Entanglement entropy and spectra of the one-dimensional Kugel-Khomskii model

We study the quantum entanglement of the spin and orbital degrees of freedom in the one-dimensional Kugel-Khomskii model, which includes both gapless and gapped phases, using analytical techniques and exact diagonalization with up to 16 sites. We compute the entanglement entropy and the entanglement spectra using a variety of partitions or “cuts” of the Hilbert space, including two distinct real-space cuts and a momentum-space cut. Our results show that the Kugel-Khomski model possesses a number of new features not previously encountered in studies of the entanglement spectra. Notably, we find robust gaps in the entanglement spectra for both gapped and gapless phases with the orbital partition, and show these are not connected to each other. The counting of the low-lying entanglement eigenvalues shows that the “virtual edge” picture, which equates the low-energy Hamiltonian of a virtual edge, here one gapless leg of a two-leg ladder, to the “low-energy” entanglement Hamiltonian, breaks down for this model, even though the equivalence has been shown to hold for a similar cut in a large class of closely related models. In addition, we show that a momentum space cut in the gapless phase leads to qualitative differences in the entanglement spectrum when compared with the same cut in the gapless spin-1/2 Heisenberg spin chain. We emphasize the new information content in the entanglement spectra compared to the entanglement entropy, and using quantum entanglement, we present a refined phase diagram of the model. Using analytical arguments, exploiting various symmetries of the model, and applying arguments of adiabatic continuity from two exactly solvable points of the model, we are also able to prove several results regarding the structure of the low-lying entanglement eigenvalues.

Figure 1

The phase diagram of the one-dimensional Kugel-Khomskii model given in Eq. (1). Phase I is a fully polarized ferromagnetic (FM) state for both spin and orbital sectors. In phase II, the ground state is antiferromagnetic (AFM) for spin and FM for orbital degrees of freedom. In phase III, spin and orbital orders are reversed relative to phase II by $X\leftrightarrow Y$. Phases IV and VI are gapped dimerized phases, and phase V is a gapless phase with nontrivial spin-orbital entanglement. The SU(4) point at $X=Y=1/4$ is indicated by a “*” and the exactly solvable point $X=Y=3/4$ indicated by a + is in the non-Haldane phase with finite string order.

Figure 2

The spin-orbit entanglement entropy $S_{\mathrm{ent}}(X,Y)$. Surface plots of $S_{\mathrm{ent}}(X,Y)$ for (a) $L=8$ and (b) $L=12$ system sizes. The red and blue dots are indicative of the SU(4) symmetry ($X=Y=1/4$) and the Kolezhuk-Mikeska (KM) ($X=Y=3/4$) points, respectively. Note the two lines of discontinuous jumps in (a) in which the KM point (blue dot) lies on one of these lines. Contour plots for (c) $L=8$ and (d) $L=12$. The pink arrows represent streamlines in the direction of steepest descent.

Figure 3

(a) The excitation gap between the two lowest eigenstates for the $L=8$ chain in the ${\sum }_i{S}_i^z={\sum }_i{\tau }_i^z=0$ sector along the $X=Y$ line in the region of level crossings. Shown for qualitative comparison is $S_{\mathrm{ent}}$ in scaled units. In regions where the gap is nonzero, the linear momenta of the absolute ground state are also labeled. (b) The analogous result for $L=12$ chains.

Figure 4

(a) $S_{\mathrm{ent}}(X,Y)$ along the $X=Y$ line for chain lengths $L=8,12,16$. The data presented for $L=16$ include ground states from both time-reversal invariant linear momenta $K=0,\pi$. (b) The ground-state energy $E_0$ of $L=16$ chains from the relevant linear momenta sectors.

Figure 5

Orbital- or rung-cut entanglement spectra against linear momentum $p$ at generic points in the AFM phases for $L=12$. The $\circ$ symbols denote singlet levels. (a) $(X,Y)=(0.22,0.17)$ from the gapless V phase. There are 32 low-energy states across all $p$ that are separated by the gap $\Delta {\xi }_{\mathrm{high}}$ from a continuum of high-energy states. (b) $(X,Y)=(0.76,0.63)$ from the gapped VI phase. There are only two low-energy levels at $p=0,\pi$ which are separated by a gap $\Delta {\xi }_{\mathrm{low}}$ from a continuum of high-energy states. The position of $\Delta {\xi }_{\mathrm{high}}$ is above that of $\Delta {\xi }_{\mathrm{low}}$.

Figure 6

Momentum-resolved entanglement spectra for $L=12$ chains at points related by ${\mathbb{Z}}_2$ spin-orbital symmetry. The $\circ$ symbols denote singlet levels while the $\ominus$ symbols denote doublets. (a) $(X,Y)=(0.05,0.25)$ in phase V. (b) $(X,Y)=(0.25,0.05)$ in phase V, which is shifted by $p\to p+\pi$ relative to (a). (c) $(X,Y)=(0.62,0.62)$ in phase IV with ground-state momentum $K=0$. (d) $(X,Y)=(0.58,0.58)$ in phase IV with $K=\pi$. Note the $\pi$ shift and doublets at $p=\pm \pi /2$ are present regardless of whether or not the ground state is in phase V or IV.

Figure 7

Entanglement spectrum of an $L=12$ system from the exactly solvable KM point $X=Y=3/4$ for $K=\pi$. This point is in the non-Haldane phase with a finite string order parameter.[51] There are only two levels $\xi =\ln 2$ at $p=0,\pi$. The $K=0$ spectrum (not shown) is almost identical except the levels are shifted slightly from $\ln 2$. Away from the KM point, the spectrum becomes “dressed” with high-energy levels as shown in Fig. 5b where there is a finite entanglement gap $\Delta {\xi }_{\mathrm{low}}$ that separates the two sets of energies.

Figure 8

Entanglement spectra at the $X=Y=1/4$ SU(4) point at various system sizes. The $\circ$ symbols denote singlet levels while the $\ominus$ symbols denote doublets. (a) $L=8$. (b) $L=12$. (c) $L=16$. The doublets at $p=\pm \pi /2$ for $L=12$ are due to the ground-state momentum $K=\pi$. In all cases, the total number of low-lying energies below the entanglement gap obey the empirical relation $2^{L/2-1}$. (d) A finite size scaling analysis of the entanglement gap $\Delta {\xi }_{\mathrm{high}}$ taken at $p=0$. The positive intercept suggests that the gap remains open in the thermodynamic limit.

Figure 9

The (nonmomentum resolved) entanglement spectrum along the $X=Y$ symmetric line for sizes $L=8,12$, and 16. The blue (dark) points correspond to spectra from ground states with momentum $K=0$ and the green (light) to $K=\pi$. In (a) and (b), the absolute numerical ground state is used to derive an ES. Due to computing limitations, our data points for $L=16$ are more sparse and we have plotted separately the ES for $K=0$ and $\pi$ sector ground states in (c) and (d). In phase V, the absolute ground state is mostly the $K=0$ one [see Fig. 4b]. Overlaid as the red (solid) line is $S_{\mathrm{ent}}(X,X)$ in arbitrary units. Marked out in (a) is the high entanglement gap $\Delta {\xi }_{\mathrm{high}}$ that separates the set of $2^{L/2-1}$ low-energy levels from the rest of the entanglement spectrum. Also marked in (a) is the low entanglement gap $\Delta {\xi }_{\mathrm{low}}$ that separates two distinguished lowest levels of the gapped AFM phase from the rest of the spectrum. The region where the entanglement entropy rapidly drops lies in the region where the transition from the gapless region V into the gapped regions IV and VI in Fig. 1 occurs.

Figure 10

The (nonmomentum resolved) entanglement spectrum along the $X+Y=0.4$ line for $L=16$. The red (solid) line is the entanglement entropy. For $X\lesssim 0.06$ and $X\gtrsim 0.36$, the structure of the entanglement spectra collapses as these states have trivial entanglement. (They are the FM phases II and III of Fig. 1.) For $0.06\lesssim X\lesssim 0.36$, the entanglement gaps $\Delta {\xi }_{\mathrm{high}}$ and $\Delta {\xi }_{\mathrm{low}}$ are seen. This path corresponds to a slice through region V.

Figure 11

Correlations functions taken from the SU(4) symmetric point ground state for $L=16$. (a) The entanglement spectrum where the blue $\circ$ symbols denote the $N\left(16\right)=128$ levels below the entanglement gap $\Delta {\xi }_{\mathrm{high}}$. The two lowest levels with the greatest Schmidt weight are marked by the solid red $•$ symbols. These levels are adiabatically connected to the two levels of the KM spectrum. (b) The spin-spin ground-state correlation function. (c) The dimer-dimer ground-state correlation function. In (b) and (c), the open circles $\circ$ are computed using only levels below $\Delta {\xi }_{\mathrm{high}}$, while the solid $•$ use only the two lowest levels of (a).

Figure 12

Momentum-resolved entanglement spectra for $L=12$ chains at the KM point with $J_{\mathrm{rung}}$ coupling (a) $J_{\mathrm{rung}}=0.001$. (b)$J_{\mathrm{rung}}=0.01$. (c) $J_{\mathrm{rung}}=0.1$. (d) $J_{\mathrm{rung}}=0.5$. In (d), the spectrum resembles the energy spectrum of a Heisenberg spin-1/2 chain and the ES of a two-leg ladder without the four spin or dimer-dimer interaction term. The degeneracies of the levels are denoted by the symbols displayed in the legend.

Figure 13

Momentum-resolved entanglement spectra for $L=12$ chains at the SU(4) point with $J_{\mathrm{rung}}$ coupling (a) $J_{\mathrm{rung}}=0.15$. (b)$J_{\mathrm{rung}}=0.2$. (c) $J_{\mathrm{rung}}=0.25$. (d) $J_{\mathrm{rung}}=0.35$. The degeneracies of the levels are denoted by the symbols displayed in the legend.

Figure 14

The entanglement spectrum at the KM point plotted vs system size. The black dots mark the degeneracy of each level. For eight sites, there is a residual effective coupling between virtual edge excitations in $H_{\mathrm{ent}}$ that splits the fourfold degenerate states into singlets and triplets. For 12 sites where the edges are more spatially separated, the fourfold or twofold degenerate per edge is restored to a good approximation. The eigenvalues are equal up to the third decimal place.

Figure 15

The mean number of eigenvalues, $n\left(\lambda \right)$, of the reduced density matrix larger than a given eigenvalue, $\lambda$, plotted as a function of $z=2\sqrt{b\ln [{\lambda }_{\max }/\lambda )}]$ in Eq. (24) for $l=4$ and $L=12$. The blue circles represent data for $X=0.25$. The lowest level at $z=0$ is nondegenerate and the next level is 15-fold degenerate. The green squares represent data for $X=0.2$. The lowest level at $z=0$ is nondegenerate and the next level is ninefold degenerate. The red line is the analytical prediction.[81]

Figure 16

The momentum cut entanglement spectrum at the SU(4) symmetric point for $L=12$ in the sector $(N_A,{\check{N}}_A)=(3,3)$. The ground state possesses momentum $K=\kappa =L/2$ in units of $2\pi /L$. The count of the lowest levels from high $M_A$ (right) is 1,2,5,10 and from low $M_A$ (left) is 1,2,5,8. The floating numbers in the plot denote the total number of levels directly below. There are two pairs of levels that are almost degenerate at $M_A=27$ and not resolved at this scale. The spectrum does not exhibit an entanglement gap despite being in the gapless phase V described by a sum of two SU(2)${}_2$ WZW theories or a single SU(4)${}_1$ WZW theory.

Figure 17

The total weight of the occupation basis coefficients taken from the ground state at the SU(4) point. The system size is $L=8$ and the ground-state momentum is $K=0$. The distribution of weights in distributed across three sectors, $M_{\mathrm{tot}}=2{\kappa }^2,2{\kappa }^2\pm L$, where $\kappa =L/2=4$.

Figure 18

Entanglement spectra arranged in ascending order taken from the SU(4) symmetric point of an $L=8$ chain in the sector $(N_A,{\check{N}}_A)=(2,2)$. Shown by the boxes ($\square$) is the spectrum from first blocking ${\rho }_{\mathrm{red}}$ and the crosses ($+$) the spectrum without blocking ${\rho }_{\mathrm{red}}$. The two spectra agree at the lowest energies and differ at the highest levels $\xi \gtrsim 40$.

Figure 19

The momentum cut entanglement spectrum of the Kugel-Khomskii chain at the SU(4) symmetric point for $L=8$. The spectrum is taken from into $M_A$ blocks of ${\rho }_{\mathrm{red}}$ in the sector $(N_A,{\check{N}}_A)=(2,2)$. The count of the lowest levels from high $M_A$ (right) is 1,2,5 and from low $M_A$ (left) it is 1,1,3.