Infinite matrix product states for $(1+1)$-dimensional gauge theories

This paper introduces a link-enhanced matrix product operator (LEMPO) construction that enables the study of abelian and non-abelian lattice gauge theories on infinite one-dimensional lattices by representing their Hamiltonians in a local, translation-invariant form, as demonstrated through applications to the Schwinger model and adjoint QCD$_2$.

The Gist

Imagine you are trying to solve a massive, complex jigsaw puzzle. But this isn't a normal puzzle; it's a puzzle representing the fundamental forces of the universe (specifically, how particles interact with forces like electricity and the strong nuclear force). The pieces are constantly shifting, and the rules are incredibly strict.

For decades, physicists have struggled to solve these puzzles, especially when they involve gauge theories (the math behind how forces work). The problem is that the number of possible configurations is so huge that even the world's fastest supercomputers get overwhelmed. It's like trying to count every grain of sand on every beach on Earth simultaneously.

This paper introduces a brilliant new tool called LEMPOs (Link-Enhanced Matrix Product Operators) to help solve these puzzles. Here is how it works, explained through simple analogies.

1. The Problem: The "Infinite" Puzzle

In the old way of doing this, physicists tried to simplify the puzzle by "gauge-fixing." Imagine you have a long line of people holding hands (representing particles and forces). To make the math easier, you decide to tell everyone, "Just hold hands with the person to your left, and ignore the right."

This simplifies the math, but it breaks the symmetry of the line. Suddenly, the person at the far left is holding a different kind of hand than the person at the far right. The "infinite" nature of the universe is lost, and you can't easily study what happens in the middle of a very long line.

2. The Solution: The "Smart" Rope (MPS)

Physicists use a technique called Matrix Product States (MPS). Think of this as a long, flexible rope made of many small knots.

• The Knots: Each knot represents a particle in the universe.
• The String: The string connecting the knots represents the "virtual" information shared between particles.

In a standard rope, the string only connects the knots to their neighbors. But in this new method, the authors realized that the string itself holds secret information about the forces (the "gauge fields") that we usually try to ignore.

3. The Innovation: The "Link-Enhanced" Tool (LEMPO)

The authors invented a new tool called a LEMPO.

• The Old Tool (MPO): Imagine you have a tool that can only touch the knots (the particles). If you want to measure the tension in the string between two knots, you have to calculate it by looking at every single knot from the very beginning of the rope to the end. It's slow and messy.
• The New Tool (LEMPO): This tool is special because it can reach inside the string (the virtual space) between the knots.

The Analogy:
Imagine a train where the cars are the particles.

• Old Method: To check the speed of the engine (the force), you have to walk from the last car to the first car, counting every wheel.
• LEMPO Method: The engine has a direct wire running through the floor of every car. You can just tap the wire in the middle of the train, and it instantly tells you the speed of the engine.

Because LEMPOs can touch the "wires" (the virtual spaces) directly, they can represent the laws of physics locally (right where the action is) and symmetrically (the rules are the same everywhere). This means we can now study an infinite train without worrying about the ends.

4. What Did They Do With It?

The authors used this new tool to solve two famous "toy models" of the universe:

1. The Schwinger Model: This is a simplified version of how electrons and light interact. It's like a 1D version of our real world. They used LEMPOs to calculate the mass of particles and how they stick together with extreme precision, matching known mathematical answers perfectly.
2. Adjoint QCD2: This is a more complex model involving "strong" forces (like those holding protons together). They studied how these forces behave when the particles are massless versus when they have mass. They discovered new details about how these forces create "strings" that hold particles together, confirming some theories and refining others.

5. Why Does This Matter?

Think of this as upgrading from a hand-drawn map to a GPS.

• Before: We could only study small, finite chunks of the universe, and we had to make messy approximations.
• Now: With LEMPOs, we can simulate an infinite universe on a computer. We can see the "big picture" without the distortion of edges.

This is a huge step forward because it allows physicists to study real-time dynamics (how things move and change over time) and confinement (why we never see a single quark floating around alone) with a level of precision that was previously impossible.

Summary

The paper presents a new mathematical "super-tool" (LEMPO) that lets physicists look inside the invisible "strings" connecting particles. By doing so, they can simulate the universe as an infinite, perfectly symmetrical system, solving complex puzzles about how matter and forces interact with unprecedented accuracy. It's like finally finding the instruction manual for the universe's most complicated game.

Technical Summary

1. Problem Statement

The paper addresses the challenge of simulating non-perturbative phenomena in (1+1)-dimensional lattice gauge theories (LGTs), such as color confinement and mass gap generation. While Hamiltonian lattice gauge theory avoids the sign problem and allows for real-time dynamics, it suffers from the exponential scaling of the Hilbert space with lattice size.

Existing tensor network approaches (specifically Matrix Product States, or MPS) face two main limitations when applied to gauge theories:

1. Gauge Fixing Strategies: Traditional methods often eliminate gauge fields via Gauss's law, resulting in non-local interactions in the matter Hamiltonian. This destroys manifest translation invariance, making it impossible to use infinite MPS (uMPS) algorithms like VUMPS (Variational Uniform Matrix Product States).
2. Link Truncation Strategies: Alternative methods introduce extra physical sites for gauge fields but require manual truncation of the infinite-dimensional gauge Hilbert space. Furthermore, the constraints required to maintain gauge invariance in these setups often do not generalize easily from Abelian to Non-Abelian theories.

The goal is to develop a method that preserves locality, manifest translation invariance, and gauge invariance simultaneously, allowing for the study of both Abelian and Non-Abelian theories on infinite lattices without manual Hilbert space truncation.

2. Methodology: Link-Enhanced Matrix Product Operators (LEMPOs)

The authors introduce a novel construction called Link-Enhanced Matrix Product Operators (LEMPOs). This method relies on Symmetric Matrix Product States (sMPS) and generalizes standard MPOs to act on both physical and virtual spaces.

Key Technical Components:

• Symmetric MPS (sMPS): The authors utilize sMPS where the MPS tensors obey local symmetry constraints. They identify these constraints with the Gauss law of the lattice gauge theory.

  • In an sMPS, the virtual indices carry representations of the gauge group $G$.
  • The local charge conservation condition at a site (virtual charge + physical charge = virtual charge) mirrors the Gauss law constraint ($L_n - L_{n-1} = Q_n$).
  • Consequently, the virtual space of the sMPS naturally encodes the state of the gauge field, effectively "reintroducing" the gauge degrees of freedom that are usually gauge-fixed away.

• Link-Enhanced MPOs (LEMPOs): Standard MPOs act only on physical sites. A LEMPO is constructed to act on both the physical sites and the virtual bonds (links) between sites.

  • Physical Part: Represents the matter Hamiltonian terms (e.g., fermion hopping, mass terms).
  • Link Part: Represents the gauge kinetic energy terms (e.g., $L_n^2$). Instead of summing charges non-locally to find the electric field, the LEMPO acts directly on the virtual indices of the sMPS tensors.
  • Mechanism: The electric field operator $L_n$ is identified with the charge operator $Q_n$ acting on the virtual space. Since the virtual space carries the gauge representation, acting with $Q_n$ on the virtual bond is equivalent to acting with the electric field on the link.

• Infinite Lattice Application (uMPS): Because the LEMPO construction maintains manifest translation invariance and locality, it can be directly applied to uniform MPS (uMPS). This allows the use of the VUMPS algorithm to find ground states and the quasiparticle ansatz to find excitation spectra on infinite lattices, avoiding finite-size boundary effects.

• Dynamic Cutoff: Unlike methods requiring a hard cutoff on the gauge field Hilbert space, the virtual bond dimensions in the sMPS are adjusted dynamically by the optimization algorithm (DMRG/VUMPS). The algorithm naturally selects the relevant gauge field representations needed for the ground state.

3. Key Contributions

1. LEMPO Formalism: The development of a general framework for representing lattice gauge theory Hamiltonians as LEMPOs. This unifies the treatment of Abelian and Non-Abelian theories under a single symmetric MPS framework.
2. Infinite Lattice Capability: Demonstrating that gauge theories can be solved on infinite lattices with full translation invariance, a feat previously difficult or impossible for generic gauge groups using tensor networks.
3. Abelian to Non-Abelian Generalization: Showing that the method works seamlessly for $U(1)$ (Schwinger model) and $SU(N_c)$ (Adjoint QCD$_2$) without needing distinct constraint formulations.
4. Implementation: Integration of LEMPOs into the Julia package MPSKit.jl, providing a practical tool for high-precision numerical studies.

4. Numerical Results

A. The Schwinger Model (U(1) Gauge Theory)

The authors benchmarked their method on the massive and massless Schwinger model.

• Precision: They achieved extremely high precision, matching exact analytical results for the massless case ($m=0$).
  • Chiral condensate: $\langle \bar{\psi}\psi \rangle / g = -0.159926(3)$ (vs. exact $-0.1599288...$).
  • Schwinger boson mass: $M_S/g = 0.564191(3)$ (vs. exact $0.5641896...$).
• Continuum Extrapolation: By comparing with strong-coupling expansions, they demonstrated that their method allows for precise extrapolation to the continuum limit ($a \to 0$) using Padé approximants and polynomial fits.
• Spectrum: They mapped the bound state spectrum as a function of fermion mass $m$ and theta-angle $\theta$. They observed the formation of a two-soliton continuum as $\theta \to \pi$, consistent with theoretical expectations.

B. Adjoint QCD$_2$ (SU(2) and SU(3) Gauge Theories)

The method was applied to Adjoint QCD$_2$ with $N_c=2$ and $N_c=3$, a theory with rich vacuum structure and supersymmetry at specific mass values.

• String Tension: They calculated the fundamental string tension $\sigma$.
  • Confirmed $\sigma = 0$ at $m=0$ (due to non-invertible symmetries/perimeter law).
  • Confirmed the linear rise of $\sigma$ with mass and agreement with weak-coupling expansions at large $m$.
• Supersymmetry: They identified the supersymmetric point at $m_{SUSY} = g\sqrt{N_c/2\pi}$.
  • In the trivial flux tube sector ($p=0$), they observed boson-fermion degeneracy at $m_{SUSY}$.
  • In non-trivial flux tube sectors ($p \neq 0$), they confirmed the gapless nature (Goldstino mode) due to spontaneous supersymmetry breaking.
• Phase Diagram: They mapped the phase diagrams for $N_c=2$ and $N_c=3$, identifying topological phases separated by gapless points and confirming the existence of first-order phase transitions in the $N_c=3$ case.
• Comparison: Their results significantly improved upon previous numerical studies (e.g., DLCQ, finite MPS) in terms of precision and the ability to access the continuum limit.

5. Significance and Future Directions

• Paradigm Shift: This work establishes that gauge fields do not need to be eliminated or manually truncated to be simulated efficiently with tensor networks. Instead, the gauge structure is naturally encoded in the virtual symmetries of the MPS.
• Scalability: The ability to use infinite lattices (uMPS) removes finite-size artifacts and allows for direct access to the thermodynamic limit, which is crucial for studying phase transitions and confinement.
• Future Applications:
  • Real-time Dynamics: The framework is compatible with time-evolution algorithms (TDVP), enabling the study of scattering and parton distribution functions.
  • Higher Dimensions: The authors propose a generalization to (2+1) dimensions using Projected Entangled Pair States (PEPS), though they note significant computational challenges regarding inner products and orthogonality of gauge states.
  • Complex Vacua: The method is poised to explore theories with complex vacuum structures (e.g., $SU(N_c)$ with $N_c > 3$) and non-invertible symmetries.

In summary, the paper presents a robust, general, and high-precision framework for solving (1+1)-dimensional gauge theories using infinite tensor networks, bridging the gap between condensed matter tensor network methods and high-energy physics lattice gauge theory.