Parton Distribution Functions in the Schwinger model from Tensor Network States

This paper proposes and demonstrates a method using tensor network states within the Hamiltonian formalism to accurately compute parton distribution functions for the vector meson in the massive Schwinger model directly in Minkowski space, thereby overcoming the limitations of Euclidean lattice calculations and offering a pathway for quantum simulations.

The Gist

The Big Picture: Seeing Inside the Proton

Imagine a proton (the building block of atoms) not as a solid marble, but as a bustling, chaotic city. Inside this city, tiny particles called quarks and gluons (collectively called "partons") are zooming around at nearly the speed of light.

Physicists want to know the "traffic report" of this city: How many quarks are there? How fast are they going? Which direction are they facing? This map is called a Parton Distribution Function (PDF).

For decades, trying to draw this map has been like trying to take a photo of a speeding race car using a camera that only works in slow motion. The math required to describe these particles usually forces scientists to work in a "Euclidean" world (a mathematical trick where time is treated like a fourth dimension of space). But in this trick-world, you can't see the "light-speed" dynamics you need to understand the PDF.

The Problem: The "Light-Speed" Barrier

To get the real traffic report, you need to look at the particles along a "light cone" (a path moving at the speed of light).

• The Old Way: Scientists tried to build the map using the slow-motion camera (Euclidean lattice QCD). They had to take a snapshot, then use a complex translator to guess what the fast-motion version looked like. It's like trying to guess the plot of a movie by only looking at the credits. It's hard, and errors pile up.
• The New Idea: What if we could just watch the movie in real-time? That's what this paper proposes.

The Solution: Tensor Networks as a "Digital LEGO Set"

The authors used a powerful mathematical tool called Tensor Networks (TN). Think of a Tensor Network as a highly efficient way to organize a massive amount of information, similar to how LEGO bricks can build complex structures without needing a billion individual bricks.

In the world of quantum physics, the "LEGO bricks" are entanglement (a spooky connection between particles).

• The Challenge: Usually, as particles interact and move, they get "entangled" in a way that makes the LEGO structure explode in size, becoming impossible to calculate on a computer.
• The Breakthrough: The authors realized that for this specific problem (calculating the PDF), the LEGO structure stays manageable. They didn't need a supercomputer the size of a planet; they could do it with a standard supercomputer.

The Experiment: The "Toy City" (The Schwinger Model)

Testing this on a real proton (which is incredibly complex) is too hard right now. So, the team built a Toy City.

• The Model: They used the Schwinger Model. Imagine a 1-dimensional world (a straight line) instead of our 3D world. It's a simplified version of the strong force that holds protons together.
• Why do this? Even though it's a toy, it has the same "rules of the road" as the real thing (confinement, mass generation). If they can solve the traffic report for this toy city, they prove the method works for the real city.

How They Did It: The "Step-by-Step" Dance

Here is the clever trick they used to simulate "real-time" without getting stuck:

1. The Wilson Line: In the math, the PDF requires a "Wilson line," which is like a string connecting two points in space-time. In the real world, this string moves at the speed of light.
2. The Zig-Zag: Instead of trying to move the string instantly, they moved it in a zig-zag pattern on their digital grid.
   • Step 1: Move forward in time (let the system evolve).
   • Step 2: Move forward in space (shift the string).
   • Repeat: Do this over and over.
3. The Static Charge: To make this work mathematically, they introduced "static charges" (like invisible anchors) that move along with the string. This allows the computer to calculate the path of the string without needing to simulate the impossible speed of light directly.

The Results: A Clear Map

By using this method, they successfully calculated the PDF for their toy city.

• The Outcome: They got a smooth, accurate curve showing how momentum is shared between the particles.
• The Surprise: The results were real and positive (which they must be to represent probability). Previous attempts using similar methods sometimes gave "negative probabilities," which makes no physical sense. Their method fixed this.
• The Future: They showed that as they made the "toy city" bigger and the grid finer (approaching the "continuum limit"), the results stabilized. This proves the method is robust.

Why This Matters

This paper is a proof of concept.

• For Theory: It shows we can calculate these difficult "light-speed" properties directly, without needing the slow-motion translator.
• For Technology: The method they used (Tensor Networks) is very similar to what Quantum Computers will use in the future. They are essentially showing the blueprint for how a future quantum computer could solve the proton's internal structure.

In a nutshell: The authors built a digital LEGO model of a simplified universe, taught it how to dance in real-time using a zig-zag step, and successfully mapped out the internal traffic of a particle. This paves the way for us to finally understand the messy, fast-moving interior of the protons that make up our world.

Technical Summary

1. Problem Statement

Parton Distribution Functions (PDFs) are fundamental observables describing the non-perturbative internal structure of hadrons (quarks and gluons momentum distributions). Calculating PDFs from first principles in Quantum Chromodynamics (QCD) is notoriously difficult due to their definition involving light-front (LF) matrix elements with Wilson lines along the light cone.

• The Challenge: Standard Lattice QCD calculations are performed in Euclidean spacetime (imaginary time) using path integrals. This formulation does not provide direct access to real-time dynamics or light-cone correlations, which are essential for PDFs.
• Limitations of Previous Approaches:
  • Euclidean Methods: Approaches like Large-Momentum Effective Theory (LaMET) or quasi-PDFs require matching Euclidean results to Minkowski space, introducing systematic uncertainties and requiring high momentum.
  • Hamiltonian Methods: Previous attempts to calculate PDFs directly in Minkowski space using the Hamiltonian formalism relied on few-fermion approximations, small lattices, or exact diagonalization. These methods are not scalable and lack reliable control over systematic errors.

2. Methodology

The authors propose a novel strategy combining the Hamiltonian formalism with Tensor Network (TN) methods, specifically Matrix Product States (MPS), to simulate real-time evolution in Minkowski space.

A. The Model: Massive Schwinger Model

The study utilizes the Schwinger model (1+1 dimensional Quantum Electrodynamics) as a toy model. It shares key features with QCD, including confinement, dynamical mass generation, and asymptotic freedom, making it an ideal testing ground for new algorithms before applying them to 3+1D QCD.

• Hamiltonian Formulation: The model is discretized on a lattice using staggered fermions in the temporal gauge ($A_0=0$). Through a Jordan-Wigner transformation, fermions are mapped to spins, and gauge fields are integrated out using Gauss's law, resulting in a spin Hamiltonian with static charges.

B. Implementing the Light-Front Wilson Line

The core innovation is the implementation of the light-front Wilson line ($W$) required for the PDF definition (Eq. 3) within the TN framework:

• Stepwise Evolution: Instead of a continuous light-cone path, the Wilson line is approximated by a "zigzag" evolution on the lattice.
  • Time Steps: Evolution by $\Delta t$ using the standard Hamiltonian $e^{-iH\Delta t}$.
  • Spatial Steps: Evolution along the spatial direction is achieved by shifting the electric flux. This is mathematically equivalent to introducing a static charge that moves along the light cone.
• Physical Subspace: By integrating out gauge degrees of freedom, the calculation is performed entirely in the physical subspace. The Wilson line is realized by dynamically moving a static charge $q_k$ along the lattice, effectively "dragging" the flux string.

C. Tensor Network Algorithm

• State Representation: The ground state (vacuum) and the first excited state (vector meson) are approximated using MPS with a bond dimension $D$.
• Time Evolution: The time evolution operator is applied using a second-order Suzuki-Trotter decomposition. The electric flux terms are truncated to a maximum change $L_{cut}$ (implemented as a Matrix Product Operator, MPO).
• PDF Extraction: The PDF is obtained by calculating the matrix element of the Wilson line operator between hadron states. This involves a discrete Fourier transform of the time-evolved matrix elements $M(\Delta z)$.

3. Key Contributions

1. First Direct Minkowski Calculation with TNs: This is the first demonstration of calculating PDFs directly in Minkowski space using Tensor Networks, bypassing the need for Euclidean-to-Minkowski matching.
2. Controlled Systematic Errors: Unlike previous Hamiltonian approaches, this method allows for systematic control of errors via:
   • Bond Dimension ($D$): Controlling entanglement truncation.
   • Trotter Step ($N_\tau$): Controlling time-discretization errors.
   • Flux Cutoff ($L_{cut}$): Controlling electric flux truncation.
   • Continuum Limit ($x \to \infty$): Extrapolating lattice spacing $a \to 0$.
   • Thermodynamic Limit ($V \to \infty$): Extrapolating volume effects.
3. Validation of Light-Cone Physics: The study demonstrates that TNs can successfully capture light-cone dynamics and the associated growth of entanglement entropy without becoming computationally intractable for this specific observable.

4. Results

The authors computed the fermion PDF for the vector meson in the Schwinger model for various fermion masses ($\tilde{m}$).

• Continuum Extrapolation: Results were extrapolated to the continuum limit ($x \to \infty$) and, for the benchmark case $\tilde{m}=10$, to the infinite volume limit.
• Physical Constraints:
  • The resulting PDFs are real, non-negative for $0 \le \xi \le 1$ and vanish for $|\xi| > 1$, satisfying the probabilistic interpretation of the parton model.
  • The distribution is antisymmetric ($f(\xi) = -f(-\xi)$), consistent with charge conjugation symmetry.
  • The average momentum fraction carried by fermions and antifermions is $\langle \xi \rangle = 0.50 \pm 0.02$, confirming that the hadron's momentum is equally shared.
• Mass Dependence:
  • For small masses, the PDF is broad.
  • As mass increases, the peak at $\xi = 0.5$ sharpens.
  • In the limit of infinite mass, the PDF approaches a delta function at $\xi=0.5$.
  • In the massless limit, it approaches a step function.
• Comparison: The results show excellent agreement with previous light-front calculations (Mo and Perry) in shape but correct the unphysical negative values found in those earlier approximations.

5. Significance and Future Outlook

• Proof of Concept for QCD: This work establishes that Tensor Networks are a viable tool for calculating real-time, light-cone observables in gauge theories. It proves that the "entanglement barrier" does not prevent the calculation of PDFs in the Schwinger model.
• Path to Higher Dimensions: While the Schwinger model is 1+1D, the methodology (stepwise Wilson line evolution and static charge handling) is adaptable to higher dimensions using Projected Entangled Pair States (PEPS) or other TN architectures.
• Quantum Computing Synergy: The Hamiltonian formulation used here is directly compatible with quantum simulation. The MPS representation of the initial state and the stepwise evolution of the Wilson line can be mapped to quantum circuits, offering a clear roadmap for future quantum simulations of PDFs on quantum hardware.
• Error Control: The rigorous error analysis sets a new standard for Hamiltonian lattice calculations, demonstrating that systematic errors can be quantified and reduced to achieve high-precision continuum results.

In summary, this paper bridges the gap between non-perturbative lattice gauge theory and real-time parton physics, providing a robust, first-principles framework for calculating PDFs that is scalable and compatible with emerging quantum technologies.