Information Hierarchy in Many-Body Berry Phase

Abstract: Many-body topology is a central concept in modern theories of solids, and identifying effective degrees of freedom that capture it is important both fundamentally and practically. This work studies the extent to which geometric information of an interacting many-body ground state can be inferred from a finite number of local correlations. Starting from the Resta formula, $ z=\left\langle \exp!\left(\frac{2πi}{L}\hat X\right)\right\rangle$, we view $\log z$ as the cumulant generating function and establish a generic information hierarchy across cumulant orders. We show that, for an $N$-particle system, even complete knowledge of all density correlators up to order $N-1$ does not, in general, uniquely determine the Berry phase $γ=\operatorname{Im}\log z \, (\mathrm{mod}\ 2π)$. In the thermodynamic limit, the statement becomes stronger: no finite set of local correlators suffices to determine the global holonomy. We also identify two exceptional yes-go cases in which the hierarchy is broken. First, for quasi-free models, all cumulants are determined by the particle two-point correlation function. Second, symmetry-enforced constraints can reduce the infinite cumulant sum entering $\log z$ to finite information. The argument is analytic and does not rely on a specific microscopic Hamiltonian. Our results clarify a limitation of approaches based on local degrees of freedom for many-body holonomy and provide a minimal framework for distinguishing when global holonomies are encoded in local correlations and when they are not. We also comment on the possibility of analogous hierarchies in other contexts, such as the quantum marginal problem in quantum information theory and many-body scattering problems. Finally, we discuss implications for future numerical work, including machine-learning approaches to the search for topological phases.

• The paper shows that finite local correlators do not suffice to determine the many-body Berry phase, necessitating the full cumulant hierarchy for global holonomy.
• It employs the Resta formula and cumulant expansion to reveal that up to N-1 point correlators cannot capture the nonlocal topological invariant in interacting systems.
• The study identifies exceptions in quasi-free and symmetry-enforced settings, offering guidance for computational, functional, and machine learning approaches to topological phases.

Information Hierarchy in the Many-Body Berry Phase: Summary and Implications

Introduction and Motivation

The paper "Information Hierarchy in Many-Body Berry Phase" (2606.02155) addresses the problem of reconstructing global geometric/topological information (specifically, the many-body Berry phase) from finite local correlations in interacting many-body quantum systems. The motivation stems from the central role played by topological invariants—often encoded as Berry phases—in characterizing the global properties and responses of quantum materials, and their frequent correlation with macroscopically measurable quantities such as polarization.

The work explores to what extent the geometric phase $\gamma$ (as defined via the Resta formula) is determined by local expectation values and connected correlators and clarifies intrinsic limitations of such reconstructions. The results have significant implications for ab initio computational schemes, functional theory approaches, and machine learning frameworks targeting topological order.

Resta Formula and Cumulant Expansion Framework

The Berry phase for a many-body ground state $|\Psi\rangle$ on a finite periodic lattice of size $L$ is operationalized using Resta's expectation value of the LSM ("Lieb-Schultz-Mattis") unitary: $$z = \langle \Psi|e^{i \alpha_L \hat X}|\Psi\rangle, \qquad \alpha_L = \frac{2\pi}{L}, \;\; \hat X = \sum_j j \hat n_j$$ The phase $\gamma = \operatorname{Im}\log z \pmod{2\pi}$ provides an efficient route to compute the Abelian Berry phase associated with adiabatic $U(1)$ flux insertion.

The key technical device is recognizing $\log z$ as a cumulant-generating function in the operator $\hat X$: $$\log z = \sum_{p\ge 1} \frac{(i \alpha_L)^p}{p!} \kappa_p$$ where the $p$-th order cumulant $|\Psi\rangle$0 depends on up to $|\Psi\rangle$1-point connected density correlators. Thus, the Berry phase is, in principle, a function of the full hierarchy of cumulants (or, equivalently, all local correlators).

Main Results: Order-by-Order Information Hierarchy

The central theoretical result is a model-independent no-go theorem: For a generic $|\Psi\rangle$2-particle ground state, the full set of all $|\Psi\rangle$3-point density correlators with $|\Psi\rangle$4 does not in general determine the Berry phase. In the thermodynamic limit, this strengthens: no finite set of local density correlators suffices to determine the global holonomy encoded by $|\Psi\rangle$5.

This is explicitly visualized with analytical examples, e.g., the $|\Psi\rangle$6 ring at half-filling, where two states can have identical one- and two-point correlators, and hence identical cumulants up to $|\Psi\rangle$7, but different $|\Psi\rangle$8 and thus distinct Berry phases (see Eqs. (24)--(30)). The analysis demonstrates that the difference between Berry phases persists at higher orders and is not guaranteed to vanish even if all lower-order local data are fixed.

The paper generalizes this to arbitrary system size, rigorously showing that the cumulant expansion does not truncate for generic interacting many-body ground states. Higher-order cumulants receive extensive combinatorial enhancement as $|\Psi\rangle$9 increases (because of the $L$0 scaling in $L$1-point correlators), and thus the full infinite hierarchy is necessary.

The authors further show that scaling arguments relying on the smallness of $L$2 ($L$3) are invalidated by the polynomial growth of cumulant contributions with $L$4; higher order terms are not suppressed.

Exceptional Scenarios: Yes-go Cases

Two situations are identified where the information hierarchy is bypassed, and global holonomy is accessible from finite local data:

1. Quasi-free (Gaussian) States: For noninteracting (quadratic) Hamiltonians, all higher-order density cumulants are determined by the two-point function due to Wick's theorem. The Berry phase then becomes a (nonlinear) function of the single-particle correlation matrix, as detailed via determinant formulas and explicit diagrammatic factorization.
2. Symmetry-enforced Constraints: Strong symmetries (e.g., inversion, time reversal) impose global constraints across all cumulant orders, reducing the necessary information for $L$5 to finite group data (e.g., particle number and lattice size in the inversion-symmetric case). The Berry phase becomes quantized (e.g., $L$6 invariants for time-reversal, quantization up to $L$7 for inversion), and hence is fixed from combinatorial or symmetry labels rather than the details of the correlator hierarchy.

The paper provides analytic demonstrations for both cases, including explicit manipulation of inversion symmetry constraints and the closure of the cumulant hierarchy for fermion Gaussian states.

Relation to Quantum Marginal Problem and Information Theory Perspective

The findings are situated within a broader information-theoretic landscape: Determining whether local reduced data specifies a unique global pure state is an instance of the quantum marginal problem—and, as established for Berry phases, the answer is generically negative without extra global consistency constraints.

The analogy extends to topological entanglement entropy and multipartite irreducible correlations, emphasizing that nonlocal invariants cannot be extracted from finite subsystems in general. This also connects to classical results about characteristic function polynomials and the structure of probability distributions, referencing Marcinkiewicz-type theorems.

Implications for Computation, Simulation, and Machine Learning

The limitations uncovered by the information hierarchy have direct practical and methodological consequences:

• Numerical Approaches: Methods relying on finite local correlation constraints (including many variational and functional paradigms) are insufficient for determining many-body Berry phases outside Gaussian/symmetric "yes-go" regimes. Accurate calculation or learning of the Berry phase in general interacting systems necessitates algorithms (e.g., DMRG, QMC, or quantum circuits) that access global wavefunction information or compute many-body Wilson loops directly.
• Machine Learning: Approaches that attempt to predict topological phases or invariants from finite-order correlators are subject to the same barrier—they cannot infer nonlocal holonomies unless restricted (by design or data) to symmetry-protected or quasi-free cases or unless genuinely nonlocal descriptors are used as input features. Nonlinearity in network architectures does not overcome this obstruction.
• Ab Initio and Functional Theories: The results clarify why reduced-density-matrix functionals can succeed (or fail) for topological quantities depending on the extent to which the system is symmetry-constrained or close to noninteracting reference points.

Theoretical Outlook

The conceptual framework advanced by the paper paves the way for further investigations into the information-theoretic structure of other topological invariants beyond the Berry phase, including many-body polarization, quadrupole moments, and scattering phase shifts in quantum field theory. The direct application to the quantum marginal and $L$8-representability problems offers a fertile avenue for cross-disciplinary techniques.

Potential future development includes the design of hybrid approaches that combine local and nonlocal information, systematic cataloging of symmetry-protected collapse of the information hierarchy, and new algorithms for extracting holonomy data amenable to large-scale numerical computation or quantum simulation.

Conclusion

This study establishes that, generically, the many-body Berry phase is not determined by any finite truncation of local correlators; the full set of correlated cumulants is necessary except in special symmetry-protected or quasi-free settings. These findings demarcate the boundaries of applicability for local and reduced-information methodologies in the study of interacting topological phases, clarify the role of symmetry breaking/formation in information reduction, and provide guidance for computational, functional, and machine learning approaches to many-body holonomy and topology.