Post-doctoral researcher in theoretical physics

Department of Physics & Research and Education Center for Natural Sciences

Keio University

34^{th} Building, office 404

Hiyoshi 3-14-1, Yokohama, Kanagawa 223-8522 JAPAN

on this page: Publications – Popular – PhD thesis – MSc thesis

other pages: CV

My papers on: arXiv • Inspire • Google Scholar • ResearcherID • ORCID

**Stability and absence of a Tower of States in ferrimagnets**L. Rademaker, A.J. Beekman, J. van Wezel, arXiv:1909.11381

Systems with broken symmetry and a finite spontaneous magnetization are shown to have a degenerate, stable exact ground state and no tower of low-lying states.**Introduction to spontaneous symmetry breaking**A.J. Beekman, L. Rademaker, J. van Wezel, SciPost Phys. Lect. Notes 11 (2019), arXiv:1909.01820

Lecture notes with a modern development on spontaneous symmetry breaking, focussing on singular limits and finite-size systems, including Nambu--Goldstone modes, topological defects and gauge fields.**Theory of generalized Josephson effects**A.J. Beekman, arXiv:1907.13284

The Josephson effect, the flow of DC current without external potential difference due to a difference in order parameter values, is generalized to any kind of spontaneous symmetry breaking.**Charged and neutral fixed points in the O(N)+O(N)-model with Abelian gauge fields**A.J. Beekman, G. Fejos, Phys. Rev. D, 100, 016005 (2019), arXiv:1903.05331

Using a functional renormalization group method, we find that at least three charged fixed points exist in a two-scalar modes coupled Abelian gauge fields. This implies that the quantum solid-to-hexatic phase transition is second-order.**Dual gauge field theory of quantum liquid crystals in three dimensions**A.J. Beekman, J. Nissinen, K. Wu, J. Zaanen, Phys. Rev. B 96, 165115 (2017) (Editors' suggestion), arXiv:1703.03157

Combining the results for quantum liquid crystals in two dimensions and vortex–boson duality in three dimensions, we develop the dual theory of the solid-to-liquid crystal quantum phase transition in 3+1D. Phonons dualize into two-form gauge fields, and dislocations are string worldsheets with a Burgers vector topological charge. Although this mathematical structure is rather different from vector gauge fields, the phenomenology of the quantum liquid crystal phases is very similar to the situation in two dimensions. Rotational Goldstone modes emerge in any plane with fully restored translational symmetry.**Dual gauge field theory of quantum liquid crystals in two dimensions**A.J. Beekman, J. Nissinen, K. Wu, K. Liu, R.-J. Slager, Z. Nussinov, V. Cvetkovic, J. Zaanen, Phys. Rep. 683, 1–110 (2017), arXiv:1603.04254

An extenstive review of using a Abelian-Higgs (vortex–boson) type duality to describe dislocation-mediated quantum melting of solids into liquid crystals. Phonons are dualized into*stress photons*: gauge fields mediating stress interactions between dislocations. Upon losing translational rigidity, these stress photons become gapped by the Anderson–Higgs mechanism. Consecutive melting results in a solid → smectic → nematic → superfluid hierarchy. A rotational Goldstone mode emerges in the nematic phase. We also couple in the electromagnetic field, with explicit prediction for optical conductivity, electron loss etc.**Criteria for the absence of quantum fluctuations after spontaneous symmetry breaking**A.J. Beekman, Ann. Phys. 361, 461–489 (2015), arXiv:1408.1691

The Heisenberg ferromagnet is a very peculiar system featuring spontaneous symmetry breaking: its classical groundstate unusually is an exact eigenstate of the Hamiltonian. I identify in fact seven special features and show how they are related, and how they can be generalized to other systems using symmetry algebra notions. It turns out one should consider*all*possible order parameter operators to get a complete picture of the quantum fluctuations and of the low-energy spectrum.**Photodrive of magnetic bubbles via magnetoelastic waves**N. Ogawa, W. Koshibae, A.J. Beekman, N. Nagaosa, M. Kubota, M. Kawasaki and Y. Tokura, PNAS 112(29), 8977–8981 (2015)

My colleague Ogawa manipulates magnetic bubbles by illuminating them with*off-resonant*laser light. He makes them appear, move, extend and coalesce. We were able to give an nice intuitive explanation for these phenomena: the light excites magnetoelastic waves, and those combined elastic–spin waves couple to the domain walls of the bubbles,*more strongly at higher curvature*.**Theory of magnon-skyrmion scattering in chiral magnets**J. Iwasaki, A.J. Beekman and N. Nagaosa, Phys. Rev. B 89, 064412 (2014), arXiv:1309.2361

Studying the elementary process of scattering of magnons by a single skyrmion in chiral magnets, we find strong skew-scattering that is wavenumber dependent. In turn the skyrmion moves in the opposite direction. Our analysis shows that the skyrmion acts as a fictious magnetic field on the magnons related to its Berry phase, and that the backaction can be viewed as elastic scattering when taking into account the peculiar, non-Newtonian momentum of the skyrmion.**Deconfining the rotational Goldstone mode: the superconducting nematic liquid crystal in 2+1D**A.J. Beekman, K. Wu, V. Cvetkovic and J. Zaanen, Phys. Rev. B 88, 024121 (2013), arXiv:1301.7329

In solids only translational Goldstone modes, phonons, appear even though both translational and rotational symmetry is broken. We show how the rotational modes are confined in the solid and get dynamically deconfined when melting into an isotropic liquid crystal. Surpisingly, in the dual language, the medium for the rotational modes is the condensate of dislocations itself.**Type-II Bose-Mott insulators**A.J. Beekman and J. Zaanen, Phys. Rev. B 86, 125129 (2012) (Editors' suggestion), arXiv:1207.0286

In type-II superconductors, a magnetic field penetrates in the form of quantized flux lines. Using the duality techniques of "Condensing Nielsen–Olesen strings ...", the Bose-Mott insulator is a dual superconductor, where now not magnetic field but electric current is expelled by the condensate. Consequently, we predict the existence of a regular lattice of quantized current lines in such systems under externally applied current.**The emergence of gauge invariance: the stay-at-home gauge versus local–global duality**J. Zaanen and A.J. Beekman, Ann Phys 327(4):1146–1161 (2012), arXiv:1108.2791

Whenever there is a conserved current, the conservation law can be explicitly enforced by expressing it as the curl of a gauge field, giving rise to an emergent gauge symmetry. Conversely, in any state where dynamic fluctuations freeze out, there is another emergent gauge symmetry related to local number conservation, called stay-at-home gauge. We show that these two seemingly separate emerging symmetries are actually two sides of the same coin. This is applied to quantum elasticity, where we find that a relativistic quantum nematic is the realization of linearized gravity.**Electrodynamics of Abrikosov vortices: the field theoretical formulation**A.J. Beekman and J. Zaanen, Front. Phys. 6(4):357–369 (2011), arXiv:1106.3946

Even though vortices in type-II superconductors have been known and studied for over 50 years, the electrodynamical phenomena such as dynamic screening and radiation of moving vortices have always been treated as separate individual issues. Using the vortex worldsheet formalism, we can capture all magnetic*and*electric relations in a single equation. As an interesting elaboration, we also derive the electrodynamics of two-form sources, where the Maxwell field strength itself obtains a gauge freedom.**Condensing Nielsen-Olesen strings and the vortex-boson duality in 3+1 and higher dimensions**A.J. Beekman, D. Sadri and J. Zaanen, New. J. Phys. 13:033004 (2011), arXiv:1006.2267

By representing vortices as fluctuating fields, one can describe an order–disorder phase transition as the proliferation of vortices. This was established in 2+1 dimensions by Hagen Kleinert and many others over 20 years ago, because then vortices are exactly like point particles, and can therefore be handled by common quantum field theory techniques. In higher dimensions, the vortices are extened objects, and up to now it was not known how to formalize their collective behaviour. From the known physics of the superfluid to Bose-Mott insulator transition, we argue how such a theory must arise. The simple results are directly applicable to other phase transitions.

- (Dutch)
**Vortexdualiteit**Nederlands Tijdschrift voor Natuurkunde, Dec 2012 PDF (1.0MB) - (Dutch)
**100 jaar supergeleiding**at sargasso.nl and sciencepalooza.nl part 1, part 2

In my dissertation at Leiden University under supervision of Jan Zaanen, I explore the implications of regarding a topological defect line in 3+1 dimesions as a world sheet.
From the Goldstone mode in a symmetry-broken state, the dynamics are obtained by a duality transformation, where each world sheet component has
a precise physical meaning. Furthermore, the order-to-disorder (quantum) phase transition is now viewed as the
*proliferation of topological defects*, and this well-known *vortex–boson duality* is generalized to 3+1
dimensions. The highlight result is the prediction of vortex lines of quantized electric current in charged Bose-Mott
insulators, with possible relevance to underdoped cuprate superconductors

» Vortex duality in higher dimensions (pdf, 4.4 MB) — also available at Leiden University Library

To obtain my Master's degree (Universiteit van Amsterdam, Institute for Theoretical Physics), I looked into the earlier work of my supervisor Sander Bais and some of his PhD students, in which they explored the extension of group symmetry to so-called quantum doubles in 2+1-dimensional systems. In this way, the quantum numbers of topological excitations are taken into account as well as those of the regular (gauge) particles. Furthermore, this formalism provides directly for a description of the braid properties of the particles. Such structure can for example arise in gauge theories where the original symmetry is broken down to a finite group.

A next step was to look at the possible condensate phases in such systems; that is, we imagine the new ground state of the system to be one filled with background particles, all represented by a certain state vector of one of the irreducible representations of the symmetry algebra. Because of the braiding with the condensate particles, not every excitation in the new phase can exist freely, and some of them will be ‘confined’.

In my thesis work, I have calculated through almost all possible condensate states in models where the original symmetry is that of an even dihedral group, the symmetry group of a regular *n*-gon, with *n* even. This leads to a redefinition of the braiding properties of the particles that exist freely in the condensate state. Furthermore, in some condensates the remaining symmetry algebra turns out to be much larger than one would naively expect from just looking at the breaking of the topological and regular parts of the quantum double.

» Quantum double symmetries of the even dihedral groups and their breaking (pdf, 1.0 MB)