arXiv:2401.10563

Comment on "Machine Learning the Operator Content of the Critical Self-Dual Ising-Higgs Gauge Model'', arXiv:2311.17994v1

Claudio Bonati, Andrea Pelissetto C, Ettore Vicari

Published Jan 19, 2024• 0 citations

Abstract

We critically discuss the results reported in arXiv:2311.17994v1 by L. Oppenheim, M. Koch-Janusz, S. Gazit, and Z. Ringel, on the multicritical behavior of the three-dimensional Ising-Gauge model at the multicritical point. We argue that their results do not contradict the theoretical scenario put forward in ``Multicritical point of the three-dimensional ${\mathbb Z}_2$ gauge Higgs model'', Phys. Rev. B 105, 165138 (2022), arXiv:2112.01824, that predicted a multicritical behavior controlled by the stable $XY$ fixed point of an effective three-dimensional ${\mathbb Z}_2\oplus {\mathbb Z}_2$ Landau-Ginzburg-Wilson $Φ^4$ field theory. Actually, their results, as well as all numerical results reported so far in the literature, are consistent with a multicritical $XY$ scenario.

DOI: 10.48550/arxiv.2401.10563

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Comment on ‘Mathematical structure of the three-dimensional (3D) Ising model’∗

Jacques H. H. Perk

The review paper by Zhang Zhi-Dong (Zhang Z D 2013 Chin. Phys. B 22 030513, arXiv:1305.2956) contains many errors and is based on several earlier works that are equally wrong.

Jan 1, 201313 citations

arXiv:2401.10563

Comment on "Machine Learning the Operator Content of the Critical Self-Dual Ising-Higgs Gauge Model'', arXiv:2311.17994v1

Claudio Bonati, Andrea Pelissetto C, Ettore Vicari

Published Jan 19, 2024• 0 citations

arXiv HTML PDF

Abstract

{\mathbb Z}_2

gauge Higgs model'', Phys. Rev. B 105, 165138 (2022), arXiv:2112.01824, that predicted a multicritical behavior controlled by the stable

XY

fixed point of an effective three-dimensional

{\mathbb Z}_2\oplus {\mathbb Z}_2

Landau-Ginzburg-Wilson

Φ^4

field theory. Actually, their results, as well as all numerical results reported so far in the literature, are consistent with a multicritical

XY

scenario.

DOI: 10.48550/arxiv.2401.10563

Related papers

Machine learning the operator content of the critical self-dual Ising-Higgs lattice gauge theory

Lior Oppenheim, Maciej Koch-Janusz, Snir Gazit +1 more

Understanding critical phenomena is of central importance to condensed-matter and high-energy physics. Such an understanding is reflected in our ability to sort observables based on their degeneracy, symmetries, and power-law decays. Here, we study such critical properties of the Ising-Higgs gauge theory in ( 2 + 1 ) D along the self-dual line which have recently been a subject of debate. Using machine learning techniques, we determine the low-energy operator content of the associated field theory. Our approach enables us to largely refute the existence of an emergent current operator and with it the standing conjecture that this transition is of the X Y * universality class. We contrast these results with the ones obtained for the ( 2 + 1 ) D Ashkin-Teller transverse field Ising model where we find the expected current operator. Our numerical technique extends the recently proposed real-space mutual information allowing us to extract subleading nonlinear operators. This allows a controlled and computationally scalable approach to target the conformal field theory spectrum. and discern universality classes beyond ( 1 + 1 ) D from Monte Carlo data. Published by the American Physical Society 2024Understanding critical phenomena is of central importance to condensed-matter and high-energy physics. Such an understanding is reflected in our ability to sort observables based on their degeneracy, symmetries, and power-law decays. Here, we study such critical properties of the Ising-Higgs gauge theory in ( 2 + 1 ) D along the self-dual line which have recently been a subject of debate. Using machine learning techniques, we determine the low-energy operator content of the associated field theory. Our approach enables us to largely refute the existence of an emergent current operator and with it the standing conjecture that this transition is of the X Y * universality class. We contrast these results with the ones obtained for the ( 2 + 1 ) D Ashkin-Teller transverse field Ising model where we find the expected current operator. Our numerical technique extends the recently proposed real-space mutual information allowing us to extract subleading nonlinear operators. This allows a controlled and computationally scalable approach to target the conformal field theory spectrum. and discern universality classes beyond ( 1 + 1 ) D from Monte Carlo data. Published by the American Physical Society 2024

arXiv:2311.17994Physical Review ResearchDec 26, 202410 citations

Analytic framework for self-dual criticality in Z k gauge theory with matter

Zhengyan Darius Shi, Arkya Chatterjee

The deconfined phase of ( $2+1$ )-dimensional ${\mathbb{Z}}_{k}$ gauge theory exhibits topological order, with $e$ and $m$ anyons that have a $2\ensuremath{\pi}/k$ braiding phase. Proliferating either $e$ or $m$ drives Higgs or confinement transitions, respectively. At the multicritical point where these transitions meet, the theory enjoys an additional duality symmetry that exchanges $e$ and $m$ anyons. This symmetry forces anyons with nontrivial braiding to close their gaps simultaneously, giving rise to a critical theory that mixes strong interactions with mutual statistics. We propose an effective $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ gauge theory with a mutual Chern-Simons term at level $k$ to describe the vicinity of the multicritical point for $k\ensuremath{\ge}4$. The emergence of a global $\mathrm{U}{(1)}^{\mathrm{top}}\ifmmode\times\else\texttimes\fi{}\mathrm{U}{(1)}^{\mathrm{top}}$ symmetry at the critical point imposes powerful constraints on universal properties of the phase transition. In particular, we show that (1) the lattice magnetic flux operator embeds as a conserved U(1) current with protected scaling dimension; (2) the first-order line emanating from the critical point for $k=2$ disappears generically for sufficiently large $k$ ; (3) the correlation length exponent approaches that of the three-dimensional XY model with corrections of order $1/{k}^{2}$ in the large- $k$ limit. These predictions can be tested in near-term numerical simulations and pave the way for a more general exploration of topological quantum criticality enriched with anyon-permuting symmetries.The deconfined phase of ($2+1$)-dimensional ${\mathbb{Z}}_{k}$ gauge theory exhibits topological order, with $e$ and $m$ anyons that have a $2\ensuremath{\pi}/k$ braiding phase. Proliferating either $e$ or $m$ drives Higgs or confinement transitions, respectively. At the multicritical point where these transitions meet, the theory enjoys an additional duality symmetry that exchanges $e$ and $m$ anyons. This symmetry forces anyons with nontrivial braiding to close their gaps simultaneously, giving rise to a critical theory that mixes strong interactions with mutual statistics. We propose an effective $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ gauge theory with a mutual Chern-Simons term at level $k$ to describe the vicinity of the multicritical point for $k\ensuremath{\ge}4$. The emergence of a global $\mathrm{U}{(1)}^{\mathrm{top}}\ifmmode\times\else\texttimes\fi{}\mathrm{U}{(1)}^{\mathrm{top}}$ symmetry at the critical point imposes powerful constraints on universal properties of the phase transition. In particular, we show that (1) the lattice magnetic flux operator embeds as a conserved U(1) current with protected scaling dimension; (2) the first-order line emanating from the critical point for $k=2$ disappears generically for sufficiently large $k$; (3) the correlation length exponent approaches that of the three-dimensional XY model with corrections of order $1/{k}^{2}$ in the large-$k$ limit. These predictions can be tested in near-term numerical simulations and pave the way for a more general exploration of topological quantum criticality enriched with anyon-permuting symmetries.

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Comment on "A structural test for the conformal invariance of the critical 3d Ising model" by S. Meneses, S. Rychkov, J. M. Viana Parente Lopes and P. Yvernay. arXiv:1802.02319

Bertrand Delamotte, Matthieu Tissier, Nicolás Wschebor

In a recent preprint [ArXiv 1802.02319], Meneses et al. challenge our proof that scale invariance implies conformal invariance for the three-dimensional Ising model [B. Delamotte, M. Tissier and N. Wschebor, Phys. Rev. E 93 (2016), 012144.]. We refute their arguments. We also point out a mistake in their one-loop calculation of the dimension of the vector operator $V_μ$ of lowest dimension which is not a total derivative.In a recent preprint [ArXiv 1802.02319], Meneses et al. challenge our proof that scale invariance implies conformal invariance for the three-dimensional Ising model [B. Delamotte, M. Tissier and N. Wschebor, Phys. Rev. E 93 (2016), 012144.]. We refute their arguments. We also point out a mistake in their one-loop calculation of the dimension of the vector operator $V_μ$ of lowest dimension which is not a total derivative.

arXiv:1802.07157Feb 20, 20180 citations

Comment on "Evidence for Supersymmetry in the Random-Field Ising Model at D=5''

Ivan Balog, Gilles Tarjus, Matthieu Tissier

In a recent letter, Fytas et al. [Phys. Rev. Lett. 122, 240603 (2019)] study the critical point of the equilibrium random-field Ising model (RFIM) in $D=5$ by means of state-of-art zero-temperature lattice simulations. We show that their numerical results concerning the virtual absence of violation of supersymmetry and the characterization of the main corrections to scaling are in full agreement with our predictions by means of a nonperturbative functional renormalization group and with the existence of a critical dimension $D_{DR}\approx 5.1$ above which supersymmetry and dimensional reduction are valid at the fixed point and below which they break down.In a recent letter, Fytas et al. [Phys. Rev. Lett. 122, 240603 (2019)] study the critical point of the equilibrium random-field Ising model (RFIM) in $D=5$ by means of state-of-art zero-temperature lattice simulations. We show that their numerical results concerning the virtual absence of violation of supersymmetry and the characterization of the main corrections to scaling are in full agreement with our predictions by means of a nonperturbative functional renormalization group and with the existence of a critical dimension $D_{DR}\approx 5.1$ above which supersymmetry and dimensional reduction are valid at the fixed point and below which they break down.

arXiv:1906.10058Jun 24, 20192 citations

Topological multicritical point in the phase diagram of the toric code model and three-dimensional lattice gauge Higgs model

I. S. Tupitsyn, Alexei Kitaev, Nikolay V. Prokof'ev +1 more

We construct a mapping between the two-dimensional toric code model in external magnetic fields, ${h}_{z}$ and ${h}_{x}$ , and the three-dimensional classical Ising system with plaquette interactions, which is equivalent to the three-dimensional ${\mathbb{Z}}_{2}$ gauge Higgs model with anisotropy between the imaginary time and spatial directions. The isotropic limit of the latter model was studied using Monte Carlo simulations on large (up to ${60}^{3}$ ) lattices in order to determine the stability of the topological phase against generic magnetic field perturbations and to resolve fine details of the phase diagram. We find that the topological phase is bounded by second-order transition lines, which merge into a first-order line at what appears to be a multicritical point arising from the competition between the Higgs and confinement transitions in the ${\mathbb{Z}}_{2}$ gauge system. An effective field theory for this type of multicritical point (if one actually exists) is not known. Our results have potential applications to frustrated magnets, quantum computation, lattice gauge models in particle physics, and critical phenomena.We construct a mapping between the two-dimensional toric code model in external magnetic fields, ${h}_{z}$ and ${h}_{x}$, and the three-dimensional classical Ising system with plaquette interactions, which is equivalent to the three-dimensional ${\mathbb{Z}}_{2}$ gauge Higgs model with anisotropy between the imaginary time and spatial directions. The isotropic limit of the latter model was studied using Monte Carlo simulations on large (up to ${60}^{3}$) lattices in order to determine the stability of the topological phase against generic magnetic field perturbations and to resolve fine details of the phase diagram. We find that the topological phase is bounded by second-order transition lines, which merge into a first-order line at what appears to be a multicritical point arising from the competition between the Higgs and confinement transitions in the ${\mathbb{Z}}_{2}$ gauge system. An effective field theory for this type of multicritical point (if one actually exists) is not known. Our results have potential applications to frustrated magnets, quantum computation, lattice gauge models in particle physics, and critical phenomena.

arXiv:0804.3175Physical Review BAug 17, 2010195 citations

Comment on ‘Mathematical structure of the three-dimensional (3D) Ising model’∗

Jacques H. H. Perk

The review paper by Zhang Zhi-Dong (Zhang Z D 2013 Chin. Phys. B 22 030513, arXiv:1305.2956) contains many errors and is based on several earlier works that are equally wrong.

Jan 1, 201313 citations

{\mathbb Z}_2

gauge Higgs model'', Phys. Rev. B 105, 165138 (2022), arXiv:2112.01824, that predicted a multicritical behavior controlled by the stable

XY

fixed point of an effective three-dimensional

{\mathbb Z}_2\oplus {\mathbb Z}_2

Landau-Ginzburg-Wilson

Φ^4

field theory. Actually, their results, as well as all numerical results reported so far in the literature, are consistent with a multicritical

XY

scenario.

The deconfined phase of (

2+1

)-dimensional

{\mathbb{Z}}_{k}

gauge theory exhibits topological order, with

e

and

m

anyons that have a $2\ensuremath{\pi}/k$ braiding phase. Proliferating either

e

m

drives Higgs or confinement transitions, respectively. At the multicritical point where these transitions meet, the theory enjoys an additional duality symmetry that exchanges

e

and

m

anyons. This symmetry forces anyons with nontrivial braiding to close their gaps simultaneously, giving rise to a critical theory that mixes strong interactions with mutual statistics. We propose an effective $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ gauge theory with a mutual Chern-Simons term at level

k

to describe the vicinity of the multicritical point for $k\ensuremath{\ge}4$. The emergence of a global $\mathrm{U}{(1)}^{\mathrm{top}}\ifmmode\times\else\texttimes\fi{}\mathrm{U}{(1)}^{\mathrm{top}}$ symmetry at the critical point imposes powerful constraints on universal properties of the phase transition. In particular, we show that (1) the lattice magnetic flux operator embeds as a conserved U(1) current with protected scaling dimension; (2) the first-order line emanating from the critical point for

k=2

disappears generically for sufficiently large

k

; (3) the correlation length exponent approaches that of the three-dimensional XY model with corrections of order

1/{k}^{2}

in the large-

k

limit. These predictions can be tested in near-term numerical simulations and pave the way for a more general exploration of topological quantum criticality enriched with anyon-permuting symmetries.

V_μ

of lowest dimension which is not a total derivative.

In a recent letter, Fytas et al. [Phys. Rev. Lett. 122, 240603 (2019)] study the critical point of the equilibrium random-field Ising model (RFIM) in

D=5

by means of state-of-art zero-temperature lattice simulations. We show that their numerical results concerning the virtual absence of violation of supersymmetry and the characterization of the main corrections to scaling are in full agreement with our predictions by means of a nonperturbative functional renormalization group and with the existence of a critical dimension

D_{DR}\approx 5.1

above which supersymmetry and dimensional reduction are valid at the fixed point and below which they break down.

We construct a mapping between the two-dimensional toric code model in external magnetic fields,

{h}_{z}

and

{h}_{x}

, and the three-dimensional classical Ising system with plaquette interactions, which is equivalent to the three-dimensional

{\mathbb{Z}}_{2}

gauge Higgs model with anisotropy between the imaginary time and spatial directions. The isotropic limit of the latter model was studied using Monte Carlo simulations on large (up to

{60}^{3}

) lattices in order to determine the stability of the topological phase against generic magnetic field perturbations and to resolve fine details of the phase diagram. We find that the topological phase is bounded by second-order transition lines, which merge into a first-order line at what appears to be a multicritical point arising from the competition between the Higgs and confinement transitions in the

{\mathbb{Z}}_{2}

gauge system. An effective field theory for this type of multicritical point (if one actually exists) is not known. Our results have potential applications to frustrated magnets, quantum computation, lattice gauge models in particle physics, and critical phenomena.