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Lee-Yang theorems for $q$ -deformed oscillators and doubly-scaled SYK

This paper proves that the vacuum partition function of $q$ -deformed oscillators and the annealed partition function of the doubly-scaled Sachdev-Ye-Kitaev model satisfy the Lee-Yang property by demonstrating that the former belongs to the Laguerre-Pólya class through the analysis of $q$ -deformed matching polynomials, thereby rigorously establishing a zero-free strip for the quenched SYK partition function.

Original author: Arpon Basu, Siddhant Midha

Published 2026-11-03

📖 8 min read🧠 Deep dive

CC BY 4.0

Original author: Arpon Basu, Siddhant Midha

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 author. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of theoretical physics, there exists a class of systems known as disordered systems, where the rules governing how particles interact are not fixed but are instead drawn from a random distribution. Imagine a room full of people where every pair of strangers has a different, randomly assigned rule for how they shake hands; the overall behavior of the room emerges from this chaotic web of connections. One of the most famous examples of such a system is the Sachdev-Ye-Kitaev model, a mathematical construct used by physicists to study how complex quantum systems behave when they contain many interacting parts. Understanding these systems is crucial because they serve as simplified laboratories for exploring the nature of black holes, the behavior of strange metals, and the fundamental limits of how quantum information is processed. A central question in studying these systems is whether they undergo a phase transition, a sudden and dramatic change in their state, much like water freezing into ice. To answer this, scientists look at a specific mathematical object called a partition function, which acts as a master key to the system's thermodynamic properties. If the zeros of this function—the points where the value drops to nothing—lie in specific regions of the mathematical plane, it reveals whether the system is stable or if it is on the brink of a phase change.

For decades, a powerful idea known as the Lee-Yang property has guided physicists in this search. This concept suggests that for certain types of physical systems, the zeros of the partition function are not scattered randomly but are confined to a single, straight line in the complex plane. When this happens, it guarantees that the system does not experience a phase transition at any finite temperature, meaning its behavior remains smooth and predictable. While this property has been proven for many ordered systems, such as magnets where every atom aligns with its neighbors, it has remained a mystery for disordered systems like the Sachdev-Ye-Kitaev model. The randomness of the interactions makes the mathematics incredibly difficult, and previous attempts to understand the zeros of this model's partition function relied on approximations or simulations that could not provide a rigorous proof.

In a recent study, researchers Arpon Basu and Siddhant Midha from Princeton University have finally solved this long-standing puzzle for the Sachdev-Ye-Kitaev model in a specific regime known as the doubly-scaled limit. They proved that the partition function for this disordered system behaves with the same elegant order as the simpler, ordered systems, but specifically in the thermodynamic limit where the system size grows infinitely large. Specifically, they demonstrated that all the zeros of the partition function lie exactly on the imaginary axis in this limit, a result that confirms the system is free of phase transitions at any finite temperature as it approaches infinite size. This finding is not just a mathematical curiosity; it provides a rigorous foundation for understanding why these complex quantum systems are so computationally tractable, explaining why algorithms can efficiently simulate them despite their inherent randomness.

To reach this conclusion, the authors did not attack the complex Sachdev-Ye-Kitaev model directly. Instead, they turned to a simpler, related mathematical object called a q-deformed oscillator. This oscillator is a simplified version of a quantum system where the rules for how particles are created and destroyed are slightly modified by a parameter, denoted as q, which controls the degree of deformation. When this parameter is set to one, the system behaves like a standard boson; when set to minus one, it behaves like a fermion; and for values in between, it occupies a middle ground that captures the essential features of the disordered system. The researchers focused on the "vacuum partition function" of this oscillator, a specific mathematical expression that describes the system's state when it is empty. They set out to prove that this expression also possesses the Lee-Yang property, meaning its zeros are all purely imaginary.

The path to this proof required the team to bridge the gap between abstract algebra and combinatorics, the study of counting and arrangement. They discovered that the vacuum partition function could be understood as a limit of a sequence of polynomials, which are expressions built from variables raised to various powers. These polynomials, which they called q-deformed matching polynomials, are constructed by counting the ways to pair up elements in a set, but with a twist: every time two pairs cross each other in a specific diagrammatic representation, the count is weighted by the deformation parameter. The researchers proved that for any value of the parameter between minus one and one, these polynomials have a remarkable property: all their roots are real numbers. This is a significant result in its own right, as it connects the random crossings of these diagrams to a very structured geometric property of the resulting equations.

By establishing that these polynomials are "real-rooted," the team could use a powerful theorem from complex analysis to show that the limit of these polynomials—the vacuum partition function itself—must also have its zeros confined to a specific line. In this case, because of the way the polynomials were constructed, the zeros of the partition function lie on the imaginary line. This confirmed a long-standing conjecture that the partition function of the q-deformed oscillator has no zeros off the imaginary axis. Furthermore, they were able to precisely locate the zero closest to the real axis, showing that it stays at a safe distance, which is crucial for understanding the system's stability.

With this rigorous proof in hand, the researchers then connected their findings back to the original Sachdev-Ye-Kitaev model. They showed that in the doubly-scaled limit, where the number of particles and the complexity of their interactions grow together in a specific way, the behavior of the disordered system is indistinguishable from that of the q-deformed oscillator. This allowed them to transfer their results directly: since the oscillator's partition function has all its zeros on the imaginary axis, the Sachdev-Ye-Kitaev model's partition function must do the same in the thermodynamic limit. This means that as the system grows infinitely large, it retains a "zero-free strip" around the real axis where the partition function never vanishes.

The implications of this zero-free strip are profound for both physics and computer science. In the context of the Sachdev-Ye-Kitaev model, it rigorously proves that the system does not undergo a phase transition at any finite temperature in the thermodynamic limit, confirming earlier numerical observations and large-scale calculations. It is important to note that this behavior is an asymptotic property; for finite system sizes, such as n=8 and k=4, the partition function can indeed possess zeros with non-zero real parts, indicating that the Lee-Yang property emerges only as the system becomes infinitely large. More broadly, this result explains why these disordered quantum systems are so amenable to efficient simulation. The absence of zeros in a wide region around the real axis in the thermodynamic limit implies that the system's properties can be computed using standard mathematical techniques without running into the singularities that usually make such problems intractable. This provides a unifying mathematical principle that explains the algorithmic success seen in recent years for simulating these chaotic quantum systems.

The researchers also explored the combinatorial consequences of their findings. The fact that the polynomials are real-rooted implies a specific type of smoothness in the sequence of numbers that count the crossing patterns of the diagrams. This smoothness, known as log-concavity, is a property that often appears in nature and mathematics when systems are stable and well-behaved. By proving this property for the q-deformed moments, the team added a new layer of understanding to the combinatorial structures underlying quantum physics. They also showed that this elegant behavior is strictly limited to the range of the deformation parameter between minus one and one; outside this range, the mathematical structure breaks down, and the zeros can wander off the line, signaling a loss of stability.

In summary, this work represents a significant step forward in understanding the mathematical architecture of disordered quantum systems. By proving that the partition function of the Sachdev-Ye-Kitaev model satisfies the Lee-Yang property in the thermodynamic limit, the authors have provided a rigorous explanation for the system's stability and computational tractability. They achieved this by uncovering a deep connection between the random crossings of combinatorial diagrams and the geometric location of the roots of polynomials, showing that even in a system defined by randomness, there is an underlying order that keeps the physics predictable in the limit of infinite size. This discovery not only settles a specific conjecture about the q-deformed oscillator but also offers a powerful new tool for analyzing other disordered systems, suggesting that the Lee-Yang property might be a more universal feature of quantum matter than previously thought.

Technical Summary: Lee-Yang Theorems for q-deformed Oscillators and Doubly-Scaled SYK

Problem Statement
This paper investigates the distribution of zeros of partition functions in two distinct but connected contexts: the vacuum partition function of the $q$ -deformed oscillator and the annealed/quenched partition functions of the Sachdev-Ye-Kitaev (SYK) model in the doubly-scaled limit.

The primary object of study is the vacuum partition function of the $q$ -deformed oscillator, defined as $L_q(z) := \langle 0 | e^{zT_q} | 0 \rangle$ , where $T_q = a_q + a_q^*$ is the transfer operator satisfying the $q$ -deformed commutation relation $a_q a_q^* - q a_q^* a_q = \text{Id}$ . The coefficients of $L_q(z)$ are the $q$ -matching moments $M_\ell(q)$ , which count perfect matchings weighted by their crossing numbers. While it was known that for $q=1$ (bosonic) and $q=-1$ (fermionic), the partition functions have no zeros or purely imaginary zeros respectively, the behavior for intermediate $q \in [-1, 1)$ was an open conjecture (Okuniev, 2023). Specifically, it was conjectured that $L_q(z)$ satisfies the Lee-Yang property: all its zeros lie on the imaginary axis.

A secondary motivation is the connection between these oscillators and the SYK model. In the doubly-scaled limit ( $n, k \to \infty$ with $k^2/n \to \alpha$ ), the disorder-averaged (annealed) SYK partition function is approximated by the $q$ -deformed oscillator with $q = e^{-2\alpha}$ . The paper seeks to rigorously establish the Lee-Yang property for the SYK model in this limit and derive consequences for the zero-freeness of the quenched (disorder-specific) partition function.

Methodology
The authors employ a multi-faceted approach combining combinatorics, complex analysis, and statistical mechanics:

Real-Rootedness of $q$ -Deformed Matching Polynomials:
The core technical innovation is proving that the $q$ -deformed matching polynomials, defined as $\mu_{n,q}(z) = \sum_{\ell=0}^{\lfloor n/2 \rfloor} (-1)^\ell \binom{n}{2\ell} M_\ell(q) z^{n-2\ell}$ , are real-rooted for all $q \in [-1, 1]$ .
- Differential Equations: The authors derive a partial differential equation (PDE) relating the $q$ -derivative and $z$ -derivative of $\mu_{n,q}$ using the Touchard–Riordan formula for $M_\ell(q)$ . This PDE governs the evolution of the roots of $\mu_{n,q}$ as $q$ varies.
- Root Repulsion Analysis: By analyzing the differential equation, they show that the roots of $\mu_{n,q}$ experience a "repulsion" that prevents them from colliding and moving off the real axis as $q$ moves from $1$ (where $\mu_{n,1}$ are Hermite polynomials) down to $-1$. This establishes that $\mu_{n,q}$ remains real-rooted for all $q \in [-1, 1]$ .
- Operator Theory: An alternative proof is provided using the theory of real-stability preserving operators, expressing $\mu_{n,q}$ as the output of a differential operator acting on a known real-rooted polynomial.
Limiting Behavior and Lee-Yang Property:
Using Hurwitz's theorem, the authors show that $L_q(iz)$ is the locally uniform limit of the real-rooted polynomials $\mu_{n,q}$ . Consequently, $L_q(iz)$ is real-rooted, implying $L_q(z)$ has purely imaginary zeros. They further establish that these zeros are simple and symmetric, allowing for a Hadamard product representation.
SYK Connection and Zero-Freeness:
- Annealed Limit: By comparing the moments of the SYK Hamiltonian to the $q$ -oscillator moments using graph polynomials (specifically multivariate Tutte polynomials and crossing graphs), the authors prove that the annealed SYK partition function converges locally uniformly to a rescaled $L_q$ . This transfers the Lee-Yang property to the thermodynamic limit of the SYK model.
- Nearest Root Analysis: The location of the nearest zero to the real axis is determined using cumulant expansions and properties of Tutte polynomials, yielding a precise bound on the root location as $q \to 1$ .
- Quenched Zero-Freeness: Using Gaussian Poincaré inequalities and concentration bounds, the authors show that the quenched partition function concentrates around the annealed average. Combined with the zero-freeness of the annealed limit in a specific strip, this proves that the quenched partition function is zero-free in a strip of constant width around the real axis with high probability.

Key Contributions and Results

Theorem 1.1 (Lee-Yang Property for $q$ -Oscillator): Proves that for any real $q \in [-1, 1)$ , the partition function $L_q(z)$ is an entire function with infinitely many simple zeros, all lying on the imaginary axis. This rigorously confirms the conjectures of Okuniev [Oku23].
Theorem 1.3 (Location of Nearest Zero): Establishes that the root $t_1(q)$ closest to the real axis satisfies $t_1(q) > \zeta^*/\sqrt{1-q}$ , where $\zeta^* \approx 1.3255$ . The map $q \mapsto \sqrt{1-q} \cdot t_1(q)$ is non-increasing.
Theorem 1.7 (Real-Rootedness of $\mu_{n,q}$ ): Proves that the $q$ -deformed matching polynomials are real-rooted for all $q \in [-1, 1]$ . This is a generalization of the Heilmann-Lieb theorem (which covers $q=1$ ) and the Rademacher case ( $q=-1$ ).
Theorem 1.4 (Asymptotic Lee-Yang for SYK): Shows that in the doubly-scaled limit, the annealed SYK partition function converges to a function whose zeros are purely imaginary.
Theorem 1.6 (Zero-Freeness of Quenched SYK): Demonstrates that for sufficiently large $n$ and $k \ge \omega(\sqrt{n/\log n})$ , the quenched SYK partition function is zero-free in a strip $|\text{Re } z| \le n^{0.08}, |\text{Im } z| \le \zeta^*$ with high probability.
Combinatorial Consequences: The real-rootedness of $\mu_{n,q}$ implies that the normalized $q$ -matching moments form a log-concave sequence and a Pólya-frequency sequence (Theorems 1.9 and 9.2).

Significance and Claims
The paper claims to provide a rigorous mathematical foundation for the Lee-Yang property in the context of $q$ -deformed oscillators and the doubly-scaled SYK model.

Unification of Models: It bridges the gap between combinatorial matching polynomials, $q$ -Gaussian distributions, and the physics of the SYK model.
Algorithmic Implications: The authors highlight that the zero-freeness of the partition function (specifically in a strip around the real axis) is a unifying principle for algorithmic tractability. Citing Barvinok's interpolation method, they note that the established zero-free strip implies that the free energy and thermal expectation values of the SYK model can be computed deterministically in polynomial time, complementing recent algorithmic results.
Rigorous Complement: The results on the quenched SYK partition function rigorously complement large- $k$ replica calculations (specifically [ZK26]), confirming the predicted width of the zero-free strip.
Limitations: The paper explicitly notes that the Lee-Yang property (imaginary-rootedness) holds only in the thermodynamic limit ( $n \to \infty$ ). For finite system sizes (e.g., $n=8, k=4$ ), the annealed partition function may possess non-imaginary zeros, distinguishing the SYK model from deterministic models like the Ising model where the property holds for all $n$ .

The work does not propose new experimental setups but rather provides a theoretical framework that explains the structural properties of the SYK model's partition function, linking them to deep results in the theory of real-rooted polynomials and complex analysis.

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