⚛️ quantum physics

Measurement-Only Dynamical Phase Transitions in Spin-1 Chains

This paper demonstrates that forced local projections in an interacting spin-1 chain starting from an AKLT state drive the system into three distinct dynamical phases—trivial product, dimerized valence-bond-solid, and interwoven Haldane chains—via continuous transitions characterized by nonlocal string order and entanglement scaling.

Original authors: Kemal Aziz, Haining Pan, J. H. Pixley

Published 2026-09-24

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CC BY 4.0

Original authors: Kemal Aziz, Haining Pan, J. H. Pixley

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 quantum world, the behavior of matter is often dictated by how its tiny building blocks are arranged and how they talk to one another. For decades, physicists have been fascinated by a specific kind of arrangement called a symmetry-protected topological phase. Think of this as a state of matter that is held together not by the usual forces of attraction or repulsion, but by a hidden, global rule that prevents the system from falling apart into a simpler, disordered state. A famous example of this is the Haldane phase found in chains of atoms with a specific type of magnetic spin. These chains are robust; they resist small disturbances and maintain a unique, ordered structure that is different from a standard magnet or a random collection of atoms. Understanding these phases is crucial because they represent a new way to organize information, potentially leading to more stable quantum computers. However, a major question has remained: what happens when we stop letting these systems evolve naturally and instead force them to change by constantly measuring them?

Traditionally, scientists believed that measuring a quantum system would simply collapse its delicate state into a random outcome, destroying any complex order. But recent theories suggested that if you measure a system in a very specific, controlled way, you might actually be able to steer it into entirely new states of matter, or even drive it through a dramatic transformation known as a phase transition. This idea shifts the role of measurement from a passive observer to an active driver of change. Researchers at Rutgers University and the Flatiron Institute set out to test this idea using a chain of spin-one particles, a system that is known to host the famous Haldane phase. They wanted to see if they could use a series of forced measurements to push this ordered chain into different shapes, and to understand exactly how the transition from one shape to another occurs.

The team designed a digital simulation where they could apply a sequence of "filters" to the quantum chain. Instead of letting the system evolve on its own, they forced it to conform to specific patterns by repeatedly projecting it onto certain states. Imagine a chain of atoms where each link can be in one of three states. The researchers started with the chain in its natural, ordered Haldane state. Then, they introduced a competition. They applied a filter that tried to keep the chain in its original state, but they also applied other filters that tried to force the chain into different, simpler arrangements. One filter tried to turn every atom into a single, uniform state. Another tried to pair up neighboring atoms into distinct, alternating bonds. A third tried to link atoms that were two steps apart, creating a double-layered structure. By adjusting the probability of applying these different filters, the researchers could watch the chain transform in real-time.

What they found was a rich landscape of possibilities. When they pushed the chain toward the uniform state, the original order simply vanished, and the chain became a featureless, disconnected product of individual atoms. This was a smooth transition where the hidden order faded away without any new local pattern taking its place. However, when they pushed the chain toward the paired-bond state, the transformation was different. The original order didn't just disappear; it was actively replaced by a new, explicit pattern where atoms locked into pairs. This was a clear shift from one type of order to another, driven entirely by the choice of measurement.

The most surprising discovery came when they competed the original state against the double-layered structure. Instead of a single, sharp jump from one state to the other, the chain entered a strange, intermediate zone. In this middle ground, the chain did not settle into either of the two competing orders. Instead, it remained in a critical, fluctuating state where the connections between atoms stretched and shifted in a way that defied simple classification. In this region, the entanglement between the atoms grew in a specific, logarithmic way, suggesting that the system was caught in a delicate balance between two different worlds. This intermediate phase was not just a brief moment of confusion; it appeared to be a distinct, extended regime where the system behaved differently than in either of the stable states on either side.

The researchers used sophisticated computer simulations to track these changes, measuring how the atoms were correlated over long distances and how much information was shared between different parts of the chain. They found that the transition points were sharp and well-defined, occurring at specific probabilities of applying the filters. For the transition to the uniform state, the critical point was reached when the filter was applied about half the time. For the transition to the paired state, it happened when the filter was applied roughly one-third of the time. The most complex behavior occurred in the middle, where the system hovered between the two competing structures, showing signs of a critical phase that persisted over a range of conditions.

These results are significant because they show that measurement alone, without any traditional forces or energy inputs, can be used to control and manipulate the fundamental order of a quantum system. The study demonstrates that by carefully choosing which measurements to perform, scientists can drive a system from a topological state into a trivial one, or into a new type of ordered state. It also reveals that the path between these states is not always a direct line; sometimes, the system must pass through a complex, critical region where the rules of order are suspended. This work extends our understanding of how quantum matter behaves under observation, suggesting that the act of measurement is a powerful tool for engineering new states of matter, potentially offering new ways to prepare and protect quantum information in future technologies.

Technical Summary: Measurement-Only Dynamical Phase Transitions in Spin-1 Chains

Problem Statement
This study investigates the dynamical competition between distinct ground states in an interacting spin-1 chain under "measurement-only" dynamics. Unlike standard unitary evolution, which preserves symmetry and topological indices within bounded-depth circuits, the authors explore nonunitary dynamics driven by forced, normalized local projections. The central problem is to determine how an interacting bosonic Symmetry-Protected Topological (SPT) phase—the Affleck-Kennedy-Lieb-Tasaki (AKLT) Haldane state—responds when subjected to competing local projectors that favor different phases: a topologically trivial product state, an explicitly dimerized valence-bond-solid (VBS) state, and a topologically trivial next-nearest-neighbor (NNN) AKLT state. The work addresses whether such nonunitary protocols can drive transitions between these phases and whether they can generate or destroy topological order and entanglement structures.

Methodology
The authors define three distinct measurement-only protocols on a chain of $L$ spin-1 degrees of freedom (qutrits) arranged in staircase circuit geometries. The dynamics are modeled as a sequence of local updates where, at each step, a specific local projector is applied with a certain probability $p$ , and the resulting state is renormalized. This "forced" projection mimics imaginary-time evolution (MITE) without sampling measurement outcomes via the Born rule.

The three competing models are:

SPT-to-Topologically Trivial (TT): The nearest-neighbor (NN) AKLT projector competes with a single-site $S^z_i = 0$ projector. The latter favors a featureless product state $|0\rangle^{\otimes L}$ , breaking SU(2) symmetry down to a $Z_2 \times Z_2$ subgroup.
SPT-to-Dimer: The NN AKLT projector competes with a projector on odd bonds that forces a singlet (dimer) state, explicitly breaking translation symmetry to favor a VBS state.
NN-to-NNN AKLT: The NN AKLT projector competes with an NNN AKLT projector. The NNN limit decouples the system into two interwoven AKLT chains on even and odd sublattices. Under the diagonal $Z_2 \times Z_2$ symmetry preserved by the mixed dynamics, this NNN state is topologically trivial, though it retains entanglement structure.

Simulation Techniques

Matrix Product States (MPS): Used for the SPT–TT and SPT–dimer models, where entanglement remains manageable (area law).
Exact Statevector Evolution: Required for the NN–NNN AKLT model due to the presence of a critical phase with high entanglement that causes MPS bond dimensions to grow exponentially. This limits simulations to system sizes $L \leq 16$ .
Diagnostics: The phases and transitions are characterized using:
- Nonlocal String Order Parameters (SOP): To detect hidden antiferromagnetic order and SPT phases.
- Local Dimer Order: To detect explicit symmetry breaking.
- Entanglement Scaling: Half-cut bipartite entanglement entropy ( $S(L/2)$ ) and its variance across trajectories.
- Finite-Size Scaling: Analysis of order parameters and entanglement to extract critical exponents ( $\nu, \beta, z$ ) and locate critical points ( $p_c$ ).

Key Results

SPT to Topologically Trivial Product State:
- A continuous transition occurs at a critical probability $p_{TT,c} \approx 0.50$ .
- Below $p_c$ , the system remains in the Haldane SPT phase with finite string order and area-law entanglement ( $S \approx 2$ ).
- Above $p_c$ , the system collapses into a disentangled product state with vanishing string order.
- At the critical point, string order decays algebraically, and entanglement grows logarithmically with system size ( $S \sim \alpha \ln L$ ) with a coefficient $\alpha \approx 0.48$ . The dynamical exponent is $z \approx 1.16$ .
SPT to Explicitly Dimerized VBS:
- A continuous transition occurs at $p_{D,c} \approx 0.34$ .
- The transition is marked by the vanishing of NN string order and the emergence of staggered dimer order.
- Critical behavior includes logarithmic entanglement growth with a smaller coefficient ( $\alpha \approx 0.37$ ) compared to the SPT–TT transition, and a dynamical exponent $z \approx 1.08$ .
NN AKLT to NNN AKLT:
- This model exhibits a more complex phase diagram featuring two continuous transitions separated by an extended critical regime.
- The NN AKLT phase (topologically nontrivial) transitions to a critical phase at $p^{(2)}_{nnn,c} \approx 0.28$ .
- The critical phase transitions to the NNN AKLT phase (topologically trivial under diagonal symmetry) at $p^{(1)}_{nnn,c} \approx 0.64$ .
- Within the intermediate regime, both NN and NNN string order parameters vanish algebraically with system size, and entanglement grows logarithmically with a significantly larger coefficient ( $\alpha \approx 2.2 - 2.6$ ) than in the other transitions.
- The bond dimension required to simulate this regime grows exponentially, confirming the presence of high entanglement consistent with a gapless critical phase.

Significance and Claims
The paper claims to extend the study of monitored quantum dynamics and measurement-induced phase transitions (MIPTs) from free-fermion systems to interacting bosonic SPT phases. The primary contributions are:

Control of Topological and VBS Order: The study demonstrates that local, nonunitary projections can deterministically drive transitions between distinct topological and symmetry-breaking phases, effectively controlling the order of the system.
Critical Regime in Competing SPTs: The identification of an extended critical regime separating two distinct AKLT-like states (one topological, one trivial) suggests that competing nonunitary projectors can stabilize gapless phases with logarithmic entanglement scaling, distinct from the area-law phases found at the endpoints.
Universality and Exponents: The calculated critical exponents and dynamical exponents ( $z \approx 1$ ) are consistent with conformally invariant critical percolation limits, suggesting a degree of universality in measurement-induced transitions even in interacting bosonic systems.
Limitations: The authors modestly note that while finite-size signatures ( $L \leq 16$ ) strongly support the existence of the extended critical regime in the NN–NNN model, the thermodynamic persistence of this phase is not rigorously established due to simulation constraints.

The work concludes that measurement-only dynamics provides a robust framework for exploring the interplay between topology, symmetry breaking, and entanglement in quantum many-body systems, revealing rich phase structures driven purely by the competition of local projective measurements.

Technical Summary: Measurement-Only Dynamical Phase Transitions in Spin-1 Chains

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