⚛️ quantum physics

Criticality enabled long-range order in a U(1)-symmetric spin-1 Heisenberg chain with biquadratic interactions

This paper presents numerical evidence supporting Nahum's theoretical prediction that a U(1)-symmetric spin-1 Heisenberg chain with biquadratic interactions exhibits spontaneous long-range order at a critical point between two quasi-long-range-ordered phases, characterized by a magnetization exponent of $\beta\approx0.29$ and a dynamical critical exponent of $z\approx2$ .

Original authors: Natalia Chepiga, Fabian H. L. Essler

Published 2026-09-28

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CC BY 4.0

Original authors: Natalia Chepiga, Fabian H. L. Essler

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

In the quiet world of quantum physics, there is a long-standing rule about how matter behaves when it is squeezed into a single, thin line. For decades, scientists believed that if you took a chain of tiny magnetic atoms and cooled it down to its lowest possible energy state, the atoms could never agree on a single direction to point. Even if they wanted to, the frantic, jittery motion of quantum mechanics would prevent them from settling into a neat, ordered pattern. This idea, known as the Mermin-Wagner theorem, suggested that true order was impossible in one-dimensional systems unless the atoms were locked into place by a very specific, rigid symmetry. It was a comforting certainty: in a line, chaos always wins. However, a new theoretical proposal challenged this deep-seated belief. It suggested that under very specific conditions, right at the precise moment a system changes from one type of disorder to another, the atoms might suddenly snap into a perfect, long-range alignment. This would be a rare instance where the act of changing phases itself creates a stable, ordered state, defying the usual laws that keep one-dimensional chains in a state of flux.

A team of researchers at Oxford University set out to test this bold prediction using a powerful computer simulation. They did not build a physical chain of atoms in a lab; instead, they constructed a detailed digital model of a specific type of magnetic chain, known as a spin-1 chain, where each link has a specific set of magnetic properties. They programmed the simulation to explore how these chains behave when the strength of the interactions between the atoms is carefully adjusted. Their goal was to watch for the moment the system transitions between two different "liquid-like" states of magnetism. In one state, the atoms have no overall magnetic direction, while in the other, they possess a small, steady magnetic pull. The researchers were looking for the narrow boundary where these two states meet, hoping to see if the atoms would spontaneously align in a way that had never been observed before.

The simulation revealed that the theoretical prediction was correct. As the researchers tuned the interaction strength to the exact point where the two magnetic states meet, the atoms did indeed break their usual chaotic behavior. They spontaneously organized themselves into a state of true long-range order, pointing in a coordinated direction across the entire chain. This happened specifically in the plane perpendicular to the chain's axis, a phenomenon that had been predicted by a theory developed by a physicist named Adam Nahum. The researchers found that the strength of this new order grew in a very specific way as they approached the transition point. By measuring how the magnetization changed, they calculated a value of approximately 0.29, which matches almost perfectly with the theoretical prediction of 0.28. This close agreement suggests that the underlying theory describing this strange new state of matter is accurate.

Beyond simply observing the order, the team investigated the deeper mechanics of how this transition works. They looked at how the system responds to changes in size and how the energy of the atoms behaves as they approach the critical point. They found that a key property describing the stiffness of the magnetic chain, known as the Luttinger parameter, grew infinitely large as the system neared the transition. This divergence is exactly what the theory predicted would happen to allow the atoms to lock into a rigid, ordered pattern. Furthermore, they measured how fast disturbances travel through the chain near this critical point. They found that the speed of these disturbances corresponds to a value of roughly two, which is distinct from the value of one found in typical magnetic liquids. This difference confirms that the transition is governed by a unique set of physical rules, different from the standard behaviors seen in other quantum systems.

The study also uncovered a curious side effect that occurs just before the system reaches this ordered state. In the region where the atoms have a small magnetic pull but have not yet fully aligned, the system tends to separate into distinct domains. Imagine a crowd of people where some are leaning slightly left and others slightly right; instead of mixing evenly, they might cluster into groups of left-leaning and right-leaning individuals. The researchers saw this same "phase separation" in their simulation, where regions of different magnetic strengths formed distinct blocks within the chain. These blocks are separated by sharp boundaries, and the energy required to create these boundaries is finite, meaning they are stable features of the system rather than fleeting fluctuations. This behavior helps explain why the system is so sensitive to the transition point and provides a clearer picture of the complex landscape the atoms navigate before settling into their final, ordered state.

The researchers were careful to rule out other possibilities that might have explained their results. They checked to ensure that the order they saw was not an artifact of the finite size of their simulation or a result of the specific boundaries they used. By running the simulation with different chain lengths and different boundary conditions, they confirmed that the long-range order was a genuine property of the infinite system, not a result of the computer model. They also compared their findings against previous numerical studies that had suggested a different value for the speed of disturbances in the system. Their results clearly disagreed with those earlier estimates, showing that the speed is closer to two rather than one and a half. This discrepancy highlights the importance of their more precise simulation techniques and suggests that the earlier understanding of this specific transition was incomplete.

Ultimately, this work provides strong evidence that the strange, counter-intuitive behavior predicted by theory can actually occur in nature. It shows that the rigid rules forbidding order in one-dimensional systems have a loophole: at the precise moment of a phase transition, the system can spontaneously generate a stable, long-range pattern. The researchers did not just observe this phenomenon; they measured its characteristics with high precision, confirming that the mathematical description of the transition is correct. While this is a fundamental discovery about the nature of matter and does not immediately lead to a new technology, it reshapes our understanding of what is possible in the quantum world. It proves that even in the most constrained environments, where chaos usually reigns, there are moments of perfect harmony waiting to be found if one knows exactly where to look. The study stands as a testament to the power of combining advanced theory with rigorous numerical simulation to explore the frontiers of physics, turning a theoretical curiosity into a confirmed reality.

Technical Summary: Criticality Enabled Long-Range Order in a U(1)-Symmetric Spin-1 Heisenberg Chain

Problem Statement
Standard theoretical consensus, rooted in the Mermin-Wagner-Hohenberg theorem, posits that continuous symmetries cannot be spontaneously broken in the ground states of one-dimensional quantum spin chains with local interactions, except when the order parameter commutes with the Hamiltonian (e.g., the spin-1/2 ferromagnet). However, a recent renormalization-group (RG) analysis by Nahum [13] challenged this view. Nahum proposed that true long-range order (LRO) in the $x$ - $y$ plane of a $U(1)$ -symmetric spin chain can emerge spontaneously at a zero-temperature phase transition separating two distinct quasi-long-range-ordered (QLRO) Luttinger liquid phases. This "Nahum transition" is predicted to occur in a field theory describing a perturbed $SO(3)$ ferromagnet, where the dynamical critical exponent is $z \approx 2$ and the magnetization exponent is $\beta \approx 0.28$ . The central problem addressed by this paper is to verify these field-theoretic predictions through extensive numerical simulations of a microscopic lattice model.

Methodology
The authors investigate the anisotropic spin-1 Heisenberg ferromagnet with $U(1)$ symmetry, defined by the Hamiltonian:
$H = -J \sum_{j} \mathbf{S}_j \cdot \mathbf{S}_{j+1} + g_1 \sum_{j} (S^z_j)^2 + g_2 \sum_{j} (S^z_j)^2(S^z_{j+1})^2$
where $g_1$ represents single-ion anisotropy and $g_2$ is a biquadratic coupling. The study employs the Density Matrix Renormalization Group (DMRG) method using variational Matrix Product States (MPS). Simulations are performed with both Open Boundary Conditions (OBC) and Periodic Boundary Conditions (PBC) on chains up to $N=1500$ sites (OBC) and $N=240$ sites (PBC).

Key numerical techniques include:

Phase Diagram Mapping: Systematically varying $g_1$ at fixed $g_2$ to locate phase boundaries between the saturated ferromagnet, the magnetized Luttinger liquid, and the unmagnetized Luttinger liquid.
Entanglement Analysis: Calculating bipartite entanglement entropy (BEE) to extract the central charge $c$ and analyzing the entanglement spectrum to determine the Luttinger parameter $K$ .
Correlation Functions: Computing spin-spin correlation functions ( $C^{zz}$ and $C^{xy}$ ) to test finite-size scaling against Luttinger liquid predictions.
Excitation Spectra: Targeting multiple low-lying energy states to determine the dynamical critical exponent $z$ and the scaling of energy gaps near the transition.

Key Contributions and Results

Verification of the Phase Diagram:
The numerical results confirm the existence of three distinct phases predicted by field theory:
- A saturated ferromagnetic phase ( $g_1 < g_{c1}$ ).
- A magnetized Luttinger liquid phase ( $g_{c1} < g_1 < g_{c2}$ ).
- An unmagnetized Luttinger liquid phase ( $g_1 > g_{c2}$ ).
  The transition between the two Luttinger liquids corresponds to the Nahum transition.
Magnetization Exponent ( $\beta$ ):
In the magnetized Luttinger liquid phase, the ground state magnetization per site vanishes as the system approaches the Nahum transition ( $g_1 \to g_{c2}$ ). The authors find a critical exponent $\beta \approx 0.29$ (for $g_2=1$ ), which is consistent with the $\epsilon$ -expansion prediction of $\beta \approx 0.28$ . The deviation from the theoretical value is attributed to proximity to the $SU(2)$ -invariant point ( $g_1=g_2=0$ ) where scaling corrections are significant.
Divergence of the Luttinger Parameter ( $K$ ):
As the system approaches the transition from either Luttinger liquid side, the Luttinger parameter $K$ diverges. The authors establish the scaling relation $K \propto |g_1 - g_c|^{\nu(1-z)}$ . The numerical data supports this divergence with an exponent consistent with the field theory prediction $\nu(1-z) \approx -0.55$ . This divergence implies that the transverse correlation exponent ( $1/2K$ ) vanishes at the transition, signaling the onset of true long-range order in the $x$ - $y$ plane.
Dynamical Critical Exponent ( $z$ ):
By analyzing the finite-size scaling of low-lying excitation energies, the authors determine the dynamical critical exponent. The results yield $z \approx 2$ at the transition. This finding supports Nahum's field theory prediction ( $z \approx 1.992$ ) and contradicts a previous numerical estimate of $z \approx 1.51$ based on dynamical structure factors.
Phase-Separated States:
In the magnetized Luttinger liquid phase, the authors identify low-lying excited states that exhibit phase separation. These states consist of ferromagnetic domains separated by domain walls (DWs). Unlike the ground state, these states possess a finite energy gap in the thermodynamic limit. The authors suggest these states can be described by an inhomogeneous Luttinger liquid with position-dependent velocity and Luttinger parameter.

Significance
The paper provides strong numerical evidence supporting the existence of a criticality-enabled long-range order transition in a 1D quantum spin system with short-range interactions. The results validate Nahum's field-theoretic framework, confirming that:

True LRO can emerge at a phase transition between two QLRO phases without the order parameter commuting with the Hamiltonian.
The critical exponents ( $\beta$ , $z$ , $\nu$ ) derived from the $\epsilon$ -expansion accurately describe the microscopic lattice model.
The dynamical exponent $z \approx 2$ distinguishes this transition from standard Luttinger liquid transitions ( $z=1$ ).

The work resolves a theoretical controversy regarding the possibility of spontaneous symmetry breaking in 1D systems and characterizes the unique critical behavior of the Nahum transition, including the specific nature of the divergence in the Luttinger parameter and the existence of phase-separated excitations.

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