Modular Hamiltonians: Definitions and Domains

A modular Hamiltonian is the self-adjoint generator of modular flow for a specified algebra and state. In a finite tensor factor, a one-sided generator can be written as $-\log\rho_A$. A sharp local algebra in continuum QFT generally has neither a reduced density matrix nor a trace, so the definition that survives is spectral: construct $\Delta$ from the algebra–state pair and set $K=-\log\Delta$ on its natural domain.

Required background. Use bounded and unbounded operators for domains and closure, self-adjointness for unitary generation, standard form for cyclic and separating vectors, relative entropy in QFT for state comparison, and factorization failure to avoid assuming a local density matrix.

The chapter’s structure map separates intrinsic modular data from geometric special cases. Its comparison table and validity guide record the hypotheses that must accompany any local or regulated formula.

Three generators that should not be conflated

Let a regulated bipartite Hilbert space be $\mathcal H_A\otimes\mathcal H_{A^c}$ and let $\rho_A$ be faithful. The one-sided entanglement Hamiltonian is

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

With the convention used throughout this chapter,

$$\sigma_s(A)=\rho_A^{is}A\rho_A^{-is} =e^{-isK_A}Ae^{isK_A}.$$

Replacing $K_A$ by $K_A+c\mathbf1$ rescales the unnormalized operator $e^{-K_A}$ and shifts $\log Z$; it changes neither the normalized state nor the adjoint flow.

Suppose a purification has Schmidt rank $r$,

$$\lvert\Omega\rangle =\sum_{i=1}^{r}\sqrt{p_i}\, \lvert i\rangle_A\lvert i\rangle_{A^c}, \qquad p_i>0.$$

Restrict both tensor factors to their $r$-dimensional Schmidt supports. On those supports, the modular operator of the left algebra is

$$\Delta_\Omega=\rho_A\otimes\rho_{A^c}^{-1}, \qquad K_\Omega=-\log\Delta_\Omega =K_A\otimes\mathbf1-\mathbf1\otimes K_{A^c}.$$

The inverse is not defined on any extra null subspace of $\rho_{A^c}$. The two-sided generator annihilates $\Omega$ and implements the same flow on left operators as $K_A$. Witten 2018, § IV.A, pp. 17–20, eqs. (4.1)–(4.32) derives these statements and shows why full Schmidt rank is precisely the cyclic-and-separating condition.

A relative modular operator compares two states on one algebra. Its logarithm is not obtained by subtracting two noncommuting one-sided logarithms; relative modular operators and cocycles keep the left and right actions explicit.

Spectral definition and natural domain

For a cyclic, separating vector $\Omega$ of a von Neumann algebra $\mathcal A$, Tomita–Takesaki theory supplies an injective positive self-adjoint operator $\Delta_\Omega$. If $E_\Delta(\lambda)$ is its projection-valued spectral measure, then

$$K_\Omega=-\log\Delta_\Omega =-\int_{(0,\infty)}\log\lambda\,dE_\Delta(\lambda).$$

The corresponding domain is

$$\mathcal D(K_\Omega)= \left\{\xi\in\mathcal H: \int_{(0,\infty)}\lvert\log\lambda\rvert^2 \,d\langle\xi,E_\Delta(\lambda)\xi\rangle<\infty\right\}.$$

Although $K_\Omega$ can be unbounded, $\Delta^{is}=e^{-isK_\Omega}$ is unitary for every real $s$. Thus $\Delta^{is}A\Delta^{-is}$ may be meaningful while $K_\Omega\xi$ or $[K_\Omega,A]\xi$ is not. Differentiation at $s=0$ requires a common invariant domain on which the generator and operator products are defined.

A safe hierarchy is therefore:

1. use $\Delta^{is}A\Delta^{-is}$ for exact real modular time;
2. use spectral integrals for functions of $K_\Omega$;
3. differentiate only on a stated common invariant domain;
4. treat a local or bilocal integral for $K$ as an additional theorem or approximation, never as the definition.

Support, faithfulness, and the continuum limit

For matrices, faithfulness means $\rho>0$. If $p=s(\rho)$ is the support projection, $-\log\rho$ is an operator on $p\mathcal H$, and modular statements belong to the reduced algebra $p\mathcal A p$. Assigning $+\infty$ to $-\log0$ is useful in extended-valued variational formulas, but it does not create an operator on the discarded null space.

In continuum QFT, the vacuum is commonly cyclic and separating for a local algebra even though that algebra is type III. There is then no trace-class $\rho_A$ whose logarithm generates the intrinsic flow; Witten 2018, § VI.E, pp. 29–31 explains the type-III obstruction. A UV regulator may produce $\rho_{A,\epsilon}$, but its entropy, additive normalization, and one-sided generator contain regulator-dependent boundary terms. The algebraic automorphism group can nevertheless have a continuum limit.

The best-known controlled geometric case is the vacuum Rindler wedge. For the right wedge and $\sigma_s=\operatorname{Ad}\Delta^{is}$, the Bisognano–Wichmann normalization is a boost of rapidity $-2\pi s$. In a regulated one-sided description on $t=0$, $x^1>0$,

$$K_A=2\pi\int_{x^1>0}d^{d-1}x\;x^1T_{00}(0,\mathbf x)+c.$$

The original scalar theorem and its domain hypotheses are given in Bisognano and Wichmann 1975, Theorem 1 and §§ V–VI, pp. 985–1007. The local integral is special because the theorem supplies a geometric action; it is not representative of a generic region or state.

A two-mode Gaussian benchmark and a wedge control

Use the correlation convention

$$C_{ij}=\operatorname{Tr}(\rho_A c_j^\dagger c_i).$$

For a faithful number-conserving fermionic Gaussian state on finitely many modes,

$$\rho_A=Z^{-1}\exp(-c^\dagger h c), \qquad h=\log\!\left[(\mathbf1-C)C^{-1}\right].$$

With the alternative convention $C_{ij}=\langle c_i^\dagger c_j\rangle$, the right-hand side gives $h^T$. Peschel 2003, eqs. (5)–(12), pp. L205–L207 derives the correlation-matrix reconstruction.

Take the reproducible two-mode input

$$C=\begin{pmatrix} 1/2&1/4\\ 1/4&1/2 \end{pmatrix}.$$

Its eigenvalues are $\nu_+=3/4$ and $\nu_-=1/4$, so

$$\epsilon_+=-\log3, \qquad \epsilon_-=\log3, \qquad Z=(1+e^{-\epsilon_+})(1+e^{-\epsilon_-})=\frac{16}{3}.$$

The single-particle eigenvalues $\epsilon_a$ are not the many-body spectrum. For occupations $n_a\in\{0,1\}$,

$$\kappa_{n_+n_-}=\log Z+n_+\epsilon_++n_-\epsilon_-$$

are the four eigenvalues of $K_A$. The two-sided modular spectrum is

$$\operatorname{spec}(K_\Omega) =\{\kappa_{\mathbf n}-\kappa_{\mathbf m}\} =\{-2\log3,-\log3,0,\log3,2\log3\},$$

and $\operatorname{spec}(\Delta_\Omega)=\{3^k:k=-2,-1,0,1,2\}$. Every operator domain is the full finite-dimensional space. If each $\nu_a$ is known within $\eta$ and remains in $[\delta,1-\delta]$, then

$$\lvert\delta\epsilon_a\rvert \leq\frac{\eta}{\delta(1-\delta)}+O(\eta^2).$$

Here $\delta=1/4$, giving the first-order bound $16\eta/3$. This eigenvalue tolerance is the benchmark’s numerical control parameter.

Property	Two-mode regulator	Vacuum wedge
Algebra and state	$M_4(\mathbb C)$ with faithful Gaussian $\rho_A$	Wedge algebra with the vacuum
Intrinsic generator	Finite matrix $K_\Omega$	$K_W=-\log\Delta_W$ on the boost-generator domain
Spectrum	Five displayed level differences	Spectrum of the boost representation; not inferred from a finite matrix
Normalization	$\operatorname{Tr}\rho_A=1$ fixes $\log Z$	The theorem fixes the factor $2\pi$; no local trace fixes an additive density-matrix constant
Control	Eigenvalue tolerance $\eta$ and gap $\delta$ from $0,1$	Wightman, vacuum, covariance, spectrum, and wedge-locality hypotheses

The table compares structures, not a claimed numerical continuum extrapolation. A lattice calculation would also need a sequence of regions and cutoffs, plus convergence evidence for the observables being compared.

Failure injections: support and algebra dependence

First replace the benchmark by $C_0=\operatorname{diag}(1,1/2)$. The first mode is certainly occupied, so the full Fock-space density matrix has zero eigenvalues. Its formal entanglement energy tends to $-\infty$, and $-\log\rho_A$ has no finite value on the null sector. The reduced support still carries a faithful state and a valid modular flow; the full-space logarithm does not.

Second keep a faithful qubit density matrix $\rho=\operatorname{diag}(p,1-p)$ fixed but change the algebra. On $M_2(\mathbb C)$,

$$\sigma_s(A)=\rho^{is}A\rho^{-is}$$

acts nontrivially on off-diagonal matrix units when $p\neq1/2$. On the diagonal subalgebra every $A$ commutes with $\rho$, so $\sigma_s(A)=A$. The same matrix $\rho$ therefore does not determine modular flow until the algebra and its representation have also been named.

Common pitfalls

Writing $K=-\log\rho_A$ before naming a factorization. A sharp continuum region is assigned an algebra, not automatically a tensor factor. Use a density matrix only after stating a type-I or regulated setting.

Dropping domains because the exponential is unitary. Stone’s theorem makes $e^{-isK}$ globally bounded; it does not make $K\xi$ or $[K,A]\xi$ meaningful for every vector.

Treating additive constants like support projections. Constants cancel from adjoint flow. Removing a null space changes the representation on which the logarithm exists.

Exercises

1. Starting from $K_A=-\log\rho_A$, derive the chapter’s modular-flow sign and show explicitly that $K_A+c\mathbf1$ gives the same adjoint action.

Solution

Functional calculus gives $\rho_A^{is}=e^{is\log\rho_A}=e^{-isK_A}$. Hence

$$\rho_A^{is}A\rho_A^{-is}=e^{-isK_A}Ae^{isK_A}.$$

After the shift,

$$e^{-is(K_A+c\mathbf1)}A e^{is(K_A+c\mathbf1)} =e^{-isc}e^{-isK_A}A e^{isK_A}e^{isc},$$

and the scalar phases cancel. Likewise $e^{-(K_A+c\mathbf1)}=e^{-c}e^{-K_A}$, so normalization removes the same scalar.

2. For the two-mode benchmark, compute the four probabilities of the occupation configurations and verify both the $K_A$ spectrum and the five possible differences in $K_\Omega$.

Solution

The occupations of the diagonal modes are independent with probabilities $\nu_+=3/4$ and $\nu_-=1/4$. In the order $(n_+,n_-)=(0,0),(1,0),(0,1),(1,1)$, the probabilities are

$$\frac{3}{16},\quad\frac{9}{16},\quad\frac{1}{16},\quad\frac{3}{16}.$$

Taking minus logarithms gives

$$\log\frac{16}{3},\quad \log\frac{16}{9},\quad \log16,\quad \log\frac{16}{3},$$

which agree with $\log Z+n_+\epsilon_++n_-\epsilon_-$. Pairwise differences are integer multiples of $\log3$ from $-2\log3$ through $2\log3$, as stated.

3. Let $\rho=\operatorname{diag}(p,1-p)$ with $0<p<1$ and $p\neq1/2$. Compare modular flow on $M_2(\mathbb C)$ with its restriction to the diagonal algebra. Then take $p\to1$ and identify the first hypothesis that fails.

Solution

For the matrix unit $E_{01}$,

$$\sigma_s(E_{01}) =\left(\frac{p}{1-p}\right)^{is}E_{01},$$

so the full matrix algebra has nontrivial flow. Every diagonal matrix commutes with $\rho$, so the restricted flow is the identity. As $p\to1$, $\rho$ loses faithfulness: its support becomes the span of $\lvert0\rangle$, and $\rho^{-1}$ and $-\log\rho$ cease to exist on the orthogonal null vector. The support-reduced one-dimensional algebra still has the trivial modular flow.

References

• Bisognano, Joseph J., and Eyvind H. Wichmann. “On the Duality Condition for a Hermitian Scalar Field.” Journal of Mathematical Physics 16 (1975): 985–1007. DOI.
• Peschel, Ingo. “Calculation of Reduced Density Matrices from Correlation Functions.” Journal of Physics A: Mathematical and General 36 (2003): L205–L208. DOI; arXiv.
• Witten, Edward. “Notes on Some Entanglement Properties of Quantum Field Theory.” Reviews of Modern Physics 90 (2018): 045003. DOI; Accepted manuscript PDF; arXiv.