Entanglement, anomalies, and Mathisson’s helices

We study the physical properties of a length-torsion functional which encodes the holographic entanglement entropy for ($1+1$)-dimensional theories with chiral anomalies. Previously, we have shown that its extremal curves correspond to the mysterious Mathisson’s helical motions for the centroids of spinning bodies. We explore the properties of these helices in domain-wall backgrounds using both analytic and numerical techniques. Using these insights we derive an entropic $c$-function $c_{\mathrm{Hel}}(\ell )$ which can be succinctly expressed in terms of Noether charges conserved along these helical motions. While for generic values of the anomaly there is some ambiguity in the definition of $c_{\mathrm{Hel}}(\ell )$, we argue that at the chiral point this ambiguity is absent.

Figure 1

In the Poincaré patch of asymptotically ${\mathrm{AdS}}_3$ spacetimes, we study curves which admit a tip $z_{*}$, i.e., a point where $\dot{z}=0$ and $\ddot{z}<0$. We show two curves with identical $z_{*}$ but different ${\zeta }_{*}$: this corresponds to a rigid boost, and hence the end points of the red curve lie on the $t=0$ line, while the blue curve end points are boosted. In numerical solutions, boundary quantities, such as $\ell$ and the boost $\kappa$, are computed at $z(\pm s_{\epsilon })=\epsilon$.

Figure 2

Example of escape regions: in the tip parameter space $(z_{*},k_{*}^1)$, we show which values allow for a boundary-reaching solution. The blue/yellow color coding stands for the $I_a$ and $I_c$ helix types, described in (2.15) and (2.16). In both plots we set $L_{\mathrm{UV}}/L_{\mathrm{IR}}=1/4$, $\mathfrak{m}=1$, ${\zeta }_{*}=0$, ${\dot{\zeta }}_{*}=0$. We used the interpolating function $f(z)$ in (c1). The left panel has $\mathfrak{s}=2$, while the right one has $\mathfrak{s}=1$. Since for the latter $\mathfrak{m}/\mathfrak{s}=1/L_{\mathrm{UV}}$, it represents the chiral point of the theory: notice the absence of type $I_a$ asymptotic behavior.

Figure 3

Graphical representation of a family of Mathisson helices, solutions of Eqs. (5.4) and (5.5). The color of each curve ranges from red to blue as $z_{*}$ is increased. This family is such that the boundary interval boost, $\kappa$, vanishes (i.e., the end points lie on the $t=0$ line) and the asymptotic total curvature $k_{\mathrm{FS}}^2$ is zero on both end points. These curves are not geodesics, and they are nonplanar: the vertical view on the right shows this unequivocally. Note that the boundary conditions $\kappa =0$ and ${\lambda }_{+}=1$ can be reached only with a specific choice of tip values ${\zeta }_{*}$ and $k_{*}^1$. This choice changes nontrivially for different $z_{*}$.

Figure 4

We take the escape region in the left panel of Fig. 2 and plot logarithmic contour lines $\ell (z_{*},k_{*}^1)$ (on the left) and $\widehat{\mathcal{F}}(z_{*},k_{*}^1)$ (on the right). For each point within the region, we carefully tuned ${\zeta }_{*}$ so that each Mathisson helix ends at the boundary with $\kappa =0$, as in Fig. 3. Within the escape region we also show two black lines, as exemplification of possible choices for curve prescriptions. The dashed (lower) black line corresponds to a prescription which as we change $\ell$ is also monotonic in $\widehat{\mathcal{F}}$. On the other hand, the dotted black line is an unsuitable choice because for increasing $\ell$, the value of $\widehat{\mathcal{F}}$ decreases (this happens where the line crosses the same contour twice).

Figure 5

The escape region at the chiral point, taken from the right panel of Fig. 2, with logarithmic contour lines of $\ell (z_{*},k_{*}^1)$ (on the left) and $\widehat{\mathcal{F}}(z_{*},k_{*}^1)$ (on the right). The vertical axis has been rescaled with respect to Fig. 2. The dot-dashed black line in the right panel shows the critical contour discussed in the main text. The value of $\widehat{\mathcal{F}}$ along this line is precisely the infrared CFT value of the central charge: therefore, monotonicity requires that any asymptotic prescription must lie below this limiting contour.

Figure 6

As in Fig. 5, we show the contour plots of $\ell$ and $\widehat{\mathcal{F}}$ within the escape regions at the chiral point, now for the domain wall metric (c2). From panels (a) to (d) we increase $L_{\mathrm{UV}}/L_{\mathrm{IR}}$ from 2 to 5. In all cases there is a critical contour: the only commonly valid prescription for the c-function is to choose asymptotically geodesic helices.