Schwinger boson theory of ordered magnets

The Schwinger boson theory provides a natural path for treating quantum spin systems with large quantum fluctuations. In contrast to semiclassical treatments, this theory allows us to describe a continuous transition between magnetically ordered and spin liquid states, as well as the continuous evolution of the corresponding excitation spectrum. The square lattice Heisenberg antiferromagnet is one of the first models that was approached with the Schwinger boson theory. Here we revisit this problem to reveal several subtle points that were omitted in previous treatments and that are crucial to further develop this formalism. These points include the freedom for the choice of the saddle point (Hubbard-Stratonovich decoupling and choice of the condensate) and the $1/N$ expansion in the presence of a condensate. A key observation is that the spinon condensate leads to Feynman diagrams that include contributions of different order in $1/N$, which must be accounted for to get a qualitatively correct excitation spectrum. We demonstrate that a proper treatment of these contributions leads to an exact cancellation of the single-spinon poles of the dynamical spin structure factor, as expected for a magnetically ordered state. The only surviving poles are the ones arising from the magnons (two-spinon bound states), which are the true collective modes of an ordered magnet.

Figure 1

Single-spinon spectrum of a square lattice antiferromagnet in the absence of a symmetry-breaking field (a), in the presence of a symmetry-breaking field $h$ linearly coupled with the staggered magnetization (b), and with an additional symmetry-breaking field $t$ that couples to a single-spinon nearest-neighbor hopping (c), where $h,t\sim 1/{\mathcal{N}}_s$.

Figure 2

Diagrammatic representation of the four contributions to the dynamical spin susceptibility ${\chi }_{\mathrm{SP}}^{\mu \nu }\left(q\right)$ at the SP level corresponding to the four terms of Eq. (79). The full (dashed) line represents the propagator of the noncondensed (condensed) spinons. The open circles represent the external vertices of the theory introduced in Eq. (76).

Figure 3

Diagrammatic representation of the four contributions to the polarization operator $\mathrm{\Pi }(\mathit{q},i{\omega }_n)$ corresponding to the four terms of Eq. (102). The full circles represent the internal vertices of the theory introduced in Eq. (25).

Figure 4

$1/N$ diagrams of dynamical spin susceptibility. The wavy line refers to the RPA propagator $\frac{1}{N}D(\mathit{q},\omega )$.

Figure 5

Vicinity of the zero of ${\tilde{D}}_{11}(\mathit{q},\omega )$ (gray area) where the RPA propagator displays anomalous large-$N$ scaling, namely $d{\tilde{D}}_{11}{/d\omega |}_{\omega ={\varepsilon }_{\mathit{q}n}}\propto N$, as indicated by the red line.

Figure 6

Remaining diagrams of nominal order $1/N$ (a1), (b1), (c1) and their corresponding counterdiagrams (a2), (b2), (c2). (d1) Arbitrary diagram of the $1/N$-expansion and its counterdiagram (d2). The thick solid line represents the full single-spinon propagator including contributions from condensed and noncondensed spinons.

Figure 7

(a) Magnetic susceptibility including renormalized external/internal vertices (gray/black shaded areas), spinon propagator (double solid line), and RPA propagator (double wavy line). (b) Counterdiagram that cancels out the single-spinon poles of the diagram shown in panel (a). (c) Renormalized propagator of the fluctuation fields.

Figure 8

Recovery of semiclassical limit ($S\to \infty$) with the Schwinger boson theory by including fluctuations around the SP solution $|G_3\rangle$ for a decoupling parameter $\alpha =0.5$. Transverse (a) and longitudinal (b) DSSF for an arbitrarily chosen momentum $\mathit{q}=(1.25,2.09)$ in r.l.u.

Figure 9

Transverse dynamical spin structure factor ($S=1/2$). Parts (a)–(d) show the SP result for the simple BEC solution represented by the state $|G_3\rangle$ and $\alpha =0,0.25,0.4,0.5$, while (e)–(h) include the $1/N$ correction described in Fig. 4. Note that $S^{xx}(\mathit{q},\omega )=S^{yy}(\mathit{q},\omega )$. The green dots indicate the magnon dispersion. Inset: high-symmetry path of the horizontal axis.

Figure 10

Longitudinal dynamical spin structure factor ($S=1/2$). Parts (a)–(d) show the SP result for the simple BEC solution represented by the state $|G_3\rangle$ and $\alpha =0,0.25,0.4,0.5$, while (e)–(h) include the $1/N$ correction described in Fig. 4. Inset: high-symmetry path of the horizontal axis.

Figure 11

Total dynamical spin structure factor $S(\mathit{q},\omega )={\sum }_{\mu }{S}^{\mu \mu }(\mathit{q},\omega )$ ($S=1/2$). Parts (a)–(d) show the result for the SP solution $|G_3\rangle$ and $\alpha =0,0.25,0.4,0.5$, and (e)–(h) include the $1/N$ correction described in Fig. 4. The green dots indicate the magnon dispersion. Inset: high-symmetry path of the horizontal axis.

Figure 12

Linear extrapolation of the integral over frequency and momentum of the dynamical spin structure factor after correcting the simple BEC SP result [see Eq. (71)] with the contribution described in Fig. 4. The panels (a)–(c) correspond to extrapolations as a function of the width $\eta$ for $\alpha =0.25$ and finite lattices of ${\mathcal{N}}_s=144$, 576, and 1026, respectively. Panel (d) shows the finite-size scaling and the extrapolation to the thermodynamic limit of the extrapolated ($\eta \to 0$) results obtained in the previous panels. Panel (e) shows the final result after performing the $\eta \to 0$ and ${\mathcal{N}}_s\to \infty$ extrapolations for different values of $\alpha$.