Modular Hamiltonians for future-perturbed states

TL;DR

This work develops a perturbative framework for the modular Hamiltonian $\overset{\leftrightarrow}{H}$ in a 2D CFT split into left and right wedges, perturbed by a local operator in the future wedge. By inserting the perturbation inside correlation functions and implementing a precise analytic continuation of complex modular flow, the authors derive an operator expression for the first-order change $\delta \overset{\leftrightarrow}{H}$ that includes a local future-wedge piece plus contact terms, and show that the KMS conditions are satisfied. A key outcome is writing $\delta \overset{\leftrightarrow}{H} = -i\lambda [E, \overset{\leftrightarrow}{H}^{(0)}]$, with $E$ defined via a Green’s-function-like integral; in many cases $E$ reduces to the perturbing operator ${\cal O}_2$, leading to significant simplifications. The analysis validates the construction through KMS analyticity and modular symmetry, clarifies the role of singularities and contact terms, and points to future directions including all-orders conjectures and extensions to smeared perturbations and past-wedge insertions.

Abstract

We develop a perturbative understanding of the modular Hamiltonian for a 2D CFT, divided into left and right half-spaces, with a weak local perturbation inserted in the future wedge. A formal perturbation series for the modular Hamiltonian is available, but must be properly interpreted in quantum field theory. We work inside correlation functions with spectator operators, and introduce a prescription for defining complex modular flow via analytic continuation to properly resolve singularities. From the correlators, we extract an operator expression for the modular Hamiltonian. It takes the form of a local operator in the future wedge plus contact terms with an unconventional singularity structure. Thanks to this structure the KMS conditions are satisfied, which independently establishes the validity of the results. Similar techniques apply to perturbations inserted in the past wedge. We mention various future directions, including an all-orders speculation for the excited state modular Hamiltonian.

Figures (7)

• Figure 1: The correlator is singular (i) when ${\cal O}_3$ is null separated from the perturbing operator ${\cal O}_2$, (ii) when ${\cal O}_3$ lies on a space-like hyperbola that passes through ${\cal O}_2$ or its CPT conjugate $\widetilde{{\cal O}_2}$, (iii) when ${\cal O}_1$ and ${\cal O}_3$ are connected by a null ray that bounces off the space-like hyperbola in $F$, (iv) when ${\cal O}_1$ and ${\cal O}_3$ are connected by a null ray that bounces off the space-like hyperbola in $P$.
• Figure 2: Light cone structure for the correlator $\langle \lbrace {\cal O}(\xi_1^+,\xi_1^-), {\cal O}^{-0}(\xi_2^+,\xi_2^-) \rbrace \rangle$. The operators commute on the dashed null ray but not on the solid null ray.
• Figure 3: Light cone structure associated with the second attempt at $\delta H_A^{(1)}$. Since it involves an integral of ${\cal O}_2^{-0}$ over a spacelike hyperbola in the past Rindler wedge, obtained by boosting the CPT conjugate point $(-\xi_2^+,-\xi_2^-)$, it is guaranteed to commute with local operators in the left Rindler wedge.
• Figure 4: Top panel: the starting point $r = 0$, showing some generic fixed and mobile poles as well as the pole at $w = -1$. The fixed poles and the pole at $w = -1$ are shown in red. The mobile poles, shown in black, are labeled A, B, C, D. The mobile poles rotate clockwise in $\delta H_A^{(1)}$. Middle panel: continuation to $r = \pi - \delta$. Mobile poles of type C get wrapped by the integration contour. The contour around C can be deformed to a $\wedge$-shaped form in the lower half plane, parametrized by $w = e^{\pm i \delta} x$ for $x \in {\mathbb R}$. The $\wedge$-shaped contour has the advantage that (after closing the contour at infinity) it only encircles mobile poles of type $C$, while avoiding all other fixed and mobile poles. Moreover it does this in a way that is universal, meaning the contour is independent of the positions of the spectator operators. Bottom panel: we can strip off the spectator operators to obtain a three-part contour integral expression for $\delta H_A^{(1)}$.
• Figure 5: Contour of $\delta H_A^{(1)}$ (left) and $\delta H_A^{(2)}$ (right). The potential pinch singularities as $r \rightarrow \pi$ are resolved by the CFT Wightman prescription, so as $r \rightarrow \pi$ the contours can be subtracted. This gives a contour that encircles the $w=-1$ pole, which produces the commutator in (\ref{['OneSideCommutator']}).