_1$. The points at which the solid lines become dashed (marked by vertical arrows) indicate an instability in the flow and express the onset of long-range magnetic order. A smooth flow down to $\mathrm{\Lambda }\to 0$ (black curves) indicates paramagnetic behavior.

Figure 6

The normalized susceptibility profiles in the first Brillouin zone, a rhombic dodecahedron, for the $S=1/2J_1-J_2-J_3$ Heisenberg model on a bcc lattice with FM $J_1$ for the magnetically ordered states. The plots are calculated for the coupling parameters quoted in the caption of Fig. 5. The corresponding $\mathrm{\Lambda }$ values are given by the respective points of instabilities as indicated by the vertical arrows in the left panel of Fig. 5.

Figure 7

The normalized susceptibility profiles in the first Brillouin zone, a rhombic dodecahedron, for the $S=1/2J_1-J_2-J_3$ Heisenberg model on a bcc lattice with AF $J_1$ for the magnetically ordered states. The plots are calculated for the coupling parameters quoted in the caption of Fig. 5. The corresponding $\mathrm{\Lambda }$ values are given by the respective points of instabilities as indicated by the vertical arrows in the right panel of Fig. 5.

Figure 8

The normalized paramagnetic susceptibility profile evaluated at the end of the flow for (a) ferromagnetic $J_1$ and (b) antiferromagnetic $J_1$ for the spin-$1/2$ bcc Heisenberg model. In both plots the susceptibility profile is calculated at $(J_2/|J_1|,J_3/|J_1\left|\right)=(0.8,0.3)$. Note the softer peaks as compared to the sharp peaks in the similar plots for the ordered states.

Figure 9

Main panel: Ground state energies per site for the $S=1/2J_1-J_2$ Heisenberg model on a bcc lattice with FM $J_1$. For the $(\pi ,\pi ,\pi )$ state we use ED for a finite lattice of $N=36$ and the CCM for an infinite lattice. The crossing of the energies of the (0,0,0) and $(\pi ,\pi ,\pi )$ states determines the first-order transition between both phases (see Table 3). Inset: CCM result for the magnetic order parameter (sublattice magnetization) $M$ as well as ED and CCM results for the nearest-neighbor (1-NN), next-nearest-neighbor (2-NN), and third-nearest-neighbor (3-NN) equal-time spin-spin correlation functions of the $(\pi ,\pi ,\pi )$ phase (ED, symbols; CCM, lines). The black vertical dotted line indicates the first-order transition, where the $(\pi ,\pi ,\pi )$ state gives way to the ferromagnetic ground state. Although the $(\pi ,\pi ,\pi )$ state ceases to be the ground state beyond the phase boundary and the correlators change their nature to FM ones, we are still able to trace the $(\pi ,\pi ,\pi )$ state correlators into the FM phase as shown in the inset.

Figure 10

The susceptibility $\chi \left(\mathbf{k}\right)$ (in units of $1/|J_1|$) as a function of the wave vector $\mathbf{k}$ for different values of $J_2$, with both FM (left panel) and AF (right panel) $J_1$, along the high symmetry lines in the momentum space. The characteristic points of our choice are $\text{I}=(0,0,0),\text{II}=(0,0,\pi ),\text{III}=(2\pi ,2\pi ,2\pi ),\text{IV}=(2\pi ,2\pi ,0),\text{V}=(\pi ,\pi ,\pi )$. The susceptibility is plotted for $\mathrm{\Lambda }/|J_1|\simeq 4/\pi$, which corresponds to a temperature $T/|J_1|\simeq 2$. Note that the ordering wave vector $\mathbf{q}=(0,0,0)$ is equivalent to $\mathbf{q}=(2\pi ,2\pi ,0)$. Similarly, the ordering wave vector $\mathbf{q}=(2\pi ,0,0)$ is equivalent to $\mathbf{q}=(2\pi ,2\pi ,2\pi )$.

Figure 11

Variation of $q$ for the incommensurate spiral phases $(q,0,0),(q,q,0),(2\pi -q,0,0)$, and $(2\pi -q,q,0)$ along three different cuts in the $J_2-J_3$ plane of the Heisenberg model on the bcc lattice. Circles (triangles) denote the case of FM (AF) $J_1$. The corresponding classical values are shown by gray lines.

Figure 12

Critical temperatures $T_c$ of the $J_1-J_2$ bcc Heisenberg model as a function of the frustrating AF $J_2$ coupling. Left panel: Curie temperatures $T_C/\left|J_1\right|$ for $J_1$ FM. Right panel: Néel temperatures $T_N/J_1$ for $J_1$ AF. Please refer to Table 4 for numerical values.

Figure 13

QMC estimates for the Binder ratio $U_4$ and the ratio of the finite-size correlation length over the lattice size $\xi /L$ for a FM Heisenberg model on the bcc lattice, and the corresponding quantities for the AF model, $U_4^{\mathrm{AF}}$ and ${\xi }^{\mathrm{AF}}/L$. The temperature $T$ is expressed in units of $|J_1|$.