Auxiliary-fermion approach to critical fluctuations in the two-dimensional quantum antiferromagnetic Heisenberg model

The nearest-neighbor quantum antiferromagnetic (AF) Heisenberg model for spin-$1∕2$ on a two-dimensional square lattice is studied in the auxiliary-fermion representation. Expressing spin operators by canonical fermionic particles requires a constraint on the fermion charge $Q_i=1$ on each lattice site $i$, which is imposed approximately through the thermal average. The resulting interacting fermion system is first treated in mean-field theory (MFT), which yields an AF ordered ground state and spin waves in quantitative agreement with conventional spin-wave theory. At finite temperature a self-consistent approximation beyond mean field is required in order to fulfill the Mermin-Wagner theorem. We first discuss a fully self-consistent approximation, where fermions are renormalized due to fluctuations of their spin density, in close analogy to FLEX. While static properties like the correlation length, $\xi \left(T\right)\propto \exp (aJ∕T)$, come out correctly, the dynamical response lacks the magnon-like peaks which would reflect the appearance of short-range order at low $T$. This drawback, which is caused by overdamping, is overcome in a “minimal self-consistent approximation” (MSCA), which we derive from the equations of motion. The MSCA features dynamical scaling at small energy and temperature and is qualitatively correct both in the regime of order-parameter relaxation at long wavelengths $\lambda >\xi$ and in the short-range-order regime at $\lambda <\xi$. We also discuss the impact of vertex corrections and the problem of pseudo-gap formation in the single-particle density of states due to long-range fluctuations. Finally we show that the (short-range) magnetic order in MFT and MSCA helps to fulfill the constraint on the local fermion occupancy.

Figure 1

Diagrammatic representation of $\Sigma ,\Pi ,\Delta ⟨S^z⟩$ in the mean-field approximation, Sec. 3A. The thin dashed line is the bare interaction $J$ in Eq. (8c); the thick dashed line the renormalized one, Eq. (30), and the full lines are the Green’s function of pseudo fermions, Eq. (13) in the mean-field approximation, Eq. (15). The dots represent Pauli matrices $\times 1∕2$. A closed fermion loop requires a trace in spin space.

Figure 2

The self-consistent approximation for the paramagnetic phase at $T>0$ discussed in Sec. 4. Full lines denote the fermion propagator $\overline{G}$ first introduced in Eq. (13); the thick dashed line is the renormalized interaction $D$ defined in Eq. (30); the thin dashed line represents the bare interaction $J$. Shown are the fermion self-energy $\overline{\Sigma }$, Eq. (34), the irreducible bubble $\Pi$, Eq. (19), and the renormalized interaction $D$. Dots are Pauli matrices $\times 1∕2$. In the paramagnetic phase the traces in spin space are easily performed, leading to the set of Eqs. (35).

Figure 3

Self-energy diagrams of the ladder type for the exchange particle-hole (“RVB/flux”) channel (top row) and the particle-particle (pairing) channel (bottom row). The contribution of ${\overline{\Sigma }}^{ex}$ and ${\overline{\Sigma }}^{pp}$ has almost no effect on the results; see the text, Sec. 4.

Figure 4

Main figure: Dynamical structure factor from the numerical solution of the self-consistent approximation, Eqs. (35). Plotted is the scaling function $\Phi (k\xi ,\omega ∕{\omega }_0)$ at low energies $\sim {\omega }_0$. $\Phi$ is shown for fixed wave vectors $k=0,{\xi }^{-1},2{\xi }^{-1},5{\xi }^{-1}$, each for 2 different temperatures corresponding to the correlation lengths $\xi =12.1$, 101.2. The curves lie almost on top of each other, indicating that dynamical scaling holds for $\xi >10$. Inset: Analytically calculated scaling function $\Phi (k\xi ,\omega ∕{\omega }_0)$, Eq. (53), from the static approximation.

Figure 5

Spectral function $\rho \left(\omega \right)$ of the fermion propagator Eq. (35e). Full line: numerical solution of Eqs. (35). Dashed line: static approximation, Eq. (44). The arrows indicate delta peaks, representing $\rho \left(\omega \right)$ in the antiferromagnetic phase at $T=0$ in the mean-field theory, Eq. (17).

Figure 6

Top: Irreducible bubble $\widehat{\Pi }$ including the vertex corrections from the conserving approximation-scheme. Bottom: Bethe-Salpeter equation for the vertex function $\widehat{\Lambda }$. Lines are those introduced in Fig. 2, calculated self-consistently from the corresponding equations (35).

Figure 7

The minimal self-consistent approximation (MSCA) for the disordered phase at $T>0$ discussed in Sec. 5, Eqs. (60). The full line denotes the fermion propagator, Eq. (60e), which contains the self-energy $\Sigma$, Eq. (60d). The thick dashed line is the renormalized interaction $D$, Eq. (60c). Note that in contrast to the fully self-consistent approximation shown in Fig. 2, the fermion propagator in the self-energy $\Sigma$ as well as one of the fermion lines in the irreducible bubble $\Pi$, Eq. (60a), remain unrenormalized (bare). These are given by $G^0\left(i\omega \right)=1∕i\omega$. In the MSCA the only self-consistent quantity is the susceptibility $\chi$, Eq. (60b), that enters $D$.

Figure 8

Main figure: Correlation length $\xi \left(T\right)$ of the minimal self-consistent approximation (MSCA). Shown is the data from the numerical solution of Eqs. (60) and the function $\xi \left(T\right)=0.362(J∕T)\exp (0.359J∕T)$ resulting from a linear regression of the data, plotted as $\left[t(\ln \left(\xi \right)+\ln \left(t\right))\right]$, to the function $[a+t\ln \left(b\right)]$, $t=T∕J$ for $T⩽0.18J$. Inset: Energy scale ${\omega }_0\left(T\right)$ of the MSCA, extracted from the numerical solution of Eqs. (60) using Eq. (54). Shown is the numerical data for ${\omega }_0\left(T\right)\xi \left(T\right)$ and the function ${\omega }_0\xi =0.690T$, obtained from a linear regression for $T⩽0.18J$. The extrapolation to $T=0$ is ${\omega }_0\left(0\right)\xi \left(0\right)=0.00003J$.

Figure 9

Main figure: Dynamical structure factor from the numerical solution of the MSCA, Eqs. (60), at low energies $\sim {\omega }_0$. Plotted is the scaling function $\Phi (k\xi ,\omega ∕{\omega }_0)$ as defined in Eq. (55). $\Phi$ is shown for fixed wave vectors $k=0,{\xi }^{-1},2{\xi }^{-1},5{\xi }^{-1}$, each for 3 different temperatures corresponding to $\xi =3.6$, 44.4, 403.4. The curves for $\xi =44.4$, 403.4 are almost indistinguishable. For wave vectors $k>{\xi }^{-1}$ a propagating “spin-wave” mode becomes visible. Inset: The scaling function $\Phi (k\xi ,\omega ∕{\omega }_0)$ of the analytical calculation described in Sec. 5C, Eq. (85), for $p=0.25$.

Figure 10

Main figure: Numerical spectral function $\rho \left(\omega \right)$ of the fermion (full line) in the MSCA, Eqs. (60), in the high-energy range $\sim J$. The dashed line is the high-energy approximation, Eq. (77), from the analytical calculation. Inset: Numerical $\rho \left(\omega \right)$ at low energies $\sim {\omega }_0$; shown is the scaling function $\overline{\rho }(\omega ∕{\omega }_0)=(3JT∕{\omega }_0)\rho \left(\omega \right)$ introduced in Eq. (74). The curves for $\xi =44.4$ and 403.4 (dotted and full line, respectively) are almost identical. The dashed line is $\overline{\rho }(\omega ∕{\omega }_0)$ from the analytical calculation, Eq. (74).

Figure 11

Influence of constraint fluctuations on the local spin moment $S_{\mathit{loc}}^{\mathit{st}}=⟨{\left(S_i^x\right)}^2⟩$, calculated numerically using Eq. (B1). Shown is $4S_{\mathit{loc}}^{\mathit{st}}$ as a function of the correlation length $\xi \left(T\right)$ for the MSCA (crosses) and the fully self-consistent approximation (diamonds). For $\xi >10$ the MSCA reaches 85% of the exact value $4S_{\mathit{loc}}^{\mathit{st}}=\frac{4}{3}S(S+1)=1$. The dashed line indicates the high-$T$ limit $1∕2$.

Figure 12

Diagrammatic representation of Eq. (D7), which has been iterated. All diagrams corresponding to vertex corrections have been omitted; see the text. Thin and thick continuous lines denote Green’s functions $\tilde{G}$ and $G$, respectively. The dashed line is $-V$, fermion loops contain a — sign. The exchange diagrams, obtained via $1^{\prime }\leftrightarrow 2^{\prime }$, are not shown.