Entanglement spectra between coupled Tomonaga-Luttinger liquids: Applications to ladder systems and topological phases

We study the entanglement spectrum (ES) and entropy between two coupled Tomonaga-Luttinger liquids (TLLs) on parallel periodic chains. This problem gives access to the entanglement properties of various interesting systems, such as spin ladders as well as two-dimensional topological phases. By expanding interchain interactions to quadratic order in bosonic fields, we are able to calculate the ES for both gapped and gapless systems using only methods for free theories. In certain gapless phases of coupled nonchiral TLLs, we interestingly find an ES with a dispersion relation proportional to the square root of the subsystem momentum, which we relate to a long-range interaction in the entanglement Hamiltonian. We numerically demonstrate the emergence of this unusual dispersion in a model of hard-core bosons on a ladder. In gapped phases of coupled nonchiral TLLs, which are relevant to spin ladders and topological insulators, we show that the ES consists of linearly dispersing modes, which resembles the spectrum of a single-chain TLL but is characterized by a modified TLL parameter. Based on a calculation for coupled chiral TLLs, we are also able to provide a very simple proof for the correspondence between the ES and the edge-state spectrum in quantum Hall systems consistent with previous numerical and analytical studies.

Figure 1

“Cut and glue” approach (Ref. [50]) applied to a quantum Hall state. (a) We define the system on a cylinder of circumference $L$ and aim to study the entanglement between two halves of the cylinder, $A$ and $B$. (b) To this end, we physically cut the system into $A$ and $B$, obtaining gapless modes described by chiral TLLs, $H_R$ and $H_L$, on the new edges. We then glue them along the edges by switching on an interaction that couples $A$ and $B$. (c) The entanglement problem of the quantum Hall state can then be reduced to the problem of entanglement between the two coupled chiral TLLs.

Figure 2

Entanglement excitation spectra $\{{\xi }_i-{\xi }_0\}$ plotted against the subsystem momentum $k$ in the ladder model (94) with $t=1$, $V=-1$, $t^{\prime }=0$, and varying $U$ at the $1/3$-filling. (a) The spectrum for $U=2$ and $L=15$. (b) The lowest branch of the spectrum for $U=0.2$ (red), $1.0$ (green), and $2.0$ (blue) from top to bottom. The data for different system sizes $L=9$ (square), 12 (circle), and 15 (triangle) are plotted together. Horizontal lines indicate the energies of the flat dispersion, $w$ in Eq. (87d), predicted by field theory in Sec. 3c4. They are evaluated to be $w=7.79$, $4.59$, and $3.13$ for $U=0.2$, $1.0$, and $2.0$, respectively, using the TLL parameters $K_{\pm }$ determined in Ref. [65].

Figure 3

Entanglement excitation spectra $\{{\xi }_i-{\xi }_0\}$ in the ladder model (94) with $t=1$, $V=-1$, $U=2$, and varying $t^{\prime }$ at the $1/3$-filling. (a) The spectrum for $t^{\prime }=0.5$, and $L=15$. (b) Squared excitation energies $\left\{{({\xi }_i-{\xi }_0)}^2\right\}$ of the lowest branch for $t^{\prime }=0.1$ (red), $0.3$ (green), $0.5$ (blue), and $1.0$ (magenta) from top to bottom. The data for different system sizes $L=9$ (square), 12 (circle), and 15 (triangle) are plotted together. The linear lines are drawn by connecting the origin with the data points at the smallest $\left|k\right|\ne 0$. The dashed line in (a) is obtained by simply taking the square root of the linear line in (b).

Figure 4

The slope $4v_{\mathrm{e}}$ of the linear relation ${({\xi }_i-{\xi }_0)}^2\propto \left|k\right|$ found in Fig. 3, plotted against the interchain tunneling $t^{\prime }$. Logarithmic scales are taken for both axes. We fix $U=2$ and plot the data for the system sizes $L=9$ (square), 12 (circle), and 15 (triangle). For each $L$, we determine the slope $4v_{\mathrm{e}}$ by using the data at $k=2\pi /L$ as shown by lines in Fig. 3. We extrapolate the data of $v_{\mathrm{e}}$ to the limit $L\to \infty$ by fitting with a linear function in $1/L$; the obtained values are indicated by cross symbols. We fit the $L=\infty$ data by a power function (as shown by a dashed line), obtaining $4v_{\mathrm{e}}=5.48{\left(t^{\prime }\right)}^{-0.544}$. The exponent $0.544$ of this function is close to the expected value $1/[2-1/\left(2K_{-}\right)]=0.558$.