Modular Hamiltonians in QFT

• Modular Hamiltonians are operators defined as -ln(ρ) that quantify the entanglement structure and modular flow in designated regions of quantum field theory.
• They admit explicit local representations in symmetric cases, exemplified by the boost generator for the Rindler wedge and integrals over null cuts derived via the Bisognano–Wichmann theorem and QNEC.
• Their applications span holographic bulk reconstruction, entanglement entropy calculations, and insights into quantum error correction and operator growth.

A modular Hamiltonian is, for a given region in quantum field theory (QFT), the operator $K = -\ln \rho$ associated with the reduced density matrix $\rho$ for that region. It plays a central role in the study of entanglement, relative entropy, and the structure of quantum field theory, with deep connections to bulk reconstruction in holography, algebraic quantum field theory, and quantum information theory. In general QFTs, modular Hamiltonians are highly nonlocal, but for certain regions (notably the Rindler wedge or regions bounded by null cuts in the vacuum) they admit concrete and explicit local representations.

1. Fundamental Definition and Context

For a relativistic QFT, consider a quantum state $|\Psi\rangle$ and a spatial region $A$. The reduced density matrix $\rho_A$ is constructed by tracing out the complement $\bar{A}$. The modular Hamiltonian $K_A$ is defined by

$\rho_A = e^{-K_A}$

with conventions often choosing $K_A$ to be traceless on the supporting Hilbert space. The operator $K_A$ generates the so-called modular flow, a one-parameter group of automorphisms acting on operators localized in $A$: $O(s) = e^{isK_A}O e^{-isK_A}$ where $s$ is modular “time”. In high-energy physics, modular Hamiltonians encode deep properties—Tomita-Takesaki theory, entanglement structure, modular inclusions—and provide powerful constraints via quantum information-theoretical inequalities (Abt et al., 2018, Koeller et al., 2017).

2. Explicit Formulas in Special Cases: Rindler Wedge and Null Cuts

Rindler Wedge and the Bisognano–Wichmann Theorem

In the vacuum of any relativistic QFT, the modular Hamiltonian for the right Rindler wedge ($x^1 \geq 0$ at $t=0$) takes a simple, local form (“boost generator”): $$K_{\mathrm{Rindler}} = 2\pi \int_{x^1\geq0} d^{d-1}x\, x^1 T_{00}(t=0, x)$$ where $T_{00}$ is the local energy density. This is a direct consequence of the Bisognano-Wichmann theorem.

Arbitrary Null Cuts

For regions bounded by an arbitrary smooth function $v=V(y)$ of the transverse coordinates $y$ along a null plane $u=0$, the vacuum-subtracted modular Hamiltonian generalizes as follows (Koeller et al., 2017, Casini et al., 2017, Balakrishnan et al., 2020): $$\Delta K = 2\pi \int d^{d-2}y \int_{v=V(y)}^\infty (v - V(y))\,T_{vv}(u=0, v, y)\,dv$$ Here, $T_{vv}$ is the null-null component of the stress tensor. This formula is proven by invoking the Quantum Null Energy Condition (QNEC) and its saturation properties in the vacuum, both in free and large-$N$ (holographic) theories (Koeller et al., 2017).

General Properties

• Locality: For wedges and null-plane cuts, $K$ is explicitly local and linear in the stress tensor.
• Integration via QNEC: The derivation exploits the operator equality $K''(y) = 2\pi\, T_{vv}(y)$, connecting the shape second derivative of $K$ to the null energy.
• Generalizations: Through conformal mappings, these forms extend to spherical regions or causal diamonds in CFTs.

3. Algebraic and Markovian Structure

Modular Hamiltonians associated with null-plane cuts possess an infinite-dimensional Lie algebra structure:

• Ray-wise Virasoro Algebras: For each generator along transverse coordinates, the algebra of operators $O_f(x_\perp)=\int du\, f(u) T_{uu}(u, x_\perp)$ closes under commutation with structure constants mirroring the Virasoro algebra (up to central terms).
• Markov Property: For vacuum modular Hamiltonians on overlapping null-plane regions, the identity holds:

$$\hat{H}_{\gamma_1} + \hat{H}_{\gamma_2} - \hat{H}_{\gamma_1 \cap \gamma_2} - \hat{H}_{\gamma_1 \cup \gamma_2} = 0$$

thereby ensuring saturation of strong sub-additivity and the strong super-additivity of the relative entropy. The vacuum state is a quantum Markov state for these algebras (Casini et al., 2017).

• Modular Flow as Null Reparametrization: Modular flow acts as a local affine reparameterization of the null coordinate, matching the classical geometric expectation for boost or conformal Killing flows.

4. Connections to Quantum Information and Holography

Modular Hamiltonians underpin relative entropy and information-theoretic distances in QFT:

• First Law of Entanglement: At linear order, the change in entanglement entropy $\Delta S$ under a small variation of parameters is matched by the expectation value of the modular Hamiltonian:

$\Delta S = \Delta \langle K \rangle$

as a consequence of the first law of entanglement (Abt et al., 2018). For highly symmetric regions (balls in CFT vacuum, intervals in 2d CFT, thermal states), the modular Hamiltonian contains no higher-order corrections and is fully determined by $\Delta S$.

• Beyond the First Law & Entanglement Plateaux: For nontrivial regions—such as two intervals saturating the Araki-Lieb inequality—modular Hamiltonians acquire genuinely nonlocal contributions, and cannot be written solely in terms of the entanglement entropy. Higher-order corrections become operationally crucial in the computation of the quantum Fisher information metric and relative entropy (Abt et al., 2018).
• Holographic Duals and the JLMS Proposal: Modular Hamiltonians in AdS/CFT are dual to bulk gravitational charges associated with extremal (Ryu-Takayanagi) surfaces; the JLMS formula (Faulkner et al., 2016):

$$K^{\rm CFT}_A = \frac{\mathrm{Area}[\tilde{A}]}{4G_N} + K^{\rm bulk}_{\mathcal{M}(A)} + O(G_N)$$

holds for regions $A$ in the large-$N$ (holographic) semiclassical regime, linking boundary modular Hamiltonians to the area and bulk modular generator in the entanglement wedge.

5. Excited States and Endpoint Contributions

In excited states or deformed regions, modular Hamiltonians deviate from their vacuum forms:

• First-Order Corrections: For states obtained by acting with unitary perturbations, $|\Psi\rangle = U|0\rangle$, the modular Hamiltonian receives a bulk “commutator” term and, for operators with modular weight $n \geq 2$, an additional endpoint contribution localized at the entangling surface (Rindler horizon) (Kabat et al., 2020):

$$\delta H^{\rm endpoint}_A = -2\pi \lambda (n-1) \int_{x_\perp} \int_{0}^\infty dx^+ J^{(n)}(x^+, x^- = 0, x_\perp), \;\;\; n \geq 2$$

• Interpretation: These endpoint contributions quantify the nontrivial flow of degrees of freedom across the entangling boundary for high-weight operators, while scalar and current operators (modular weight $n=0,1$) induce no such term (Kabat et al., 2020).

6. Applications: Correlation Functions, Entropy, and Complexity

• Entanglement Entropy and Spectrum: In Gaussian theories, modular Hamiltonians are quadratic in fields, with their spectrum encoding the full entanglement properties. For 2d CFTs, the kernel for a single interval is explicitly local, while for multiple intervals or in the presence of defects/boundaries, nonlocal (bi-local) terms appear, often coupling fields at conjugate points (Jia, 2023, Mintchev et al., 2020, Mintchev et al., 2020).
• Complexity via Modular Evolution: The evolution generated by modular Hamiltonians can be analyzed using Krylov (Lanczos) techniques—yielding a notion of complexity and operator growth tied with the entanglement spectrum and revealing universal chaotic behavior (modular Lyapunov exponent $\lambda^{\rm mod}_L = 2\pi$) (Caputa et al., 2023).

7. Implications, Limitations, and Outlook

• Physical Interpretation and Limitations: The locality of the modular Hamiltonian for wedges and null cuts reflects maximal symmetry and saturation of QNEC. For arbitrary shapes or excited states, modular Hamiltonians become highly nonlocal, encoding subtle quantum correlations and operator algebras that go beyond simple geometric flow. A general non-perturbative proof of vacuum QNEC saturation for null cuts remains open (Koeller et al., 2017).
• Connections to Bulk Reconstruction: Modular Hamiltonians, via their commutants, can be used to localize and reconstruct bulk operators from boundary data (HKLL and extensions), providing explicit algorithms to recover bulk fields using intersecting modular Hamiltonians and their algebras (Kabat et al., 2018, Kabat et al., 2017).
• Quantum Error Correction and Bulk Locality: The structure of modular Hamiltonians in entanglement plateaux and their nonlocal corrections are closely tied to the emergence of bulk locality, the encoding of bulk information in boundary degrees of freedom, and the quantum error-correcting properties of holography (Abt et al., 2018).

In summary, modular Hamiltonians are central objects in QFT and holography—providing exact local generators in highly symmetric situations, encoding quantum information-theoretic structure, and serving as diagnostic tools for bulk emergence, entanglement structure, and the deep algebraic workings of quantum theories of spacetime (Koeller et al., 2017, Casini et al., 2017, Kabat et al., 2018, Abt et al., 2018, Kabat et al., 2020, Jia, 2023, Faulkner et al., 2016).