🔬 condensed matter

Emergence of a monopole phase in the $J_1{-}J_2$ Heisenberg model on the triangular lattice for small magnetic fields

Using variational Monte Carlo and field-theory approaches, this study reveals that a condensate of gapless monopoles emerges as a stable phase with finite scalar chirality in the $J_1{-}J_2$ Heisenberg model on a triangular lattice under small magnetic fields, particularly around the $J_2/J_1=1/8$ ratio where a spin-liquid phase was previously suspected.

Original authors: Sasank Budaraju, Shi Feng, Josef Willsher, Johannes Knolle, Frank Pollmann, Federico Becca

Published 2026-07-17

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CC BY 4.0

Original authors: Sasank Budaraju, Shi Feng, Josef Willsher, Johannes Knolle, Frank Pollmann, Federico Becca

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where tiny magnets, called spins, live on a flat, triangular playground. In most materials, these spins are good neighbors; they like to line up neatly, either all pointing the same way or in a perfect, repeating pattern. But on a triangular grid, things get messy. If one spin points up, its two neighbors want to point down to be "happy," but they can't both do it at the same time because they are also neighbors with each other. This is called "frustration," and it's like trying to settle a three-way argument where no one can compromise. When this frustration is strong enough, the spins refuse to pick a single pattern at all. Instead, they dance in a chaotic, fluid state known as a "quantum spin liquid." It's a mysterious state of matter where the spins never freeze, even at absolute zero, and they behave as if they are connected by invisible, magical threads. Scientists are obsessed with finding these liquids because they might hold the secrets to super-fast computers and new kinds of physics.

Now, imagine you are a scientist trying to figure out what happens to this chaotic dance when you turn on a strong magnetic field, like a giant magnet hovering over the playground. Does the field force the spins to line up and stop dancing? Or does the liquid find a new, weird way to flow? This is the question tackled in a new study by Sasank Budaraju and their team. They used powerful computer simulations to watch how these frustrated spins behave on a triangular lattice when they are pushed by a magnetic field. They were particularly interested in a specific "sweet spot" where the spins are most frustrated, hoping to see if a strange, invisible particle called a "monopole" could form a stable phase. Think of a monopole not as a magnet with just one pole (which doesn't exist in normal life), but as a tiny, swirling knot of magnetic energy that can pop in and out of existence in this quantum liquid.

The researchers built a detailed map, or "phase diagram," showing which state the spins prefer depending on how strong the magnetic field is and how much frustration exists between them. They compared several different "costumes" the spins could wear: a "Y" shape where three spins point in different directions, a "stripe" pattern, and the exotic "monopole" phase. Their simulations revealed that for a specific range of frustration (where the ratio of interactions is between 0.1 and 0.16) and for small magnetic fields, the spins don't just line up. Instead, they settle into a fascinating "monopole phase." In this state, the spins develop a finite amount of "scalar chirality," which is a fancy way of saying they twist around each other in a specific, handed way, like a corkscrew. Crucially, the team found that while these spins are twisting, they are not forming a rigid, ordered pattern across the whole material.

This is a big deal because some earlier theories suggested that if monopoles formed, they would force the spins to line up in a neat, predictable pattern, breaking the liquid's freedom. However, this paper suggests the opposite: the monopole phase can exist as a stable, gapless state where the spins remain fluid and disordered, even while twisting. The team's computer models show that this monopole phase is the most stable option in that specific region of the map, beating out the more traditional "Y" and "stripe" patterns. They confirmed this by checking the "static structure factor," a mathematical tool that acts like a camera to see if there is a repeating pattern. For the monopole phase, the camera showed no sharp, repeating spots, meaning there is no long-range magnetic order, just a swirling, chaotic dance that is stable under the magnetic field.

The study also looked at how this phase transitions into others. When the magnetic field gets stronger, the monopole phase doesn't fade away gently; it hits a wall and suddenly jumps to a different state, like the "Y" phase, in what is called a "first-order transition." The researchers also checked their work against other advanced simulation methods and found their results matched up well, giving them confidence that their "monopole" picture is a solid description of what's happening. While they admit their simulation was done on a finite-sized grid (like a small patch of the playground) and not the infinite real world, the trends they saw suggest that this monopole phase is a real, robust possibility for quantum materials. It represents a natural evolution of the quantum spin liquid when you turn on a magnetic field, offering a new chapter in our understanding of how these frustrated magnets behave when pushed to their limits.

Technical Summary: Emergence of a Monopole Phase in the $J_1-J_2$ Heisenberg Model

Problem Statement
The paper investigates the ground-state phase diagram of the $J_1-J_2$ Heisenberg model on a triangular lattice under an external Zeeman field ( $H$ ). While the zero-field regime near $J_2/J_1 = 1/8$ is widely debated as a candidate for a $U(1)$ Dirac quantum spin liquid (QSL), the fate of this state under finite magnetic fields remains unclear. Previous theoretical works, including linear spin-wave theory and density-matrix renormalization group (DMRG) simulations, have suggested the existence of exotic phases below the $m=1/3$ magnetization plateau that are not captured by semi-classical approximations. Specifically, there is a need to determine whether the Dirac spin liquid remains stable, transitions to conventional magnetic orders (such as the "Y" phase or "up-up-down" phase), or evolves into a distinct state involving monopole condensation.

Methodology
The authors employ the Variational Monte Carlo (VMC) approach using Gutzwiller-projected fermionic partons. The spin operators are rewritten in terms of fermionic partons, and a Gutzwiller projector ( $P_G$ ) enforces the single-occupancy constraint, while a magnetization projector ( $P_m$ ) fixes the total $S^z$ .

The study constructs and compares variational wave functions for several competing phases:

Y Phase: A semi-classical state with three-sublattice order, characterized by non-uniform real hopping amplitudes and site-dependent fictitious magnetic fields.
Umbrella Phase: A state with translationally invariant hopping and a rotating fictitious field.
Canted-Stripe Phase: A two-sublattice state with modulated real hopping and a specific rotating fictitious field.
Monopole Phase: Constructed by adding a finite density of magnetic fluxes to the underlying Dirac spin liquid ansatz. This state breaks lattice translational symmetry, introduces complex hopping phases, and generates Landau levels in the fermionic spectrum. Crucially, for a given magnetization $m$ , this ansatz has no free variational parameters.

The authors optimize the variational parameters for the ordered states to minimize energy on an $18 \times 18$ cluster and compare these energies against the parameter-free monopole states. Additionally, a field-theory approach incorporating a self-consistent random-phase approximation (RPA) of low-lying spin fluctuations is used to corroborate the numerical findings.

Key Results

Phase Diagram: The authors map the phase diagram for $0 \le J_2/J_1 \le 0.2$ $0 \leq J_{2} / J_{1} \leq 0.2$ and $0 \le H/J_1 \le 2$ $0 \leq H / J_{1} \leq 2$ .
- For $J_2/J_1 \lesssim 1/6$ , a continuous transition is observed from a gapless "Y" phase to a gapped "up-up-down" (UUD) phase, consistent with spin-wave theory.
- Monopole Phase Stability: In the highly frustrated regime around $J_2/J_1 = 1/8$ , a "monopole phase" is stabilized over a significant region of small magnetic fields. This phase exhibits lower variational energy than the semi-classical Y and canted-stripe orders in the range $0.1 \le J_2/J_1 \le 0.16$ .
- Transition Nature: The transition from the monopole phase to the Y phase is found to be first-order.
Scalar Chirality: The monopole phase possesses a finite scalar spin chirality ( $\chi \neq 0$ ), whereas the Y phase is coplanar and has zero chirality.
Absence of Transverse Order: Analysis of the static structure factor $S_{xy}(q)$ reveals that while the Y and canted-stripe phases exhibit sharp Bragg peaks indicating long-range order, the monopole phase does not. The in-plane structure factor $S_{xy}(K)/N$ scales to zero in the thermodynamic limit, suggesting the absence of transverse magnetic order despite the finite monopole density.
Scaling: The monopole phase shows a specific scaling behavior for the structure factor relative to magnetization and system size, described by an empirical law with exponent $\sigma \approx 1.42$ .

Significance and Claims
The paper claims that the monopole phase represents the natural evolution of the $U(1)$ Dirac spin liquid when an external magnetic field is applied. The primary significance of the work lies in:

Stabilization of Monopoles: Demonstrating that a condensate of monopoles (gapless gauge excitations of the zero-field spin liquid) can be stabilized in a finite region of the phase diagram, rather than immediately triggering confinement or conventional magnetic order.
Benchmarking: Providing a critical benchmark for future studies, showing that simple variational ansätze can capture the essential physics of the model, achieving energies comparable to DMRG with significantly fewer parameters.
Agreement with Theory: The results align well with recent spin-wave theory and DMRG calculations regarding the extent of phases and the nature of transitions, supporting the existence of an exotic phase below the $m=1/3$ plateau.
Theoretical Validation: The stability of the monopole phase is confirmed by a complementary field-theory approach, and the boundary between the monopole and Y phases is obtained without free parameters.

The authors note that while the phase diagram is derived from finite-cluster calculations without size-scaling analysis (leading to an underestimation of the zero-field spin-liquid region compared to previous work), the emergence of the monopole phase as the optimal variational state in the frustrated regime is a robust finding.

Technical Summary: Emergence of a Monopole Phase in the J1−J2J_1-J_2J1​−J2​ Heisenberg Model

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Technical Summary: Emergence of a Monopole Phase in the $J_1-J_2$ Heisenberg Model