Finite temperature entanglement negativity in conformal field theory

TL;DR

This work resolves the incorrect naive approach to finite-temperature entanglement negativity in 1D conformal field theories by showing that the correct computation requires a four-point twist-field correlator with auxiliary insertions and a careful replica-limit order. The authors derive a universal finite-temperature negativity for a single interval, incorporating a nontrivial function of the full operator content and revealing cancellations that remove unphysical high-temperature divergences. They also provide semi-infinite and boundary-adapted generalizations, and confirm key predictions with numerical results from the critical harmonic chain, demonstrating a universal scaling for the subtracted negativity in terms of $ ext{ℓ}T$. The findings deepen the understanding of quantum entanglement in mixed states and highlight the essential role of operator content beyond the central charge in finite-temperature CFT entanglement measures.

Abstract

We consider the logarithmic negativity of a finite interval embedded in an infinite one dimensional system at finite temperature. We focus on conformal invariant systems and we show that the naive approach based on the calculation of a two-point function of twist fields in a cylindrical geometry yields a wrong result. The correct result is obtained through a four-point function of twist fields in which two auxiliary fields are inserted far away from the interval, and they are sent to infinity only after having taken the replica limit. In this way, we find a universal scaling form for the finite temperature negativity which depends on the full operator content of the theory and not only on the central charge. In the limit of low and high temperatures, the expansion of this universal form can be obtained by means of the operator product expansion. We check our results against exact numerical computations for the critical harmonic chain.

Figures (4)

• Figure 1: Partial transposition and ${\rm Tr} (\rho^{T_A})^n$ of one interval $A$ at finite temperature. Top: simple arrows indicate that in ${\rm Tr} (\rho^{T_A})^n$ one passes from the $j$-th copy to the $(j+1)$-th one by following them through the cut (depending on the verse of the arrow). Middle: deforming the dashed green line as indicated, part of it becomes a cut extending along the whole cylinder and parallel to its axis, while the remaining part merges with the red segment. Bottom: The double arrow denotes a double jump, from the $j$-th copy to the $(j+2)$-th one, following it through the cut. The $j$-th copy is sewed to the $(j+1)$-th one through the dashed line. Because of this connection, the $n_e$-sheeted Riemann surface occurring in ${\rm Tr} (\rho^{T_A} )^{n_e}$ does not factorise into the product of two identical $(n_e/2)$-sheeted Riemann surfaces.
• Figure 2: The deformation procedure described for Fig. \ref{['fig_argument']} (with the same notations) performed at zero temperature, i.e. on the sphere. In this case the cut given by the dashed green line can be shrunk to a point and it annihilates, leaving only the cut through $A$, which connect the $j$-th copy to the $(j+2)$-th one. Thus, at zero temperature, the $n_e$-sheeted Riemann surface occurring in ${\rm Tr} (\rho^{T_A} )^{n_e}$ factorizes into the product of two identical $(n_e/2)$-sheeted Riemann surfaces.
• Figure 3: Logarithmic negativity $\mathcal{E}$ for a subsystem composed of $\ell$ contiguous sites embedded in a critical harmonic chain of length $L$ with Dirichlet boundary condition. Left: $\mathcal{E}$ for a large range of $\ell T$. For all finite $\ell$, $\mathcal{E}$ vanishes for $T\geq T_{\rm sd}(\ell,L)$ (sudden death). Right: Subtracted negativity $\mathcal{E}_s \equiv \mathcal{E}(T)- \mathcal{E}(0)$ for small values of $\ell T$, i.e. for $T\ll T_{\rm sd}$.
• Figure 4: Subtracted logarithmic negativity $\mathcal{E}_s(T) \equiv \mathcal{E}(T)- \mathcal{E}(0)$ as in Fig. \ref{['fig neg chain 1']}. After shifting the temperature by $1/(2L)$, the data collapse on a master curve for temperatures smaller than those when finite size effects become important. Notice that the shift $1/(2L)$ vanishes in the thermodynamic limit.