Entanglement Entropy of Fermi Liquids via Multidimensional Bosonization

The logarithmic violations of the area law, i.e., an “area law” with logarithmic correction of the form $S\sim L^{d-1}\log L$, for entanglement entropy are found in both 1D gapless fermionic systems with Fermi points and high-dimensional free fermions. This paper shows that both violations are of the same origin, and that, in the presence of Fermi-liquid interactions, such behavior persists for 2D fermion systems. In this paper, we first consider the entanglement entropy of a toy model, namely, a set of decoupled 1D chains of free spinless fermions, to relate both violations in an intuitive way. We then use multidimensional bosonization to rederive the formula by Gioev and Klich [D. Gioev and I. Klich, Entanglement Entropy of Fermions in Any Dimension and the Widom Conjecture, Phys. Rev. Lett. 96, 100503 (2006).] for free fermions through a low-energy effective Hamiltonian and explicitly show that, in both cases, the logarithmic corrections to the area law share the same origin: the discontinuity at the Fermi surface (points). In the presence of Fermi-liquid (forward-scattering) interactions, the bosonized theory remains quadratic in terms of the original local degrees of freedom, and, after regularizing the theory with a mass term, we are able to calculate the entanglement entropy perturbatively up to second order in powers of the coupling parameter for a special geometry via the replica trick. We show that these interactions do not change the leading scaling behavior for the entanglement entropy of a Fermi liquid. At higher orders, we argue that this should remain true through a scaling analysis.

Popular Summary

In the fascinating world of quantum mechanics, particles that are separated in space-time can be correlated. Quantum entanglement refers to such counter-intuitive phenomena, and can in fact be made quantitatively tangible by appropriate measures. The commonly used measure is the so-called entanglement entropy which tells us how much quantum randomness there is in a subsystem of a larger one due to the quantum entanglement between the subsystem and the rest of the larger one when the state of the larger system has no randomness. Entanglement entropy has emerged as an encompassing concept that is important to several seemingly unrelated fields, black-hole physics, quantum information science, and quantum many-body physics (or condensed matter physics), and has been shown to follow elegant “area laws”: The entropy scales with the size of the boundary (“surface area”) of the subsystem, either simply linearly or almost linearly with a small logarithmic correction. In this theoretical paper, we reveal the origins of the logarithmic corrections seen in a number of quantum many-body systems and show that they are common.

The “area law” involving only linear scaling has an intuitive interpretation: When the coupling between the quantum particles is short-ranged, only particles within this range near the boundaries are “entangled” and contribute to the entanglement entropy. But, where do the logarithmic corrections come from? Two systems of particular interest to condensed matter physics that show logarithmic corrections are one-dimensional chains of fermions with gapless electronic structures and also noninteracting fermions in higher dimensions. Using a technique called multidimensional bosonization, we show that the logarithmic corrections in both systems share a common origin: the singularity at the Fermi surface.

Furthermore, generalizing the above work, we also investigate more interesting and difficult systems of Fermi liquids (i.e., fermions with interactions) by combining the bosonization technique with the replica trick. Our results strongly suggest that the “area law” of a Fermi liquid has the same leading scaling behavior as that of a free Fermi gas with the same Fermi surface.

We believe that our work not only represents much-needed advance in the current understanding of the “area laws,” but also offers possibilities of many future explorations, including investigations of non-Fermi liquids and of entanglement.

Figure 1

The toy model both in real space and in momentum space. (a) A set of parallel decoupled $1d$ chains of spinless free fermions (dashed lines); the subsystem division is represented by the solid lines, both convex and concave geometries. (b) Fermi surfaces of the toy model.

Figure 2

Patching of the Fermi surface. The low-energy theory is restricted to within a thin shell about the Fermi surface with a thickness $\lambda \ll k_F$, in the sense of renormalization. The thin shell is further divided into $N$ different patches; each has a transverse dimension ${\Lambda }^{d-1}$ where $d=2$, 3 is the space dimensions. The dimensions of the patch satisfy three conditions: (1) $\lambda \ll \Lambda$ minimizes interpatch scattering; (2) $\Lambda \ll k_F$ and ${\Lambda }^2/k_F\ll \lambda$ together makes the curvature of the Fermi surface negligible. (a): Division of a 2D Fermi surface into $N$ patches. Patch $\mathit{S}$ is characterized by the Fermi momentum ${\mathit{k}}_{\mathit{S}}$. (b): A patch for $d=3$. The patch has a thickness $\lambda$ along the normal direction and a width $\Lambda$ along the transverse direction(s).

Figure 3

Origin of the bosonic commutator of patch density operators illustrated for $d=2$. As shown in Eq. (7), the commutator is reduced to computing the difference of occupied states, i.e., the area difference below the Fermi surface, between the two $\theta$ functions ($\theta (\mathit{S};\mathit{k}-\mathit{q})-\theta (\mathit{S};\mathit{k}+\mathit{p})$). The solid box indicates the original patch, or $\theta (\mathit{S};k)$. The red line shows the Fermi surface. Both $\theta (\mathit{S};\mathit{k}-\mathit{q})$ and $\theta (\mathit{S};\mathit{k}+\mathit{p})$ are denoted by dashed boxes. The occupied part in $\theta (\mathit{S};\mathit{k}+\mathit{q})$ is denoted by blue, that of $\theta (\mathit{S};\mathit{k}-\mathit{q})$ is denoted by red, and the overlapping region is denoted by yellow. Subtracting the remaining blue area from the red, we obtain that $\theta (\mathit{S};\mathit{k}-\mathit{q})$ occupies $(\Lambda -q_{\mathit{S}\perp })q_{{\widehat{\mathit{n}}}_{\mathit{S}}}$ more states, which gives us the commutator.

Figure 4

Path integral representation of the reduced density matrix. (a) When we sew $\varphi (x{)}^{\prime }=\varphi (x{)}^{\prime \prime }$ together for all $x$’s, we get the partition function $Z$. (b) When we sew only $x\mathrm{\in ̸}A$ together, we get ${\rho }_A$.

Figure 5

Formation of the $n$-sheeted Riemann surface in the replica trick. By sewing $n$ copies of the reduced density matrices together, one obtains the replica partition function $Z_n$. In the zero temperature limit, $\beta \to \infty$, each cylinder representing one copy of ${\rho }_A$ becomes an infinite plane. Those $n$-planes sewed together form a $n$-sheeted Riemann surface in Fig. 5 which can be simply realized by enforcing a $2n\pi$ periodicity on the angular variable of the polar coordinates of the $(1+1)d$ plane instead of the usual $2\pi$ one. (a): $n$ copies of the reduced density matrices. For clarity only $n=2$ is shown. (b): Visualization of a $n$-sheeted Riemann surface.

Figure 6

The half-cylinder geometry and equivalence of boundary conditions in $\widehat{x}\mathrm{\text{−}}\widehat{\tau }$ and ${\widehat{\mathit{n}}}_{\mathit{S}}\mathrm{\text{−}}\widehat{\tau }$ planes. The system is infinite in the $\widehat{x}$ direction while obeys periodic boundary condition along the $\widehat{y}$ direction with length $L$. The system is cut along the $\widehat{y}$ axis so that we are computing the entanglement entropy between the two half planes. (a): The half-cylinder geometry. (b): The projection of ${\mathit{r}}_{\mathit{S}}$ onto the $\widehat{x}\mathrm{\text{−}}\widehat{\tau }$ plane. Consider polar coordinates of an arbitrary ${\widehat{\mathit{n}}}_{\mathit{S}}\mathrm{\text{−}}\widehat{\tau }$ plane (the blue plane). Since the polar coordinates in the $\widehat{x}\mathrm{\text{−}}\widehat{\tau }$ plane satisfy the $2n\pi$ periodic boundary condition, consider the one-to-one projection of the vector ${\mathit{r}}_{\mathit{S}}$ onto the $\widehat{x}\mathrm{\text{−}}\widehat{\tau }$ plane. Consider, if we move the vector in the $\widehat{x}\mathrm{\text{−}}\widehat{\tau }$ plane around the origin $n$ times (the red circle). Because of the one-to one mapping, ${\mathit{r}}_{\mathit{S}}$ should also move around the origin $n$ times (the blue “circle”; it is actually an eclipse), thus obeys the $2n\pi$ periodicity as well.