Modular transport in two-dimensional conformal field theory

TL;DR

This work analyzes modular transport in a two-dimensional conformal field theory with U(1)×U(1) symmetry, focusing on interval bipartitions on the line and circle. By introducing spacetime-dependent chiral velocities and chemical potentials, the authors derive modular evolutions for chiral fields and currents, construct conserved charges, and formulate modular continuity equations. They compute modular correlators, demonstrate a thermal character via a KMS temperature ˜β=1, and establish modular Johnson–Nyquist-like noise and a fluctuation–dissipation relation for currents, extending to both infinite and finite volume. The results reveal ballistic modular transport governed by curl-free currents and provide a framework for exploring modular transport in more general CFTs and potentially holographic settings.

Abstract

We study the quantum transport generated by the bipartite entanglement in two-dimensional conformal field theory at finite density with the $U(1) \times U(1)$ symmetry associated to the conservation of the electric charge and of the helicity. The bipartition given by an interval is considered, either on the line or on the circle. The continuity equations and the corresponding conserved quantities for the modular flows of the currents and of the energy-momentum tensor are derived. We investigate the mean values of the associated currents and their quantum fluctuations in the finite density representation, which describe the properties of the modular quantum transport. The modular analogues of the Johnson-Nyquist law and of the fluctuation-dissipation relation are found, which encode the thermal nature of the modular evolution.

Figures (11)

• Figure 1: Modular trajectories generated by either the modular Hamiltonian (\ref{['fmh']}) (solid lines) or the modular momentum (\ref{['full-mod-momentum']}) (dashed lines), obtained from (\ref{['mod-traj-tau-line']}) and (\ref{['mod-traj-mom-gs']}) respectively. The coloured squares denote the images through the modular conjugation (\ref{['inversion-xt']}) of the spacetime points corresponding to the dots having the same colour. The dot dashed segments identify the partition $\mathcal{D}_A = \mathcal{D}_{\textrm{\tiny R}} \cup \mathcal{D}_{\textrm{\tiny L}} \cup \mathcal{D}_{\textrm{\tiny F}} \cup \mathcal{D}_{\textrm{\tiny P}}$ of the diamond $\mathcal{D}_A$.
• Figure 2: The modular hyperbolae $\mathcal{I}_{_P}$ in (\ref{['mod-hyper']}) correspond to the red, blue and green arcs, while the modular hyperbolae $\widetilde{ \mathcal{I} }_{_P}$ in(\ref{['mod-hyper-P-evo']}) is made by the purple, orange and dark yellow arcs. The asymptotes of $\mathcal{I}_{_P}$ and $\widetilde{ \mathcal{I} }_{_P}$ are the orange and blue thin straight lines respectively.
• Figure 3: Vector fields for the mean values of the charge currents (\ref{['mc1x']}) and (\ref{['mc1t']}), whose potential is the first expressions in (\ref{['potentials-W-line']}). The interval is $A=[0,1]$ on the line and the CFT has $c=1$, $\kappa = 3$ and either equal chemical potentials $\mu_{+} = \mu_{-} = 0.52$ (left panels) or different chemical potentials $\mu_{+} = 0.52$ and $\mu_{-} = 0.22$ (right panels). In the top panels the initial point $(x,t) \in \mathcal{D}_A \cup \mathcal{R}_A$, while in the bottom panels the vector field $\boldsymbol{j} (x,t)$ is extended to the entire Minkowski spacetime.
• Figure 4: Vector fields for the mean values of the helicity currents (\ref{['mc10x']}) and (\ref{['mc10t']}), whose potential is the second expressions in (\ref{['potentials-W-line']}), for either equal (left panel) or different (right panel) chemical potentials, in the same setup of Fig. \ref{['fig:j-curr']}.
• Figure 5: Vector fields for the mean values of the energy density currents (\ref{['mc11x']}) and (\ref{['mc11t']}), whose potential is the first expressions in (\ref{['potential-W-energy']}), for either equal (left panel) or different (right panel) chemical potentials, in the same setup of Fig. \ref{['fig:j-curr']}.