Revisiting the $J_1$-$J_2$ Heisenberg Model on a Triangular Lattice: Quasi-Degenerate Ground States and Phase Competition

Using state-of-the-art matrix product state simulations, this study challenges the conventional view that the two nearly degenerate ground states of the spin-1/2 triangular-lattice $J_1$-$J_2$ Heisenberg model represent distinct topological sectors of a gapped $\mathbb{Z}_2$ spin liquid, by demonstrating that they exhibit significant differences in both static correlations and dynamical excitations.

The Gist

Imagine a crowded dance floor where everyone is trying to hold hands with their neighbors, but the room is shaped like a triangle. This is the Triangular Lattice Heisenberg Model.

In this quantum dance, every "dancer" (an electron spin) wants to hold hands with its neighbor in the opposite direction (one up, one down). But because the floor is a triangle, if Dancer A holds hands with Dancer B (who is "down"), and Dancer B holds hands with Dancer C (who must be "up"), Dancer C is now stuck. They can't hold hands with Dancer A without breaking the rule. This is called frustration.

For a long time, physicists thought that in the middle of this frustration, the dancers would give up on forming any pattern and just spin randomly in a chaotic, fluid state called a Quantum Spin Liquid (QSL). It was believed that this liquid state was a "topological" secret code, meaning it had two hidden versions (like a left-handed and right-handed version of the same liquid) that looked identical from the outside but were fundamentally different inside.

The Plot Twist: The "Twins" Aren't Identical

In this new paper, the researchers decided to take a closer look at these two "hidden versions" (called the Even and Odd sectors) using a super-powerful computer simulation (Matrix Product States). They expected to find two identical twins, just wearing different invisible hats.

Instead, they found that the twins are actually completely different people wearing different outfits.

Here is what they discovered, using some simple analogies:

1. The Static Snapshot (The "Group Photo")

Imagine taking a photo of the dance floor to see how the dancers are holding hands.

• The "Even" Twin: This group looks like a calm, fluid crowd. Their hand-holding patterns are spread out evenly, looking like a smooth, disordered liquid. This matches the theory of a U(1) Dirac Spin Liquid (a very specific, exotic type of quantum fluid).
• The "Odd" Twin: This group looks suspiciously like they are about to start a rigid line dance! Their hand-holding patterns are almost identical to the "120-degree order" phase (the phase where everyone is perfectly organized). They are much more uniform and isotropic (the same in all directions) than the Even twin.

The Analogy: It's like looking at two groups of people in a room. You expect them to be the same. But one group is sitting in a relaxed, chaotic circle (the Liquid), while the other is standing in a formation that looks exactly like a military parade (the Ordered Phase), even though they are supposed to be in the "liquid" zone.

2. The Sound Check (The "Dynamical Structure Factor")

Next, the researchers listened to the "music" of the system—how the spins vibrate and move when excited.

• The "Even" Twin: The music is distributed across the whole room. The energy is shared between different points in the dance floor, consistent with a fluid state.
• The "Odd" Twin: The music is concentrated in one specific spot (the "K" point), just like it is in the perfectly ordered military parade phase. It's as if the "liquid" group suddenly started humming the exact same tune as the "ordered" group.

The Big Conclusion

The old theory was: "We see two states that are almost the same energy. They must be the two topological secrets of a Quantum Spin Liquid."

The new paper says: "No, that's wrong. These two states are not just different 'secrets' of the same liquid. They are actually two different phases of matter fighting for dominance."

• One state (Even) might actually be the true Quantum Spin Liquid.
• The other state (Odd) looks like it's trying to be an ordered magnet but is stuck in a "near-miss" state because of the frustration.

Why Does This Matter?

Think of it like trying to find a hidden treasure map. For years, explorers thought the map had two identical paths leading to the same chest (the topological sectors). This paper reveals that one path leads to a treasure chest (the Spin Liquid), but the other path leads to a completely different destination (a magnetic order).

This changes how scientists understand these materials. It suggests that the "Quantum Spin Liquid" we thought we found might be hiding right next to a magnetic phase, and they are constantly competing to see which one wins. It also warns scientists that when they see two similar-looking states in a computer simulation, they can't just assume they are the same thing; they need to listen to the "music" and look at the "photos" to see if they are actually different species entirely.

In short: The researchers found that the "ghost twins" of the quantum world aren't ghosts at all—they are two very different neighbors arguing over who gets to live in the house.

Technical Summary

Here is a detailed technical summary of the paper "Revisiting the J1-J2 Heisenberg Model on a Triangular Lattice: Quasi-Degenerate Ground States and Phase Competition."

1. Problem Statement

The spin-1/2 antiferromagnetic Heisenberg model on a triangular lattice with nearest-neighbor ($J_1$) and next-nearest-neighbor ($J_2$) couplings is a paradigmatic system for studying frustrated quantum magnetism.

• Context: For small $J_2/J_1$, the system exhibits $120^\circ$magnetic order. For large$J_2/J_1$, it transitions to a stripe-ordered phase.
• The Gap: In the intermediate regime ($0.07 \lesssim J_2/J_1 \lesssim 0.15$), extensive numerical studies suggest a Quantum Spin Liquid (QSL) phase. However, the nature of this QSL is debated, with two main candidates: a gapped$\mathbb{Z}_2$spin liquid and a gapless$U(1)$ Dirac spin liquid.
• The Specific Puzzle: Density Matrix Renormalization Group (DMRG) studies on finite-width cylinders have consistently identified two nearly degenerate ground states (referred to as "even" and "odd" sectors) in this regime.
  • Standard Interpretation: These states were commonly interpreted as distinct topological sectors of a gapped $\mathbb{Z}_2$ spin liquid, which should be indistinguishable by local observables in the thermodynamic limit.
  • The Question: Do these states truly represent a single topological phase, or do they signify a more complex phase competition or a different physical origin?

2. Methodology

The authors employed state-of-the-art numerical techniques to investigate the static and dynamical properties of these two sectors at the representative coupling ratio $J_2/J_1 = 0.125$.

• Model & Geometry: The Hamiltonian was studied on a triangular lattice wrapped into a cylinder with periodic boundary conditions along the circumference ($L_y = 6$) and open boundaries along the length ($L_x = 36$), denoted as YC6-36.
• Simulation Technique:
  • Matrix Product States (MPS): Used for ground state optimization.
  • Symmetry: Exploited non-Abelian SU(2) symmetry using the QSpace tensor library to handle the high entanglement and frustration efficiently.
  • Algorithm: Single-site DMRG with controlled bond expansion. The simulations were run until the bond dimension reached $D^* = 3000$ multiplets, then compressed to $D^* = 1000$ for analysis.
  • State Identification: The authors observed a reproducible "energy drop" during the DMRG sweeping routine at $D^* \approx 2048$. The state before the drop is labeled $|\psi_{\text{even}}\rangle$, and the state after is $|\psi_{\text{odd}}\rangle$. These states are nearly orthogonal ($F \sim 10^{-4}$).
• Dynamical Calculations:
  • Method: The Tangent Space Krylov (TaSK) method was used to compute the Dynamical Structure Factor (DSF), $S(\mathbf{k}, \omega)$.
  • Advantage: Unlike time-evolution methods, TaSK projects the Hamiltonian onto the tangent space of the MPS, allowing for high-resolution spectral functions at low frequencies without the need for time-stepping.

3. Key Contributions

1. SU(2)-Symmetric MPS Simulations: The study maintains full SU(2) symmetry throughout the simulation, avoiding the symmetry breaking often required in flux-insertion methods to access the "odd" sector.
2. Direct Comparison of Sectors: The authors provide the first direct, high-resolution comparison of the static and dynamical properties of the even and odd sectors within the same framework.
3. Rejection of the $\mathbb{Z}_2$ Topological Sector Hypothesis: By demonstrating that the two states have distinct local order parameters and dynamical responses, the paper argues against the interpretation that they are merely topological sectors of a gapped $\mathbb{Z}_2$ spin liquid.

4. Key Results

A. Static Correlations

• Equal-Time Structure Factor (ETSF):
  • Even Sector: Shows broadened features at both $K$ and $M$ points in the Brillouin zone with comparable intensity. This profile is consistent with a gapless $U(1)$ Dirac spin liquid.
  • Odd Sector: Shows broadened features but with a distinct intensity profile. The maxima at $M$ points are less pronounced, and the weight is more concentrated near the $K$ point, resembling the proximate $120^\circ$ ordered phase.
• Nearest-Neighbor (NN) Spin Correlations:
  • Even Sector: Exhibits a "staggering" pattern in bond strengths (similar to the stripe phase but with smaller amplitude differences), indicating anisotropy.
  • Odd Sector: Shows isotropic bond correlations, closely matching the values found in the $120^\circ$ordered phase ($J_2=0$).
  • Conclusion: The two sectors are not locally indistinguishable, contradicting the expectation for topological sectors of a gapped phase.

B. Dynamical Structure Factor (DSF)

• Even Sector: The spectral weight is distributed relatively uniformly, peaking at both $K$ and $M$ points with comparable intensity. This supports the $U(1)$ Dirac QSL scenario.
• Odd Sector: The spectral weight is drastically different, concentrating almost entirely at the $K$ point. This distribution is reminiscent of the Goldstone modes found in the magnetically ordered $120^\circ$ phase, suggesting the odd sector may be a proximate ordered state rather than a distinct QSL.

5. Significance and Outlook

• Redefining the Phase Diagram: The results suggest that the "two-sector" structure observed in previous DMRG studies does not necessarily signal a gapped $\mathbb{Z}_2$ spin liquid. Instead, it likely indicates a competition between a $U(1)$ Dirac spin liquid (even sector) and a state proximate to the $120^\circ$ order (odd sector).
• Stability of the Odd Sector: The odd sector appears to be the stable ground state for the YC6 geometry (lower energy by 1.1%), raising questions about whether the true thermodynamic ground state is a QSL or a symmetry-broken state.
• Methodological Impact: The work highlights the importance of using SU(2)-symmetric MPS and high-resolution dynamical methods (TaSK) to distinguish between competing quantum phases that appear degenerate in energy but differ fundamentally in their local and dynamical properties.
• Future Directions: The authors call for systematic studies on wider cylinders and different geometries to determine if the odd sector remains the ground state in the thermodynamic limit and to clarify the true nature of the intermediate phase.

In summary, this paper challenges the prevailing view of the intermediate phase in the triangular lattice $J_1-J_2$ model, providing strong evidence that the observed quasi-degenerate states are physically distinct phases rather than topological sectors of a single gapped spin liquid.