Nonlocal Modular Hamiltonians and Limits of Explicit Control

Outside exceptional vacuum wedges, conformal balls, and a few other symmetric settings, a modular generator is generally state dependent and nonlocal. “Known perturbatively” is therefore meaningful only after naming a regulator, an operator domain, a small parameter, an error metric, and the spectral region on which the estimate remains finite.

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.

The chapter overview places these approximations in the route from standard pairs to geometric flow, compares representative modular flows and their boundaries, and gives a checklist for deciding when a modular-flow claim is licensed.

Nonlocality and the algebraic object

With a UV regulator or a split inclusion, a region may have a density matrix and

$$K_A=-\log\rho_A.$$

In an unregulated continuum QFT, a sharp local algebra is typically type III and has no trace-class $\rho_A$. The intrinsic object is the modular operator $\Delta_{A,\Omega}$ in a standard representation, with generator $-\log\Delta_{A,\Omega}$. A regulated one-sided expression is useful only if its relation to that algebraic object and its cutoff dependence are stated.

In a regulated operator basis, spatial nonlocality may appear as

$$\begin{aligned} K_A={}& \int_A d^{d-1}x\;f^\mu(x)T_{0\mu}(x)\\ &+\int_A d^{d-1}x\,d^{d-1}y\; h_{ab}(x,y)\mathcal O_a(x)\mathcal O_b(y) +\text{higher multilocal terms}. \tag{1} \end{aligned}$$

The split into local, bilocal, and higher terms depends on the regulated operator basis. What is basis independent is that the adjoint flow need not act as a pointwise spacetime transformation.

Gaussian free fermions: conventions and support

For a gauge-invariant, number-conserving fermionic Gaussian state, define the one-particle correlation matrix by

$$C_{ij}=\langle c_j^\dagger c_i\rangle. \tag{2}$$

If $0<C<\mathbf1$ on the retained one-particle space, then

$$K_A=\sum_{i,j}c_i^\dagger h_{ij}c_j+c_0, \qquad h=\log\!\left[(\mathbf1-C)C^{-1}\right]. \tag{3}$$

This index order matters. Peschel instead writes $\widetilde C_{ij}=\langle c_i^\dagger c_j\rangle=C_{ji}$; in that convention the corresponding statement is

$$h^{\mathsf T} =\log\!\left[(\mathbf1-\widetilde C)\widetilde C^{-1}\right],$$

as in Peschel 2003, equations (11)–(12). Omitting the transpose is harmless for a real symmetric correlator but wrong in a general complex basis.

If $C$ has eigenvalue $0$ or $1$, (3) is an unbounded limit. One must restrict to the support on which $0<C<1$ or retain the divergent entanglement energies explicitly. If anomalous correlators

$$F_{ij}=\langle c_i c_j\rangle$$

are nonzero, (3) is incomplete: the quadratic generator includes pairing terms and is determined by a Nambu or Majorana covariance matrix Peschel 2003, equations (13)–(15).

Logarithms and a spectral error bound

Let a faithful regulated state vary as $\rho(\lambda)=\rho_0+\lambda\,\delta\rho+O(\lambda^2)$. The Fréchet derivative of $K=-\log\rho$ is

$$\delta K =-\int_0^\infty d\beta\; (\rho_0+\beta)^{-1}\delta\rho (\rho_0+\beta)^{-1}. \tag{4}$$

In an eigenbasis $\rho_0|m\rangle=p_m|m\rangle$,

$$(\delta K)_{mn} =-\frac{\log p_m-\log p_n}{p_m-p_n} (\delta\rho)_{mn}, \tag{5}$$

with coincident limit $-(\delta\rho)_{nn}/p_n$. Equations (4)–(5) are the standard resolvent and divided-difference forms of a matrix-function derivative Higham 2008, §3.2, especially equations (3.11)–(3.16).

If $p_{\min}=\min\operatorname{spec}\rho_0>0$, the operator norm obeys

$$\|\delta K\|_2 \le \|\delta\rho\|_2 \int_0^\infty\frac{d\beta}{(p_{\min}+\beta)^2} =\frac{\|\delta\rho\|_2}{p_{\min}}. \tag{6}$$

The estimate is explicit and reproducible, but its ceiling diverges as $p_{\min}\to0$. This bound therefore supplies no uniform operator-norm control in a continuum or large-volume limit. Such a limit needs either a sharper operator-norm estimate or a declared weaker topology—often matrix elements on an energy-bounded domain, quadratic forms, or modular-frequency-smeared correlators.

Formal modular-time expansions are not automatically controlled

The logarithm can also be organized as integrals of perturbations transported by a reference modular flow. Schematically,

$$\delta K =\int_{-\infty}^{\infty}ds\; g(s)\,\sigma_s^{(0)}(\delta X), \tag{7}$$

where the distribution $g$ and the zero-modular-frequency term depend on the preparation and convention. Contact prescriptions, the sign of modular time, and Fourier normalization must be fixed before two forms of (7) can be compared.

The excited-state series of Sárosi and Ugajin is explicitly presented as a formal expansion, not a convergent expansion for arbitrary states Sárosi–Ugajin 2018, abstract and §2. Their §6 identifies settings that may supply control—small subsystems, large-$N$ counting subject to multitrace caveats, or a small mixture parameter—but each application must still bound its own remainder Sárosi–Ugajin 2018, §6. Formal modular ordering alone supplies no universal error bar.

For a small deformation $\zeta^\mu$ of a vacuum half-space, a concrete QFT benchmark is

$$\begin{aligned} UK_AU^\dagger={}&K_{A_0} -2\pi\int_{\mathcal H_+}\zeta^+T_{++} +2\pi\int_{\mathcal H_-}\zeta^-T_{--}\\ &+2\pi\int_{A_0}\zeta^\nu [K_{A_0},T_{0\nu}] +O(\zeta^2), \tag{8} \end{aligned}$$

where $U$ identifies the deformed and undeformed regulated Hilbert spaces. Equation (8), including its inward-deformation convention, is Faulkner–Leigh–Parrikar–Wang 2016, §1.1, equations (12)–(14). It controls only first order, and the source explains the entangling-surface cutoff in Appendix A. It does not license replacing a generic modular Hamiltonian by a local stress-tensor integral.

First application: a reproducible free-field benchmark

Use a number-conserving free-fermion regulator and retain two normalized wave-packet modes supported in the deformed region. Suppose the computed restricted correlator is

$$C(\varepsilon)= \begin{pmatrix} 1/4&\varepsilon\\ \varepsilon&3/4 \end{pmatrix}, \qquad r(\varepsilon)=\sqrt{\frac1{16}+\varepsilon^2}. \tag{9}$$

Here $\varepsilon$ is the dimensionless first-order response of the two retained modes to the deformation. This benchmark begins after the geometric calculation has supplied (9); it tests the logarithm and modular flow without pretending that every deformation has this two-mode response.

The eigenvalues of $C$ are $1/2\pm r$. Faithfulness of both $C$ and $1-C$ requires

$$|\varepsilon|<\frac{\sqrt3}{4}\approx0.433013. \tag{10}$$

Define

$$L(\varepsilon) =\log\frac{1/2+r}{1/2-r}.$$

With $\sigma_x,\sigma_z$ the Pauli matrices, exact functional calculus gives

$$h(\varepsilon) =\frac{L}{4r}\,\sigma_z -\frac{L\varepsilon}{r}\,\sigma_x. \tag{11}$$

The first-order approximation is

$$h^{(1)}(\varepsilon) =\log3\,\sigma_z -4\varepsilon\log3\,\sigma_x. \tag{12}$$

Thus the first nonlocal correction is the hopping term

$$\delta K^{(1)} =-4\varepsilon\log3\, (c_1^\dagger c_2+c_2^\dagger c_1). \tag{13}$$

Use the spectral-norm remainder

$$R_h(\varepsilon) =\|h(\varepsilon)-h^{(1)}(\varepsilon)\|_2 =\sqrt{ \left(\frac{L}{4r}-\log3\right)^2 +\left(-\frac{L\varepsilon}{r} +4\varepsilon\log3\right)^2 }. \tag{14}$$

The benchmark values below follow directly from (9)–(14); “relative remainder” means $R_h/(4|\varepsilon|\log3)$, the remainder divided by the norm of the retained nonlocal correction.

ε	Smallest eigenvalue of C or 1 − C	Remainder norm	Relative remainder
0.02	0.249201	0.000754	0.00858
0.05	0.245049	0.004825	0.02196
0.10	0.230742	0.020886	0.04753
0.20	0.179844	0.110496	0.12572
0.40	0.028301	1.463293	0.83247

For a normalized one-particle smearing $f\in\mathbb C^2$, set $c(f)=\sum_i f_i^*c_i$. The CAR give $\|c(f)\|=\|f\|_2$ and

$$[K,c(f)]=-c(hf).$$

Consequently,

$$\|[K-K^{(1)},c(f)]\| \le R_h\|f\|_2, \tag{15}$$

and the exact and truncated modular flows satisfy the Duhamel bound

$$\|\sigma_s(c(f))-\sigma_s^{(1)}(c(f))\| \le |s|R_h\|f\|_2. \tag{16}$$

Equations (14)–(16) state the domain, norm, deformation parameter, and modular-time dependence explicitly. No continuum claim is being made: the result controls only these two retained modes at the chosen regulator.

Adversarial test: drive the spectrum to its support boundary

Increasing $|\varepsilon|$ in (9) simultaneously strengthens the nonlocal coupling and worsens spectral conditioning. At $|\varepsilon|=\sqrt3/4$, the smallest eigenvalue reaches zero, $L(\varepsilon)$ diverges, and neither (11) nor the operator-norm estimate has a finite full-space limit. Beyond that point $C$ is not a valid fermionic correlation matrix because an eigenvalue lies outside $[0,1]$.

The table shows that the first-order approximation is already poor at $\varepsilon=0.40$: its remainder is about $83\%$ of the retained correction. The strongest surviving statements are the exact finite-mode formula (11) for the open interval (10), and possibly matrix-element statements on a restricted support. The perturbative claim (12)–(16) must be withdrawn when the chosen tolerance is exceeded; ultraviolet refinement requires a separate convergence study rather than extrapolation from this table.

A four-part error budget

A publication-level nonlocal calculation should report separately:

• state or shape truncation: omitted powers of the declared small parameter;
• spectral conditioning: distance of $C$, $\rho$, or $\Delta$ from its support boundary;
• operator truncation: omitted bilocal, pairing, or higher multilocal structures;
• regulator removal: stability under lattice spacing, split distance, volume, and mode-cutoff refinement.

Agreement with entropy alone is not a flow test: an operator with zero expectation value can leave the entropy unchanged while changing commutators and modular evolution.

Common pitfalls

Using the wrong Gaussian index convention. With $C_{ij}=\langle c_j^\dagger c_i\rangle$, equation (3) has no transpose. With Peschel’s reversed indices, it does.

Calling a formal modular-time series controlled. Name the regime that suppresses higher orders and exhibit a remainder in the topology actually used.

Keeping an operator-norm claim as the support gap closes. Equations (6) and (14) display the divergence. Change the domain or topology, or stop the claim.

Exercises

1. Derive the transpose in Peschel’s convention

Diagonalize a Hermitian one-particle matrix $h$ and prove that

$$\widetilde C_{ij}=\langle c_i^\dagger c_j\rangle$$

obeys $h^{\mathsf T}=\log[(1-\widetilde C)\widetilde C^{-1}]$.

Solution

Write $h=U\epsilon U^\dagger$ and $c_i=\sum_\alpha U_{i\alpha}d_\alpha$. For the Gaussian density operator proportional to $e^{-c^\dagger hc}$,

$$\langle d_\alpha^\dagger d_\beta\rangle =\delta_{\alpha\beta}\frac1{1+e^{\epsilon_\alpha}}.$$

Therefore

$$\widetilde C_{ij} =\sum_\alpha U_{i\alpha}^*U_{j\alpha} \frac1{1+e^{\epsilon_\alpha}} =\left[(1+e^h)^{-1}\right]_{ji}.$$

In matrix notation,

$$\widetilde C^{\mathsf T}=(1+e^h)^{-1}.$$

Solving for $h$ gives

$$h=\log[(1-\widetilde C^{\mathsf T}) (\widetilde C^{\mathsf T})^{-1}],$$

and transposing yields the claimed formula. For complex $U$, dropping the transpose changes the matrix; it is not merely notation.

2. Prove the resolvent ceiling

Starting from (4), prove (6) and state exactly why it gives no uniform continuum estimate when $p_{\min}\to0$.

Solution

Because $\rho_0\ge p_{\min}\mathbf1$,

$$\|(\rho_0+\beta)^{-1}\|_2 \le\frac1{p_{\min}+\beta}.$$

Submultiplicativity and (4) give

$$\|\delta K\|_2 \le\|\delta\rho\|_2 \int_0^\infty\frac{d\beta}{(p_{\min}+\beta)^2} =\frac{\|\delta\rho\|_2}{p_{\min}}.$$

If a regulator sequence has $p_{\min}\to0$, the right-hand side diverges even when $\|\delta\rho\|_2$ stays fixed. Hence the estimate supplies no regulator-independent operator-norm control. A weaker, explicitly declared topology may still converge, but that is a different claim requiring its own proof.

3. Reproduce the two-mode error and flow bound

At $\varepsilon=0.10$, compute $r$, the support gap, $L$, and $R_h$. Then bound the flow error at $|s|=2$ for $\|f\|_2=1$.

Solution

Equations (9)–(11) give

$$r=\sqrt{0.0625+0.01}=0.269258, \qquad p_{\rm gap}=0.5-r=0.230742,$$

and

$$L=\log\frac{0.769258}{0.230742}=1.204128.$$

Substitution in (14) gives

$$R_h=0.020886.$$

The commutator error for a normalized smearing is at most $0.020886$. Equation (16) gives

$$\|\sigma_s(c(f))-\sigma_s^{(1)}(c(f))\| \le2R_h =0.041772$$

at $|s|=2$. This is a finite-mode operator-norm statement; it does not bound omitted regulator modes or the $O(\zeta^2)$ geometric remainder that led to the input correlator.

References

• Faulkner, Thomas, Robert G. Leigh, Onkar Parrikar, and Huajia Wang. “Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition.” Journal of High Energy Physics 2016, no. 9 (2016): 038. DOI. Open HTML, equations (12)–(14).
• Higham, Nicholas J. Functions of Matrices: Theory and Computation. Philadelphia: Society for Industrial and Applied Mathematics, 2008. DOI.
• Peschel, Ingo. “Calculation of Reduced Density Matrices from Correlation Functions.” Journal of Physics A: Mathematical and General 36 (2003): L205–L208. DOI. Open preprint.
• 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. Open HTML.