Absence of fractal quantum criticality in the quantum Newman-Moore model

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Absence of fractal quantum criticality in the quantum Newman-Moore model

Raymond Wiedmann, Lea Lenke, Matthias Mühlhauser, and Kai Phillip Schmidt

Phys. Rev. Research 6, 013191 – Published 22 February 2024

Abstract

The quantum phase transition between the low-field fracton phase with type-II fracton excitations and the high-field polarized phase is investigated in the two-dimensional self-dual quantum Newman-Moore model. We apply perturbative and numerical linked-cluster expansions to calculate the ground-state energy per site in the thermodynamic limit, revealing a level crossing at the self-dual point. In addition, high-order series expansions of the relevant low-energy gaps are determined using perturbative continuous unitary transformations indicating no gap closing. Our results therefore predict a first-order phase transition between the low-field fracton and the high-field polarized phase at the self-dual point.

Received 17 February 2023
Accepted 9 January 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.013191

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Fractons Magnetism Quantum phase transitions Topological order

Ising model Series expansions & exact enumeration

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalQuantum Information, Science & TechnologyStatistical Physics & Thermodynamics

Authors & Affiliations

Raymond Wiedmann^1,2, Lea Lenke ¹, Matthias Mühlhauser¹, and Kai Phillip Schmidt ¹

¹Department of Physics, Staudtstraße 7, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), 91058 Erlangen, Germany
²Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany

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Vol. 6, Iss. 1 — February - April 2024

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Images

Figure 1
Low-field limit $Γ = 0$ [high-field limit $J = 0$ ] of the qNM on the triangular lattice with spins-1/2 indicated as black arrows and three-spin interactions acting on downward-pointing triangles $▿$ is illustrated in (a) [in (b)]. (a) In the low-field limit $Γ = 0$ the fully polarized state with spins pointing in the $z$ direction is one ground state, so that all pseudospins on the centers of downward-pointing triangles point also upward. A single spin-flip changes the eigenvalue of the three attached pseudo-spins on downward-pointing triangles, which is illustrated in red. The three-spin-flip excitation on the next larger Sierpinski triangle is shown in blue. (b) In the high-field limit $J = 0$ the fully polarized state with spins pointing in the $x$ direction is the unique ground state. Fully mobile excitations in the high-field limit are associated with flipping the three spins on a downward-pointing triangle (shown in red) as well as on the first and second nontrivial Sierpinski triangle illustrated in blue and orange, respectively.
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Figure 2
Series expansion of the ground-state energy per site $E_{0} / (Γ N)$ up to order 26 in $J / Γ$ for the low- and high-field phase. The individual orders of the series are shown with increasing opacity. The highest order is further highlighted in bold as indicated in the legend. By construction, the series intersect in all orders at the self-dual point $J / Γ = 1$ marked by the vertical dashed line (left inset zoom close to the self-dual point). Note that the energies from numerical linked cluster expansions are in quantitative agreement with the ones from the series expansions on the displayed energy resolution. The right inset shows the angle $β$ between the low- and high-field energy expression from the perturbative (empty circles) and numerical linked-cluster expansions (filled circles) as a function of $1 / order$ .
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Figure 3
Critical exponent $α$ from various biased DLog Padé approximants of the second derivative of $E_{0} / (Γ N)$ with respect to $J / Γ$ displayed as a function of the order of the DLog Padé approximant. The results are grouped in families ([L, $L + C$ ] with order $2 L + C$ ) and shown for $| C | \leq 5$ . The inset plot shows the average value for each order with the sample standard deviation. For the lowest-order value only a single value is available, marked by a circle.
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Figure 4
Low-energy excitation gaps $Δ_{1}$ of the 1QP, $Δ_{2}$ of the 2QP, and $Δ_{3}^{(1)}$ and $Δ_{3}^{(2)}$ of the 3QP sector are shown as a function of $J / Γ$ using average and standard deviation of nondefective DLog Padé approximants. The vertical dashed line indicates the self-dual point.
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