Partial time-reversal transformation and entanglement negativity in fermionic systems

The partial transpose of density matrices in many-body quantum systems, in which one takes the transpose only for a subsystem of the full Hilbert space, has been recognized as a useful tool to diagnose quantum entanglement. It can be used, for example, to define the (logarithmic) negativity. For fermionic systems, it has been known that the partial transpose of Gaussian fermionic density matrices is not Gaussian. In this work, we propose to use partial time-reversal transformation to define (an analog of) the entanglement negativity and related quantities. We demonstrate, for the symmetry-protected topological phase realized in the Kitaev chain, the conventional definition of the partial transpose (and hence the entanglement negativity) fails to capture the formation of the edge Majorana fermions, while the partial time reversal computes the quantum dimension of the Majorana fermions. Furthermore, we show that the partial time reversal of fermionic density matrices is Gaussian and can be computed efficiently. Various results (both numerical and analytical) for the entanglement negativity using the partial time reversal are presented for (1+1)-dimensional conformal field theories, and also for fermionic disordered systems (random single phases).

Figure 1

Path-integral representation of Rényi entropy for (a) multisheet and (b) single-sheet space-time manifold.

Figure 2

(a) Path-integral representation of moments of the partial time reversal ${\mathcal{E}}_n$ (49). (b), (c) Equivalent space-time manifold of ${\mathcal{E}}_n$ for two adjacent intervals which is a torus ${\mathrm{\Sigma }}_g$ of genus $g$ when $n$ is odd and ${\mathrm{\Sigma }}_g$ with an additional pinched torus when $n$ is even.

Figure 3

Space-time manifold of $\text{Tr}\left[{\rho }_A^{R_1}{\rho }_A^{R_1†}\right]$ or $\text{Tr}\left[{\left({\rho }_A^{R_1}\right)}^2\right]$ for two disjoint intervals (a) and two adjacent intervals (b). In each panel, equivalent loops appear in the same color. (c) Space-time manifold of ${\rho }_A^{R_1}{\rho }_A^{R_1†}$ or ${\left({\rho }_A^{R_1}\right)}^2$ for two adjacent intervals as a generator of ${\mathcal{E}}_n$ shown in Figs. 2 and 2.

Figure 4

Path-integral representation of Rényi entropy [(a) and (b)], partial TR (49) [(c) and (d)], in terms of $n$ decoupled partition functions with twist defects. (a), (c) Space-time picture; (b), (d) MPS representation ${\lambda }_k=e^{i2\pi k/n}$ and ${\tilde{\lambda }}_k=e^{i\delta }{\lambda }_k^{*}$ where $\delta =\pi$ or $\frac{\pi (n-1)}{n}$ for $n$ even or odd.

Figure 5

Logarithmic negativity of the Kitaev chain as a function of $\mu$ for two adjacent intervals with equal length $\ell$. (a) Comparison of different definitions of the partial transpose. From the legend, first curve (blue circles) is computed for the bosonic many-particle density matrix according to Eq. (4) in the Ising chain with periodic boundary condition. Second (red crosses) and third (green upward triangles) curves are computed for the fermionic many-particle density matrix according to the rules introduced in Ref. [54] and our paper Eq. (9), respectively. Fourth curve (orange downward triangles) is computed using the free-fermion formula (34). For all curves in this panel, we put $L=4\ell =8$. (b) Logarithmic negativity of the partial TR as computed in Eq. (34) for large systems $\ell =L/4$. All the data for fermionic chains are shown for antiperiodic boundary condition.

Figure 6

Logarithmic negativity of SSH model as a function of $t_2$ (60) for two adjacent intervals with equal length $\ell$. (a) Comparison of different definitions of the partial transpose. See the caption of Fig. 5 for details of what each curve represents. For all curves in this panel, we put $L=4\ell =8$ (16 fermion sites). (b) Logarithmic negativity of the partial TR as computed in Eq. (34) for large systems $\ell =L/4$. All the data for fermionic chains are shown for antiperiodic boundary condition.

Figure 7

Logarithmic negativity of two adjacent intervals evaluated by Eq. (34) at the criticality, (a) Kitaev chain (58) at $\mu =-2t$ where $c=\frac{1}{2}$, and (b) SSH model (60) at $t_2=t_1$ where $c=1$. The solid curves are the analytical results (87). The data are shown for a closed chain with antiperiodic boundary condition.

Figure 8

The random-singlet phase of disordered spin chains.

Figure 9

(a) Logarithmic negativity of the random XX chain versus subsystem length $\overline{\mathcal{E}}=c_1\left(\delta \right)ln\ell +k$ for clean limit $\delta =0$ and strong disorder $\delta =2$. (b) The slope $c_1$ as a function of disorder strength $\delta$ for various system sizes [see legend of (a)]. The average is over $2\times {10}^4$ realizations and error bars are smaller than symbol sizes.

Figure 10

Probability distribution $P\left(\mathcal{E}\right)$ of the entanglement negativity for the random XX chain with $L=8\ell =400$. We use ${10}^5$ samples to make these histograms.