# Magnetic Monopoles, Dyons and Confinement in Quantum Matter

## Abstract

**:**

## 1. Magnetic Monopoles

## 2. Effective Electromagnetic Action for Quantum Materials

## 3. Compact Effective Action of Granular Insulators

## 4. Quantum Wires and Josephson Junction Chains

## 5. The Superconductor-to-Superinsulator Transition in Quantum Films

## 6. Confinement and Superinsulation

## 7. Dyons, Oblique Confinement and the Pseudogap State

## 8. The Role of Disorder

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Goddard, P.; Olive, D.I. Magnetic monopoles in gauge field theories. Rep. Prog. Phys.
**1978**, 41, 1357–1437. [Google Scholar] [CrossRef] [Green Version] - Polyakov, A.M. Compact gauge fields and the infrared catastrophe. Phys. Lett.
**1975**, 59, 82–84. [Google Scholar] [CrossRef] - Polyakov, A.M. Fields and Strings; Harwood Academic Publisher: Chur, Switzerland, 1987. [Google Scholar]
- Diamantini, M.C.; Trugenberger, C.A.; Vinokur, V.M. Confinement and asymptotic freedom with Cooper pairs. Comm. Phys.
**2018**, 1, 77. [Google Scholar] [CrossRef] [Green Version] - Negele, J.W.; Orland, H. Quantum Many-Particle Physics; Addison-Wesley Publishing Company: Redwood City, CA, USA, 1988. [Google Scholar]
- Fazio, R.H.; van der Zant, H. Quantum phase transitions and vortex dynamics in superconducting networks. Phys. Rep.
**2001**, 355, 235–334. [Google Scholar] [CrossRef] [Green Version] - Arutyunov, K.Y.; Golubev, D.S.; Zaikin, A.D. Superconductivity in one dimension. Phys. Rep.
**2008**, 464, 1–70. [Google Scholar] [CrossRef] [Green Version] - Coleman, S. Aspects of Symmetry; Cambridge University Press: Cambridge, UK, 1985. [Google Scholar]
- Golubev, D.S.; Zaikin, A.D. Quantum tunnelling of the order parameter in superconducting nanowires. Phys. Rev.
**2001**, B64, 014504. [Google Scholar] [CrossRef] [Green Version] - Minnhagen, P. The two-dimensional Coulomb gas, vortex unbinding and superfluid-superconducting films. Rev. Mod. Phys.
**1987**, 59, 1001–1066. [Google Scholar] [CrossRef] - Wilczek, F. Disassembling Anyons. Phys. Rev. Lett.
**1992**, 69, 132–135. [Google Scholar] [CrossRef] - Deser, S.; Jackiw, R.; Templeton, S. Three-dimensional massive gauge theories. Phys. Rev. Lett.
**1982**, 48, 975. [Google Scholar] [CrossRef] [Green Version] - Diamantini, M.C.; Sodano, P.; Trugenberger, C.A. Gauge theories of Josephson junction arrays. Nucl. Phys.
**1996**, B474, 641–677. [Google Scholar] [CrossRef] - Diamantini, M.C.; Mironov, A.Y.; Postolova, S.M.; Liu, X.; Hao, Z.; Silevitch, D.M.; Vinokur, V.M. Bosonic topological intermediate state in the superconductor-insulator transition. Phys. Lett.
**2020**, A384, 126570. [Google Scholar] [CrossRef] - Trugenberger, C.; Diamantini, M.C.; Poccia, N.; Nogueira, F.S.; Vinokur, V.M. Magnetic monopoles and superinsulation in Josephson junction arrays. Quant. Rep.
**2020**, 2, 388–399. [Google Scholar] [CrossRef] - Vinokur, V.M. Superinsulator and quantum synchronization. Nature
**2008**, 452, 613–615. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Baturina, T.I.; Vinokur, V.M. Superinsulator-superconductor duality in two dimensions. Ann. Phys.
**2013**, 331, 236–257. [Google Scholar] [CrossRef] [Green Version] - Trugenberger, C.A. Superinsulators, Bose Metals, High-T
_{c}Superconductors: The Quantum Physics of Emergent Magnetic Monopoles; World Scientific: Singapore, 2022. [Google Scholar] - Parra, C.; Niestemski, F.C.; Contryman, A.W.; Giraldo-Gallo, P.; Geballe, T.H.; Fisher, I.R.; Manoharan, H.C. Signatures of two-dimensional superconductivity emerging within a three-dimensional host superconductor. Proc. Natl. Acad. Sci. USA
**2021**, 118, e2017810118. [Google Scholar] [CrossRef] - Diamantini, M.C.; Trugenberger, C.A.; Vinokur, V.M. Quantum magnetic monopole condensate. Comm. Phys.
**2021**, 4, 25. [Google Scholar] [CrossRef] - Quevedo, F.; Trugenberger, C.A. Phases of antisymmetric tensor field theories. Nucl. Phys.
**1997**, B501, 143–172. [Google Scholar] [CrossRef] [Green Version] - Polyakov, A. Confining strings. Nucl. Phys.
**1997**, B486, 23–33. [Google Scholar] [CrossRef] [Green Version] - Tinkham, M. Introduction to Superconductivity; Dover Publications: New York, NY, USA, 1996. [Google Scholar]
- Diamantini, M.C.; Postolova, S.V.; Mironov, A.Y.; Gammaitoni, L.; Strunk, C.; Trugenberger, C.A.; Vinokur, V.M. Direct probe of the interior of an electric pion in a Cooper pair superinsulator. Nat. Comm. Phys
**2020**, 3, 142. [Google Scholar] [CrossRef] - Mironov, A.; Diamantini, M.C.; Trugenberger, C.A.; Vinokur, V.M. Relaxation electrodynamics of superinsulators. Sci. Rep.
**2022**, 12, 19918. [Google Scholar] [CrossRef] - Wilczek, F. Two applications of axion electrodynamics. Phys. Rev. Lett.
**1987**, 58, 1799–1802. [Google Scholar] [CrossRef] [PubMed] - Metliski, M.A.; Kane, C.L.; Fisher, M.P.A. Bosonic topological insulator in three dimensions and the statistical Witten effect. Phys. Rev.
**2013**, B88, 035131. [Google Scholar] [CrossRef] [Green Version] - Witten, E. Dyons of Charge θ/2π. Phys. Lett.
**1979**, 86, 283–287. [Google Scholar] [CrossRef] [Green Version] - Cardy, J.L.; Rabinovici, E. Phase structure of Z(p) models in presence of theta parameter. Nucl. Phys.
**1982**, B205, 1–16. [Google Scholar] [CrossRef] - Diamantini, M.C.; Quevedo, F.; Trugenberger, C.A. Confining Strings with Topological Term. Phys. Lett
**1997**, B396, 115–121. [Google Scholar] [CrossRef] [Green Version] - Diamantini, M.C.; Trugenberger, C.A.; Vinokur, V.M. Topological nature of high-temperature superconductivity. Adv. Quantum Technol.
**2021**, 4, 2000135. [Google Scholar] [CrossRef] - Wilczek, F. Fractional Statistics and Anyon Superconductivity; World Scientific: Singapore, 1990. [Google Scholar]
- Moy, B.; Goldman, H.; Sohal, R.; Fradkin, E. Theory of oblique topological insulators. arXiv
**2022**, arXiv:2206.07725. [Google Scholar] - Campi, G.; Bianconi, A.; Poccia, N.; Bianconi, G.; Barba, L.; Arrighetti, G.; Ricci, A. Inhomogeneity of charge-density-wave order and quenched disorder in a high-T
_{c}superconductor. Nature**2015**, 525, 359–362. [Google Scholar] [CrossRef] [Green Version] - Campi, G.; Bianconi, A. Functional nanoscale phase separation and intertwined order in quantum complex materials. Condens. Matter
**2021**, 6, 40. [Google Scholar] [CrossRef] - Pelc, D.; Vučković, M.; Grbić, M.S.; Požek, M.; Yu, G.; Sasagawa, T.; Barišić, N. Emergence of superconductivity in the cuprates via a universal percolation process. Nature Comm.
**2018**, 9, 4327. [Google Scholar] [CrossRef] [Green Version] - Diamantini, M.C.; Trugenberger, C.A.; Vinokur, V.M. Effective magnetic monopole mechanism for localized electron pairing in HTS. Front. Phys.
**2022**, 10, 909310. [Google Scholar] [CrossRef] - Mukhin, S. Euclidean Q-balls of fluctuating SDW/CDW in the nested Hubbard model of high-T
_{c}superconductors as the origin of pseudogap and superconducting behaviour. Condens. Matter**2022**, 7, 31. [Google Scholar] [CrossRef] - Proust, C.; Taillefer, L. The Remarkable Underlying Ground States of Cuprate Superconductors. Annu. Rev. Condesne Matter
**2019**, 10, 409–429. [Google Scholar] [CrossRef] [Green Version] - Xia, J.; Schemm, E.; Deutscher, G.; Kivelson, S.A.; Bonn, D.A.; Hardy, W.N.; Kapitulnik, A. Polar Kerr effect measurements of YBa2Cu3O6+x: Evidence for broken symmetry near the pseudogap temperature. Phys. Rev. Lett
**2008**, 100, 127002. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhou, P.; Chen, L.; Liu, Y.; Sochnikov, I.; Bollinger, A.T.; Han, M.G.; Natelson, D. Electron pairing in the pseudogap state revealed by shot noise in copper oxide junctions. Nature
**2019**, 527, 493–496. [Google Scholar] [CrossRef] - Barisic, N.; Chan, M.K.; Greven, M. Universal sheet resistance and revised phase diagram of the cuprate high-temperature superconductors. Proc. Natl. Acad. Sci. USA
**2013**, 110, 12235. [Google Scholar] [CrossRef] [Green Version] - Chalker, J.T.; Coddington, P.D. Percolation, quantum tunnelling and the integer Hall effect. J. Phys.
**1988**, C21, 2665. [Google Scholar] [CrossRef] - Sato, Y. Thermodynamic evidence for a nematic phase transition at the onset of the pseudogap in YBa2Cu3Oy. Nat. Phys.
**2017**, 13, 1074. [Google Scholar] [CrossRef] [Green Version] - Legros, A. Universal T-linear resistivity and Planckian limit in overdoped cuprates. Nature Physics
**2019**, 15, 142. [Google Scholar] [CrossRef] [Green Version] - Diamantini, M.C.; Trugenberger, C.A.; Bollinger, A.T.; Vinokur, V.M.; Bozovic, I. Topological model of the pseudogap state: Experimental signatures. Front. Phys.
**2022**, 9, 756760. [Google Scholar] [CrossRef] - Finkel’shtein, A.M. Superconducting transition temperature in amorphous films. JETP Lett.
**1987**, 45, 46–49. [Google Scholar] - Ovadia, M.; Sacépé, B.; Shahar, D. Electron-phono decoupling in disordered insulators. Phys. Rev. Lett.
**2009**, 102, 176802. [Google Scholar] [CrossRef] [PubMed] - Tamir, I. Excessive noise as a test for many-body localization. Phys. Rev.
**2019**, B99, 035135. [Google Scholar] [CrossRef] [Green Version] - Diamantini, M.C.; Trugenberger, C.A.; Vinokur, V.M. Superconductor-to-insulator transition in absence of disorder. Phys. Rev.
**2021**, B103, 174516. [Google Scholar] [CrossRef] - James, A.J.; Konik, R.M.; Robinson, N.J. Nonthermal states arising from confinement in one and two dimensions. Phys. Rev. Lett.
**2019**, 122, 130603. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mazza, P.P.; Perfetto, G.; Lerose, A.; Collura, M.; Gambassi, A. Suppression of transport in non- disordered quantum spin chains due to confined excitations. Phys. Rev.
**2019**, B99, 180302. [Google Scholar] [CrossRef]

**Figure 1.**A magnetic monopole with its Dirac string, an infinitely thin and long solenoid bringing in the magnetic flux from infinity. The Dirac string is a coordinate singularity and can be displaced to another position by a gauge transformation.

**Figure 4.**A (non-relativistic) magnetic monopole instanton tunnelling between the one-vortex sector and the zero-vortex sector in a 2D quantum film.

**Figure 5.**Short magnetic dipoles when the vortices have tension, panel (

**a**); free magnetic monopoles when the vortices become tensionless Dirac strings, panel (

**b**).

**Figure 6.**Logarithmic plot of the sheet resistance of a NbTiN film as a function of $1/T$. The dashed straight line corresponds to the usual activated behaviour of an insulator. The data show a hyperactivated behaviour fitting the divergent BKT behaviour [10], with ${T}_{c}=0.062{\phantom{\rule{4pt}{0ex}}}^{\circ}K$ without an applied magnetic field and with ${T}_{c}=0.175{\phantom{\rule{4pt}{0ex}}}^{\circ}K$ at $B=0.3$ T. From [17], ©Elsevier (2013).

**Figure 7.**The I(V) curves of a superinsulating NbTiN quantum film at 50 mK, clearly showing the two kinks and three regimes corresponding to the electric Meissner state, the mixed state and the normal insulating state. From [24], Creative Commons Attribution 4.0.

**Figure 8.**The transition from hyperactivated to metallic behaviour of superinsulators as the sample size is decreased, showing asymptotic freedom in the electric pion interior. From [24], Creative Commons Attribution 4.0.

**Figure 9.**Dynamic response of a NbTiN quantum film. Panel (

**a**): the difference between the normal insulator at 300 mK and the superinsulator at 20 mK. Panel (

**b**): the scaling of the shift time ${t}_{\mathrm{sh}}$ as a function of the reduced voltage $(V-{V}_{\mathrm{c}1})$ (with ${V}_{\mathrm{c}1}$ denoted by ${V}_{p}$ here). The two different critical exponents correspond to jumps from the Meissner state to the mixed state and from the Meissner state to the normal insulator, respectively. From [25], Creative Commons Attribution 4.0.

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**MDPI and ACS Style**

Trugenberger, C.A.
Magnetic Monopoles, Dyons and Confinement in Quantum Matter. *Condens. Matter* **2023**, *8*, 2.
https://doi.org/10.3390/condmat8010002

**AMA Style**

Trugenberger CA.
Magnetic Monopoles, Dyons and Confinement in Quantum Matter. *Condensed Matter*. 2023; 8(1):2.
https://doi.org/10.3390/condmat8010002

**Chicago/Turabian Style**

Trugenberger, Carlo A.
2023. "Magnetic Monopoles, Dyons and Confinement in Quantum Matter" *Condensed Matter* 8, no. 1: 2.
https://doi.org/10.3390/condmat8010002