Li-Haldane Conjecture in Topological States

• Li-Haldane Conjecture is the proposal that the low-energy entanglement spectrum of a topological state mirrors its physical boundary excitations.
• It establishes a bulk-edge correspondence by relating reduced density matrix eigenvalues to gapless edge modes, as seen in fractional quantum Hall systems.
• Extensions include analytical proofs, finite-size refinements, and generalizations to gapless free-fermion models with implications for experimental entanglement Hamiltonian reconstruction.

The Li-Haldane conjecture is the proposal that the low-lying entanglement spectrum of a topological quantum state contains the universal structure of its physical boundary excitation spectrum. In the standard formulation, one writes the reduced density matrix of a subsystem as $\rho_A = e^{-H_E}$, with $H_E$ the entanglement Hamiltonian, and interprets the eigenvalues of $H_E$ as an entanglement spectrum whose low-energy part mirrors edge physics and thereby diagnoses topological order (Zache et al., 2021). Originally formulated in the context of fractional quantum Hall states, the conjecture has subsequently been analytically substantiated for broad classes of model states, refined by finite-size counting results, extended to critical free-fermion systems in arbitrary dimensions, and also challenged in settings where the entanglement Hamiltonian acquires relevant long-range or environment-dependent structure not captured by a simple virtual-edge picture (Chandran et al., 2011, Hermanns et al., 2010, Guo et al., 24 Sep 2025, Li et al., 2023, Liu et al., 2023).

1. Core statement and conceptual content

The classic statement, as summarized in later work, is that the entanglement spectrum of a gapped topological phase corresponds to its physical boundary excitation spectrum, in particular that low-energy entanglement levels reflect edge states (Guo et al., 24 Sep 2025). In a closely related formulation, if $\rho_A = \exp(-H_E)$, then the low-lying part of the spectrum of $H_E$ mirrors the physical edge spectrum of the phase (Bietenholz et al., 2016). This elevates the entanglement spectrum from a scalar diagnostic, such as entanglement entropy, to a spectroscopic object that can encode gapless boundary modes, symmetry-protected degeneracies, and other universal data not accessible from local order parameters alone (Bietenholz et al., 2016).

The conceptual picture is that a bipartition creates a “virtual” boundary. The entanglement Hamiltonian associated with that cut can then inherit the same universal structure that governs an actual edge. In later experimentally oriented work, this perspective is connected to the quasi-local structure of entanglement Hamiltonians and to the Bisognano-Wichmann theorem, which motivates viewing $H_E$ as a spatially deformed Hamiltonian localized near the cut (Zache et al., 2021). In that framework, the conjecture is not merely a numerical pattern in spectra but a structural statement relating reduced-density-matrix eigenvalues to edge conformal field theory data.

A recurring implication across the literature is that the conjecture is a bulk diagnostic of topological order. Because the spectrum is extracted from a bulk wavefunction without introducing an actual open boundary, it realizes a form of entanglement-based bulk-edge correspondence (Chandran et al., 2011). This role becomes especially pronounced in later generalizations, where the entanglement spectrum is used to detect nontrivial topology even when conventional topological invariants are ill-defined at criticality (Guo et al., 24 Sep 2025).

2. Fractional quantum Hall origin and analytic substantiation

The fractional quantum Hall setting provided the original arena in which the conjecture became influential. In this context, the orbital entanglement spectrum (OES) is obtained by partitioning single-particle orbital space, tracing out one part, and diagonalizing the reduced density matrix. The low-lying OES was conjectured to reproduce the counting of edge conformal field theory excitations. A major analytic substantiation was then given for model fractional quantum Hall states that are unique zero modes of pseudopotential Hamiltonians (Chandran et al., 2011).

That substantiation proceeds by relating two entanglement spectra. The particle entanglement spectrum (PES), obtained by partitioning particles rather than orbitals, has counting identical, up to accidental degeneracies, to the counting of bulk quasiholes. The OES, by contrast, is associated with edge physics. The central result is a one-to-one map between the thermodynamic-limit counting of the PES and the OES, thereby giving a microscopic realization of bulk-edge correspondence in entanglement spectra (Chandran et al., 2011).

The algebraic mechanism is encoded in clustering operators,

$$D_\beta |\psi\rangle = \left( \sum_{l_1,\ldots,l_k=0}^{N_\phi} d_{\beta-\sum_{j=1}^{k}l_j}\prod_{j=1}^{k} d_{l_j}\right)|\psi\rangle=0,$$

which arise from conformal field theory operator expansions and constrain the model wavefunctions (Chandran et al., 2011). Using these constraints, the counting of OES eigenvalues in the thermodynamic limit is shown to equal the counting of bulk quasiholes, and the latter equals the counting of edge modes at a hard-wall boundary placed on the sample (Chandran et al., 2011). The result is notable for applying not only to canonical gapped model states but also to conformal-field-theory states likely bulk gapless, such as the Gaffnian wavefunction (Chandran et al., 2011).

In this line of work, the Li-Haldane conjecture becomes more than a heuristic comparison of low-lying levels. It is recast as an exact counting correspondence between bulk quasihole sectors and edge-mode sectors, mediated by the entanglement structure of the ground state (Chandran et al., 2011).

3. Finite-size refinements and sector-resolved diagnostics

Subsequent work sharpened the conjecture by showing that finite-size deviations from asymptotic edge counting are themselves structured. For Laughlin states at filling $\nu=1/m$, the counting of levels in the orbital entanglement spectra of finite-sized droplets was conjectured to be described by Haldane statistics of particles in a box of finite size (Hermanns et al., 2010). In this formulation, the number of non-zero reduced-density-matrix eigenvalues at fixed angular-momentum shift is determined by $(1,m)$-admissible occupation patterns rather than by a generic finite-size correction.

The counting is expressed through

$$N_s(l_A,N_A,\Delta L_z) = \min[N(l_A,N_A,\Delta L_z),\, N(l_B,N_B,\Delta L_z)],$$

with

$H_E$0

and the extended orbital length $H_E$1 determined differently for bosons and fermions (Hermanns et al., 2010). In this refinement, the universal part of the finite-size entanglement spectrum encodes not only edge-mode counting but also the statistical parameter $H_E$2, allowing extraction of the boson compactification radius $H_E$3, which is inaccessible in the thermodynamic-limit counting alone (Hermanns et al., 2010).

A different refinement arises in non-chiral topological phases with strong time-reversal breaking. In an $H_E$4 spin liquid PEPS with $H_E$5 topological order and nine sectors, some entanglement spectra on long cylinders can appear chiral in a subset of sectors and precisely follow Li-Haldane state counting of a truly chiral phase, even though the state is non-chiral and has zero chiral central charge (Arildsen et al., 2022). The mechanism is a pronounced velocity hierarchy between left- and right-moving branches, with $H_E$6, so that only primary states from the fast branch contribute at low entanglement energies while the slow branch contributes a full tower of descendants (Arildsen et al., 2022).

This yields a precise caution. Limited entanglement-spectrum data, especially in the topologically trivial sector, can reproduce the expected chiral counting and lead to misidentification. Only a full sector-resolved analysis across all nine topological sectors reveals the non-chiral doubled structure (Arildsen et al., 2022). The conjecture therefore remains useful, but its operational content depends strongly on which sectors are examined.

4. Generalization to critical free-fermion systems

A major recent development is the extension of the Li-Haldane correspondence from gapped to gapless free-fermion systems at topological criticality. In “Generalized Li-Haldane Correspondence in Critical Free-Fermion Systems” (Guo et al., 24 Sep 2025), an exact relation is established between the bulk entanglement spectrum and the boundary energy spectrum of critical free-fermion systems protected by global on-site symmetries, in arbitrary spatial dimensions.

For a quadratic Majorana Hamiltonian,

$H_E$7

the reduced density matrix of a subsystem is written as

$H_E$8

The entanglement Hamiltonian is constructed from the bulk single-particle Hamiltonian by the flattening formula

$H_E$9

and its single-particle eigenvalues $H_E$0 are related to the entanglement-spectrum levels $H_E$1 by

$H_E$2

Within this framework, $H_E$3 is the entanglement-spectrum signature of a zero mode (Guo et al., 24 Sep 2025).

The principal theorem is that zero modes in the boundary energy spectrum of a critical topological free-fermion system correspond exactly to zero-mode levels at $H_E$4 in the bulk entanglement spectrum (Guo et al., 24 Sep 2025). Equivalently, the number of entanglement-spectrum levels at $H_E$5 equals the number of robust edge zero modes at the open boundary. This furnishes a bulk diagnostic of topological criticality without introducing open boundaries.

The paper further reports explicit numerical illustrations in one-dimensional AIII systems, two-dimensional Chern insulators, and three-dimensional class DIII topological superconductors, together with persistence under symmetry-preserving disorder and under moderate interactions in appropriate symmetry classes, where the correspondence is verified with DMRG (Guo et al., 24 Sep 2025). This suggests that the Li-Haldane idea can survive far beyond its original gapped fractional quantum Hall setting, provided the relevant structural hypotheses are satisfied.

5. Limits of validity, counterexamples, and revised mechanisms

Several later works emphasize that the Li-Haldane correspondence should not be interpreted as an unrestricted identification of the full entanglement Hamiltonian with a physical edge Hamiltonian. In a two-dimensional quantum antiferromagnetic bilayer Heisenberg model, large-scale simulation of the entanglement Hamiltonian showed that the entanglement spectrum can resemble that of a single-layer Heisenberg model while the entanglement Hamiltonian itself contains relevant long-range interactions that qualitatively alter its physics (Li et al., 2023). The decisive evidence is a finite-temperature phase transition of the entanglement Hamiltonian, which would be forbidden by the Mermin-Wagner theorem for a short-range two-dimensional Heisenberg Hamiltonian (Li et al., 2023).

In that analysis, the Li-Haldane-Poilblanc conjecture is said to ignore necessary corrections to the entanglement Hamiltonian. The low-lying spectrum may still show a superficial resemblance to edge excitations, but critical and thermodynamic properties can differ because the operator content includes relevant long-range terms (Li et al., 2023). The distinction between spectrum shape and full Hamiltonian equivalence is therefore essential.

A more direct challenge comes from the two-dimensional AKLT model on the square-octagon lattice with a tunable boundary. There, quantum Monte Carlo extraction of large-scale entanglement spectra yields counterexamples in which the low-lying entanglement spectrum does not always show behavior similar to the energy spectrum on the virtual boundary, and can even resemble the edge spectrum when the edge is gapped (Liu et al., 2023). Perturbing only the environment can gap the entanglement spectrum even when the isolated subsystem boundary would be gapless (Liu et al., 2023).

To explain such cases, the work advances the “wormhole mechanism” in the replica path integral of the reduced density matrix. In that picture, worldlines can traverse the entanglement cut through wormhole-like shortcuts, and the relative weight of bulk and edge contributions is governed by the effective path cost, schematically summarized by a bulk-to-edge cost ratio proportional to $H_E$6 in the ground-state limit (Liu et al., 2023). Within this framework, the Li-Haldane conjecture emerges as a particular case in a limit where edge paths dominate; outside that limit, the entanglement spectrum can be controlled by environment and coupling effects (Liu et al., 2023). The scope of the conjecture is thus constrained, but not negated: it is situated inside a broader mechanism for entanglement-spectrum formation.

6. Relation to Haldane’s conjecture, later terminology, and experimental probes

The Li-Haldane conjecture is historically and conceptually related to, but distinct from, Haldane’s conjecture for spin chains. Haldane’s 1983 statement for one-dimensional antiferromagnetic Heisenberg chains is that half-integer spins are gapless and integer spins have a finite gap,

$H_E$7

with the low-energy mapping to the $H_E$8 nonlinear sigma model and $H_E$9 (Bietenholz et al., 2016). Later work interprets the integer-spin Haldane chain as a symmetry-protected topological phase with nontrivial edge states and a characteristic entanglement spectrum, while half-integer chains are gapless and do not realize the same SPT order (Bietenholz et al., 2016). In that sense, the Li-Haldane conjecture uses entanglement data to reveal the same topological structures that Haldane’s original conjecture exposed through spectral gaps and topological terms.

The later literature does not use the label uniformly. Some works refer to generalized Haldane conjectures for $\rho_A = \exp(-H_E)$0 or $\rho_A = \exp(-H_E)$1 spin chains as “generalized Haldane (Li-Haldane) conjecture” (Gozel et al., 2020), and other works invoke a “Haldane-charge conjecture” whose topological protection is connected to the Pollmann et al. conjecture about Haldane phases and edge states (Nonne et al., 2011). This suggests that “Li-Haldane” can denote either the original entanglement-spectrum conjecture or a broader family of topological conjectures associated with Haldane phases in later usage.

Experimentally, the entanglement-spectrum conjecture has become a target of direct probing. For an integer quantum Hall state of non-interacting fermions on a two-dimensional lattice and for a one-dimensional interacting fermionic SPT phase, entanglement Hamiltonian tomography and quantum variational learning were proposed as routes to reconstruct $\rho_A = \exp(-H_E)$2 and measure its spectrum (Zache et al., 2021). In the free-fermion case, the entanglement Hamiltonian can be written as

$\rho_A = \exp(-H_E)$3

while more generally a Bisognano-Wichmann-type ansatz of the form

$\rho_A = \exp(-H_E)$4

is used (Zache et al., 2021). The reconstructed spectra reproduce chiral counting in the integer quantum Hall example and the expected fourfold ground-state manifold in the interacting SPT chain, thereby furnishing an experimental route to Li-Haldane spectroscopy (Zache et al., 2021).

Taken together, these developments place the Li-Haldane conjecture at the intersection of topological order, entanglement diagnostics, field-theoretic bulk-edge correspondence, and quantum simulation. Its strongest formulations are now analytic in several important settings, its finite-size and sector-dependent subtleties are better understood, and its limitations have become part of a more general theory of entanglement-spectrum formation.