Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2307.16655

Calculating composite-particle spectra in Hamiltonian formalism and demonstration in 2-flavor QED$_{1+1\text{d}}$

Etsuko Itou, Akira Matsumoto, Yuya Tanizaki

Published Jul 31, 2023• 0 citations
arXivHTMLPDF

Abstract

We consider three distinct methods to compute the mass spectrum of gauge theories in the Hamiltonian formalism: (1) correlation-function scheme, (2) one-point-function scheme, and (3) dispersion-relation scheme. The first one examines spatial correlation functions as we do in the conventional Euclidean Monte Carlo simulations. The second one uses the boundary effect to efficiently compute the mass spectrum. The third one constructs the excited states and fits their energy using the dispersion relation with selecting quantum numbers. Each method has its pros and cons, and we clarify such properties in their applications to the mass spectrum for the 2-flavor massive Schwinger model at $m/g=0.1$ and $θ=0$ using the density-matrix renormalization group (DMRG). We note that the multi-flavor Schwinger model at small mass $m$ is a strongly coupled field theory even after the bosonizations, and thus it deserves to perform the first-principles numerical calculations. All these methods mostly agree and identify the stable particles, pions $π_a$ ($J^{PG}=1^{-+}$), sigma meson $σ$ ($J^{PG}=0^{++}$), and eta meson $η$ ($J^{PG}=0^{--}$). In particular, we find that the mass of $σ$ meson is lighter than twice the pion mass, and thus $σ$ is stable against the decay process, $σ\to ππ$. This is consistent with the analytic prediction using the WKB approximation, and, remarkably, our numerical results are so close to the WKB-based formula between the pion and sigma-meson masses, $M_σ/M_π=\sqrt{3}$.

DOI
10.48550/arxiv.2307.16655

Related papers

Calculating composite-particle spectra in Hamiltonian formalism and demonstration in 2-flavor QED1+1d

Etsuko Itou, Akira Matsumoto, Yuya Tanizaki

Abstract We consider three distinct methods to compute the mass spectrum of gauge theories in the Hamiltonian formalism: (1) correlation-function scheme, (2) one-point-function scheme, and (3) dispersion-relation scheme. The first one examines spatial correlation functions as we do in the conventional Euclidean Monte Carlo simulations. The second one uses the boundary effect to efficiently compute the mass spectrum. The third one constructs the excited states and fits their energy using the dispersion relation with selecting quantum numbers. Each method has its pros and cons, and we clarify such properties in their applications to the mass spectrum for the 2-flavor massive Schwinger model at m/g = 0.1 and θ = 0 using the density-matrix renormalization group (DMRG). We note that the multi-flavor Schwinger model at small mass m is a strongly coupled field theory even after the bosonizations, and thus it deserves to perform the first-principles numerical calculations. All these methods mostly agree and identify the stable particles, pions πa (JPG = 1−+), sigma meson σ (JPG = 0++), and eta meson η (JPG = 0−−). In particular, we find that the mass of σ meson is lighter than twice the pion mass, and thus σ is stable against the decay process, σ → ππ. This is consistent with the analytic prediction using the WKB approximation, and, remarkably, our numerical results are so close to the WKB-based formula between the pion and sigma-meson masses, Mσ/Mπ = $$ \sqrt{3} $$ 3 .

Journal of High Energy Physics•Nov 30, 2023•14 citations

DMRG study of the theta-dependent mass spectrum in the 2-flavor Schwinger model

Etsuko Itou, Akira Matsumoto, Yuya Tanizaki

A bstract We study the θ -dependent mass spectrum of the massive 2-flavor Schwinger model in the Hamiltonian formalism using the density-matrix renormalization group (DMRG). The masses of the composite particles, the pion and sigma meson, are computed by two independent methods. One is the improved one-point-function scheme, where we measure the local meson operator coupled to the boundary state and extract the mass from its exponential decay. Since the θ term causes a nontrivial operator mixing, we unravel it by diagonalizing the correlation matrix to define the meson operator. The other is the dispersion-relation scheme, a heuristic approach specific to Hamiltonian formalism. We obtain the dispersion relation directly by measuring the energy and momentum of the excited states. The sign problem is circumvented in these methods, and their results agree with each other even for large θ . We reveal that the θ -dependence of the pion mass at m/g = 0 . 1 is consistent with the prediction by the bosonized model. We also find that the mass of the sigma meson satisfies the semi-classical formula, M σ /M π = $$ \sqrt{3} $$ 3 , for almost all region of θ . While the sigma meson is a stable particle thanks to this relation, the eta meson is no longer protected by the G -parity and becomes unstable for θ ≠ 0.

arXiv:2407.11391•Journal of High Energy Physics•Sep 24, 2024•14 citations

Three ways of calculating mass spectra for the 2-flavor Schwinger model in the Hamiltonian formalism

Akira Matsumoto, Etsuko Itou, Yuya Tanizaki

We propose three independent methods to compute the hadron mass spectra of gauge theories in the Hamiltonian formalism. The determination of hadron masses is one of the key issues in QCD, which has been precisely calculated by the Monte Carlo method in the Lagrangian formalism. We confirm that the mass of hadrons can be calculated by examining correlation functions, the one-point function, or the dispersion relation in Hamiltonian formalism. These methods are suitable for quantum computation and tensor network approaches. The methods are demonstrated with the tensor network (DMRG) in the 2-flavor Schwinger model, which shares important properties with QCD. We show that the numerical results are consistent with each other and with the analytic prediction of the bosonization technique. We also discuss their efficiency and potential applications to other models.

arXiv:2401.07556•International Symposium on Lattice Field Theory (LATTICE)•Jan 4, 2024•0 citations

Computing theta-dependent mass spectrum of the 2-flavor Schwinger model in the Hamiltonian formalism

Akira Matsumoto, Etsuko Itou, Yuya Tanizaki

We compute the $\theta$-dependent mass spectrum of the 2-flavor Schwingr model using the tensor network (DMRG) in the Hamiltonian formalism. The pion and the sigma meson are identified as stable particles of the model for nonzero $\theta$ whereas the eta meson becomes unstable. The meson masses are obtained from the one-point functions, using the meson operators defined by diagonalizing the correlation matrix to deal with the operator mixing. We also compute the dispersion relation directly by measuring the energy and momentum of the excited states, where the mesons are distinguished by the isospin quantum number. We confirmed that the meson masses computed by these methods agree with each other and are consistent with the calculation by the bosonized model. Our methods are free from the sign problem and show a significant improvement in accuracy compared to the conventional Monte Carlo methods. Furthermore, at the critical point $\theta = \pi$, the mesons become almost massless, and the one-point functions reproduce the expected CFT-like behavior. This talk is based on the paper [1].

arXiv:2501.18960•International Symposium on Lattice Field Theory (LATTICE)•Feb 3, 2025•1 citation

The two-flavor Schwinger model at 50: Solving Coleman's puzzles

Gabriel Cuomo, Ross Dempsey, Andrei Katsevich +4 more

In his 1976 paper "More about the massive Schwinger model", Coleman introduced $1+1$-dimensional Quantum Electrodynamics coupled to two charged massive fermions. By applying Abelian bosonization, he elucidated much of the physics of this two-flavor Schwinger model, but he listed three puzzles at the end of his paper. We present new analytical and numerical calculations to solve Coleman's three puzzles and thereby deepen our understanding of this model. These puzzles pertain to the theory with equal fermion masses at $θ= 0$ and at $θ= π$, as well as the size of isospin-breaking effects when the fermion masses are unequal. For the puzzle at $θ= π$, the solution is related to the structure of the zero-temperature phase diagram arXiv:2305.04437: for equal fermion masses $m$, the model exhibits spontaneous breaking of charge conjugation symmetry and absence of confinement for any value of the gauge coupling $g$, so that there is a smooth interpolation from weak to strong coupling. Using two-loop Renormalization Group and integrability methods, we show that the mass gap behaves as $\sim m e^{-0.111 g^2/m^2}$ in the strong coupling regime $m\ll g$. Our numerical results using the lattice Hamiltonian are in good agreement with this behavior. For the puzzle at $θ= 0$, the solution is related to a level crossing between two isosinglet particles with different discrete quantum numbers; we demonstrate the necessity of such a crossing by comparing integrability and weak coupling calculations, and we also exhibit the crossing numerically. Finally, we provide a new estimate for the size of isospin-breaking effects caused by different fermion masses at strong coupling.

hep-thCross-listedarXiv:2605.08042•May 11, 2026

Fermion discretization effects in the two-flavor lattice Schwinger model: A study with matrix product states

Tim Schwägerl, Karl Jansen, Stefan Kühn

We present a comprehensive tensor network study of staggered, Wilson, and twisted mass fermions in the Hamiltonian formulation, using the massive two-flavor Schwinger model as a benchmark. Particular emphasis is placed on twisted mass fermions, whose properties in this context have not been systematically explored before. We confirm the expected O ( a ) improvement in the free theory and observe that this improvement persists in the interacting case. By leveraging an electric-field-based method for mass renormalization, we reliably tune to maximal twist and establish the method’s applicability in the two-flavor model. Once mass renormalization is included, the pion mass exhibits rapid convergence to the continuum limit. Finite-volume effects are addressed using two complementary approaches: dispersion relation fits and finite-volume scaling. Our results show excellent agreement with semiclassical predictions and reveal a milder volume dependence for twisted mass fermions compared to staggered and Wilson discretizations. In addition, we observe clear isospin-breaking effects, suggesting intriguing parallels with lattice QCD. These findings highlight the advantages of twisted mass fermions for Hamiltonian simulations and motivate their further exploration—particularly in view of future applications to higher-dimensional lattice gauge theories.

arXiv:2509.02329•Physical review. D/Physical review. D.•Oct 16, 2025•1 citation
Calculating composite-particle spectra in Hamiltonian formalism and demonstration in 2-flavor QED1+1d_{1+1\text{d}}1+1d​
We consider three distinct methods to compute the mass spectrum of gauge theories in the Hamiltonian formalism: (1) correlation-function scheme, (2) one-point-function scheme, and (3) dispersion-relation scheme. The first one examines spatial correlation functions as we do in the conventional Euclidean Monte Carlo simulations. The second one uses the boundary effect to efficiently compute the mass spectrum. The third one constructs the excited states and fits their energy using the dispersion relation with selecting quantum numbers. Each method has its pros and cons, and we clarify such properties in their applications to the mass spectrum for the 2-flavor massive Schwinger model at m/g=0.1m/g=0.1m/g=0.1 and θ=0θ=0θ=0 using the density-matrix renormalization group (DMRG). We note that the multi-flavor Schwinger model at small mass mmm is a strongly coupled field theory even after the bosonizations, and thus it deserves to perform the first-principles numerical calculations. All these methods mostly agree and identify the stable particles, pions πaπ_aπa​ (JPG=1−+J^{PG}=1^{-+}JPG=1−+), sigma meson σσσ (JPG=0++J^{PG}=0^{++}JPG=0++), and eta meson ηηη (JPG=0−−J^{PG}=0^{--}JPG=0−−). In particular, we find that the mass of σσσ meson is lighter than twice the pion mass, and thus σσσ is stable against the decay process, σ→ππσ\to ππσ→ππ. This is consistent with the analytic prediction using the WKB approximation, and, remarkably, our numerical results are so close to the WKB-based formula between the pion and sigma-meson masses, Mσ/Mπ=3M_σ/M_π=\sqrt{3}Mσ​/Mπ​=3​.
Abstract We consider three distinct methods to compute the mass spectrum of gauge theories in the Hamiltonian formalism: (1) correlation-function scheme, (2) one-point-function scheme, and (3) dispersion-relation scheme. The first one examines spatial correlation functions as we do in the conventional Euclidean Monte Carlo simulations. The second one uses the boundary effect to efficiently compute the mass spectrum. The third one constructs the excited states and fits their energy using the dispersion relation with selecting quantum numbers. Each method has its pros and cons, and we clarify such properties in their applications to the mass spectrum for the 2-flavor massive Schwinger model at m/g = 0.1 and θ = 0 using the density-matrix renormalization group (DMRG). We note that the multi-flavor Schwinger model at small mass m is a strongly coupled field theory even after the bosonizations, and thus it deserves to perform the first-principles numerical calculations. All these methods mostly agree and identify the stable particles, pions πa (JPG = 1−+), sigma meson σ (JPG = 0++), and eta meson η (JPG = 0−−). In particular, we find that the mass of σ meson is lighter than twice the pion mass, and thus σ is stable against the decay process, σ → ππ. This is consistent with the analytic prediction using the WKB approximation, and, remarkably, our numerical results are so close to the WKB-based formula between the pion and sigma-meson masses, Mσ/Mπ = 3 \sqrt{3} 3​ 3 .
A bstract We study the θ -dependent mass spectrum of the massive 2-flavor Schwinger model in the Hamiltonian formalism using the density-matrix renormalization group (DMRG). The masses of the composite particles, the pion and sigma meson, are computed by two independent methods. One is the improved one-point-function scheme, where we measure the local meson operator coupled to the boundary state and extract the mass from its exponential decay. Since the θ term causes a nontrivial operator mixing, we unravel it by diagonalizing the correlation matrix to define the meson operator. The other is the dispersion-relation scheme, a heuristic approach specific to Hamiltonian formalism. We obtain the dispersion relation directly by measuring the energy and momentum of the excited states. The sign problem is circumvented in these methods, and their results agree with each other even for large θ . We reveal that the θ -dependence of the pion mass at m/g = 0 . 1 is consistent with the prediction by the bosonized model. We also find that the mass of the sigma meson satisfies the semi-classical formula, M σ /M π = 3 \sqrt{3} 3​ 3 , for almost all region of θ . While the sigma meson is a stable particle thanks to this relation, the eta meson is no longer protected by the G -parity and becomes unstable for θ ≠ 0.
We propose three independent methods to compute the hadron mass spectra of gauge theories in the Hamiltonian formalism. The determination of hadron masses is one of the key issues in QCD, which has been precisely calculated by the Monte Carlo method in the Lagrangian formalism. We confirm that the mass of hadrons can be calculated by examining correlation functions, the one-point function, or the dispersion relation in Hamiltonian formalism. These methods are suitable for quantum computation and tensor network approaches. The methods are demonstrated with the tensor network (DMRG) in the 2-flavor Schwinger model, which shares important properties with QCD. We show that the numerical results are consistent with each other and with the analytic prediction of the bosonization technique. We also discuss their efficiency and potential applications to other models.
We compute the θ\thetaθ-dependent mass spectrum of the 2-flavor Schwingr model using the tensor network (DMRG) in the Hamiltonian formalism. The pion and the sigma meson are identified as stable particles of the model for nonzero θ\thetaθ whereas the eta meson becomes unstable. The meson masses are obtained from the one-point functions, using the meson operators defined by diagonalizing the correlation matrix to deal with the operator mixing. We also compute the dispersion relation directly by measuring the energy and momentum of the excited states, where the mesons are distinguished by the isospin quantum number. We confirmed that the meson masses computed by these methods agree with each other and are consistent with the calculation by the bosonized model. Our methods are free from the sign problem and show a significant improvement in accuracy compared to the conventional Monte Carlo methods. Furthermore, at the critical point θ=π\theta = \piθ=π, the mesons become almost massless, and the one-point functions reproduce the expected CFT-like behavior. This talk is based on the paper [1].
In his 1976 paper "More about the massive Schwinger model", Coleman introduced 1+11+11+1-dimensional Quantum Electrodynamics coupled to two charged massive fermions. By applying Abelian bosonization, he elucidated much of the physics of this two-flavor Schwinger model, but he listed three puzzles at the end of his paper. We present new analytical and numerical calculations to solve Coleman's three puzzles and thereby deepen our understanding of this model. These puzzles pertain to the theory with equal fermion masses at θ=0θ= 0θ=0 and at θ=πθ= πθ=π, as well as the size of isospin-breaking effects when the fermion masses are unequal. For the puzzle at θ=πθ= πθ=π, the solution is related to the structure of the zero-temperature phase diagram arXiv:2305.04437: for equal fermion masses mmm, the model exhibits spontaneous breaking of charge conjugation symmetry and absence of confinement for any value of the gauge coupling ggg, so that there is a smooth interpolation from weak to strong coupling. Using two-loop Renormalization Group and integrability methods, we show that the mass gap behaves as ∼me−0.111g2/m2\sim m e^{-0.111 g^2/m^2}∼me−0.111g2/m2 in the strong coupling regime m≪gm\ll gm≪g. Our numerical results using the lattice Hamiltonian are in good agreement with this behavior. For the puzzle at θ=0θ= 0θ=0, the solution is related to a level crossing between two isosinglet particles with different discrete quantum numbers; we demonstrate the necessity of such a crossing by comparing integrability and weak coupling calculations, and we also exhibit the crossing numerically. Finally, we provide a new estimate for the size of isospin-breaking effects caused by different fermion masses at strong coupling.
We present a comprehensive tensor network study of staggered, Wilson, and twisted mass fermions in the Hamiltonian formulation, using the massive two-flavor Schwinger model as a benchmark. Particular emphasis is placed on twisted mass fermions, whose properties in this context have not been systematically explored before. We confirm the expected O ( a ) improvement in the free theory and observe that this improvement persists in the interacting case. By leveraging an electric-field-based method for mass renormalization, we reliably tune to maximal twist and establish the method’s applicability in the two-flavor model. Once mass renormalization is included, the pion mass exhibits rapid convergence to the continuum limit. Finite-volume effects are addressed using two complementary approaches: dispersion relation fits and finite-volume scaling. Our results show excellent agreement with semiclassical predictions and reveal a milder volume dependence for twisted mass fermions compared to staggered and Wilson discretizations. In addition, we observe clear isospin-breaking effects, suggesting intriguing parallels with lattice QCD. These findings highlight the advantages of twisted mass fermions for Hamiltonian simulations and motivate their further exploration—particularly in view of future applications to higher-dimensional lattice gauge theories.