---
title: Momentum-Resolved Entanglement Spectrum
url: https://www.emergentmind.com/topics/momentum-resolved-entanglement-spectrum-es
type: topic
---

# Momentum-Resolved Entanglement Spectrum

Momentum-resolved entanglement spectrum (ES) denotes an organization of entanglement data by a momentum quantum number, most commonly the momentum conserved **along** an entanglement cut in a translation-invariant real-space bipartition. In that setting, the reduced density matrix or its single-particle correlation operator decomposes into independent momentum sectors, yielding entanglement levels such as $\epsilon_n(k_\perp)$ or $\xi_n(k_\parallel)$. The same phrase is also used, more loosely, for entanglement spectra obtained from **momentum-space bipartitions**, and in some recent work for **momentum-resolved spectral functions of the entanglement Hamiltonian** rather than direct Schmidt levels. Across these usages, the momentum variable is valuable because it exposes boundary dispersions, spectral flow, fixed-momentum degeneracy patterns, and departures from locality or universality that are invisible in an unordered list of entanglement eigenvalues [1410.8670].

## 1. Definitions, geometries, and principal usages

In the standard condensed-matter usage, one partitions space by a cut normal to one direction while preserving translation symmetry in the transverse directions. The conserved momentum parallel to the cut, denoted $k_\perp$ or $k_y$ in two dimensions, block-diagonalizes the entanglement problem. For free fermions this yields single-particle entanglement energies in each sector, $\epsilon_{n,a}(k_\perp)$, while for interacting states on cylinders one often plots low-lying many-body entanglement levels at fixed transverse momentum and compares their counting or dispersion to edge theories [1410.8670].

A second usage concerns **momentum-space entanglement**, where the Hilbert-space partition itself is made in reciprocal space. There the ES is usually organized by the particle number or other quantum numbers remaining in one momentum subset, rather than by a conserved boundary momentum. A third, related but distinct construction is the phase-space entanglement spectrum, which interpolates continuously between position-space and momentum-space cuts by rotating the cut in phase space; this is not the standard meaning of momentum-resolved ES in topological band theory [1802.09531].

| Construction | Momentum variable | Typical output |
|---|---|---|
| Real-space cut with translation symmetry along boundary | $k_\parallel$, $k_y$, or $k_\perp$ | $\epsilon_n(k_\parallel)$, fixed-momentum level counting, spectral flow |
| Momentum-space bipartition | Momentum subset defining subsystem | ES organized by $N_A$, $M_A$, or related subsystem labels |
| Dynamical entanglement spectroscopy | Boundary momentum $q$ | Spectral intensity $G(\omega,q)$ of the entanglement Hamiltonian |
| Phase-space cut | Rotation parameter $\theta$; momentum cut at $\theta=\pi/2$ | Phase-space ES, not standard boundary-momentum-resolved ES |

For Gaussian states, the formal starting point is the reduced density matrix
\[
\rho_A=e^{-H_E},
\]
with a quadratic single-particle entanglement Hamiltonian determined by the restricted correlation matrix $C$. In the standard free-fermion formulation,
\[
h_E=\log(C^{-1}-\mathbb I), \qquad \epsilon_n=\log\frac{1-\lambda_n}{\lambda_n},
\]
where $\lambda_n$ are eigenvalues of $C$ [1410.8670]. This single-particle description underlies much of the explicit momentum-resolved literature.

## 2. Canonical real-space momentum resolution

For translation-invariant free fermions, the real-space cut and the occupied-band projector $P$ lead to the restricted correlator
\[
\hat C=RPR,
\]
or equivalently
\[
\hat C'=PRP,
\]
with identical nonzero eigenvalues because $P$ and $R$ are projectors. When translation symmetry survives along the cut, the correlation matrix decomposes into sectors labeled by conserved transverse momentum. On a cylinder in \(2+1\) dimensions, one may write \(C=\{C^{k_y}_{i,a;j,b}\}\), the many-body ground state factorizes as \(|\mathrm{GS}\rangle=\bigotimes_{k_y}|\mathrm{GS}_{k_y}\rangle\), and the entanglement energies become \(\varepsilon_n(k_y)\) [1410.8670]. This is the basic momentum-resolved ES construction.

In paired two-dimensional superfluids on an infinite cylinder, the BCS ground state likewise factorizes into \(k_y\)-sectors, and the Schmidt decomposition can be carried out sector by sector. For the model weak-pairing \(p\)-wave state, the ES in each momentum sector has only two Schmidt terms and an exact pseudo-energy branch
\[
\varepsilon(k_y)=2\ln\left(\sqrt{1+\left(\frac{k_y}{m^*\widehat{\Delta}}\right)^2}+\frac{k_y}{m^*\widehat{\Delta}}\right),
\]
with small-\(k_y\) form \(\varepsilon(k_y)\simeq 2k_y/(m^*\widehat{\Delta})\). In the weak-pairing phase of \(\ell\)-wave paired spinless fermions, the universal low-lying momentum-resolved ES contains \(|\ell|\) chiral Majorana modes; for spin-singlet even-\(\ell\) states it contains \(2|\ell|\) chiral Majorana modes [1105.4808].

Coupled Tomonaga–Luttinger liquids provide another canonical setting. Tracing out one chain preserves subsystem translation symmetry, so the ES is naturally resolved by subsystem momentum
\[
k=\frac{2\pi j}{L}.
\]
The resulting single-particle entanglement dispersion \(w_k\) depends sharply on which channels are gapped. For coupled non-chiral TLLs, both channels gapped give a linear ES, \(w_k\approx v_e|k|\); one gapless and one gapped channel give the nonanalytic law \(w_k\approx 2\sqrt{v_e|k|}\); and both channels gapless yield a flat ES [1310.0829]. This already shows that momentum-resolved ES is a diagnostic of the locality of the entanglement Hamiltonian, not merely of edge counting.

A non-topological many-body example is the two-dimensional Bose–Hubbard model on a cylinder cut across its axis. In the Mott phase, the ES is controlled by boundary-local virtual processes, but because the cut is a ring of length \(W\), these boundary excitations disperse in momentum \(k\) along the cut. For the lowest \(\delta N_A=1\) multiplet, the perturbative entanglement dispersion is
\[
\xi_{k}=\log(U^2/2)+\frac{2(2W+10)}{U^2}
-24\Big[\frac{1}{U}+\frac{2W-25}{U^3}\Big]\cos k
-\frac{68}{U^2}\cos 2k
+\frac{552}{U^3}\cos 3k,
\]
and the corresponding entanglement Hamiltonian is an effective one-dimensional tight-binding model defined on the boundary ring [1212.5634].

## 3. Asymptotic spacing, Wannier interpolation, and edge correspondence

A distinctive contribution of the free-fermion literature is that momentum resolution controls not only the **existence** of entanglement branches but also the **spacing** between adjacent entanglement levels in each momentum sector. For a translation-invariant cut, an interpolation between the entanglement correlator \(PRP\) and the Wannier operator \(PXP\) yields the asymptotic formula
\[
\epsilon_{n,a}(k_\perp)\approx [n+X_a(k_\perp)]\,f(g(k_\perp)), \qquad f(g)>2g.
\]
Here \(X_a(k_\perp)\) is the Wannier polarization in the fixed-\(k_\perp\) sector, and \(g(k_\perp)\) is the Wannier decay rate or analyticity scale set by the nearest complex-momentum gap closing [1410.8670].

This has several consequences. First, within each momentum sector the ES is asymptotically equally spaced,
\[
\epsilon_{n+1,a}(k_\perp)-\epsilon_{n,a}(k_\perp)\approx f(g(k_\perp)).
\]
Second, the spectral flow of the ES is inherited from \(X_a(k_\perp)\), so the branch motion with \(k_\perp\) is directly tied to topological pumping. Third, the spacing is controlled by complex analyticity: \(g\) is the distance from the real momentum axis to the nearest singularity of the projector \(P(k)\), equivalently the closest complex momentum at which the occupied/unoccupied gap closes [1410.8670].

The same interpolation gives a one-to-one correspondence between the entanglement spectrum and edge states. In the two-dimensional Dirac/Chern-insulator example, the Wannier polarization flows with \(k_y\), the correlator eigenvalues \(c\) show a corresponding branch crossing from \(0\) to \(1\), and the entanglement energies \(\epsilon=\log(c^{-1}-1)\) exhibit the same spectral flow, but with momentum-dependent level spacing. The shift over one \(k_y\)-period is identified with the Chern number \(C_1\) [1410.8670].

A practical implication is numerical. Since \(\lambda_n\sim e^{-f(g)n}\), larger \(g(k_\perp)\) implies faster decay of entanglement occupancies and therefore more favorable truncation in tensor-network methods. This suggests that momentum-resolved ES contains direct information about truncation error in MPS or PEPS calculations, not only about topology [1410.8670].

## 4. Nonlocal entanglement Hamiltonians and anomalous momentum dependence

Momentum-resolved ES is especially informative when the entanglement Hamiltonian is nonlocal. In coupled non-chiral TLLs, the \(\sqrt{k}\) law
\[
w_k\approx 2\sqrt{v_e|k|}
\]
cannot arise from a local one-dimensional Hamiltonian with a standard derivative expansion. It is instead traced to logarithmic long-range terms in the entanglement Hamiltonian, such as
\[
-\frac{2K_+}{\pi}\int\!\!\int dx\,dx'\,
\partial_x\theta_1(x)\,\partial_{x'}\theta_1(x')\,
\ln\left|e^{i\frac{2\pi}{L}x}-e^{i\frac{2\pi}{L}x'}\right|,
\]
or the analogous density-field kernel in the complementary partially gapped case [1310.0829]. In that setting, the momentum-resolved ES directly diagnoses whether \(H_E\) is local or long-ranged.

A related phenomenon appears in quantum Monte Carlo studies of a two-dimensional AKLT-related spin model. There the measured object is not a list of Schmidt eigenvalues \(\xi_n(q)\) but a momentum-resolved **spectral function** of the entanglement Hamiltonian,
\[
G(\tau_A,q)=\frac{1}{L}\sum_{i,j} e^{-iq_x(x_i-x_j)} \langle s_i^z(\tau)\, s_j^z(0)\rangle,
\]
analytically continued to \(G(\omega,q)\). In the AKLT phase (\(g=0.4\)) the entanglement spectrum shows a gapless two-spinon continuum resembling the expected virtual edge theory. In the Néel phase, by contrast, the low-energy entanglement spectrum becomes a sharp magnon-like branch with an “M-shaped” dispersion that is sublinear near \(q=\pi\), well fit by
\[
\omega=a|\Delta q|^s,\qquad \Delta q=q-\pi,
\]
with \(s<1\), including \(s(\infty)=0.2016(7)\) at \(g=0.7\) and \(s(\infty)=0.211(2)\) at \(g=0.8\) [2506.10078]. The interpretation is that gapless bulk modes induce relevant long-range interactions in the entanglement Hamiltonian.

The wormhole framework sharpens this conclusion. In a perturbed boundary study of the two-dimensional AKLT model, momentum-resolved ES can gap out even when the physical edge of subsystem \(A\) remains gapless, because the relevant object is a **generalized entanglement boundary** involving both \(A\), \(\bar A\), and the coupling across the cut. The replica path integral creates short worldline paths through the traced-out region, and the usual Li–Haldane correspondence emerges only as a special limit of this broader mechanism [2303.00772]. A plausible implication is that a boundary-momentum branch in the ES should not be identified automatically with the spectrum of a local physical edge Hamiltonian.

A further caution comes from the spin-\(\tfrac12\) ladder in the Haldane phase. The momentum-resolved ES there was long interpreted as a single des Cloizeaux–Pearson-like \(\sin|k|\) mode, but exact diagonalization up to 40 spins resolves two distinct low-energy branches centered at \(k=0\) and \(k=\pi\), crossing near \(k=\pi/2\). Breaking SU(2) with XXZ anisotropy then produces an entanglement quantum phase transition at \(\Delta_c^{\mathrm{ES}}\approx1\), distinct from the bulk critical point \(\Delta_c^{\mathrm{GS}}\approx1.18\) [2509.03588]. This is another case where the momentum-resolved ES reveals phase structure intrinsic to \(H_E\).

## 5. Fixed-momentum splittings, conformal sectors, and chirality diagnostics

In chiral topological phases on cylinders, the simplest momentum-resolved statement is Li–Haldane counting: at each fixed momentum, the number of low-lying ES states matches the edge conformal field theory. More refined analyses ask what determines the **splittings within a fixed momentum sector**. For real-space entanglement spectra of \((2+1)\)-dimensional chiral phases with global SU(2) symmetry, the relevant momentum is
\[
k_L=\frac{2\pi}{\ell}L_0,
\]
so states at fixed descendant level \(K\) have fixed momentum along the cut. The observed level splittings at fixed \(K\) are explained not by \(H_L\) alone but by a generalized Gibbs ensemble
\[
H_E-\mathrm{const.}=\sum_i \beta_i H^{(i)},\qquad
H^{(i)}=\frac{1}{2\pi}\int_0^\ell \Phi_i(x)\,dx,
\]
built from symmetry-allowed conserved quantities of the chiral CFT [2107.02545].

For \(\mathrm{SU}(2)_2\), some of these conserved quantities are local integrals of operators of fractional conformal dimension, notably \(G_0\), with
\[
G_0^2=L_0-\frac{c}{24}.
\]
This implies the asymptotic scaling
\[
H_E \sim \pm \beta_0 \sqrt{v\,k_L}+O(k_L)
\]
in the \(j=\tfrac12\) sector, a striking departure from simple linear dispersion [2107.02545]. The fixed-momentum splitting pattern therefore becomes a finer diagnostic than state counting alone.

The PEPS literature on \(\mathrm{SU}(3)\) spin liquids pushes this logic further. A non-chiral PEPS with \(D(\mathbb Z_3)\) topological order can show low-lying ES branches in some sectors that precisely reproduce the Li–Haldane counting of a chiral \(\mathrm{SU}(3)_1\) CFT, because its ES contains left- and right-moving branches with a strong velocity hierarchy [2207.03246]. In that regime the low-lying ES may consist of the lowest primary multiplet of a high-velocity branch tensored with the full content of a low-velocity branch, so selected sectors look effectively chiral.

For genuinely chiral \(\mathrm{SU}(3)\) PEPS, however, the momentum-resolved ES obeys a stronger constraint: conjugate irreps are exactly degenerate, because the conserved quantities that would split them—those related to the cubic Casimir and odd-dimensional \(W\)-type operators—are forbidden by symmetry. In non-chiral states, conjugate splittings are demonstrably non-vanishing [2305.13240]. This makes fixed-momentum SU(3)-multiplet structure a sharper chirality diagnostic than Li–Haldane counting by itself.

## 6. Momentum-space partitions, universality limits, and interpretive cautions

The phrase “momentum-resolved entanglement spectrum” is frequently conflated with **momentum-space entanglement spectrum**, but the distinction is substantive. In the spin-\(\tfrac12\) XXZ chain, one may bipartition momentum orbitals as
\[
A=\{m>0\},\qquad B=\{m\le 0\},
\]
and then organize the ES by subsystem particle number \(N_A\) and subsystem momentum \(M_A\) or crystal momentum \(M_{A,c}\). In that setting the entanglement gap need not close at the physical phase transition: for bosons it closes within the gapless \(c=1\) phase, while for fermions it remains open well into the Ising phase [1404.7545]. This is momentum-space bipartition, not a real-space cut resolved by boundary momentum.

Disordered free-fermion models provide another momentum-space construction. A left/right mover bipartition in momentum space reveals localization physics because disorder-induced backscattering entangles opposite-velocity sectors. Near resonant extended states of \(n\)-mer models, the momentum-space ES becomes weakly entangled, with many single-particle entanglement eigenvalues \(\zeta_i\) moving toward \(0\) or \(1\), whereas localization correlates with stronger left/right entanglement [1206.3313]. In an interacting disordered one-dimensional fermion model, the ES after positive/negative momentum or small/large momentum cuts is organized by subsystem particle number \(n\), not by a conserved subsystem momentum; it shows paired-particle and chaotic components and is used to diagnose localization and an interaction-driven transition [1607.05877].

These examples motivate a broader caution already explicit in the literature on real-space, momentum-labeled ES: low-lying ES structure is not generically universal. In free fermions with a cut, “the ES vs the momentum along the cut, \(k_y\) is a band spectrum,” yet the entanglement Hamiltonian can undergo its own phase transitions, and quasi-degeneracy counting as a function of momentum difference \(\delta k_y\) can change non-universally; in one configuration the counting is \(\{1,1,2,3,5,\ldots\}\), in another \(\{1,2,5,\ldots\}\), and by changing mode velocities “almost any sequence is possible” [1311.2946]. The safest inference is that momentum labels expose structure but do not restore universality.

A concise synthesis follows. Momentum-resolved ES is most sharply defined for real-space cuts that preserve translation symmetry along the boundary, producing entanglement levels or entanglement-Hamiltonian spectra as functions of \(k_\parallel\). In that form it diagnoses boundary dispersions, spectral flow, fixed-momentum multiplet splittings, and the locality properties of \(H_E\). But the same literature shows that \(H_E\) may be long-ranged, may undergo pseudo-transitions disconnected from the bulk, and may preserve momentum labels while deviating strongly from any local edge theory. Momentum resolution is therefore a powerful organizer of entanglement data, but not, by itself, a guarantee of edge universality or of a literal bulk-boundary correspondence [1311.2946].

Source: https://www.emergentmind.com/topics/momentum-resolved-entanglement-spectrum-es