⚛️ high-energy theory

Distinct finite-temperature phase diagrams of non-invertible Kennedy--Tasaki duals

This paper demonstrates that non-invertible Kennedy--Tasaki duals do not necessarily share identical finite-temperature phase diagrams, revealing a specific mismatch in three dimensions where a deconfinement transition in the dual $\mathbb{Z}_2$ gauge theory corresponds to an analytic paramagnetic phase in the original cluster model, thereby proving that such dualities can fail to preserve thermal order.

Original authors: Weiguang Cao, Haruki Watanabe

Published 2026-07-28

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CC BY 4.0

Original authors: Weiguang Cao, Haruki Watanabe

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 the universe as a giant, cosmic game of LEGO. Physicists love to build models of this game to understand how matter behaves, especially when it gets hot or cold. In this game, there are special rules called "symmetries" that tell us how pieces can be swapped without changing the overall picture. Usually, if you have a rule that lets you swap piece A for piece B, you can also swap piece B back to piece A. It's like a perfect mirror: what you see on the left is exactly what you see on the right, and you can always reverse the reflection.

But recently, scientists discovered a weird new kind of rule called a "non-invertible" symmetry. Think of this not as a mirror, but as a magical shredder. You can feed a specific LEGO structure into the shredder, and it comes out as a completely different, valid structure. But if you try to feed that new structure back into the shredder, it doesn't turn back into the original; it just vanishes or turns into something else entirely. This breaks the usual "mirror" logic. The big question for physicists is: if two different systems are connected by this magical shredder, do they still behave the same way when you heat them up? Do they melt at the same temperature? Do they freeze into the same shapes? This matters because understanding these connections helps us predict how materials might work in future computers or how the early universe behaved.

This paper, written by Weiguang Cao and Haruki Watanabe, tackles that exact question using a famous "shredder" known as the Kennedy–Tasaki transformation. They set up a game where they slowly morph one type of quantum material (called a "cluster model") into its "shredded" twin (a "gauge theory") by turning a dial from 0 to 1. In the world of very cold temperatures (near absolute zero), the two sides of this dial behave like perfect twins: they both have a sharp transition at the exact middle of the dial, just like a perfect mirror would predict.

However, when the authors turned up the heat to simulate finite temperatures, the twins started acting very differently. In a one-dimensional world (think of a single line of beads), the twins remained identical; both sides refused to order themselves no matter how hot it got, so their phase diagrams matched perfectly. But in a three-dimensional world (think of a block of ice), the magic broke. At the very start of the dial (where the cluster model is just a simple, disordered gas), the dual side was already undergoing a dramatic phase change called "deconfinement" at a temperature of roughly 1.31. The original side, meanwhile, stayed perfectly calm and disordered.

The authors used powerful computer simulations called Quantum Monte Carlo to watch what happened as they moved the dial. They found a "mismatch zone" where the two sides were completely different. On the cluster side, a strange "frozen wedge" appeared where the material got stuck in a specific, disordered state. On the dual gauge side, a "deconfined dome" formed, where the material behaved like a fluid of free-floating particles. These two shapes didn't overlap; they were distinct islands of behavior. The paper shows that while the mathematical "shredder" connects the two systems, it only guarantees they look the same if you ignore a huge number of hidden, chaotic states (the "kernel"). When you include those hidden states in a hot environment, they change the rules, causing the two sides to diverge.

The study concludes that non-invertible dualities are not a free pass to assume two systems are identical at high temperatures. The "shredder" only preserves the relationship for a specific, filtered version of the system. In three dimensions, the extra chaos that gets thrown away by the shredder is so energetic that it completely reshapes the phase diagram, creating a mismatch that persists over a wide range of temperatures. The authors are confident in these findings because they are backed by exact mathematical proofs at the endpoints and robust numerical simulations in the middle, showing that this isn't just a fluke of small computer models, but a real feature of how these quantum systems behave when they get warm.

Technical Summary: Distinct finite-temperature phase diagrams of non-invertible Kennedy–Tasaki duals

Problem Statement
The paper addresses a fundamental question in the thermodynamics of quantum lattice models: Do two Hamiltonians related by a non-invertible transformation necessarily share the same finite-temperature phase diagram? While invertible dualities (unitary changes of variables) preserve the partition function and transport phase boundaries exactly, recent developments have extended duality concepts to non-invertible operators, such as the Kennedy–Tasaki (KT) transformation. The KT map exchanges symmetry-protected topological (SPT) order with spontaneous symmetry breaking (SSB) at zero temperature. However, the KT map is a partial isometry that projects the Hilbert space onto a specific "all-plus" fusion sector, annihilating other charge sectors. The central problem is whether this projection preserves the bulk thermodynamic properties (free energy singularities) at finite temperature, or if the thermal occupation of the discarded sectors alters the phase diagram.

Methodology
The authors investigate this question by comparing the finite-temperature phase diagrams of a cluster-model interpolation $H(s)$ and its KT dual $\tilde{H}(s)$ in one and three spatial dimensions.

Models:
- 1D: A $Z_2 \times Z_2$ cluster chain (SPT) and its dual, two decoupled transverse-field Ising chains (SSB).
- 3D: The Raussendorf–Bravyi–Harrington (RBH) cluster model and its dual, two decoupled 3D $Z_2$ gauge theories.
Theoretical Framework: The KT map $\tilde{D}$ is defined as a partial isometry onto the all-plus sector of the fusion group. The authors derive that while the projected partition functions $Z_+(s, \beta) = \text{Tr}(P_+ e^{-\beta H})$ are identical for both sides, the unrestricted partition functions $Z(s, \beta)$ differ because the kernel of $\tilde{D}$ contains an extensive set of charge sectors in three dimensions.
Analytical Tools:
- 1D: Exact free-fermion solutions via Jordan–Wigner transformation and Toeplitz determinant analysis for magnetization.
- 3D: Quantum Monte Carlo (QMC) simulations using continuous-time algorithms (ParaToric) for the gauge side and discrete-time worldline QMC for the cluster side.
- Symmetry Analysis: Examination of one-form symmetry twists and the behavior of the conjugate order parameter under the Hellmann–Feynman theorem to distinguish continuous vs. first-order transitions.

Key Contributions and Results

Dimensional Dependence of Duality Equivalence:
- One Dimension: The phase diagrams of the cluster model and its dual are coincident. Both models remain disordered at all $T > 0$ due to the proliferation of finite-energy domain walls. The only transition is the zero-temperature critical point at $s = 1/2$ . The non-invertibility of the map does not affect the phase diagram here because thermal order is forbidden on both sides.
- Three Dimensions: The phase diagrams are inequivalent.
  - The Cluster Model (SPT side) exhibits an analytic paramagnetic free energy for all $T > 0$ . The one-form SPT order, diagnosed by a twisted membrane operator, decays with system size (area law) and does not survive at finite temperature.
  - The Dual Gauge Theory (SSB side) exhibits a genuine finite-temperature deconfinement transition. The dual model possesses a "deconfined dome" where the $Z_2$ gauge theory supports a phase with perimeter-law Wilson loops and broken one-form symmetry.
The Mismatch Mechanism:
The discrepancy arises because the KT projector $P_+$ fixes an extensive set of local one-form charges in 3D. While the projected ensembles share the same singular free energy, the unrestricted Gibbs states include the discarded charge sectors. In the gauge theory, these sectors correspond to deconfined flux loops that are suppressed by line tension below a critical temperature $T_c \approx 1.31$ (at $s=0$ ). In the cluster model, the corresponding excitations are point-like domain walls that proliferate immediately at any $T > 0$ . The thermal entropy of these extensive sectors in the gauge theory changes the dominant free energy, creating a phase transition absent in the cluster model.
Detailed Phase Structure (QMC Results):
- Cluster Side: A "self-dual frozen wedge" appears near $s=1/2$ . This is a region of ordered–ordered coexistence bounded by first-order lines, terminating at a finite-size endpoint $(s, T) \approx (0.47, 0.27)$ . This structure arises from a nearly degenerate manifold of link configurations with extensive configurational entropy.
- Gauge Side: The phase diagram features a continuous transition arm (3D Ising universality) extending from the $s=0$ deconfinement point ( $T_c \approx 1.31$ ) to the self-dual line. Near $s=1/2$ , this transitions into a weakly first-order segment (due to 4D self-duality effects at $T=0$ ) that meets the continuous arm at a tricritical point. The deconfined phase forms a dome that persists up to $T \approx 0.48$ at $s=1/2$ .
Nature of the Self-Dual Point:
At $s=1/2$ , the unitary entangler $U$ becomes a symmetry. In 1D, the transition is continuous, and $U$ remains unbroken. In 3D, the transition is first-order, and $U$ is spontaneously broken, evidenced by a discontinuity in the derivative of the free energy (latent heat/slope).

Significance
The paper establishes that non-invertible dualities, while exact at zero temperature for ground-state properties, do not automatically preserve finite-temperature phase diagrams. The equivalence holds only for the projected ensemble (all-plus sector). In dimensions where the discarded sectors carry extensive entropy (specifically 3D), the thermal occupation of these sectors can fundamentally alter the bulk free energy and create phase transitions (like deconfinement) that have no counterpart in the dual description.

The authors conclude that a thermodynamic application of non-invertible duality requires independent control over the discarded sectors (e.g., proving their contribution to $\ln Z$ is subextensive). Without such control, the fusion and intertwining relations determine projected data but do not determine physical phase boundaries. This result clarifies the limitations of extending zero-temperature duality arguments to thermal and mixed-state SPT orders.

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