Schwinger Model Research Articles

Real-time chiral dynamics at finite temperature from quantum simulation

In this study, we explore the real-time dynamics of the chiral magnetic effect (CME) at a finite temperature in the (1+1)-dimensional QED, the massive Schwinger model. By introducing a chiral chemical potential μ5 through a quench process, we drive the system out of equilibrium and analyze the induced vector currents and their evolution over time. The Hamiltonian is modified to include the time-dependent chiral chemical potential, thus allowing the investigation of the CME within a quantum computing framework. We employ the quantum imaginary time evolution (QITE) algorithm to study the thermal states, and utilize the Suzuki-Trotter decomposition for the real-time evolution. This study provides insights into the quantum simulation capabilities for modeling the CME and offers a pathway for studying chiral dynamics in low-dimensional quantum field theories.

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

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, Mσ/Mπ = 3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{3} $$\\end{document}, 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.

End-to-end complexity for simulating the Schwinger model on quantum computers

The Schwinger model is one of the simplest gauge theories. It is known that a topological term of the model leads to the infamous sign problem in the classical Monte Carlo method. In contrast to this, recently, quantum computing in Hamiltonian formalism has gained attention. In this work, we estimate the resources needed for quantum computers to compute physical quantities that are challenging to compute on classical computers. Specifically, we propose an efficient implementation of block-encoding of the Schwinger model Hamiltonian. Considering the structure of the Hamiltonian, this block-encoding with a normalization factor of O(N3) can be implemented using O(N+log2⁡(N/ε)) T gates. As an end-to-end application, we compute the vacuum persistence amplitude. As a result, we found that for a system size N=128 and an additive error ε=0.01, with an evolution time t and a lattice spacing a satisfying t/2a=10, the vacuum persistence amplitude can be calculated using about 1013 T gates. Our results provide insights into predictions about the performance of quantum computers in the FTQC and early FTQC era, clarifying the challenges in solving meaningful problems within a realistic timeframe.

Particle production and hadronization temperature in the massive Schwinger model

We study the pair production, string breaking, and hadronization of a receding electron-positron pair using the bosonized version of the massive Schwinger model in quantum electrodynamics in 1+1 space-time dimensions. Specifically, we study the dynamics of the electric field in Bjorken coordinates by splitting it into a coherent field and its Gaussian fluctuations. We find that the electric field shows damped oscillations, reflecting pair production. Interestingly, the computation of the asymptotic total particle density per rapidity interval for large masses can be fitted using a Boltzmann factor, where the temperature can be related to the hadronization temperature in QCD. Lastly, we discuss the possibility of an analog quantum simulation of the massive Schwinger model using ultracold atoms, explicitly matching the potential of the Schwinger model to the effective potential for the relative phase of two linearly coupled Bose-Einstein condensates. Published by the American Physical Society 2024

De Sitter at all loops: the story of the Schwinger model

We consider the two-dimensional Schwinger model of a massless charged fermion coupled to an Abelian gauge field on a fixed de Sitter background. The theory admits an exact solution, first examined by Jayewardena, and can be analyzed efficiently using Euclidean methods. We calculate fully non-perturbative, gauge-invariant correlation functions of the electric field as well as the fermion and analyze these correlators in the late-time limit. We compare these results with the perturbative picture, for example by verifying that the one-loop contribution to the fermion two-point function, as predicted from the exact solution, matches the direct computation of the one-loop Feynman diagram. We demonstrate many features endemic of quantum field theory in de Sitter space, including the appearance of late-time logarithms, their resummation to de Sitter invariant expressions, and Boltzmann suppressed non-perturbative phenomena, with surprising late-time features.

Locating quark-antiquark string breaking in QCD through chiral symmetry restoration and Hawking-Unruh effect

The relationship between QCD and the string model offers a valuable perspective for exploring the interaction potential between quarks. In this study, we investigate the restoration of chiral symmetry in connection with the Unruh effect experienced by accelerating observers. Utilizing the Schwinger model, we analyze the critical point at which the string or chromoelectric flux tube between quark-antiquarks breaks with increasing separation between quarks. In this study, the critical distance for quark-antiquark chromoelectric flux tube or string breaking is determined to be rc=1.294±0.040 fm. The acceleration and Unruh temperature corresponding to this critical point signify the transition of the system's chiral symmetry from a broken to a restored state. Our estimates for the critical acceleration (ac=1.14×1034 cm/s2) and Unruh temperature (Tc=0.038 GeV) align with previous studies. This analysis illuminates the interplay between chiral symmetry restoration, the Unruh effect, and the breaking of the string or chromoelectric flux tube within the context of quark interactions.

On the anomaly interpretation of amplitudes in self-dual Yang-Mills and gravity

We investigate the integrability anomalies arising in the self-dual sectors of gravity and Yang-Mills theory, focusing on their connection to both the chiral anomaly and the trace anomaly. The anomalies in the self-dual sectors generate the one-loop all-plus amplitudes of gravitons and gluons, and have recently been studied via twistor constructions. On the one hand, we show how they can be interpreted as an anomaly of the chiral U(1) electric-magnetic-type duality in the self-dual sectors. We also note the similarity, for the usual fermionic chiral anomaly, between the 4D setting of self-dual Yang-Mills and the 2D setting of the Schwinger model. On the other hand, the anomalies in the self-dual theories also resemble the trace anomaly, sharing the same type of non-local effective action. We highlight the role of a Weyl-covariant fourth-order differential operator familiar from the trace anomaly literature, which (i) explains the conformal properties of the one-loop amplitudes, and (ii) indicates how this story may be extended to non-trivial spacetime backgrounds, e.g. with a cosmological constant. Moving beyond the self-dual sectors, and focusing on the gravity case, we comment on an intriguing connection to the two-loop ultraviolet divergence of pure gravity, whereby cancelling the anomaly at one-loop eliminates the two-loop divergence for the simplest helicity amplitudes.

Real-time dynamics of the Schwinger model as an open quantum system with Neural Density Operators

Ab-initio simulations of multiple heavy quarks propagating in a Quark-Gluon Plasma are computationally difficult to perform due to the large dimension of the space of density matrices. This work develops machine learning algorithms to overcome this difficulty by approximating exact quantum states with neural network parametrisations, specifically Neural Density Operators. As a proof of principle demonstration in a QCD-like theory, the approach is applied to solve the Lindblad master equation in the 1 + 1d lattice Schwinger Model as an open quantum system. Neural Density Operators enable the study of in-medium dynamics on large lattice volumes, where multiple-string interactions and their effects on string-breaking and recombination phenomena can be studied. Thermal properties of the system at equilibrium can also be probed with these methods by variationally constructing the steady state of the Lindblad master equation. Scaling of this approach with system size is studied, and numerical demonstrations on up to 32 spatial lattice sites and with up to 3 interacting strings are performed.

Studying the phase diagram of the three-flavor Schwinger model in the presence of a chemical potential with measurement- and gate-based quantum computing

We propose an ansatz quantum circuit for the variational quantum eigensolver (VQE), suitable for exploring the phase structure of the multiflavor Schwinger model in the presence of a chemical potential. Our ansatz is capable of incorporating relevant model symmetries via constrains on the parameters, and can be implemented on circuit-based as well as measurement-based quantum devices. We show via classical simulation of the VQE that our ansatz is able to capture the phase structure of the model, and can approximate the ground state to a high level of accuracy. Moreover, we perform proof-of-principle simulations on superconducting, gate-based quantum hardware. Our results show that our approach is suitable for current gate-based quantum devices, and can be readily implemented on measurement-based quantum devices once available. Published by the American Physical Society 2024

Realtime dynamics of hyperon spin correlations from string fragmentation in a deformed four-flavor Schwinger model

Self-polarizing weak decays of Λ-hyperons provide unique insight into the role of entanglement in the fragmentation of QCD strings through measurements of the spin correlations of ΛΛ¯ pairs produced in collider experiments. The simplest quantum field theory representing the underlying parton dynamics is the four-flavor massive Schwinger model plus an effective spin-flip term, where the flavors are mapped to light (up or down) and heavy (strange) quarks and their spins. This construction provides a novel way to explore hyperon spin correlations in 1+1 dimensions. We investigate the evolution of these correlations for different string configurations that are sensitive to the rich structure of the model Hamiltonian. Published by the American Physical Society 2024

Exploring quantum entanglement in chiral symmetry partial restoration with 1+1 string model

Within the confining color flux tube picture, we assess the color electric field generated by quark-antiquark pairs within the framework of the Schwinger model and estimate its impact on the chiral condensate. We observe differences in the distribution of the color flux tube generated by quark-antiquark pairs at different separation distances, leading to discrepancies in the partial restoration of chiral symmetry. Furthermore, we suggest incorporating the color magnetic field in the calculation of chiral condensate, leading to quantum entanglement effects, and proceed to compute the entanglement entropy. We observe that the entanglement entropy increases with the distance of the color source (the separation distance between quark and anti-quark), and the magnitude of the color electric field and chiral condensate restoration after spatial integration also increases with the distance of the color source. Then, we try to provide a qualitative explanation for the existence of these phenomena.

High-Energy Collision of Quarks and Mesons in the Schwinger Model: From Tensor Networks to Circuit QED.

With the aim of studying nonperturbative out-of-equilibrium dynamics of high-energy particle collisions on quantum simulators, we investigate the scattering dynamics of lattice quantum electrodynamics in 1+1 dimensions. Working in the bosonized formulation of the model and in the thermodynamic limit, we use uniform-matrix-product-state tensor networks to construct multiparticle wave-packet states, evolve them in time, and detect outgoing particles post collision. This facilitates the numerical simulation of scattering experiments in both confined and deconfined regimes of the model at different energies, giving rise to rich phenomenology, including inelastic production of quark and meson states, meson disintegration, and dynamical string formation and breaking. We obtain elastic and inelastic scattering cross sections, together with time-resolved momentum and position distributions of the outgoing particles. Furthermore, we propose an analog circuit-QED implementation of the scattering process that is native to the platform, requires minimal ingredients and approximations, and enables practical schemes for particle wave-packet preparation and evolution. This study highlights the role of classical and quantum simulation in enhancing our understanding of scattering processes in quantum field theories in real time.

The Schwinger and chiral Schwinger models in a non-perturbative spectral regularization

We investigate the employment of a non-perturbative regularization scheme – the spectral regularization, which is based on the gauge technique, previously implemented in the context of chiral quark models – in the study of the gauge symmetry preservation within the Schwinger model and violation in the chiral Schwinger model. We show that the spectral regularization provides mathematical consistent and ambiguity free solutions for the two-point functions of both Schwinger and chiral Schwinger models in exact (1+1) dimensions, correctly displaying the gauge invariance in the Schwinger model and the axial anomaly in the chiral Schwinger model. The employment of the spectral regularization avoids any dependence on ambiguous amplitudes and/or unconventional γ5 algebra. Our results reinforce the strength of the spectral regularization as a mathematical consistent, divergence free, and ambiguity free regularization scheme that correctly implements symmetry conservation or violation in each case.

Fractal States of the Schwinger Model.

The lattice Schwinger model, the discrete version of QED in 1+1 dimensions, is a well-studied test bench for lattice gauge theories. Here, we study the fractal properties of this model. We reveal the self-similarity of the ground state, which allows us to develop a recurrent procedure for finding the ground-state wave functions and predicting ground-state energies. We present the results of recurrently calculating ground-state wave functions using the fractal Ansatz and automized software package for fractal image processing. In certain parameter regimes, just a few terms are enough for our recurrent procedure to predict ground-state energies close to the exact ones for several hundreds of sites. Our findings pave the way to understanding the complexity of calculating many-body wave functions in terms of their fractal properties as well as finding new links between condensed matter and high-energy lattice models.

Entanglement in massive Schwinger model at finite temperature and density

Entanglement in massive Schwinger model at finite temperature and density

Gauge boson mass dependence and chiral anomalies in generalized massless Schwinger models

I bosonize the position-space correlators of flavor-diagonal scalar fermion bilinears in arbitrary generalizations of the Schwinger model with nF massless fermions coupled to nA gauge bosons for nF ≥ nA. For nA = nF, the fermion bilinears can be bosonized in terms of nF scalars with masses proportional to the gauge couplings. As in the Schwinger model, bosonization can be used to find all correlators, including those that are forbidden in perturbation theory by anomalous chiral symmetries, but there are subtleties when there is more than one gauge boson. The new result here is the general treatment of the dependence on gauge boson masses in models with more than one gauge symmetry. For nA < nF, there are fermion bilinears with nontrivial anomalous dimensions and there are unbroken chiral symmetries so some correlators vanish while others are non-zero due to chiral anomlies. Taking careful account of the dependence on the masses, I show how the nA < nF models emerge from nA = nF as gauge couplings (and thus gauge boson masses) go to zero. When this is done properly, the limit of zero gauge coupling is smooth. Our consistent treatment of gauge boson masses guarantees that anomalous symmetries are broken while unbroken chiral symmetries are preserved because correlators that break the non-anomalous symmetries go to zero in the limit of zero gauge coupling.

Phase Diagram of the Two-Flavor Schwinger Model at Zero Temperature.

We examine the phase structure of the two-flavor Schwinger model as a function of the θ angle and the two masses, m_{1} and m_{2}. In particular, we find interesting effects at θ=π: along the SU(2)-invariant line m_{1}=m_{2}=m, in the regime where m is much smaller than the charge g, the theory undergoes logarithmic renormalization group flow of the Berezinskii-Kosterlitz-Thouless type. As a result, dimensional transmutation takes place, leading to a nonperturbatively small mass gap ∼e^{-Ag^{2}/m^{2}}. The SU(2)-invariant line lies within a region of the phase diagram where the charge conjugation symmetry is spontaneously broken and whose boundaries we determine numerically. Our numerical results are obtained using the Hamiltonian lattice gauge formulation that includes the mass shift m_{lat}=m-g^{2}a/4 dictated by the discrete chiral symmetry.

Training normalizing flows with computationally intensive target probability distributions

Machine learning techniques, in particular the so-called normalizing flows, are becoming increasingly popular in the context of Monte Carlo simulations as they can effectively approximate target probability distributions. In the case of lattice field theories (LFT) the target distribution is given by the exponential of the action. The common loss function's gradient estimator based on the “reparametrization trick” requires the calculation of the derivative of the action with respect to the fields. This can present a significant computational cost for complicated, non-local actions like e.g. fermionic action in QCD. In this contribution, we propose an estimator for normalizing flows based on the REINFORCE algorithm that avoids this issue. We apply it to two dimensional Schwinger model with Wilson fermions at criticality and show that it is up to ten times faster in terms of the wall-clock time as well as requiring up to 30% less memory than the reparameterization trick estimator. It is also more numerically stable allowing for single precision calculations and the use of half-float tensor cores. We present an in-depth analysis of the origins of those improvements. We believe that these benefits will appear also outside the realm of the LFT, in each case where the target probability distribution is computationally intensive.

Twirling Operations to Produce Energy Eigenstates of a Hamiltonian by Classically Emulated Quantum Simulation

We propose a simple procedure to produce energy eigenstates of a Hamiltonian with discrete eigenvalues. We use ancilla qubits and quantum entanglement to separate an energy eigenstate from the other energy eigenstates. We exhibit a few examples derived from the (1 + 1)‐dimensional massless Schwinger model. Our procedure in principle will be applicable for a Hamiltonian with a finite‐dimensional Hilbert space. Choosing an initial state properly, we can in principle produce any energy eigenstate of the Hamiltonian.

Phase diagram near the quantum critical point in Schwinger model at θ = π: analogy with quantum Ising chain

Abstract The Schwinger model, 1D quantum electrodynamics, has CP symmetry at θ = π due to the topological nature of the θ term. At zero temperature, it is known that as the fermion mass increases, the system undergoes a second-order phase transition to the CP broken phase, which belongs to the same universality class as the quantum Ising chain. In this paper, we obtain the phase diagram near the quantum critical point (QCP) in the temperature and fermion mass plane using first-principle Monte Carlo simulations, while avoiding the sign problem by using the lattice formulation of the bosonized Schwinger model. Specifically, we perform a detailed investigation of the correlation function of the electric field near the QCP and find that its asymptotic behavior can be described by the universal scaling function of the quantum Ising chain. This finding indicates the existence of three regions near the QCP, each characterized by a specific asymptotic form of the correlation length, and demonstrates that the CP symmetry is restored at any nonzero temperature, entirely analogous to the quantum Ising chain. The range of the scaling behavior is also examined and found to be particularly wide.