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arXiv:2503.12879 · Physical review. B./Physical review. B

Frustration-free fermionic systems: Finite-size scaling and band topology

Rintaro Masaoka, Seishiro Ono, Hoi Chun Po, Haruki Watanabe

Published Apr 10, 2026• 1 citation
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Abstract

Frustration-free Hamiltonians provide pivotal models for understanding quantum many-body systems. In this paper, we establish a general framework for frustration-free fermionic systems. First, we derive a necessary and sufficient condition for a free fermion model to be frustration-free. In the case of translation-invariant, noninteracting systems, we show that any band touching between the valence and conduction bands is at least quadratic. Furthermore, by extending the Gosset-Huang inequality to fermionic systems, we demonstrate that even in interacting and non-translation-invariant cases, the finite-size gap of gapless excitations scales as $O((\log L)^2/L^2)$, provided the ground-state correlation function exhibits a power-law decay. Our results provide a foundation for studying frustration-free fermionic systems, including flat-band ferromagnetism and $η$-pairing states.

DOI
10.1103/qv6d-dk21

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Frustration-free Hamiltonians provide pivotal models for understanding quantum many-body systems. In this paper, we establish a general framework for frustration-free fermionic systems. First, we derive a necessary and sufficient condition for a free fermion model to be frustration-free. In the case of translation-invariant, noninteracting systems, we show that any band touching between the valence and conduction bands is at least quadratic. Furthermore, by extending the Gosset-Huang inequality to fermionic systems, we demonstrate that even in interacting and non-translation-invariant cases, the finite-size gap of gapless excitations scales as O((log⁡L)2/L2)O((\log L)^2/L^2)O((logL)2/L2), provided the ground-state correlation function exhibits a power-law decay. Our results provide a foundation for studying frustration-free fermionic systems, including flat-band ferromagnetism and ηηη-pairing states.
We develop a general theory of frustration-free free-fermion systems and derive the necessary and sufficient conditions for such Hamiltonians. Assuming locality and translation invariance, we find that any band touching between the valence and the conduction bands is always quadratic or softer, which rules out the possibility of describing Dirac and Weyl semimetals using frustration-free local Hamiltonians. We further construct a frustration-free free-fermion model on the honeycomb lattice and show that its density fluctuations acquire an anomalous gap originating from the diverging quantum metric associated with the quadratic band-touching points. Nevertheless, an O ( 1 / L 2 ) finite-size scaling of the charge-neutral excitation gap can be verified even in the presence of interactions, consistent with the more general results we derive in an accompanying work [Masaoka , ].
Recent studies have revealed that frustration-free models, expressed as sums of finite-range interactions or hoppings, exhibit several properties markedly different from those of frustrated models. In this work, we demonstrate that, by relaxing the finite-range condition to allow for exponentially decaying hoppings, one can build gapped frustration-free systems that realize Chern insulators as well as quasidegenerate ground states with finite-size splittings. Moreover, by permitting power-law decaying hoppings, we also construct a gapless band metal whose finite-size gap scales inversely with the system size LLL. These findings serve as an important step toward clarifying the general properties of frustration-free systems and those represented by tensor network states.
This thesis studies the mathematical structure of quantum Hall transport in lattice systems, with an emphasis on relating mutiple definitions of Hall conductance in different mathematical branches: Geometrically as a bulk Chern number for a Bloch bundle; analytically as an edge spectral flow for a half-space Hamiltonian; and in operator algebra language as a many body charge transport index generated by magnetic flux insertion. The main results of the thesis reveal restrictions imposed on these indices by frustration-free structure, and improved locality estimates for interacting many body systems.The first result concerns frustration free free-Fermion (F 4 ) Hamiltonians. For translation-invariant, strictly local, gapped F4 Hamiltonians on Z 2 , we prove that the bulk index vanishes. More precisely, the two Bloch bundle associated with the image/kernel of the momentum space Hamiltonian is shown to be trivial by an algebraic argument using polynomial modules and a free resolution. Thus a frustration-free free-Fermion model cannot realize a nontrivial integer quantum Hall phase at the level of the bulk Chern index.The second new result is the corresponding edge statement. For a translation-invariant F4 Hamiltonian restricted to a half-plane, we prove that the edge index vanishes. We first identified that frustration free is equivalent to flat ground state band. Detailed analysis of spectral flow of the edge Hamiltonian shows that the edge mode is an analytic function and can be absorbed by the flat band. Consequently, the edge spectral flow across a regular energy in the bulk gap is constant and must equal its trivial value below the spectrum. This establishes, in this setting, that the frustration-free structure rules out nonzero edge transport as well as nonzero bulk transport. The third new result is an exponential clustering theorem for finite-volume quantum spin systems with a possibly degenerate and split ground-state sector. Assuming only a uniform spectral gap above a low-energy spectral patch of sufficiently small width, we prove exponential decay of correlations in the form|⟨Ω, ABΩ⟩ − ⟨Ω, AP BΩ⟩| ≤ C∥P∥1∥A∥∥B∥e −cd(X,Y ),for local observables A and B with disjoint supports. This improves the locality control needed in many body Hall conductance arguments, where previous estimates under comparable hypotheses were superpolynomial. Together, these results clarify how locality and spectral gaps constrain both free-Fermion and interacting formulations of topological transport.
In quantum many-body systems, the existence of a spectral gap above the ground state has far-reaching consequences. In this paper, we discuss “finite-size” criteria for having a spectral gap in frustration-free spin systems and their applications. We extend a criterion that was originally developed for periodic systems by Knabe and Gosset-Mozgunov to systems with a boundary. Our finite-size criterion says that if the spectral gaps at linear system size n exceed an explicit threshold of order n−3/2, then the whole system is gapped. The criterion takes into account both “bulk gaps” and “edge gaps” of the finite system in a precise way. The n−3/2 scaling is robust: it holds in 1D and 2D systems on arbitrary lattices and with arbitrary finite-range interactions. One application of our results is to give a rigorous foundation to the folklore that 2D frustration-free models cannot host chiral edge modes (whose finite-size spectral gap would scale like n−1).
This work rigorously establishes a universal lower bound z ≥ 2 for the dynamical exponent in frustration-free quantum many-body systems whose ground states exhibit power-law decaying correlations. The derivation relies on the Gosset-Huang inequality, providing a unified framework applicable across various lattice structures and spatial dimensions, independent of specific boundary conditions. Remarkably, our result can be applied to prove bounds for dynamics of classical stochastic processes. Specifically, we utilize a well-established mapping from the time evolution of local Markov processes with detailed balance to that of frustration-free quantum Hamiltonians, known as Rokhsar-Kivelson Hamiltonians. This proves z ≥ 2 for such Markov processes, which is an improvement over existing bounds. Beyond these applications, the quantum analysis of the z ≥ 2 bound is further broadened to include systems exhibiting hidden correlations, which may not be evident from purely local operators.