Nonlocal Modular Hamiltonians and Limits of Explicit Control

Outside exceptional vacuum wedges and conformal balls, modular Hamiltonians are generally nonlocal and state dependent. Explicit control then comes from spectral calculus, Gaussian reductions, perturbation theory, or bounds—not from assuming a local stress-tensor weight. A useful result must state the operator domain, the regulator, the expansion parameter, and how the neglected terms are estimated.

Required background. Spectra and projectors supply finite spectral decompositions, and modular Hamiltonian domains supplies the continuum logarithm.

Helpful background. Shape dependence supplies controlled geometric perturbations.

What nonlocality means

A spatially nonlocal generator couples operators at separated points. Schematically,

K_A= \int_A d^{d-1}x\;f^\mu(x)T_{0\mu}(x) +\int_A d^{d-1}x\,d^{d-1}y\; h_{ab}(x,y)\mathcal O_a(x)\mathcal O_b(y) +\cdots.

The first term is local; the second is bilocal; higher terms may be multilocal. The decomposition depends on the operator basis and regulator, but the failure of the adjoint flow to act as a pointwise spacetime transformation is invariant. A bilocal kernel known to first order is not the exact modular Hamiltonian.

For a finite free-fermion Gaussian state, the subsystem correlation matrix $C_{ij}=\langle c_i^\dagger c_j\rangle$ determines

K_A=c^\dagger h c+c_0, \qquad h=\log\!\left[(\mathbf1-C)C^{-1}\right].

Even when the microscopic Hamiltonian is local, the matrix function of the restricted correlator is usually dense. Its off-diagonal decay depends on the state, gap, geometry, and distance from the entangling surface. This is the cleanest finite-mode example of a controlled but nonlocal modular generator.

The structural map places Nonlocal Modular Hamiltonians and Limits of Explicit Control between intrinsic modular data and the additional hypotheses that permit a geometric interpretation.

A standard algebra-state pair determines the Tomita operator, modular conjugation, modular flow, and relative modular data; only additional covariance or conformal hypotheses turn the flow into wedge or ball geometry. — Tomita polar decomposition intrinsically produces $J$ and $\Delta$ . Wedge boosts and CFT ball flow are special consequences of locality, covariance, the spectrum condition, the vacuum, and—only for the ball—the conformal map. The diagram is schematic and not to scale.

Resolvent expansion of the logarithm

For a faithful regulated state $\rho(\lambda)=\rho_0+\lambda\,\delta\rho+\cdots$ , the Fréchet derivative of $K=-\log\rho$ is

\delta K =-\int_0^\infty d\beta\; (\rho_0+\beta)^{-1}\delta\rho(\rho_0+\beta)^{-1}.

This formula preserves noncommutative ordering. In the eigenbasis of $\rho_0$ ,

(\delta K)_{mn} =-\frac{\log p_m-\log p_n}{p_m-p_n}\, (\delta\rho)_{mn},

with the coincident limit $-(\delta\rho)_{nn}/p_n$ . The divided difference shows why replacing $\delta K$ by $-\rho_0^{-1}\delta\rho$ is valid only in a commuting direction.

At finite dimension with $p_{\min}>0$ , the resolvent gives the rough estimate

\lVert\delta K\rVert \le \frac{\lVert\delta\rho\rVert}{p_{\min}}.

The bound becomes useless as the smallest eigenvalue approaches zero—exactly what happens under many continuum or large-volume limits. Relative modular methods and matrix elements on controlled vectors may remain finite even when the operator norm does not.

Modular-time kernels

The same derivative can be expressed as an integral of perturbations transported by the reference modular flow. After separating the part commuting with $K_0$ , a typical form is

\delta K =\int_{-\infty}^{\infty}ds\; g(s)\,\sigma_s^{(0)}(\delta X),

where $g(s)$ is fixed by the divided-difference kernel and $\delta X$ depends on how the state was prepared. The kernel is a distribution near $s=0$ and requires a prescription. Different-looking formulas are equivalent only after matching the modular-flow sign, Fourier convention, contact terms, and commuting zero mode.

Excited-state calculations often organize higher orders through modular-ordered products, as in Sárosi and Ugajin 2018, §2. Shape deformations can instead move contributions to null boundaries. Neither representation makes the exact generator local; each is a controlled expansion around a reference state or region with known modular flow.

Truncation and domain control

A perturbative statement should include four separate error sources:

state truncation: the norm or matrix-element size of omitted powers of the perturbation;
spectral conditioning: sensitivity to small eigenvalues or large modular frequencies;
operator truncation: error from omitting bilocal or higher operator structures;
regulator removal: stability as lattice spacing, split distance, or mode cutoff is refined.

For unbounded QFT operators, norm control may be unavailable. One may instead prove convergence of quadratic forms on an energy-bounded dense domain, convergence of selected correlators, or convergence after modular-frequency smearing. The claim must match the topology actually controlled.

The commutator expansion

\sigma_s(A)=A-is[K,A] -\frac{s^2}{2}[K,[K,A]]+\cdots

is especially fragile. It requires analytic vectors for the adjoint action and a radius or remainder estimate. Exact unitary conjugation is safer whenever it can be evaluated.

A practical workflow

For a proposed nonlocal modular Hamiltonian:

name the algebra, reference state, and regulator;
verify faithfulness or restrict to the common support;
reproduce a commuting or Gaussian limit;
test Hermiticity and the condition $K\Omega=0$ in standard form;
check the KMS boundary relation for independent operators;
refine the cutoff and expansion order separately;
report the observable or quadratic-form norm in which the residual is small.

Agreement with entropy alone is weak: adding an operator with zero expectation value can leave the entropy unchanged while altering modular flow.

Common pitfalls

Promoting a first-order kernel to an exact generator. Label the expansion parameter and remainder. Nonlocal structures can appear at the next order even when the first correction is local.

Using an operator-norm bound after $p_{\min}\to0$ . The inverse spectral gap makes the bound divergent. Switch to relative, smeared, or energy-bounded quantities whose limit is actually controlled.

Omitting contact and zero-mode prescriptions. Modular-time kernels are distributions. Their value depends on how singularities and commuting components are separated.

Before assigning geometric meaning to this modular statement, use the validity map to check standardness, spectral domains, normalization, geometric hypotheses, and approximation control independently.

A decision map separates exact modular conclusions from failures caused by missing faithfulness, uncontrolled operator domains, absent geometric theorems, or perturbative nonlocal kernels. — A modular-flow claim is only as strong as its algebra–state standardness, spectral domain, normalization, geometric hypotheses, and approximation control. Dashed branches mark conditional steps; the terminal warnings identify common out-of-domain uses. The diagram is schematic and not to scale.

References

Sárosi, Gábor, and Tomonori Ugajin. “Modular Hamiltonians of Excited States, OPE Blocks and Emergent Bulk Fields.” Journal of High Energy Physics 2018, no. 1 (2018): 012. DOI; arXiv.