Operationally accessible entanglement of one-dimensional spinless fermions

For indistinguishable itinerant particles subject to a superselection rule fixing their total number, a portion of the entanglement entropy under a spatial bipartition of the ground state is due to particle fluctuations between subsystems and thus is inaccessible as a resource for quantum information processing. We quantify the remaining operationally accessible entanglement in a model of interacting spinless fermions on a one-dimensional lattice via exact diagonalization and the density matrix renormalization group. We find that the accessible entanglement exactly vanishes at the first-order phase transition between a Tomonaga-Luttinger liquid and phase separated solid for attractive interactions and is maximal at the transition to the charge density wave for repulsive interactions. Throughout the phase diagram, we discuss the connection between the accessible entanglement entropy and the variance of the probability distribution describing intrasubregion particle-number fluctuations.

Figure 1

Phase diagram of the $t\text{−}V$ model accompanied by pictures of candidate ground states for $N=2$ fermions on a $L=4$ site lattice with antiperiodic boundary conditions. For the purposes of measuring accessible entanglement, the lattice has been bipartitioned into spatial subregions $A$ (blue) and $\overline{A}$ (red), each of size $\ell =2$. In the limit of strong attractive interactions where $V/t\ll -2$, the particles cluster together and there are $L$ equally probable configurations corresponding to all translations of the cluster. At the first-order phase transition where $V/t=-2$, all $\left(\genfrac{}{}{0pt}{}{L}{N}\right)$ configurations are equally probable resulting in a flat state. In the TLL phase with $|V/t|<2$, particles are delocalized and we have included a characteristic state corresponding to free fermions $(V=0)$. In the limit of strong repulsive interactions where $V/t\gg 2$, fermions maximize their distance from each other resulting in a charge density wave (CDW) phase. The open and closed circles on the $V/t$ axis denote a first-order and continuous phase transition, respectively.

Figure 2

Accessible entanglement entropy $S_{\alpha }^{\mathrm{acc}}\left(\ell \right)$ for $\alpha =1,2$ in the ground state of the $t\text{−}V$ model as a function of interaction strength $V/t$ at half-filling, $N=L/2$. The top panel shows the results for an odd number of total particles: $N=11,13,15$ and the bottom, for even: $N=12,14,16$. The solid and dashed gray vertical lines indicate the locations of the known phase transitions for the model $V/t=\pm 2$. For $N=15,16$ the asymptotic results computed in Sec. 3 in the limits $V/t\to \pm \infty$ for $S_1^{\mathrm{acc}}$ are shown as solid black lines.

Figure 3

Interaction strength at which the maximum $S_1^{\mathrm{acc}}$ occurs as a function of the total number of particles $N$. Filled shapes $(N\le 15)$ were computed with exact diagonalization while the remaining symbols for $N\ge 15$ are the result of density matrix renormalization group calculations. Finite-size corrections are investigated via a two-parameter fit of $lnN$ vs $ln(V/t-2)$ and support scaling toward the infinite-size phase transition at $V/t=2$. Inset: The interaction dependence of $S_1^{\mathrm{acc}}$ for various $N$ in the neighborhood of $\frac{V}{t}{|}_{\max }$ shows an evolving peak.

Figure 4

Difference between the von Neumann and accessible entanglement entropies $\mathrm{\Delta }{S}_1=S_1-S_1^{\mathrm{acc}}$ (large circles) and the Shannon entropy of a Gaussian distribution $\frac{1}{2}ln2\pi e{\sigma }^2$ (small circles) as functions of interaction strength $V/t$. Results were determined via exact diagonalization for the ground state of Eq. (18) with $N=15,16$. The observed agreement between the large and small symbols demonstrates the rapid convergence of $P_n$ to a Gaussian distribution as seen in the inset for $V/t=-1.5$ where $\mathcal{N}(\langle n\rangle ,{\sigma }^2)$ is a normal distribution with the same mean and variance as $P_n$. Solid lines are computed from the theoretical variance of the number of fermions in region $A$ inside the Tomonaga-Luttinger liquid phase ${\sigma }^2=K{\sigma }_{\text{FF}}^2$, where $K$ is the Luttinger parameter computed via Eq. (19) and ${\sigma }_{\text{FF}}^2$ is the exact variance for free fermions ($V/t=0$).

Figure 5

Interaction dependence of the difference between the Rényi and accessible entanglement entropy $\mathrm{\Delta }{S}_{\alpha }=S_{\alpha }-S_{\alpha }^{\mathrm{acc}}$. Large symbols are computed via exact diagonalization for the ground state of the $t\text{−}V$ model at half-filling with a spatial partition corresponding to $L/2$ sites. Small symbols are the classical Rényi entropies $H_{\alpha }$ computed from $P_n$, the probability of finding $n$ particles in the spatial subregion. We observe $H_{\alpha }\le \mathrm{\Delta }{S}_{\alpha }$ with the lower bound being nearly saturated over a wide range of interactions, but is worse for even $N$ and as $\alpha$ is increased from 2 to 10. This is quantified in the inset which compares $H_{\alpha }$ computed from the exact probability distribution $P_n$ at $V/t=-1.5$ with that obtained from a Gaussian with the same mean and variance as $P_n$ for $N=16$.

Figure 6

Rescaling the effective probability distribution defined in Eq. (14) for the ground state of the $t\text{−}V$ model at half-filling demonstrates the approximate proportionality relation $P_n\sim {\left(P_{n,\alpha }\right)}^{1/\alpha }$ for interaction strengths corresponding to the charge density wave (top row, $V/t=10$), Tomonaga-Luttinger liquid (middle row, $V/t=-1.5$), and cluster phases (bottom row, $V/t=-10$) for Rényi indices $\alpha =1,2,5,10$. ${\mathcal{A}}_{\alpha }^{-1}={\sum }_n{\left(P_{n,\alpha }\right)}^{1/\alpha }$ is a normalization constant and $P_{n,\alpha }$ is defined in Eq. (14). The shape of the distributions and their connection to the physical ground states of the $t\text{−}V$ model are discussed in the text.

Figure 7

The effective probability distribution $P_{n,\alpha =2}$ for the ground state of the $t\text{−}V$ model at half-filling and for different interaction strengths $V/t$ in the TLL phase is rescaled as ${\left[P_{n,2}\right]}^K$, were $K$ is the corresponding Luttinger parameter computed from Eq. (19). While the probabilities seem to show collapse near the middle of the distribution where $n\simeq N/2$, the inset shows strong additional $K$ dependence of the probability for fixed particle number $n=8$ in $A$. As discussed in the text, this lack of collapse is due to the subleading interaction-dependent corrections to the asymptotic scaling of the variance ${\sigma }^2$ in Eq. (27).

Figure 8

The filling fraction dependence of the variance ${\sigma }^2$ (black dots) of the particle-number distribution $P_n$ compared to that computed from the difference between the (exponentiated) accessible and spatial entanglement entropies for $\alpha =1$ (large circles) and $\alpha =2$ (small circles) using Eq. (29). Exact diagonalization results in the ground state of the $t\text{−}V$ model with $V/t=-1.5$ and $L=28$ demonstrate consistency with the predicted asymptotic scaling function $\mathcal{F}(N,\ell )$ given in Eq. (27) where we have fixed $\ell =L/2$ (left panel) or set $\ell =N$ (right panel) and changed $N$ from $2\cdots 14$. In order to reduce finite-size effects, both of the partition size $\ell$ and the total number of particles $N$ are replaced by their corresponding chord length $X\left(\ell \right)=(L/\pi )sin(\pi \ell /L)$ and $X\left(N\right)$, respectively. The constants ${\mathcal{B}}_1={\left(\pi e\right)}^{-1}$ and ${\mathcal{B}}_2={\left(2\pi \right)}^{-1}$ are used to rescale the entanglement reduction to obtain a prediction for the variance ${\sigma }^2$.