🔬 condensed matter

Unconventional entanglement scaling and quantum criticality in the long-range spin-one Heisenberg chain with single-ion anisotropy

This study employs matrix-product state calculations and high-order series expansions to map the ground-state phase diagram of a long-range spin-one Heisenberg chain with single-ion anisotropy, revealing unconventional quantum criticality with continuously varying exponents and logarithmic entanglement corrections in the competition between Haldane, U(1), and SU(2) symmetry-breaking phases.

Original authors: Patrick Adelhardt, Sean R. Muleady, Kai P. Schmidt, Alexey V. Gorshkov

Published 2026-04-15

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CC BY 4.0

Original authors: Patrick Adelhardt, Sean R. Muleady, Kai P. Schmidt, Alexey V. Gorshkov

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 long line of tiny, spinning tops (atoms) arranged in a row. In the world of quantum physics, these tops don't just spin; they talk to each other. Usually, they only whisper to their immediate neighbors. But in this paper, the scientists are asking: What happens if these tops can shout across the entire line to talk to friends far away?

This paper explores a specific type of quantum chain where the "spins" (the tops) have three possible states (like a traffic light: Red, Yellow, Green) and can talk to each other over long distances, but with a twist: the strength of their shout gets weaker the farther apart they are, following a specific mathematical rule.

Here is the story of their discovery, broken down into simple concepts:

1. The Three Main Characters (The Phases)

The scientists found that depending on how much the tops "want" to be in a specific state (controlled by a knob called Anisotropy) and how far they can shout (controlled by a knob called Decay), the line of tops organizes itself into three distinct "neighborhoods":

The "Large-D" Neighborhood (The Disordered Chaos): Imagine the tops are so obsessed with being in the middle state (Yellow) that they ignore their neighbors entirely. They are all just sitting there, doing their own thing. There is no order, no pattern. It's a quiet, boring crowd.
The "Haldane" Neighborhood (The Secret Club): This is the most famous part of the story. Even though the tops look disordered, they are actually part of a secret, topological club. They are "entangled" in a way that creates invisible, protected edge effects. Think of it like a line of people holding hands in a circle, but the ends of the line are holding hands with invisible ghosts. This is a "Symmetry Protected Topological" phase—a fancy way of saying it's a special, robust state that can't be easily broken without tearing the whole system apart.
The "Symmetry Breaking" Neighborhood (The Organized March): When the tops start shouting to each other strongly over long distances, they suddenly snap into a perfect, alternating pattern (Red-Yellow-Red-Yellow). They have broken their "freedom" to choose any state and have marched into a rigid order. This is called Continuous Symmetry Breaking.

2. The Big Surprise: The "Unconventional" Criticality

Usually, when a system changes from one neighborhood to another (a "phase transition"), it behaves in a predictable, universal way. It's like water freezing into ice; the rules are the same whether you freeze a cup or a lake.

But this paper found something weird.

When the tops transition from the "Disordered Chaos" or the "Secret Club" into the "Organized March," the rules of the game change depending on how far the tops can shout.

The Analogy: Imagine a crowd of people trying to decide whether to stand up or sit down.
- In a normal crowd (short-range interactions), they only talk to the person next to them. The decision spreads like a wave.
- In this experiment, people can shout to anyone in the stadium.
- The Discovery: The scientists found that the "critical point" (the exact moment the crowd decides to stand up) doesn't have fixed rules. Instead, the "rules of the game" (called critical exponents) slide and change continuously depending on how far the shout reaches. It's as if the crowd's behavior changes its personality based on the size of the stadium.

3. The "Edge" Effect (Entanglement)

The scientists also looked at Entanglement, which is a measure of how much the tops are "connected" to each other.

In normal chains, this connection follows a simple rule (like a straight line).
In this long-range chain, they found logarithmic corrections.
The Analogy: Imagine measuring the length of a rope. Usually, if you double the rope, you double the length. But here, because the tops are shouting across the whole line, the "length" of the connection grows in a weird, curved way. It's like the rope is stretching and twisting in a way that defies simple geometry.

4. The "Boundary" Problem

One of the most practical findings is about how you look at the system.

If you look at the chain with "Open Ends" (like a line of people with no one holding the ends), the math looks one way.
If you look at it with "Periodic Ends" (like a circle where the last person holds the first person's hand), the math looks completely different.
The Lesson: The scientists realized that for these long-range systems, the way you set up the experiment (the boundary conditions) fundamentally changes the physics you observe. It's like studying a fish in a tank; if the tank is square, the fish swims one way; if it's round, the fish swims another. You can't just ignore the shape of the tank.

Why Does This Matter?

This isn't just abstract math. The paper suggests that we can build these "shouting top" chains in real life using trapped ions (atoms held by lasers) or Rydberg atoms (super-excited atoms).

The Playground: This model is a "minimal playground." It's the simplest setup to study how long-range connections (like the internet connecting people globally) interact with topology (hidden, robust structures) and symmetry (order vs. chaos).
The Future: By understanding these "unconventional" transitions, we might be able to design better quantum computers or sensors that are robust against noise, or simply understand how the universe organizes itself when things are connected over vast distances.

In a nutshell: The paper shows that when quantum particles can talk to each other from far away, the rules of how they change from chaos to order become fluid and unpredictable, changing their behavior based on the "distance" of their conversation and the "shape" of their container. It's a new, weird, and exciting chapter in the story of quantum matter.

1. Problem Statement

The paper investigates the ground-state properties and quantum criticality of a one-dimensional spin-one Heisenberg chain subject to single-ion anisotropy ( $D$ ) and staggered (unfrustrated) long-range antiferromagnetic interactions decaying as a power law ( $J(r) \propto (-1)^r r^{-\alpha}$ ).

Key challenges addressed include:

Haldane Phase Stability: How the celebrated Haldane phase (a symmetry-protected topological phase) competes with continuous symmetry breaking (CSB) phases induced by long-range interactions.
Hohenberg-Mermin-Wagner (HMW) Theorem: While the HMW theorem forbids continuous symmetry breaking in 1D short-range systems, long-range interactions can circumvent this, stabilizing ordered phases. The nature of the transitions between these ordered phases and the Haldane phase is poorly understood.
Unconventional Criticality: Understanding how the decay exponent $\alpha$ modifies critical exponents and entanglement scaling, particularly in regimes where interactions are strong enough to alter the universality class.

2. Methodology

The authors employ a multi-faceted numerical approach to ensure high precision and cross-validation:

Matrix Product States (MPS) & DMRG:
- Used to calculate ground states for finite system sizes up to $L = 340$ .
- Long-range handling: Power-law interactions are approximated by a sum of exponentials ( $\tilde{f}(r) = \sum a_l b_l^{-r}$ ) to fit the MPS formalism, with errors kept below $10^{-9}$ .
- Symmetry sectors: Calculations are performed in specific total spin sectors ( $m_z = 0$ and $m_z = 1$ ) to disentangle bulk properties from edge-state effects (crucial for the Haldane phase).
- Observables: Spin-spin correlations, string order parameters, staggered magnetization ( $M_z, M_\perp$ ), and von Neumann entanglement entropy ( $S_{VN}$ ).
Perturbative Continuous Unitary Transformations + Monte Carlo (pCUT+MC):
- Used to complement MPS results, particularly near the large- $D$ limit.
- Expands from the non-interacting limit ( $D \to \infty$ ) using the perturbation parameter $\lambda = 1/D$ .
- Calculates elementary excitation gaps and spectral weights to high orders (up to 10th order) to determine critical lines in the thermodynamic limit.
Finite-Size Scaling (FSS) & Data Collapse:
- Critical points and exponents are extracted by collapsing magnetization curves and entanglement entropy data for various system sizes.
- Boundary Condition Analysis: A novel aspect of this work is the explicit comparison between Open Boundary Conditions (OBC, used in DMRG) and Periodic Boundary Conditions (PBC, used in Exact Diagonalization for small systems) to understand scaling above the upper critical dimension.

3. Key Contributions and Results

A. Ground-State Phase Diagram

The authors map out a rich phase diagram as a function of anisotropy $D$ and decay exponent $\alpha$ :

Large-D Phase: A trivial disordered phase for large positive $D$ .
Haldane Phase: A symmetry-protected topological (SPT) phase for small $D$ and short-range interactions (large $\alpha$ ).
$Z_2$ AF Phase: Staggered antiferromagnetic order for negative $D$ .
Continuous Symmetry Breaking (CSB) Phases:
- SU(2) CSB: Occurs at $D=0$ for sufficiently small $\alpha$ (strong long-range interactions), breaking full spin rotation symmetry.
- U(1) CSB: Occurs for $D > 0$ and small $\alpha$ , breaking rotational symmetry around the $z$ -axis.

B. Unconventional Entanglement Scaling

In the CSB phases, the entanglement entropy $S_{VN}$ exhibits a logarithmic correction to the area law: $S_{VN} \sim b \ln L + c$ .

Coefficient $b$ : Unlike short-range models where $b = N_G/2$ (with $N_G$ being the number of Goldstone modes), the coefficient $b$ varies continuously with the decay exponent $\alpha$ .
Limiting Cases:
- As $\alpha \to 1$ (short-range limit), $b$ approaches the standard values ( $b \approx 0.47$ for U(1) and $b \approx 0.92$ for SU(2)).
- As $\alpha \to 0$ (all-to-all limit), $b$ approaches values predicted by collective spin models ( $b=1/2$ for U(1) and $b=1$ for SU(2)).
Significance: This confirms that long-range interactions modify the low-energy dispersion relation (sublinear dispersion), leading to non-universal, $\alpha$ -dependent entanglement scaling.

C. Critical Exponents and Universality

The paper identifies several distinct types of quantum phase transitions:

Gaussian Transition (Haldane $\leftrightarrow$ Large-D):
- A topological transition with central charge $c_{eff} = 1$ .
- The critical point is determined accurately by the crossing of entanglement entropy in different $m_z$ sectors. Long-range interactions do not alter the central charge.
Ising Transition (Haldane $\leftrightarrow$ $Z_2$ AF):
- Exhibits standard 2D Ising criticality ( $\nu \approx 1, \beta \approx 0.25$ ) for large $\alpha$ .
- Deviations occur as the transition line approaches the SU(2) CSB point at $D=0$ .
CSB Transitions (Haldane/Large-D $\leftrightarrow$ U(1)/SU(2) CSB):
- Continuously Varying Exponents: The critical exponents $\nu$ and $\beta$ vary continuously as a function of the effective decay parameter $\sigma = \alpha - d$ (where $d=1$ ).
- Large-D to U(1) CSB: The exponents follow the behavior of the long-range $O(2)$ $O (2)$ quantum rotor model.
  - For $\sigma \leq \sigma_{uc} \approx 2/3$ (above upper critical dimension), the system shows mean-field behavior.
  - For $\sigma > \sigma_{uc}$ , exponents vary continuously.
- Haldane to U(1) CSB: This transition also shows continuously varying exponents but does not collapse onto the same curve as the Large-D transition when plotted against $\sigma$ . This suggests the topological nature of the Haldane phase influences the critical theory, potentially pointing to a distinct universality class or "deconfined" criticality.

D. Boundary Condition Dependence

A major technical finding is the strong dependence of finite-size scaling on boundary conditions above the upper critical dimension ( $\sigma \leq \sigma_{uc}$ ):

OBC (DMRG): The scaling exponent $\Theta$ behaves as $\Theta = (1 - \sigma/2)/2$ .
PBC (ED): The scaling follows a different form with a pseudocritical exponent $\vartheta = 3\sigma/2$ .
Implication: Ignoring boundary conditions in long-range systems can lead to the misidentification of universality classes. The authors extend the finite-size scaling framework to account for this in quantum systems.

4. Significance and Outlook

Theoretical Impact: The work establishes that the interplay between topology (Haldane phase) and long-range interactions creates a unique critical landscape. The continuous variation of critical exponents and the distinct behavior of the Haldane-to-CSB transition challenge standard Landau-Ginzburg-Wilson paradigms.
Experimental Relevance: The model is directly realizable in current quantum simulation platforms, including:
- Trapped Ions: Capable of tuning power-law interactions over a wide range of $\alpha$ .
- Rydberg Atom Arrays: Can engineer spin-1 models with single-ion anisotropy.
- Dipolar Molecules: Naturally exhibit long-range interactions.
Future Directions: The paper suggests that these platforms can be used to experimentally probe the "unconventional criticality" identified here, specifically testing the role of topology in quantum phase transitions and verifying the boundary-condition-dependent scaling laws.

In summary, this paper provides a comprehensive numerical characterization of a spin-1 chain with long-range interactions, revealing that long-range couplings not only stabilize ordered phases in 1D but also fundamentally alter the nature of quantum criticality and entanglement scaling in ways that depend on both the interaction range and the topological nature of the disordered phase.