Excited State Modular Hamiltonian

• Excited state modular Hamiltonian is a generalized concept that extends the traditional ground state formalism to excited states, capturing nonlocal entanglement and operator dynamics.
• It employs advanced perturbative expansions, analytic continuation, and careful handling of contact terms to address singularities and maintain algebraic consistency.
• Its applications span holographic duality, quantum field theory, and quantum information, offering insights into entanglement dynamics and the evolution of complex quantum correlations.

An excited state modular Hamiltonian is the generalization of the modular Hamiltonian concept—traditionally associated with the reduced density matrix of a quantum system’s ground state—to the context where the reference state is an excited or perturbed pure state of the full system. The modular Hamiltonian $K_A$ associated to a subsystem $A$ and a given global state $\rho$ is defined as $K_A = -\log \rho_A$, with $\rho_A$ the reduced density matrix, and encodes crucial information about entanglement, correlation structure, and operator evolution within $A$. The excitation of the global state, as opposed to the ground state, leads to new forms of nonlocality, correlation, and often requires new methods for its explicit construction and analysis.

1. Operator Structure and Perturbative Expansions

The excited state modular Hamiltonian differs structurally and physically from its ground state counterpart. For ground states—especially in local or relativistic quantum field theories—$K_A$ often admits a quasi-local expression (e.g., an integral of the energy-momentum tensor for a half-space in the vacuum). In contrast, for excited states, explicit constructions rely on advanced operator techniques, analytic continuations, and, where possible, first-principle expansions.

Perturbatively, for a state prepared by perturbing the vacuum with a weak, localized operator (e.g., $|\psi\rangle = \exp(-i\lambda O)|0\rangle$), the change in the modular Hamiltonian to first order in $\lambda$ can be written as: $$\delta K = -i \lambda [E, \cdot] + \text{contact terms}$$ where $E$ is determined from the perturbing operator via path-ordered analytically-continued modular flow integrals, and the contact terms are necessary to resolve analytic ambiguities and to ensure the correct algebraic properties, especially the KMS condition for modular flow (Jiang et al., 22 Sep 2025). In cases with minimal singularity structure (for example, single local insertions away from the entanglement cut and spectator operators in generic positions), $E$ reduces to the perturbing operator itself so that $\delta K$ is simply a commutator, amounting to an infinitesimal symmetry action (e.g., a boost).

For higher-order corrections and more general states, the modular Hamiltonian can be formally expanded as a series of modular flowed insertions, often organized in terms of OPE blocks integrated along the modular flow generated by the vacuum modular Hamiltonian, with explicit recursive or perturbative structure (1705.01486, Lashkari et al., 2018, Jiang et al., 22 Sep 2025). The explicit appearance of contact terms (“endpoint contributions”)—integrals supported on the boundary of the entangling region—emerges for deformations of certain modular weight ($n \geq 2$), reflecting the capacity of some operators (e.g., the stress tensor) to “move” degrees of freedom across the interface between a region and its complement (Kabat et al., 2020).

2. Analytic Continuation and Singular Structure

Accurate definition and computation of the excited state modular Hamiltonian demand precise analytic control. Modular flow for excited states involves modular evolution by operators with complex time arguments, often requiring analytic continuation in both real and imaginary modular parameters. This leads to potential issues with singularities (pinched contours or ambiguous operator orderings), which are systematically resolved through specific $i\delta$-prescriptions for contours, ensuring all Wightman orderings and modular KMS analyticity are preserved (Jiang et al., 22 Sep 2025).

These care-intensive prescriptions guarantee that the resulting perturbed modular Hamiltonian not only implements the correct automorphism of the observable algebra (Tomita-Takesaki theory) but also preserves the fundamental cyclic-separating nature of the reference state. In summary, rigorous operator algebra and analytic continuation are indispensable for both the formal existence and explicit calculation of excited state modular Hamiltonians.

3. Endpoint and Contact Terms

A key phenomenon in excited state modular Hamiltonians is the emergence of endpoint—also known as contact—terms. Such terms are strictly localized at the entangling surface (the “endpoints” of the region $A$) and arise only when perturbing operators possess sufficient modular weight. For example, for perturbations of the vacuum modular Hamiltonian by a generator $G$ arising from integrating a local operator $J^{(n)}$ of modular weight $n \geq 2$ over a null plane: $$\delta K_\text{endpoint} = -2\pi \epsilon (n-1) \int d^{d-2}x_\perp\ f(0, x_\perp) \int_{-\infty}^\infty dx^+ J^{(n)}(x^+, 0, x_\perp),$$ otherwise zero for lower weights (Kabat et al., 2020). The endpoint terms are derived from careful reorganization of nonfactorizable operator contributions and are essential for representing how modular flow can “move” quantum degrees of freedom nonlocally.

These endpoint contributions are universal features, dictated by the structure of vacuum modular flow and the presence of operators capable of changing the subalgebra between a region and its complement. In integrable or conformal field theoretical models, these appear explicitly in series expansions or as summands in exact expressions for the modular Hamiltonian under local deformations, such as shape changes of the entangling region (Balakrishnan et al., 2020).

4. Holographic and Bulk Dual Interpretations

Within the AdS/CFT correspondence and, more generally, holographic frameworks, excited state modular Hamiltonians encode geometric features of the bulk spacetime that correspond to quantum entanglement structure at the boundary. For ground states of the boundary CFT, the modular Hamiltonian is dual to geometric evolution in the bulk—famously, the area of extremal (Ryu-Takayanagi) surfaces determines entanglement entropy.

Excitations about the vacuum correspond in the dual gravitational description to bulk perturbations, with the excited modular Hamiltonian driving linearized or backreacted metric perturbations. The commutators between the modular Hamiltonian and bulk operators generically do not vanish outside the causal wedge, reflecting that the modular Hamiltonian is a “precursor”—encoding information nonlocally in the bulk (Jafferis et al., 2014). Operator insertions along the modular flow correspond to integrals over bulk surfaces or volumes in the entanglement wedge, with analytic structure and contributions determined by the behaviour of the OPE blocks (“operator product expansion blocks”) under modular evolution (1705.01486, Das et al., 2019).

This duality manifests in the relation between quadratic corrections to the modular Hamiltonian (and associated relative entropy) and the canonical energy in the bulk, confirming a deep connection between entanglement, modular flow, and emergent spacetime geometry (1705.01486). In large-$N$ CFTs, the modular flow of coherent-excited states corresponds to canonical transformations in the bulk and the Tomita–Takesaki theory precisely maps the algebraic structure of the entanglement subregion in the boundary to the accessibility of the corresponding bulk region (2002.04637).

5. Algebraic and KMS Consistency

A robust excited state modular Hamiltonian must satisfy the abstract algebraic and analytic properties guaranteed for the vacuum case, notably modular automorphism, KMS (Kubo-Martin-Schwinger) analyticity, and symmetry under modular flow. Perturbative and all-orders constructions ensure that these properties are preserved, with all operator ambiguities and nontrivial singular structures resolved through contact terms and careful analytic continuation, as justified in detailed checks (Jiang et al., 22 Sep 2025).

The modular Hamiltonian’s action, written (to first order for future wedge perturbations) as $\delta = -i \lambda [E, \cdot]$, is shown to produce the correct automorphisms and analytic structure within correlation functions, with deviations only at higher orders or for more complicated state preparations. All calculations must validate that the modular condition remains satisfied, both at the operator and the state (vacuum or excited) level, independently establishing the rigor of the modular Hamiltonian construction.

6. Future Directions and Speculative All-Orders Constructions

The generalization of perturbative constructions to all orders presents both technical challenges and opportunities. There are indications that, at least for certain wedge-localized deformations, an exponentiated modular Hamiltonian $K = U K_0 U^\dag$ may exist, with $U = \exp(-i\lambda G)$ or, more generally, $U$ constructed from the analytically continued operator $E$ derived from the deformation. This would parallel existing all-orders constructions for modular flows of coherent states and those prepared by unitaries localized in half-spaces (Jiang et al., 22 Sep 2025, Lashkari et al., 2018, 2002.04637).

Such all-orders exponentiation would allow for a direct construction of the full Radon-Nikodym cocycle (the modular operator) for general excited states, with deep implications for the understanding of relative entropy, modular inclusions, and entanglement propagation. However, the analytic structure, necessity of contact terms, and proper operator ordering (as dictated by singularity and KMS conditions) remain open technical directions for ongoing and future work.

7. Significance and Outlook

The paper of the excited state modular Hamiltonian reveals essential features of quantum entanglement, information propagation, and operator nonlocality in many-body and quantum field systems. Beyond elucidating the structure of entanglement in excited states (as opposed to ground states), the modular Hamiltonian governs the dynamics of modular flow, operator spreading, and complexity growth. Its structure encodes information about microcanonical/thermalization phenomena, the stability of subregion duality in holography, and the fundamental organization of quantum correlations in arbitrary pure states. The ongoing development of precise analytic and algebraic techniques for its construction is therefore pivotal to advances in quantum field theory, holography, and quantum information theory.