Entanglement Spectra of Quantum Heisenberg Ladders

Bipartite entanglement measures are surprisingly useful tools to investigate quantum phases of correlated electrons. Here, I analyze the entanglement spectrum of gapped two-leg quantum Heisenberg ladders on a periodic ribbon partitioned into two identical periodic chains. The entanglement spectrum closely reflects the low-energy gapless spectrum of each individual edge. This extends the conjecture initially drawn for fractional quantum Hall systems to the field of quantum magnetism, stating a direct correspondence between the low-energy entanglement spectrum of a partitioned system and the true spectrum of the virtual edges. A mapping of the reduced density matrix to a thermodynamic density matrix is also proposed via the introduction of an effective temperature.

Figure 1

(a) Ribbon made of two coupled periodic Heisenberg chains (two-leg ladder). The partition into two identical $A$ and $B$ subsystems is made by cutting the rungs along the dashed line. (b) Phase diagram of the two-leg ladder mapped onto a circle assuming $J_{\mathrm{leg}}=\cos \theta$ and $J_{\mathrm{rung}}=\sin \theta$.

Figure 2

Various $S_n$ entanglement entropies ($n=1$ von Neumann $S_{\mathrm{VN}}$ entropy; $n=2$ and $n=\infty$ Rényi entropies) computed on $2\times L$ ladders of length up to $L=14$, normalized by $L\ln 2$ and plotted versus the angle $\theta$. Note that ${\xi }_0=S_{\infty }\equiv S_{\inf }$. Only the two right quadrants of Fig. 1b are considered. For comparison, the fluctuation of $S_A$ (normalized by $3L/4$) is also plotted (stars). The corresponding ES are shown in Fig. 3 for the values of $\theta$ marked by arrows. Inset: Effective inverse temperature ${\beta }_{\theta }$ (see text).

Figure 3

Entanglement excitation spectra versus total momenta $K$ in the chain direction for four different values of $\theta$ (shown by arrows in Fig. 2) corresponding to the rung singlet (I) phase (a),(b) and the Haldane phase (c),(d). All low-energy excitations computed on $2\times 10$, $2\times 12$, and $2\times 14$ ladders are shown by open (black) triangles, (red) squares, and (blue) circles, respectively. The lowest triplet eigenstates (for all $L$) are marked by (black) $+$ symbols and are fitted as $\Delta \xi =v|\sin (K)|$ by dashed lines. The lowest singlet eigenstates for $L=14$ are also marked by (red) $\times$ symbols.

Figure 4

(e),(f) The same as Fig. 3 for the rung singlet (II) phase (only $L=10$ and $L=14$ are shown). Here the GS is the saturated ferromagnet. The total spins $S$ of the lowest eigenstates are indicated by different symbols (and colors) and can be assigned to $m$-magnon bound states, $m=S_{\max }-S=L/2-S$. The lowest-energy excitations for $L=14$ are fitted according to $E_{\min }(K)$ (see text). In (e), the fit for $L=14$ rescaled by a factor $14/10$ (upper dotted line) also gives good agreement with the $L=10$ data.