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Bound States in 2d Yukawa Theory from Hamiltonian Truncation

This paper extends Hamiltonian truncation effective theory to two-dimensional Yukawa theory with fermions and logarithmic UV divergences, successfully demonstrating the convergence of bound state energy levels and mapping their evolution from weak to strong coupling regimes.

Original authors: Olivier Delouche, Kara Farnsworth, Francesco Riva

Published 2026-09-28

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CC BY 4.0

Original authors: Olivier Delouche, Kara Farnsworth, Francesco Riva

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 theoretical physics, there is a persistent challenge: understanding how particles behave when they are locked in a tight, intense embrace. While scientists can easily predict the behavior of particles that are far apart and moving slowly, the rules change dramatically when they are pushed together with great force. This is the realm of strong coupling, where the usual mathematical tools of approximation break down, and entirely new phenomena can emerge. To study these conditions, physicists often turn to simplified models that capture the essential drama of the interaction without the overwhelming complexity of the real world. One such model is the Yukawa theory, a framework that describes how a force-carrying particle, like a messenger, interacts with matter particles to create attraction. This interaction is fundamental to our universe, responsible for generating the masses of elementary particles and binding them into larger structures. However, calculating exactly what happens when these particles form stable, bound pairs under intense conditions has remained a difficult puzzle, particularly when trying to bridge the gap between weak interactions and the chaotic, strong-coupling regime.

A team of researchers has now taken a significant step forward in solving this puzzle by applying a powerful computational technique known as Hamiltonian truncation to the Yukawa theory in a two-dimensional setting. Imagine trying to understand the sound of a complex musical chord by listening to only the loudest notes; this is essentially what the researchers did, but with the energy states of a quantum system. They constructed a mathematical description of the system and then deliberately cut off the highest energy levels, keeping only the most accessible states to perform calculations. The problem with this approach is that ignoring the high-energy states introduces errors, much like ignoring the quietest notes in a chord changes the overall sound. To fix this, the team developed a sophisticated correction method based on effective field theory. This method allowed them to mathematically account for the influence of the discarded high-energy states, effectively "healing" the truncated calculation so that it could accurately reflect the full, complex reality of the system.

By using this refined technique, the researchers were able to track the evolution of a bound state—a pair of particles held together by the Yukawa force—as they increased the strength of the interaction from weak to strong. Their simulations revealed that the energy levels of these bound states converged smoothly and predictably as they included more states in their calculation, confirming that their correction method was working as intended. At weak interaction strengths, their results matched perfectly with existing theoretical predictions, validating their approach. More importantly, they successfully followed the bound state into the strong-coupling regime, a territory where previous methods had struggled. They found that as the interaction grew stronger, the binding energy of the pair increased in a way that aligned with expectations derived from other known limits of the theory. This work demonstrates that with the right mathematical corrections, it is possible to use truncated calculations to explore the deep, non-perturbative heart of quantum field theories, offering a reliable window into how particles bind together under extreme conditions.

Technical Summary: Bound States in 2d Yukawa Theory from Hamiltonian Truncation

Problem Statement
The paper addresses the challenge of calculating the non-perturbative spectrum of strongly coupled Quantum Field Theories (QFTs), specifically the two-dimensional Yukawa theory involving a Dirac fermion and a scalar field. While Hamiltonian truncation is a powerful variational method for studying such systems in the continuum, it historically faces an "exponential barrier" where the Hilbert space size grows too rapidly with the energy cutoff ( $E_{\text{max}}$ ) to be computationally tractable. Furthermore, standard truncation suffers from slow convergence due to omitted high-energy states and UV divergences that cannot be removed by simple normal ordering in two dimensions. The authors aim to extend the Hamiltonian Truncation Effective Theory (HTET) framework to include fermions and logarithmic UV divergences, specifically to study fermion-antifermion bound states across weak and strong coupling regimes.

Methodology
The authors employ equal-time Hamiltonian truncation combined with HTET corrections. The methodology involves the following steps:

Hamiltonian Setup: The theory is defined on a spatial circle of length $L$ . The Hamiltonian is split into a solvable free part ( $H_0$ ) containing massive scalars and fermions, and an interaction part ( $V$ ) representing the Yukawa coupling $g\phi\bar{\psi}\psi$ . The system is quantized in a Fock basis, and the Hilbert space is truncated to states with total energy $E \le E_{\text{max}}$ .
UV Renormalization: Unlike previous applications of HTET in scalar theories, the 2D Yukawa model possesses non-trivial UV divergences (vacuum energy and scalar self-energy) at order $g^2$ . The authors introduce local counterterms ( $H_{\text{CT}}$ ) to the full theory before truncation to cancel these divergences. This is achieved using a high-energy Wilsonian cutoff $\Lambda$ which is eventually taken to infinity, ensuring the effective Hamiltonian remains finite.
Effective Hamiltonian Construction (Matching): Using a Gell-Mann-Low adiabatic switching procedure, the authors match the overlap of free and interacting eigenstates between the full renormalized theory and the truncated theory. This generates an effective Hamiltonian ( $H_{\text{eff}}$ $H_{eff}$ ) acting only on the truncated subspace.
- $H_{\text{eff}}$ is expanded in powers of the interaction $V$ and inverse powers of the cutoff $1/E_{\text{max}}$ .
- The authors explicitly calculate terms up to Next-to-Leading Order (NLO) in the $1/E_{\text{max}}$ expansion. This includes local corrections to the identity and $\phi^2$ operators, as well as non-local corrections involving commutators with $H_0$ (e.g., $\{H_0, \phi^2\}$ ).
Physical Mass Renormalization: To fix the physical fermion mass non-perturbatively, an additional interaction term $\Delta V = m_{V,\psi} \int dx :\bar{\psi}\psi:$ is introduced. Counterterms are derived for this term as well, allowing the input mass parameter to be tuned to match the physical mass defined by the energy gap between the vacuum and the lowest charge-1 state.
Numerical Implementation: The corrected effective Hamiltonian is diagonalized numerically in symmetry sectors (charge 0 and 1, parity, and charge conjugation). The convergence of energy levels is tested as a function of $E_{\text{max}}$ .

Key Contributions

Extension of HTET to Fermions: The paper successfully generalizes the HTET framework to theories with fermionic degrees of freedom, deriving the necessary counterterms and effective operators for the Yukawa model.
Handling Logarithmic Divergences: The authors demonstrate how to systematically incorporate local counterterms to handle the logarithmic UV divergences specific to 2D Yukawa theory within the Hamiltonian truncation formalism.
Convergence Scaling: They show that with the inclusion of the derived $O(1/E_{\text{max}})$ and $O(1/E_{\text{max}}^2)$ corrections, the energy levels converge with the scaling predicted by power counting, significantly improving upon uncorrected truncation.
Bound State Identification: A strategy is developed to track fermion-antifermion bound states from weak to strong coupling, distinguishing them from scattering states in a finite volume.

Results

Convergence: Numerical results confirm that the corrected Hamiltonian yields energy levels that converge rapidly with increasing $E_{\text{max}}$ , following the expected $1/E_{\text{max}}^2$ scaling for the leading remaining errors.
Weak Coupling: In the weak coupling regime ( $g/m_\phi \ll 1$ ), the calculated binding energies agree well with finite-volume perturbation theory (NRPT). The authors observe that finite-volume effects can obscure the characteristic $g^4$ scaling of the binding energy at small volumes, a phenomenon they analyze in detail.
Strong Coupling and Heavy Scalar Limit: The study explores the regime where the scalar mass $m_\phi$ is large. In this limit, integrating out the scalar generates a four-fermion interaction, flowing toward the massive Thirring/Sine-Gordon limit. The results recover the expected weak-coupling behavior in this limit and outline the path to extending the calculation toward the integrable Sine-Gordon spectrum.
Finite Volume Effects: The paper provides a detailed analysis of how bound states behave in finite volume, noting that at small volumes ( $L \lesssim L_B$ ), the bound state is deformed and the infinite-volume perturbative predictions are modified.

Significance and Claims
The authors claim that this work provides a robust test of the HTET formalism in a more complex setting involving fermions and non-trivial renormalization. By successfully reproducing known perturbative results and demonstrating controlled convergence, the paper validates the method's applicability to theories that are difficult to discretize (such as those with chiral fermions) and offers a pathway to studying strongly coupled regimes where analytic solutions are unavailable. The work highlights the importance of separating scales and correctly handling UV divergences to extend Hamiltonian truncation to more realistic QFTs, including potential applications to four-dimensional gauge theories like QCD. The paper does not claim to solve the full non-perturbative spectrum of the theory in all regimes but rather establishes the methodology and demonstrates its efficacy in specific limits.

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