Gauge-covariant projected entangled paired states for interacting systems in a magnetic field

This paper introduces a gauge-covariant projected entangled-pair state (PEPS) framework that allows for the simulation of interacting particles in a magnetic field using translation-invariant tensor networks, effectively bypassing the need for gauge choices or enlarged magnetic unit cells.

The Gist

The Problem: The "Magnetic Maze"

Imagine you are trying to map out a massive, crowded dance floor (this is our quantum system). On this dance floor, there are thousands of dancers (our particles). Now, imagine that a giant, invisible magnet is placed under the floor. This magnet doesn't pull the dancers physically, but it forces them to move in swirling, circular patterns.

In physics, when we try to simulate this on a computer, we run into a massive headache called the "Gauge Problem."

To describe the magnetic field, scientists have to pick a "map" (called a gauge). But there isn't just one map. You could use a map that says "the wind blows North," or a map that says "the wind blows East but everything is tilted." Mathematically, both maps describe the same wind, but they look completely different on paper.

Because our computer simulations are very picky, if you change the map, the math becomes a nightmare. Usually, to make the math work, you have to create a "super-sized map" (a magnetic unit cell) that is much larger than the actual dance floor. This makes the computer work ten times harder, often to the point where it crashes or runs out of memory.

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The Solution: The "Universal Translator"

The researchers in this paper have invented a new way to simulate these dancers using a tool called PEPS (Projected Entangled-Pair States).

Think of PEPS as a high-tech, digital "pattern book." Instead of trying to draw every single dancer's exact position on a giant, complicated map, the researchers realized they could focus on the relationship between the dancers.

Here is their clever trick: The Virtual Flux.

Instead of changing the "map" (the gauge) every time they want to move a dancer, they added a tiny, invisible "adjustment knob" to the connections between the dancers.

The Analogy: The Synchronized Swimmers
Imagine a team of synchronized swimmers in a pool. If you want to describe their movement, you could try to track every single drop of water (the old, hard way). Or, you could just give each swimmer a tiny, invisible metronome. Even if the pool's current changes, as long as every swimmer's metronome adjusts slightly to match their neighbor, the pattern of the dance stays perfect.

In this paper, the "metronomes" are the virtual flux tensors. They allow the simulation to:

1. Ignore the Map: It doesn't matter which "gauge" (map) you pick. The "metronomes" automatically adjust the math so the result is always the same.
2. Stay Small: Because the pattern is consistent, you don't need a "super-sized map." You can use a tiny, simple map (a single unit cell) and the math still works perfectly.
3. Continuous Tuning: You can turn the magnetic field up or down smoothly, like a volume knob, rather than having to jump in huge, clunky steps.

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Why does this matter?

In the world of quantum computing and new materials, we are looking for "magic" states of matter—things like Fractional Chern Insulators (which are like super-stable, high-tech versions of the Hall Effect used in electronics).

Until now, simulating these was like trying to solve a Rubik's Cube where the colors keep changing depending on how you hold it. This paper provides a way to hold the cube so that the colors stay still, allowing scientists to finally see the patterns they are looking for.

In short: They found a way to simulate complex, swirling quantum particles without getting lost in the "mathematical fog" created by magnetic fields.

Technical Summary

Technical Summary: Gauge-Covariant Projected Entangled Pair States (PEPS)

1. Problem Statement

The study of itinerant particles (bosons or fermions) on a 2D lattice in a uniform magnetic field is a central problem in condensed matter physics, exemplified by the Harper-Hofstadter-Hubbard model. A fundamental difficulty arises from the gauge dependence of the vector potential $\mathbf{A}(\mathbf{r})$.

In a lattice setting, the magnetic field is typically implemented via the Peierls substitution, where hopping amplitudes acquire a phase $\phi_{ij}$. While the physical magnetic flux per plaquette $\phi$ is a gauge-invariant quantity, any specific choice of gauge (e.g., the Landau gauge) breaks the discrete translational symmetry of the Hamiltonian. To restore symmetry, one must define a magnetic unit cell, which, for a rational flux $\phi = 2\pi p/q$, must encompass $q$ lattice sites. This requirement imposes severe constraints on numerical methods:

• Exact Diagonalization (ED): Limited by system sizes that must be multiples of the magnetic unit cell.
• Quantum Monte Carlo (QMC): Suffers from the "sign problem" due to complex hopping phases.
• Standard Tensor Networks (MPS/PEPS): Traditional translation-invariant PEPS require large, enlarged unit cells to accommodate the magnetic flux, making simulations computationally expensive and restricting the values of $\phi$ to specific rational fractions.

2. Methodology

The authors propose a novel gauge-covariant PEPS ansatz that separates the physical variational parameters from the gauge choice.

A. The Ansatz:
Instead of using a single translation-invariant tensor $A$ for the entire lattice, the authors introduce a pattern of virtual flux tensors into the PEPS structure.

• The local PEPS tensors $A$ are kept translation-invariant and possess a local $U(1)$ charge conservation symmetry.
• The magnetic field is implemented by imprinting a pattern of $U(1)$ group actions on the virtual bonds of the PEPS.
• For a given gauge (e.g., Landau gauge), the virtual bonds are assigned phases such that the cumulative phase around any plaquette equals the physical flux $\phi$.

B. Symmetry and Gauge Covariance:
The authors prove that this ansatz is invariant under the magnetic translation group. While the wavefunction itself is not translation-invariant (due to the flux tensors), all physical expectation values of local observables remain translation-invariant. Furthermore, they demonstrate that a gauge transformation on the physical sites can be "dragged" to the virtual level, effectively reconfiguring the pattern of virtual flux tensors without changing the underlying physical state.

C. Contraction Algorithm:
Crucially, the authors show that the breaking of translational symmetry by the flux tensors does not necessitate a large-unit-cell contraction algorithm. By leveraging the $U(1)$ symmetry of the tensors, they demonstrate that the row-to-row transfer matrix can be contracted using a single-site boundary MPS algorithm. The flux $\phi$ enters the contraction only as a continuous numerical parameter, allowing for efficient optimization.

3. Key Contributions

• Gauge-Independent Framework: Developed a PEPS representation where the magnetic flux $\phi$ is a continuous parameter, bypassing the need for commensurate magnetic unit cells.
• Symmetry-Preserving Optimization: Introduced a variational algorithm that optimizes a single PEPS tensor while explicitly respecting the magnetic translation symmetry.
• Theoretical Proof of Covariance: Provided a rigorous mapping showing how gauge transformations in the physical space correspond to reconfigurations of the virtual flux pattern.
• Perturbative Justification: Showed that the ansatz is naturally motivated by second-order perturbation theory in the strong-interaction limit ($U \gg t$).

4. Results

The authors benchmarked the method using the bosonic Harper-Hofstadter-Hubbard model at unit filling and strong interaction ($U/t = 20$).

• Continuous Flux Tuning: They successfully simulated the ground state energy as a function of $\phi$ continuously from $0$ to $\pi$.
• Validation: The PEPS results showed excellent agreement with Matrix Product State (MPS) simulations performed on infinite cylinders (where $\phi$ is restricted to discrete values).
• Phase Stability: The method accurately captured the stabilization of the Mott insulating phase by the magnetic field.

5. Significance

This work represents a significant advancement in the simulation of topological and frustrated quantum many-body systems. By decoupling the gauge choice from the variational manifold, the method:

1. Enables High-Precision Simulations: Allows for the study of any flux value, not just those compatible with small unit cells.
2. Scales Efficiently: Reduces the complexity of 2D tensor network contractions in magnetic fields to a single-site problem.
3. Opens Paths to New Physics: Provides a foundation for studying Fractional Chern Insulators (FCI). The authors suggest that by fractionalizing the virtual charges on the auxiliary bonds, this PEPS framework can be used to investigate chiral topological order and the long-standing debate regarding the tensor network representation of such states.