High Energy Physics - Theory · arXiv:1308.1960 v2 · submitted 8 Aug 2013
Maxwell Theory on a Compact Manifold as a Topologically Ordered System
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“Based on these computations we claim that the Maxwell system defined on 4 torus behaves in many respects as a topologically ordered system.”
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Abstract
We study novel type of contributions to the entropy of the Maxwell system defined on a compact manifold such as torus. These new terms are not related to the physical propagating photons. Rather, these novel contributions emerge as a result of tunnelling events when transitions occur between topologically different but physical identical vacuum winding states. We compute two new (topologically protected) types of contributions to the entropy in this system resulting from this dynamics. First contribution has a negative sign, expressed in terms of the magnetic susceptibility, and it is similar in spirit to topological entanglement entropy discussed in condensed matter systems. The second contribution with a positive sign results from the emergent degeneracy which occurs when the system is placed into a background of external magnetic field. This degeneracy resembles a similar effect which occurs at \theta=\pi in topological insulators. Based on these computations we claim that the Maxwell system defined on 4 torus behaves in many respects as a topologically ordered system.
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abstract compiled · 2026-09-23 · no language-model review
How the main claim is stated: asserted 0.80
What the abstract itself states, as probabilities: claims first 0.11 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.57 · mentions code or data availability 0.01
The abstract, sentence by sentence
This table labels the abstract only (arXiv listing abstract v2). The number is the probability the classifier gave the label.
| Sentence | Role | The authors' own work | States the main conclusion | Strength |
|---|---|---|---|---|
| We study novel type of contributions to the entropy of the Maxwell system defined on a compact manifold such as torus. | contribution 0.99 | 0.89 | 0.04 | asserted 0.81 |
| These new terms are not related to the physical propagating photons. | interpretation 0.60 | 0.64 | 0.02 | asserted 1.00 |
| Rather, these novel contributions emerge as a result of tunnelling events when transitions occur between topologically different but physical identical vacuum winding states. | interpretation 0.85 | 0.48 | 0.06 | asserted 1.00 |
| We compute two new (topologically protected) types of contributions to the entropy in this system resulting from this dynamics. | contribution 0.50 | 0.96 | 0.12 | asserted 1.00 |
| First contribution has a negative sign, expressed in terms of the magnetic susceptibility, and it is similar in spirit to topological entanglement entropy discussed in condensed matter systems. | result 0.98 | 0.90 | 0.06 | asserted 0.99 |
| The second contribution with a positive sign results from the emergent degeneracy which occurs when the system is placed into a background of external magnetic field. | result 0.98 | 0.88 | 0.04 | asserted 1.00 |
| This degeneracy resembles a similar effect which occurs at \theta=\pi in topological insulators. | interpretation 0.93 | 0.22 | 0.03 | asserted 0.89 |
| Based on these computations we claim that the Maxwell system defined on 4 torus behaves in many respects as a topologically ordered system. | interpretation 1.00 | 0.78 | 0.94 | asserted 0.80 |
What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.70
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,158 input tokens, $0.0003 reported.
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How the literature uses it
11 arXiv papers cite this in 72 sentences
Papers citing it · 55 sentences without a confident label
Builds on it 1 paper
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Therefore, the partition function in the presence of the uniform magnetic field can be easily reconstructed from ([ref]), and it is given by [ref]
Dynamical Casimir Effect in a small compact manifold for the Maxwell vacuum · External static magnetic field · 0.89
Uses its method or tool 2 papers
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With these additional simplifications the topological partition function becomes [ref] :
Dynamical Casimir Effect in a small compact manifold for the Maxwell vacuum · Construction · 0.80
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We follow [ref] in our construction of the partition function $Z_top$ where it was employed for computation of the corrections to the Casimir effect due to these novel type of topological fluctuations.
Topological order and Berry connection for the Maxwell Vacuum on a four-torus · Construction · 0.99
Compares with it 1 paper
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After the expansion ([ref]) is obtained, we can substitute $τ=β/4πR$ and get [equation] We notice that similar expressions also appear in weak-coupling calculation on a different manifold [ref] .
Hagedorn transition and topological entanglement entropy · 2D modular formulae and high-temperature continuation · 0.99
Cites it as support 3 papers
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Such a similarity looks very instructive and suggestive, and further supports our arguments [ref] that the ground state of the Maxwell theory defined on a small compact manifold behaves as a TQFT.
Topological order and Berry connection for the Maxwell Vacuum on a four-torus · Introduction. Motivation. · 1.00
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In particular, $Z_top$ demonstrates the degeneracy of the system which can not be described in terms of any local operators [ref] .
Dynamical Casimir Effect in a small compact manifold for the Maxwell vacuum · Introduction. Motivation. · 0.99
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In fact, this conjecture is supported in some way by computations in a simplified model [ref] , see also [ref] with related discussions.].
Inflaton as an auxiliary topological field in a QCD-like system · Topological auxiliary field as a source of the “strange" energy · 0.99
Cites it as background 2 papers
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In section [ref], we review the construction of magnetic-type instantons and the partition function $Z_top$ in [ref] .
Aharonov-Bohm phases in a quantum LC circuit · Introduction. Motivation. · 0.83
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In context of the topological insulators and quantum Hall systems the corresponding studies have been carried out in two, three and four dimensions [ref] , see also [ref] with related discussions of the ground state in 2d Maxwell theory[Not to be confused with conventional CM notations, where it is a customary to count the spatial number of dimensions, rather than total number of dimensions.
Topological order and Berry connection for the Maxwell Vacuum on a four-torus · Berry phase in CM systems · 0.95
What it cites · 59 sentences without a confident label
Uses the method or tool of 5 papers
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We wish to derive the topological action for the Maxwell system by using the same conventional technique exploited e.g. in [ref] for the so-called “deformed QCD" and in [ref] for the Higgs model.
Superconductors are topologically ordered · Partition function in “BF" formulation · 1.00
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We wish to derive the topological action for the Maxwell system by using the same conventional technique exploited e.g. in [ref] for the so-called “deformed QCD" and in [ref] for the Higgs model.
QCD as a topologically ordered system · Partition function in “BF" formulation · 1.00
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All integrals in this partition function are gaussian and can be easily evaluated using the technique developed in [ref] .
Generalized Thirring Models · Partition function · 1.00
Compares with 5 papers
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[ref] in the presence of the electric fluxes.
Remarks on the thermodynamics and the vacuum energy of a quantum Maxwell gas on compact and closed manifolds · Degeneracy at $θ_eff=π$ · 0.95
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Therefore, these massless photons can be completely ignored in our studies of the topological properties of the partition function, similar to decoupling of the massless phonons (which are in fact responsible for the mere existence of a gap) in treatment of the topologically ordered superconductors [ref] .
Superconductors are topologically ordered · “BF" formulation of the Maxwell system · 0.95
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-lnD=- χ_mag (τ≫1 )/2 =-1/2 ⟶ D=√(e), which should be compared with conventional $D=√(m)$ for a Laughlin state in a fractional quantum Hall system with filling factor $ν=1/m$, or $D=2$ for $p+ip$ superconductor, or any other similar systems, see recent reviews [ref] for the details and original references.
Topological order: from long-range entangled quantum matter to an unification of light and electrons · Similarities and differences between TE and Topological Entanglement Entropy in CM systems · 0.81
Cites as support 2 papers
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In particular, it has been explicitly shown in [ref] that these novel terms lead to a fundamentally new contributions to the Casimir vacuum pressure, which can not be expressed in terms of conventional propagating physical degrees of freedom.
Topological Casimir effect in Maxwell Electrodynamics on a Compact Manifold · Introduction. Motivation. · 0.99
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Furthermore, the entropy ([ref]) is obviously a positively defined function at any finite $V$ and can be interpreted as the entanglement entropy [ref] .
Do gauge fields really contribute negatively to black hole entropy? · Entropy in 2d Maxwell system · 0.96
Cites as background 6 papers
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The entanglement entropy for Maxwell theory is also known, see recent paper [ref] with many references on previous works therein.
Remarks on effective action and entanglement entropy of Maxwell field in generic gauge · Topological Entropy in Maxwell theory in four dimensions · 0.98
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Furthermore, it demonstrates many features of topologically ordered systems, which were initially introduced in context of condensed matter systems, see original papers [ref] and recent reviews [ref] .
Topological BF field theory description of topological insulators · Introduction. Motivation. · 0.84
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Furthermore, it demonstrates many features of topologically ordered systems, which were initially introduced in context of condensed matter systems, see original papers [ref] and recent reviews [ref] .
Topological order: from long-range entangled quantum matter to an unification of light and electrons · Introduction. Motivation. · 0.84
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