Spin density wave order, topological order, and Fermi surface reconstruction

In the conventional theory of density wave ordering in metals, the onset of spin density wave (SDW) order coincides with the reconstruction of the Fermi surfaces into small “pockets.” We present models which display this transition, while also displaying an alternative route between these phases via an intermediate phase with topological order, no broken symmetry, and pocket Fermi surfaces. The models involve coupling emergent gauge fields to a fractionalized SDW order, but retain the canonical electron operator in the underlying Hamiltonian. We establish an intimate connection between the suppression of certain defects in the SDW order and the presence of Fermi surface sizes distinct from the Luttinger value in Fermi liquids. We discuss the relevance of such models to the physics of the hole-doped cuprates near optimal doping.

Figure 1

Schematic, minimal phase diagram of the easy-plane Hamiltonian ${\mathcal{H}}_1$ in Eq. (1.9). The vortices are the usual defects in the XY SDW order $e^{i\theta }$. The Fermi surfaces are shown in the first Brillouin zone: those in A and B are of electrons, while those in C can be either of electrons or chargons. In phase C, the single vortices in the SDW order are gapped excitations, identified as the visons of the ${\mathbb{Z}}_2$ topological order. The sketched Fermi surfaces are for hole doping with the cuprate band structure: in phases B and C only hole pockets are shown, but electron pockets will appear near the boundaries to phase A. We propose that the SU(2) spin rotation invariant analogs of phase C (discussed in Sec. 4) describe the pseudogap state in the hole-doped cuprates. Other less-correlated superconductors (such as the pnictides) are proposed to bypass phase C and evolve directly from phase B to A.

Figure 2

Lowest-energy eigenvalues of $H_{hs}$ with one holon and one spinon with total momentum $\mathbf{q}$ on a lattice of size $48\times 48$. There are ${48}^2=2304$ eigenvalues at each $\mathit{q}$, and eigenvalues above $E_{hs}=24.0$ are not shown. Note the bound states (which have charge $e$ and spin $S_z=\pm 1/2$) below the two-particle continuum. The parameter values are $\lambda =30.0,\mathrm{\Delta }=8.0,t_1=3.0,t_2=2.0,t_3=2.0,J_1=0.6,J_2=0.1,J_3=0.1$, and $\mu =0$. The energy levels shift uniformly with changes in $\mu$, and the bound state will form a pocket with electron-like quasiparticles for large enough $\mu$. There is also [15] a Fermi surface of chargons associated with the single-holon states, which are not shown above.

Figure 3

Color density plot of the energy of the lowest holon-spinon bound state in Fig. 2. Parameter values are the same. Note that there is no particular symmetry of the bound-state dispersion associated with antiferromagnetic Brillouin zone: the minimum of the dispersion is not exactly at $(\pi /2,\pi /2)$. In contrast, the holon dispersion given by Eq. (2.19) does have a minimum at $(\pi /2,\pi /2)$.

Figure 4

A gauge-fixed state, $|{\mathrm{\Psi }}_{0v}\rangle$ of two visons marked with the X's. The dotted line connects links with ${\mu }_{ij}^z=-1$. The state $|{\tilde{\mathrm{\Psi }}}_{0v}\rangle$ is obtained after the top vison encircles the site, $j$, marked with a circle.