Entanglement Spectra and Modular Spectral Data

An entanglement spectrum is the spectrum of a reduced density matrix in a chosen type-I realization: a lattice factor, a finite-mode truncation, or a split factor. It can organize useful cutoff-dependent data, but it is not an intrinsic list of eigenvalues for a sharp local algebra in continuum QFT. A continuum claim must therefore name the observable that converges, not merely display a sequence of finite matrices.

Required background. Use Relative Entropy for QFT States to distinguish density-matrix relative modular operators from their algebraic QFT counterparts and to track support assumptions. Helpful background. Gaussian correlation-matrix entropy supplies the covariance-matrix diagonalization used below.

The chapter’s structure map separates spectral data from entropy and correlation measures, its comparison table records the domain of each construction, and its validity map summarizes which choices must remain fixed in a scaling claim.

Eigenvalues, support, and additive constants

Let $\rho_A$ be a density operator on a type-I subsystem. On its support,

$$\rho_A=\sum_{\lambda_i>0}\lambda_i\lvert i\rangle\langle i\rvert, \qquad K_A=-\log\rho_A, \qquad \xi_i=-\log\lambda_i.$$

A zero eigenvalue corresponds to $\xi_i=+\infty$; it must not be silently discarded when ranks are compared. Because $\operatorname{Tr}\rho_A=1$, the normalized modular Hamiltonian $K_A$ has no freely adjustable constant. In numerical work one often diagonalizes an unnormalized entanglement Hamiltonian $H_E$ instead,

$$\rho_A=Z_E^{-1}e^{-H_E}, \qquad K_A=H_E+\log Z_E.$$

Only gaps in $H_E$ are insensitive to its additive constant. A plot must say whether it shows the normalized $\xi_i$, shifted many-body levels, or single-particle entanglement energies. The moments

$$\operatorname{Tr}\rho_A^n=\sum_i e^{-n\xi_i}$$

determine the finite-dimensional spectrum in principle, but analytic continuation or truncation can make reconstruction ill-conditioned.

Gaussian fermions: from one-particle data to the full spectrum

For a number-conserving complex-fermion Gaussian state, restrict

$$(C_A)_{mn}=\langle c_m^\dagger c_n\rangle, \qquad m,n\in A,$$

and diagonalize $C_A=U\operatorname{diag}(\nu_j)U^\dagger$, with $0\leq\nu_j\leq1$. If $f_j=\sum_m U_{mj}^*c_m$, then

$$\rho_A=\prod_j\left[(1-\nu_j)\lvert0_j\rangle\langle0_j\rvert +\nu_j\lvert1_j\rangle\langle1_j\rvert\right] =Z_E^{-1}\exp\!\left(-\sum_j\varepsilon_j f_j^\dagger f_j\right),$$

where

$$\varepsilon_j=\log\frac{1-\nu_j}{\nu_j}, \qquad \lambda_{\boldsymbol n}=\prod_j\nu_j^{n_j}(1-\nu_j)^{1-n_j}.$$

These relations are derived in Peschel 2003, Eqs. (4)–(12). A Bogoliubov Gaussian state is different: anomalous correlators are required, and the quadratic entanglement Hamiltonian contains pairing terms Peschel 2003, Eqs. (13)–(16).

As a two-mode check, take $(\nu_1,\nu_2)=(3/4,1/4)$. Then $(\varepsilon_1,\varepsilon_2)=(-\log3,\log3)$, while the normalized many-body eigenvalues are

Two-mode many-body spectrum from the one-particle occupations.

Occupation $(n_1,n_2)$	$\lambda_{n_1n_2}$	Shifted $H_E$ level
$(0,0)$	$3/16$	$0$
$(1,0)$	$9/16$	$-\log3$
$(0,1)$	$1/16$	$\log3$
$(1,1)$	$3/16$	$0$

The example also shows why “low single-particle energy” and “large many-body eigenvalue” are not synonymous until the reference occupation and normalization have been fixed.

Application: a critical free-fermion interval

Consider the half-filled infinite hopping chain and an interval of $N$ consecutive sites. Holding its physical length $\ell$ fixed makes the lattice spacing $a=\ell/N$. The restricted correlation matrix is completely reproducible:

$$(C_N)_{mn}= \begin{cases} \dfrac{\sin[k_F(m-n)]}{\pi(m-n)},&m\neq n,\\[6pt] \dfrac{k_F}{\pi},&m=n, \end{cases} \qquad k_F=\frac{\pi}{2}.$$

Diagonalizing this real symmetric matrix gives the following values. Here $\delta_N=\min_j\lvert\varepsilon_j\rvert$ is the smallest charge-changing modular gap. At half filling, particle–hole symmetry pairs $\pm\delta_N$, so the lowest neutral excitation in the fixed-charge sector costs $2\delta_N$.

Cutoff flow of the half-filled interval spectrum; logarithms are natural.

$N=\ell/a$	$a/\ell$	$\delta_N$	fixed-charge gap $2\delta_N$	$\delta_N\log(4N)$
16	$1/16$	$0.91834$	$1.83668$	$3.81927$
32	$1/32$	$0.81196$	$1.62392$	$3.93965$
64	$1/64$	$0.72794$	$1.45589$	$4.03657$
128	$1/128$	$0.65986$	$1.31972$	$4.11643$

The raw gap in the explicit calculation closes rather than stabilizes. The slowly varying last column is only a diagnostic of the expected logarithmic rescaling; a continuum comparison should use the full universal eigenvalue distribution or another declared rescaled quantity. The CFT scaling distribution is derived and tested in Calabrese and Lefevre 2008, Eqs. (2)–(7) and Fig. 3.

The $U(1)$ charge sector is not decorative metadata. With $Q_A=\sum_j f_j^\dagger f_j$, filling or emptying the mode nearest zero changes $Q_A$ and costs $\delta_N$, whereas the first neutral particle–hole excitation costs $2\delta_N$. Mixing these sectors would create a spurious factor-of-two disagreement.

Spectrum alone is not the continuum check. Form

$$h_N=\log\!\left[(I-C_N)C_N^{-1}\right], \qquad K_N=c^\dagger h_Nc+\log Z_N,$$

and compare modular-flow correlators

$$G_N(s;f,g)=\left\langle c^\dagger(f)\, e^{isK_N}c(g)e^{-isK_N}\right\rangle$$

for smooth wave packets $f,g$ supported a fixed physical distance from the endpoints. In the vacuum CFT interval, the target generator is

$$K_A^{\mathrm{CFT}} =2\pi\int_0^\ell \frac{x(\ell-x)}{\ell}\,T_{00}(x)\,dx+\text{constant},$$

whose geometric flow satisfies

$$u(x_s)=u(x)+2\pi s, \qquad u(x)=\log\frac{x}{\ell-x}.$$

The acceptance test is convergence of smeared $G_N$ to this flowed correlator together with convergence of the chosen rescaled spectral statistic. Individual hopping ranges in $h_N$ do not provide that test: the long-range lattice couplings combine nontrivially to recover the local CFT kernel Eisler, Tonni, and Peschel 2019, §§II–IV and Eqs. (43)–(46).

Adversarial test: move the cut

A cut-induced degeneracy can be exact and still nonuniversal. In the strong-dimer product state, each dimer crossed by the boundary contributes one correlation eigenvalue $\nu=1/2$. An interval whose two endpoints each cut a dimer therefore has two such modes and four equal nonzero many-body eigenvalues, $\lambda=1/4$. Translating both endpoints between dimers leaves the bulk state unchanged but crosses no dimer, so the interval state is pure and its spectrum has one nonzero eigenvalue.

This failure injection changes only the microscopic placement of the cut. A proposed continuum degeneracy must be tested along both allowed endpoint sequences as $a\to0$ and after the same symmetry resolution. If the sequences do not converge to the same declared observable, the degeneracy is boundary data, not a universal bulk invariant. In topological phases, the spectrum can still be a powerful diagnostic, but it must be combined with response, quasiparticle, and finite-size evidence; Li and Haldane 2008, pp. 010504-1–010504-4 gives the original fractional-Hall application. See also the dedicated topological-spectrum diagnostics.

What survives the type-III limit

For a sharp continuum region, the intrinsic objects are its local von Neumann algebra, the state, and the corresponding modular operator and flow. A type-III algebra does not supply a trace-class reduced density matrix with a finite or countable probability spectrum. The regulated $\rho_A^{(a)}$ is therefore a representative, not the intrinsic continuum object Witten 2018, §§3 and 6.

A defensible continuum statement must specify at least one target:

• a modular-flow correlator smeared away from the entangling surface;
• a resolvent or smeared spectral measure of a relative modular operator;
• a local entanglement-Hamiltonian kernel tested on smooth states;
• or a universal, explicitly rescaled distribution derived from a cutoff family.

Report the state, region, algebra or split factor, boundary prescription, physical size, regulator family, modular-energy normalization, symmetry sectors, numerical truncation, scaling window, and uncertainty. A finite-spectrum degeneracy by itself is never a phase classification.

Exercises

1. Support and normalization

Let $\rho=\operatorname{diag}(1/2,1/3,1/6,0)$. Find the modular energies on the support, explain the zero eigenvalue, and determine whether adding $7I$ to $K=-\log\rho$ describes the same normalized density matrix.

Solution

The finite modular energies are $(\log2,\log3,\log6)$. The last basis vector lies in the kernel of $\rho$ and has modular energy $+\infty$. The shifted operator gives

$$e^{-(K+7I)}=e^{-7}\rho.$$

After dividing by its trace one recovers $\rho$, so the shift is harmless only if the operator is being treated as an unnormalized $H_E$ with a new partition function. It is not the same normalized modular Hamiltonian: $-\log\rho$ itself has its constant fixed.

2. Reconstruct a Gaussian spectrum

Starting from independent occupations $\nu_1,\ldots,\nu_M$, derive the relation $\varepsilon_j=\log[(1-\nu_j)/\nu_j]$ and the many-body eigenvalues $\lambda_{\boldsymbol n}$.

Solution

For one mode, write

$$\rho_j=Z_j^{-1}e^{-\varepsilon_j f_j^\dagger f_j}.$$

Its empty and occupied weights are $Z_j^{-1}$ and $Z_j^{-1}e^{-\varepsilon_j}$, with $Z_j=1+e^{-\varepsilon_j}$. Therefore

$$\nu_j=\frac{e^{-\varepsilon_j}}{1+e^{-\varepsilon_j}} =\frac{1}{e^{\varepsilon_j}+1},$$

which gives the stated logarithm. Tensoring the independent modes multiplies their probabilities, so an occupation string $\boldsymbol n$ has

$$\lambda_{\boldsymbol n} =\prod_j\nu_j^{n_j}(1-\nu_j)^{1-n_j}.$$

Summing independently over every $n_j=0,1$ gives one, which verifies normalization.

3. Reproduce the cutoff table

Construct $C_N$ for $N=16,32,64$ at half filling, diagonalize it, and calculate $\delta_N$, $2\delta_N$, and $\delta_N\log(4N)$. What conclusion is ruled out by the data?

Solution

For each $N$, fill an $N\times N$ real matrix with $C_{mm}=1/2$ and

$$C_{mn}=\frac{\sin[\pi(m-n)/2]}{\pi(m-n)}$$

off the diagonal. A symmetric eigensolver gives $\nu_j$; compute $\varepsilon_j=\log[(1-\nu_j)/\nu_j]$ and take the smallest absolute value. The results are

$$\begin{array}{c|ccc} N&\delta_N&2\delta_N&\delta_N\log(4N)\\ \hline 16&0.91834&1.83668&3.81927\\ 32&0.81196&1.62392&3.93965\\ 64&0.72794&1.45589&4.03657 \end{array}$$

The unscaled gap decreases, so these data rule out convergence to a nonzero raw entanglement gap. They are compatible with slow logarithmic closing; establishing the continuum law requires larger $N$, error control, and comparison with the full CFT scaling function.

4. Diagnose a cut degeneracy

In a product of maximally entangled dimers, compare an interval that cuts two dimers with one whose endpoints lie between dimers. Which part of the calculation changes, and what would have to be shown before calling either spectrum universal?

Solution

Each crossed dimer leaves one maximally mixed mode in the interval, with $\nu=1/2$. Two crossed dimers therefore give

$$\rho_A=\frac{I_2}{2}\otimes\frac{I_2}{2}=\frac{I_4}{4},$$

so four eigenvalues equal $1/4$. If no dimer is crossed, every dimer is wholly inside or outside $A$ and $\rho_A$ is pure, with spectrum $(1,0,\ldots)$. The bulk state did not change; only the endpoint prescription did. A universality claim would require both allowed cut sequences, with fixed physical geometry and symmetry convention, to converge to the same declared continuum observable. They do not in this example.

References

• Calabrese, Pasquale, and Alexandre Lefevre. “Entanglement Spectrum in One-Dimensional Systems.” Physical Review A 78 (2008): 032329. DOI. Open preprint.
• Eisler, Viktor, Erik Tonni, and Ingo Peschel. “On the Continuum Limit of the Entanglement Hamiltonian.” Journal of Statistical Mechanics: Theory and Experiment (2019): 073101. DOI. Open preprint.
• Li, Hui, and F. Duncan M. Haldane. “Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States.” Physical Review Letters 101 (2008): 010504. DOI. Open preprint.
• Peschel, Ingo. “Calculation of Reduced Density Matrices from Correlation Functions.” Journal of Physics A: Mathematical and General 36 (2003): L205–L208. DOI. Open preprint.
• Witten, Edward. “Notes on Some Entanglement Properties of Quantum Field Theory.” Reviews of Modern Physics 90 (2018): 045003. DOI. Open preprint.