Entanglement spectra of coupled $S=\frac{1}{2}$ spin chains in a ladder geometry

We study the entanglement spectrum of spin-$1/2$ $XXZ$ ladders both analytically and numerically. Our analytical approach is based on perturbation theory starting either from the limit of strong rung coupling, or from the opposite case of dominant coupling along the legs. In the former case we find to leading order that the entanglement Hamiltonian is also of nearest-neighbor $XXZ$ form although with an, in general, renormalized anisotropy. For the cases of $XX$ and isotropic Heisenberg ladders no such renormalization takes place. In the Heisenberg case the second-order correction to the entanglement Hamiltonian consists of a renormalization of the nearest-neighbor coupling plus an unfrustrated next-nearest-neighbor coupling. In the opposite regime of strong coupling along the legs, we point out an interesting connection of the entanglement spectrum to the Lehmann representation of single-chain spectral functions of operators appearing in the physical Hamiltonian coupling the two chains.

Figure 1

Two-leg spin ladder considered in this paper. The chains are labeled by $\nu =1,2$, while the runs are labeled by $m=1,...L$. We consider the entanglement spectrum in the illustrated chain-chain bipartition.

Figure 2

Overview of the entanglement spectrum of the reduced density matrix of a single chain (cf. setup in Fig. 1) in a Heisenberg ($\Delta =1$) spin ladder for different $S_{\mathrm{tot}}^z=S_{\mathrm{tot}}$ sectors as a function of $J_{\mathrm{leg}}/J_{\mathrm{rung}}$. The system size is $L=8$. While in the upper panel the bare entanglement spectrum is shown, in the lower panels the leading asymptotic behavior in the two limiting cases $J_{\mathrm{leg}}/J_{\mathrm{rung}}\to 0$ and $J_{\mathrm{rung}}/J_{\mathrm{leg}}\to 0$ is highlighted by appropriate shifts and rescaling indicated on the $y$ axes. The filled dots in all panels denote numerical entanglement spectrum levels, while the constant lines in the lower left panel highlight twice the eigenvalues of a single $L=8$, $S=1/2$ Heisenberg chain. The gray region in the lower right panel indicates the part of the rescaled entanglement spectrum affected by the finite (double) precision arithmetic used in the numerical calculations.

Figure 3

Filled circles: numerical entanglement spectra obtained for an $L=8$ system. For each $J_{\mathrm{leg}}/J_{\mathrm{rung}}$ value the entanglement spectrum has been rescaled to lie in the interval $[0,1]$. Empty circles: rescaled energy spectrum of the analytical second-order entanglement Hamiltonian, Eq. (15).

Figure 4

Filled diamonds: entanglement spectrum prediction for an $L=8$ Heisenberg spin ladder, based on the first-order perturbation theory in the coupling between the two chains Eq. (22). Empty diamonds: $S=0$ and $S=1$ entanglement levels obtained from exact diagonalization at $J_{\mathrm{rung}}/J_{\mathrm{leg}}=0.0001$ (cf. upper panel of Fig. 2). The gray region denotes the region where the numerical entanglement spectrum is affected by the finite precision arithmetics and where no numerical data is shown.