Modular Hamiltonian in QFT & Holography

• Modular Hamiltonian is defined as the logarithm of the reduced density matrix, central to understanding entanglement and quantum information across QFT and gravity duals.
• It admits local closed-form expressions in high symmetry cases such as Rindler wedges and spherical regions, where it is expressed via energy density integrals.
• In general or excited states, its nonlocal nature necessitates perturbative and replica methods to capture complex entanglement structures and modular flow dynamics.

A modular Hamiltonian is the generator of modular flow for a specified subregion or subalgebra in quantum systems, quantum field theory (QFT), and quantum gravity. It is defined via the logarithm of the reduced density matrix associated with a spatial subregion, and plays a central role in rigorous formulations of entanglement, quantum information in field theory, and holographic duality. Although for general subregions the modular Hamiltonian is highly nonlocal and complicated, in a class of solvable situations (such as regions with special symmetry, free and Gaussian theories, or holographic duals with high symmetry), it admits compact, sometimes local representations. Modular Hamiltonians encode both the entanglement structure and the automorphism group for operator algebras attached to regions, and are fundamentally linked to Tomita–Takesaki theory.

1. Definition and Algebraic Formalism

Given a quantum system with Hilbert space $\mathcal{H}$ and global state $\rho$, the reduced density matrix $\rho_A$ for a subregion $A$ or subalgebra $\mathcal{A}$ is defined as $\rho_A = \operatorname{Tr}_{\bar{A}} \rho$. The modular Hamiltonian $K_A$ is given by

$K_A = -\log\rho_A + \text{const},$

with the constant fixed to enforce $\operatorname{Tr}(\rho_A K_A) = 0$ when necessary. In the language of operator algebras, Tomita–Takesaki theory constructs, for a von Neumann algebra $\mathcal{M}$ with a cyclic and separating state $\Omega$, the Tomita operator $S$ by $S A \Omega = A^*\Omega$ for $A \in \mathcal{M}$, with polar decomposition $S = J \Delta^{1/2}$ giving the modular conjugation $J$ and the modular operator $\Delta$. The modular Hamiltonian is then $K = -\log \Delta$ and generates the modular flow $$\sigma_t(A) = \Delta^{it} A \Delta^{-it} = e^{it K} A e^{-it K}$$ (2002.04637, Fröb, 16 Jan 2025).

In QFT, modular Hamiltonians are typically unbounded operators, and their action is well-defined on dense analytic domains. The modular Hamiltonian encodes all information about the automorphism group of $\mathcal{A}$ and is related to key quantum information-theoretic metrics such as relative entropy $$S_{\rm rel}(\rho \| \sigma) = \operatorname{Tr} (\rho \log\rho - \rho \log\sigma )$$.

2. Explicit Modular Hamiltonians: Free Theories and Symmetric Regions

For free (Gaussian or quasi-free) quantum field theories, or in situations with enhanced symmetry (notably the vacuum state restricted to Rindler wedges or spherical regions), local closed-form expressions exist for modular Hamiltonians.

Rindler wedge: In any relativistic QFT, the Bisognano–Wichmann theorem gives the vacuum modular Hamiltonian for the half-space $x^1>0$ as: $K = 2\pi \int_{x^1 > 0} d^{d-1}x\, x^1\, T_{00}(x)$ where $T_{00}$ is the local energy density (Zhang et al., 2020, Jafferis et al., 2014).

Spherical region in CFT$_d$: For a ball of radius $R$ in a CFT, conformal symmetry gives

$$K = 2\pi \int_{|\vec{x}| < R} d^{d-1}x\, \frac{R^2 - |\vec{x}|^2}{2R}\, T_{00}(\vec{x})$$

which is manifestly local (Huerta et al., 2022, Huerta et al., 2023, Piana et al., 2024). This form persists after dimensional reduction to scalar and Dirac theories restricted to one dimension, even when conformal symmetry is partially broken.

For free, massive theories, the modular Hamiltonian for general regions (e.g., double cones) is nonlocal and can only be computed in terms of integral kernels involving the two-point functions of the theory, as in the Peschel–Casini–Huerta formulas (Fröb, 16 Jan 2025, Cadamuro, 2023). For massive scalars or small-mass fermions, perturbative expansions in the mass parameter recover nonlocal contributions tied to broken conformal symmetry (Cadamuro, 2023, Cadamuro et al., 2023).

Gaussian states on the Weyl algebra: The modular Hamiltonian on a spatial region $R$ is quadratic in the fields, fully determined by the restricted two-point function, with explicit operator-theoretic expressions (Fröb, 16 Jan 2025, Longo, 2021).

3. Modular Hamiltonians in Holography and Gravity Duals

In holographic theories, modular Hamiltonians have a precise dual description in terms of gravitational charges associated with bulk isometries or asymptotic symmetries homologous to the boundary region.

In the AdS/CFT context, the covariant holographic entanglement entropy (HRT) proposal connects the entanglement entropy to the area of an extremal bulk surface. The leading-order modular Hamiltonian for a boundary region $R$ in a pure AdS/CFT vacuum is geometrically local and dual to the charge generating the associated bulk Killing vector field.

Corrections to this, and to the modular flow on operators and bulk fields, are encoded in the first law of entanglement $\delta S_A = \delta\langle K_A\rangle$ and can be computed using linear response or the Euclidean replica trick. The modular Hamiltonian acts as a quantum precursor, with nontrivial commutators with bulk operators that are spacelike-separated from $R$ except in highly symmetric cases (Jafferis et al., 2014).

Flat holography and BMSFTs: For 2D BMS-invariant field theories (BMSFTs), the vacuum modular Hamiltonian for an interval is derived algorithmically as a linear combination of BMS$_3$ generators vanishing at the endpoints: $$K_A = Q[\xi] = -\int_{\varphi_-}^{\varphi_+} d\varphi\, [T(\varphi) P(\varphi) + Y(\varphi) J(\varphi)]$$ where $T$ and $Y$ are smearing functions fixed by geometric constraints. Holographically, $K_A$ matches the gravitational charge associated to an asymptotic Killing vector in 3D generalized minimal massive gravity (GMMG), and the correspondence holds in both non-thermal and thermal states. In the Einstein gravity limit, all higher-derivative and Chern-Simons deformations decouple, and the modular Hamiltonian reduces to the pure Einstein gravity result (Setare et al., 2022).

4. Modular Hamiltonians for Excited States and Deformations

For arbitrary excited states and deformed regions, the modular Hamiltonian generally cannot be written in closed, local form. Several approaches provide perturbative control:

• Tomita–Takesaki theory constructs modular Hamiltonians for coherent excitations, yielding a "shifted" operator under a large-$N$ approximation, where the shift is essentially classical and determined by boundary sources (2002.04637, Balakrishnan et al., 2020).
• Replica/analytic continuation methods provide formulae for arbitrary matrix elements of $K = -\log\rho$ in excited CFT states, expanding around free, large-$c$, or symmetry-protected cases. These allow computation of relative entropy and the quantum Fisher information metric in terms of correlation functions on branched covering geometries (Lashkari, 2015).
• Perturbative series and analytic continuation in modular time address explicit operator forms of $K$ for deformed or nonvacuum states. The series may require regularization, analytic continuation, and proper treatment of contact terms to preserve KMS and automorphism properties (Jiang et al., 22 Sep 2025, Balakrishnan et al., 2020).

In all these cases, the first law of entanglement remains a robust linear-order relation $\delta S_A = \delta\langle K_A\rangle$, while higher-order corrections in deformations parameterize genuinely new Fisher information content, especially on entanglement plateaux where the modular Hamiltonian's nonlinear terms encode features not present in the entanglement entropy alone (Abt et al., 2018).

5. Nonlocality, Complexity, and Spectral Structure

Except for high symmetry cases, modular Hamiltonians are highly nonlocal operators. In free theories and holographic models, their structure is reflected in:

• Spectral decompositions: The entanglement spectrum (eigenvalues of $\rho_A$) directly relates to the spectrum of $K_A$, encoding all modular flows and complexity measures (Caputa et al., 2023).
• Lanczos/Krylov complexity: The evolution of states or operators under modular flow can be studied via the Lanczos tridiagonalization (Krylov) basis. The spread of a state in Krylov levels under modular evolution, known as spread complexity, exhibits universal growth (governed by a modular Lyapunov exponent $\lambda_L^{mod} = 2\pi$ in 2D CFT), with saturation controlled by the entanglement spectrum (Caputa et al., 2023).
• Nonlocal modular kernels: For generic regions or massive deformations, the modular Hamiltonian kernel is nonlocal in position space and sensitive to boundary, mass, and topology effects (Cadamuro, 2023, Cadamuro et al., 2023).

6. Applications and Holographic Generalizations

• Boundary theory diagnostics: Modular Hamiltonians provide the generator for the automorphism group acting on a region's algebra of observables. Observables such as relative entropy, mutual information, and quantum Fisher information are functionals of modular Hamiltonians.
• Bulk duals of modular Hamiltonians: In gauge/gravity duality, the modular Hamiltonian for certain regions corresponds to geometric generators in the bulk (e.g., Rindler/diamond modular Hamiltonians correspond to boost/Killing charges), while more generic regions reflect the structure of quantum extremal surfaces and bulk relative entropy (Jafferis et al., 2014).
• Flat-space and BMS-invariant models: In asymptotically flat holography, modular Hamiltonians for BMSFTs correspond exactly to gravitational charges of boundary-preserving vectors in flat or massive-gravitational 3D theories. The modular first law holds both in non-thermal and thermal settings, and all results reduce seamlessly to the Einstein gravity case under appropriate limits (Setare et al., 2022).
• Quantum Markov property and local modular Hamiltonians: In specially modified holographic settings (e.g., time-band states with entanglement wedge equal to causal wedge), the modular Hamiltonian is again local and constructed from consistency conditions on the quantum conditional mutual information (Ju et al., 18 Apr 2025).
• Lattice analogs: Discrete (lattice) Bisognano–Wichmann modular Hamiltonians provide effective descriptions for critical spin chains, giving good approximations to reduced density matrices and local observables in the scaling limit (Zhang et al., 2020).

7. Summary Table: Modular Hamiltonians in Various Contexts

Setting	Modular Hamiltonian $K$	Degree of Locality
Rindler wedge (QFT vacuum)	$2\pi \int x^1 T_{00}(x)$	Local
Ball in CFT$_d$ (vacuum)	$$2\pi \int \frac{R^2-|\vec{x}|^2}{2R}\, T_{00}(\vec{x})$$	Local
General region (free Gaussian QFT)	$K$ as quadratic nonlocal kernel via two-point function	Nonlocal
Double cone, free scalar, $m=0$	$\pi \int_{|x|<r}(r^2-x^2)T_{00}$	Local
Double cone, free scalar, $m>0$	Nonlocal, kernel via $A^{\alpha} \chi A^{\beta}$	Nonlocal
BMSFT interval (flat holography, GMMG)	$$-\int T(\varphi)P(\varphi)+Y(\varphi)J(\varphi)\,d\varphi$$	Geometric, theory-dependent
Generic/perturbed state or region	Perturbative/nonlocal, operator-valued series	Nonlocal/complex
Holographic time band (IR-modified AdS)	$\int f_T(x)\, T_{00}(x)\, dx$ (piecewise maximal parabola)	Local (for specially constructed geometry)

The modular Hamiltonian is a central object bridging quantum information, quantum field theory, and gravitational duality. Its explicit form contains nontrivial and highly theory-specific data beyond the entanglement entropy, controls the dynamics of modular flow, and encodes the emergence and structure of spacetime in holographic duals (Jafferis et al., 2014, Setare et al., 2022, Fröb, 16 Jan 2025, 2002.04637, Cadamuro, 2023).