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Lee-Yang theorem for fermions

This paper establishes a Lee-Yang zero-freeness theorem for a broad class of interacting fermion models, including the attractive and repulsive Hubbard models, which guarantees the absence of phase transitions under nonzero local external fields and enables provably efficient quantum algorithms for estimating their ground-state energies.

Original authors: Chaithanya Rayudu, Takahiro Misawa, Andrew Zhao, Jun Takahashi

Published 2026-09-22

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Chaithanya Rayudu, Takahiro Misawa, Andrew Zhao, Jun Takahashi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of physics, there is a fundamental divide between the predictable world of individual particles and the chaotic, collective behavior of trillions of them acting together. When scientists study a single electron or a lone atom, the rules are clear and the outcomes are easy to calculate. But when these particles interact in large groups, forming materials like metals or superconductors, their collective behavior becomes notoriously difficult to predict. This is the realm of many-body physics, where the whole is often greater than the sum of its parts, and where tiny changes in temperature or magnetic fields can trigger dramatic shifts in the state of matter, known as phase transitions. For decades, understanding these transitions has required powerful mathematical tools, one of the most famous being the Lee-Yang theorem. Originally developed to study magnets, this theorem acts as a mathematical lighthouse, showing that under certain conditions, the complex mathematical descriptions of a system's energy cannot have certain types of solutions. When these solutions are absent, it guarantees that the system behaves smoothly and predictably, without sudden, chaotic jumps in its properties. This insight has been a cornerstone for understanding why some materials change state while others remain stable, and it has recently begun to offer new hope for solving problems that were once thought to be impossible for computers.

For a long time, this powerful mathematical tool was limited to specific types of systems, primarily those involving magnetic spins or non-interacting particles. It remained a mystery whether the same rules applied to fermions, a class of particles that includes electrons and protons, which are the building blocks of all visible matter. Fermions are unique because they strictly avoid occupying the same space at the same time, a rule that leads to complex interactions when they are packed together in materials. Despite their importance, no one had been able to prove that the Lee-Yang theorem held true for interacting fermions, leaving a significant gap in our understanding of how these particles behave in strong magnetic fields or under the influence of external fields. A team of researchers has now closed this gap. They have proven that a broad class of interacting fermion models follows the same zero-freeness rule as the magnetic systems studied nearly seventy years ago. This means that for these specific models, the mathematical description of their energy is free from the chaotic zeros that signal a phase transition, provided a certain external field is applied.

The researchers focused on a wide variety of models that describe how electrons move and interact in solids, including the famous Hubbard models used to study high-temperature superconductors and other strongly correlated materials. They demonstrated that for these systems, if you apply a uniform external field, the mathematical function that describes the system's energy never hits zero in a specific region of the complex plane. This might sound abstract, but the physical consequence is profound: it proves that these systems cannot undergo a phase transition under those conditions. More importantly, this mathematical stability translates directly into computational power. Because the system is guaranteed to have a stable energy gap—a buffer that keeps the ground state distinct from excited states—it becomes possible to design efficient quantum algorithms to calculate the ground-state energy of these materials. This is a massive leap forward, as calculating the ground-state energy of interacting fermions is a problem that has stumped both classical and quantum computers for decades, often requiring exponential time that grows too fast to be useful.

The team showed that this result applies to several well-known physical scenarios, such as electrons moving in a lattice with attractive forces, or repulsive electrons moving on a bipartite graph (a grid that can be split into two alternating sets of points) or under a sufficiently strong magnetic field. They even extended their findings to more complex situations involving spin-orbit coupling, where the electron's spin interacts with its motion, and to models with complex phases that usually cause severe computational difficulties. In fact, they identified a specific model known as the interacting Hofstadter model, which involves electrons moving in a magnetic field with a specific pattern of phases. For this model, classical computers struggle immensely because the calculations involve complex numbers that cancel each other out in a way that creates a "sign problem," making the simulation incredibly slow and inefficient. However, the new theorem proves that a quantum computer can solve this problem efficiently, running in a time that grows polynomially with the size of the system. This suggests that quantum computers will be able to simulate these complex materials with a level of precision and speed that was previously unattainable.

Beyond the promise of faster calculations, the findings place strict limits on what kinds of exotic states of matter can exist in these systems. The researchers used their theorem to argue against the possibility of certain topological phases of matter, which are states where the material's properties are protected by its global geometry rather than local details. Specifically, they showed that if a system obeys their theorem, it cannot support a type of topological order that relies on a stable degeneracy of the ground state in the presence of a local field. This helps clarify the nature of phase transitions in materials like the honeycomb lattice, where previous numerical studies had suggested the existence of a mysterious intermediate phase. The new proof suggests that if such a phase exists, it cannot be a topologically ordered one with the specific stability properties that were hypothesized, aligning better with recent studies that favor a direct transition between different magnetic states.

The work represents a rare instance where a deep mathematical theorem provides a rigorous guarantee for the behavior of a strongly interacting quantum system, far beyond the reach of simple approximations. By proving that the Lee-Yang theorem holds for this broad class of fermionic models, the researchers have not only filled a theoretical void but also opened a clear path for efficient quantum simulation. They have shown that for these systems, the chaotic complexity that usually makes them so difficult to study is tamed by the presence of an external field, allowing for precise calculations of their fundamental properties. This bridges the gap between abstract mathematical theory and practical quantum computing, offering a concrete roadmap for exploring the behavior of electrons in the most complex materials known to science.

Technical Summary: Lee-Yang Theorem for Fermions

Problem Statement
The Lee-Yang theorem, originally established for the ferromagnetic Ising model, provides a rigorous mathematical framework for analyzing phase transitions by proving that the complex zeros of the partition function lie on specific loci (e.g., the imaginary axis) in the complex plane of the external field. This zero-freeness implies the analyticity of the free energy and the absence of phase transitions in the presence of a non-zero field. Furthermore, in theoretical computer science, this property has been leveraged to prove the existence of provably efficient classical and quantum algorithms for estimating ground-state energies.

Despite over 70 years of research, a rigorous Lee-Yang zero-freeness theorem for interacting fermionic systems has remained unavailable. While partition function zeros have been computed for specific fermionic models, no general theorem exists, even for the simplest non-interacting free-fermion models. This gap limits the ability to rigorously control strongly interacting fermionic systems far from perturbative regimes and hinders the development of efficient algorithms for a broad class of fermionic Hamiltonians.

Methodology
The authors establish a Lee-Yang theorem for a broad class of interacting fermionic Hamiltonians by combining techniques from statistical mechanics and complex analysis. The methodology proceeds in three main stages:

Definition of the Hamiltonian Class: The authors define a class of "fermionic Lee-Yang Hamiltonians" ( $H = H_0 + V$ ) where:
- $H_0$ contains quadratic terms with a positive-semidefinite (PSD) hopping matrix $A$ and an arbitrary antisymmetric pairing matrix $B$ .
- $V$ represents attractive interactions of the form $-\sum |U_{ij}|(n_i - 1/2)(n_j - 1/2)$ .
- The partition function is defined as $Z(h) = \text{Tr}[\exp(-\beta(H + h\sum n_i))]$ .
Free Fermion Analysis (Trotterized): The proof begins with the free-fermion case ( $V=0$ ). The authors analyze the Trotterized partition function $Z_\epsilon(H_0, u)$ , where the external field is discretized into complex variables $u_{i,m}$ . By rewriting the partition function as the determinant of a block matrix $D(u)$ constructed from the hopping and pairing matrices, they demonstrate that $D(u)$ is non-singular within the open unit polydisc ( $|u_{i,m}| < 1$ ). This relies on a quadratic form argument involving a specific metric $\Gamma$ and the property that the evolution operator $W = e^{-\epsilon Q}$ is contractive under this metric when $A$ is PSD.
Extension to Interacting Systems (Asano Contractions): To extend the result to interacting systems, the authors employ Asano contractions. They express the Trotterized partition function of the full Hamiltonian $Z_\epsilon(H, u)$ as the Schur product (Hadamard product) of two polynomials: the free-fermion partition function $Z_\epsilon(H_0, u)$ and the interaction-only partition function $Z_\epsilon(V, u)$ .
- $Z_\epsilon(H_0, u)$ is shown to be zero-free in the unit polydisc (Theorem 4).
- $Z_\epsilon(V, u)$ is identified as a product of classical ferromagnetic Ising partition functions, which are known to be zero-free in the unit polydisc by the classical Lee-Yang theorem (Lemma 5).
- Using a lemma by Hinkkanen regarding the Schur product of zero-free polynomials, the authors prove that $Z_\epsilon(H, u)$ is also zero-free in the open unit polydisc.
Continuum Limit and Transformation: By applying Hurwitz's theorem, the zero-freeness of the Trotterized polynomial implies the zero-freeness of the exact partition function $Z(h)$ for $\text{Re}(h) > 0$ . A particle-hole transformation ( $a \to a^\dagger$ ) is then used to extend the zero-freeness region to the strip $\text{Re}(h) \in [-\lambda_{\max}(A), 0]$ .

Key Contributions and Results

Theorem 2 (Zero-Freeness): For any Hamiltonian in the defined class, all zeros of the partition function $Z(h)$ in the complex $h$ -plane satisfy $\text{Re}(h) \in [-\lambda_{\max}(A), 0]$ . Specifically, if the hopping matrix $A=0$ , all zeros lie strictly on the imaginary axis. This rigorously establishes the nonexistence of phase transitions in these models under a non-zero local external field.
Theorem 3 (Spectral Gap and Algorithmic Efficiency): The zero-freeness implies a field-induced spectral gap. Specifically, for $h > 0$ , the Hamiltonian $H + h\sum n_i$ has a spectral gap of at least $h/8$ above the ground state. Consequently, the authors prove the existence of a provably efficient quantum algorithm to estimate the ground-state energy of these Hamiltonians to within additive precision $\epsilon$ , with runtime scaling polynomially in the number of modes $N$ , the operator norm $\|H\|_{op}$ , and $1/\epsilon$ .
Physical Model Unification: The authors demonstrate that this class, up to unitary equivalence (Bogoliubov transformations), encompasses several physically significant models:
- The attractive Hubbard model in a pairing field.
- The repulsive Hubbard model on bipartite graphs (in a staggered Zeeman field).
- The repulsive Hubbard model with a sufficiently strong uniform Zeeman field (even on non-bipartite graphs).
- The interacting Hofstadter model with time-reversal invariant hopping.

Significance and Claims

The paper claims that this work provides the first rigorous Lee-Yang zero-freeness theorem for interacting fermions. The significance is twofold:

Physical Insight: It provides a unified analytic framework to understand the absence of phase transitions in a wide variety of strongly correlated fermionic systems under external fields. It specifically addresses constraints on topological order, arguing that gapped topologically ordered phases with stable ground-state degeneracy cannot exist in these systems under a non-zero field, as the Lee-Yang spectral gap prevents the energy splitting from vanishing in the thermodynamic limit.
Algorithmic Implications: The work identifies a large class of fermionic models that are computationally tractable for quantum computers. Unlike previous results limited to free fermions or specific 1D chains, this class includes strongly interacting, higher-dimensional models (like the interacting Hofstadter model) that are known to be classically hard due to the fermionic sign problem. The authors note that while some of these models (e.g., attractive Hubbard) admit sign-problem-free classical Monte Carlo representations, others (e.g., repulsive Hubbard with complex hopping) do not. The Lee-Yang property thus offers a new pathway for efficient quantum simulation where classical methods fail or are not known to be efficient.

The authors remain modest regarding classical algorithms, noting that while the Lee-Yang property implies zero-freeness, it does not automatically guarantee that classical Markov chains mix in polynomial time for these fermionic systems, unlike the spin systems where such guarantees have been established. The primary algorithmic contribution is the rigorous proof of efficiency for quantum algorithms.

Technical Summary: Lee-Yang Theorem for Fermions

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