Magnetic Translation Group Research Articles

Topological heavy fermions in magnetic field

The recently introduced topological heavy fermion model (THFM) provides a means for interpreting the low-energy electronic degrees of freedom of the magic angle twisted bilayer graphene as hybridization amidst highly dispersing topological conduction and weakly dispersing localized heavy fermions. In order to understand the Landau quantization of the ensuing electronic spectrum, a generalization of THFM to include the magnetic field B is desired, but currently missing. Here we provide a systematic derivation of the THFM in B and solve the resulting model to obtain the interacting Hofstadter spectra for single particle charged excitations. While naive minimal substitution within THFM fails to correctly account for the total number of magnetic subbands within the narrow band i.e., its total Chern number, our method—based on projecting the light and heavy fermions onto the irreducible representations of the magnetic translation group— reproduces the correct total Chern number. Analytical results presented here offer an intuitive understanding of the nature of the (strongly interacting) Hofstadter bands.

Hom-Lie-Virasoro symmetries in Bloch electron systems and quantum plane in tight binding models

We discuss the Curtright-Zachos (CZ) deformation of the Virasoro algebra and its extensions in terms of magnetic translation (MT) group in a discrete Bloch electron system, so-called the tight binding model (TBM), as well as in its continuous system. We verify that the CZ generators are essentially composed of a specific combination of MT operators representing deformed and undeformed U(1) translational groups, which determine phase factors for a ⁎-bracket commutator. The phase factors can be formulated as a ⁎-ordered product of the commutable U(1) operators by interpreting the AB phase factor of discrete MT action as fluctuation parameter q of a quantum plane. We also show that some sequences of TBM Hamiltonians are described by the CZ generators.

Symmetry indicators in commensurate magnetic flux

We derive a framework to apply topological quantum chemistry in systems subject to magnetic flux. We start by deriving the action of spatial symmetry operators in a uniform magnetic field, which extends Zak's magnetic translation groups to all crystal symmetry groups. Ultimately, the magnetic symmetries form a projective representation of the crystal symmetry group. As a consequence, band representations acquire an extra gauge invariant phase compared to the non-magnetic theory. Thus, the theory of symmetry indicators is distinct from the non-magnetic case. We give examples of new symmetry indicators that appear at $\pi$ flux. Finally, we apply our results to an obstructed atomic insulator with corner states in a magnetic field. The symmetry indicators reveal a topological-to-trivial phase transition at finite flux, which is confirmed by a Hofstadter butterfly calculation. The bulk phase transition provides a new probe of higher order topology in certain obstructed atomic insulators.

Narrow bands in magnetic field and strong-coupling Hofstadter spectra

We develop a new, efficient, and general method to determine the Hofstadter spectrum of isolated narrow bands. The method works for topological as well as for trivial narrow bands by projecting the zero $\mathbf{B}$-field hybrid Wannier states -- which are localized in one direction and Bloch extended in another direction -- onto a representation of the magnetic translation group in the Landau gauge. We then apply this method to find, for the first time, the Hofstadter spectrum for the exact single particle charged excitations in the strong coupling limit of the magic angle twisted bilayer graphene at the charge neutrality point and at $|\nu|=2$ down to low magnetic fields when the flux through the moir\'e unit cell is only $\sim 1/25$ of the electronic flux quantum i.e. $\sim 1$T at the first magic angle. The resulting spectra provide a means to investigate Landau quantization of the quasiparticles even if their dispersion is interaction induced.

Magnetic Bloch theorem and reentrant flat bands in twisted bilayer graphene at 2π flux

Bloch's theorem is the centerpiece of topological band theory, which itself has defined an era of quantum materials research. However, Bloch's theorem is broken by a perpendicular magnetic field, making it difficult to study topological systems in strong flux. For the first time, moir\'e materials have made this problem experimentally relevant, and its solution is the focus of this work. We construct gauge-invariant irreps of the magnetic translation group at $2\pi$ flux on infinite boundary conditions, allowing us to give analytical expressions in terms of the Siegel theta function for the magnetic Bloch Hamiltonian, non-Abelian Wilson loop, and many-body form factors. We illustrate our formalism using a simple square lattice model and the Bistritzer-MacDonald Hamiltonian of twisted bilayer graphene, obtaining reentrant ground states at $2\pi$ flux under the Coulomb interaction.

Reentrant Correlated Insulators in Twisted Bilayer Graphene at 25T (2π Flux).

Twisted bilayer graphene (TBG) is remarkable for its topological flat bands, which drive strongly interacting physics at integer fillings, and its simple theoretical description facilitated by the Bistritzer-MacDonald Hamiltonian, a continuum model coupling two Dirac fermions. Because of the large moiré unit cell, TBG offers the unprecedented opportunity to observe reentrant Hofstadter phases in laboratory-strength magnetic fields near 25T. This Letter is devoted to magic angle TBG at 2π flux where the magnetic translation group commutes. We use a newly developed gauge-invariant formalism to determine the exact single-particle band structure and topology. We find that the characteristic TBG flat bands reemerge at 2π flux, but, due to the magnetic field breaking C_{2z}T, they split and acquire Chern number ±1. We show that reentrant correlated insulating states appear at 2π flux driven by the Coulomb interaction at integer fillings, and we predict the characteristic Landau fans from their excitation spectrum.

Theory of Hofstadter superconductors

We study mean-field states resulting from the pairing of electrons in time-reversal broken fractal Hofstadter bands, which arise in two-dimensional lattices where the unit cell traps magnetic flux $\Phi = (p/q)\Phi_0$ comparable to the flux quantum $\Phi_0 = h/e$. It is established that the dimension and degeneracy of the irreducible representations of the magnetic translation group (MTG) furnished by the charge 2e pairing fields have different properties from those furnished by single particle Bloch states, and in particular are shown to depend on the parity of the denominator $q$. We explore this symmetry analysis to formulate a Ginzburg-Landau theory describing the thermodynamic properties of Hofstadter superconductors at arbitrary rational flux $\Phi = (p/q)\Phi_0$ in terms of a multicomponent order parameter that describes the finite momentum pairing of electrons across different Fermi surface patches. This phenomenological theory leads to a rich phase diagram characterized by different symmetry breaking patterns of the MTG, which can be interpreted as distinct classes of vortex lattices. A class of $\mathbb{Z}_q$-symmetric Hofstadter SCs is identified, in which the MTG breaks down to a $\mathbb{Z}_q$ subgroup. We study the topological properties of such $\mathbb{Z}_q$-symmetric Hofstadter SCs and show that the parity of the Chern numbers is fixed by the parity of $q$. We identify the conditions for the realization of Bogoliubov Fermi surfaces in the presence of parity and MTG symmetries, establishing a novel topological invariant capturing the existence of such charge-neutral gapless excitations. Our findings, which could bear relevance to the description of re-entrant superconductivity in moir\'e systems in the Hofstadter regime, establish Hofstadter SC as a fertile setting to explore symmetry broken and topological orders.

Hofstadter Topology: Noncrystalline Topological Materials at High Flux.

The Hofstadter problem is the lattice analog of the quantum Hall effect and is the paradigmatic example of topology induced by an applied magnetic field. Conventionally, the Hofstadter problem involves adding ∼10^{4} T magnetic fields to a trivial band structure. In this Letter, we show that when a magnetic field is added to an initially topological band structure, a wealth of possible phases emerges. Remarkably, we find topological phases that cannot be realized in any crystalline insulators. We prove that threading magnetic flux through a Hamiltonian with a nonzero Chern number or mirror Chern number enforces a phase transition at fixed filling and that a 2D Hamiltonian with a nontrivial Kane-Mele invariant can be classified as a 3D topological insulator (TI) or 3D weak TI phase in periodic flux. We then study fragile topology protected by the product of twofold rotation and time reversal and show that there exists a higher order TI phase where corner modes are pumped by flux. We show that a model of twisted bilayer graphene realizes this phase. Our results rely primarily on the magnetic translation group that exists at rational values of the flux. The advent of Moiré lattices renders our work relevant experimentally. Due to the enlarged Moiré unit cell, it is possible for laboratory-strength fields to reach one flux per plaquette and allow access to our proposed Hofstadter topological phase.

Simulating infinite vortex lattices in superfluids

We present an efficient framework to numerically treat infinite periodic vortex lattices in rotating superfluids described by the Gross–Pitaevskii theory. The commonly used split-step Fourier (SSF) spectral methods are inapplicable to such systems as the standard Fourier transform does not respect the boundary conditions mandated by the magnetic translation group. We present a generalisation of the SSF method which incorporates the correct boundary conditions by employing the so-called magnetic Fourier transform. We test the method and show that it reduces to known results in the lowest-Landau-level regime. While we focus on rotating scalar superfluids for simplicity, the framework can be naturally extended to treat multicomponent systems and systems under more general ‘synthetic’ gauge fields.

From doubled Chern–Simons–Maxwell lattice gauge theory to extensions of the toric code

We regularize compact and non-compact Abelian Chern–Simons–Maxwell theories on a spatial lattice using the Hamiltonian formulation. We consider a doubled theory with gauge fields living on a lattice and its dual lattice. The Hilbert space of the theory is a product of local Hilbert spaces, each associated with a link and the corresponding dual link. The two electric field operators associated with the link-pair do not commute. In the non-compact case with gauge group R, each local Hilbert space is analogous to the one of a charged “particle” moving in the link-pair group space R2 in a constant “magnetic” background field. In the compact case, the link-pair group space is a torus U(1)2 threaded by k units of quantized “magnetic” flux, with k being the level of the Chern–Simons theory. The holonomies of the torus U(1)2 give rise to two self-adjoint extension parameters, which form two non-dynamical background lattice gauge fields that explicitly break the manifest gauge symmetry from U(1) to Z(k). The local Hilbert space of a link-pair then decomposes into representations of a magnetic translation group. In the pure Chern–Simons limit of a large “photon” mass, this results in a Z(k)-symmetric variant of Kitaev’s toric code, self-adjointly extended by the two non-dynamical background lattice gauge fields. Electric charges on the original lattice and on the dual lattice obey mutually anyonic statistics with the statistics angle 2πk. Non-Abelian U(k) Berry gauge fields that arise from the self-adjoint extension parameters may be interesting in the context of quantum information processing.

Energy bands for finite two-dimensional systems in a quantised magnetic field: the symmetry of the model

Analysis of properties of an electron (electrons) in a periodic two- dimensional potential and quantised magnetic field is presented. The key role in this description is played by the symmetry of the model determined by the magnetic trans- lation group, together with the symmetric and unitary group according to the duality of Weyl. The magnetic translation group is described in detail along with its irre- ducible representations which form the Brillouin zone in magnetic field. Together with the other two mentioned groups, it allows to characterize the system provid- ing some good quantum numbers. These groups facilitate determination of the band structure through the diagonalisation of the eigenvalue problem in the base adapted to the considered symmetry. The difference between the Brillouin zone with and without magneticfieldispointedout.Discussionisconcentrated onthefinitetwo-dimensional systems, closed by use of Born-von Karman boundary conditions.

Band structure, Brillouin zone, and condensation of states for an itinerant electron in a magnetic quantum dot

The model of a hopping electron on a two-dimensional lattice placed in a quantised magnetic field is considered. The symmetry of the system is described by the magnetic translation group. Its maximal abelian subgroup defines magnetic Brillouin zone while irreducible representations determine an energy band structure. It turns out, due to an equivalence relation between irreducible representations, that for the characterization of the energy band it is sufficient to use only some quasimomenta from magnetic Brillouin zone. These quasimomenta constitute the magnetic band Brillouin zone.

On Quantum Mechanics with a Magnetic Field on ℝ n and on a Torus $\mathbb{T}^{n}$ , and Their Relation

We show in elementary terms the equivalence in a general gauge of a U(1)-gauge theory of a scalar charged particle on a torus T^n = R^n/L to the analogous theory on R^n constrained by quasiperiodicity under translations in the lattice L. The latter theory provides a global description of the former: the quasiperiodic wavefunctions defined on R^n play the role of sections of the associated hermitean line bundle E on T^n, since also E admits a global description as a quotient. The components of the covariant derivatives corresponding to a constant (necessarily integral) magnetic field B = dA generate a Lie algebra g_Q and together with the periodic functions the algebra of observables O_Q . The non-abelian part of g_Q is a Heisenberg Lie algebra with the electric charge operator Q as the central generator; the corresponding Lie group G_Q acts on the Hilbert space as the translation group up to phase factors. Also the space of sections of E is mapped into itself by g in G_Q . We identify the socalled magnetic translation group as a subgroup of the observables' group Y_Q . We determine the unitary irreducible representations of O_Q, Y_Q corresponding to integer charges and for each of them an associated orthonormal basis explicitly in configuration space. We also clarify how in the n = 2m case a holomorphic structure and Theta functions arise on the associated complex torus. These results apply equally well to the physics of charged scalar particles on R^n and on T^n in the presence of periodic magnetic field B and scalar potential. They are also necessary preliminary steps for the application to these theories of the deformation procedure induced by Drinfel'd twists.

Nambu quantum mechanics on discrete 3-tori

We propose a quantization of linear, volume preserving, maps on the discrete and finite 3-torus represented by elements of the group . These flows can be considered as special motions of the Nambu dynamics (linear Nambu flows) in the three-dimensional toroidal phase space and are characterized by invariant vectors a of . We quantize all such flows, which are necessarily restricted on a planar two-dimensional phase space, embedded in the 3-torus, transverse to the vector a. The corresponding maps belong to the little group of , which is an subgroup. The associated linear Nambu maps are generated by a pair of linear and quadratic Hamiltonians (Clebsch–Monge potentials of the flow) and the corresponding quantum maps realize the metaplectic representation of on the discrete group of three-dimensional magnetic translations, i.e. the non-commutative 3-torus with a deformation parameter the Nth root of unity. Other potential applications of our construction are related to the quantization of deterministic chaos in turbulent maps as well as to quantum tomography of three-dimensional objects.

Multielectron irreducible representations of the magnetic translation group

AbstractThe symmetry of a 2D Bloch electron in the magnetic field is described by the magnetic translation groups (MTG). Their irreducible representations (IR), parameterized by numbers k1, k2 and s = 1, classify the states of an electron. This classification is performed by using the symmetry adapted bases, i.e. the bases of irreducible representations generated with the help of projection operator technique.The multielectron system will be described by appropriate tensor products of these “one electron” IRs. From the other hand the same description should be obtained by consideration of irreducible representations of MTG with the parameter s identified as the number of electrons in the Landau level. These methods of classification of states are discussed for finite dimensional planar rectangular lattices (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The structure of magnetic translation group for a finite system

The symmetry group (magnetic translation group) of a Bloch electron in a finite two-dimensional lattice in a perpendicular magnetic field is discussed. The details of the structure of this group depend on a gauge of a vector potential of a magnetic field. For two gauges, symmetric and Landau, the group elements are presented in terms of generators, and the relation between the group properties and the number of magnetic fluxes through the elementary cell of the lattice is pointed out.

Discrete accidental symmetry for a particle in a constant magnetic field on a torus

A classical particle in a constant magnetic field undergoes cyclotron motion on a circular orbit. At the quantum level, the fact that all classical orbits are closed gives rise to degeneracies in the spectrum. It is well-known that the spectrum of a charged particle in a constant magnetic field consists of infinitely degenerate Landau levels. Just as for the 1 / r and r 2 potentials, one thus expects some hidden accidental symmetry, in this case with infinite-dimensional representations. Indeed, the position of the center of the cyclotron circle plays the role of a Runge–Lenz vector. After identifying the corresponding accidental symmetry algebra, we re-analyze the system in a finite periodic volume. Interestingly, similar to the quantum mechanical breaking of CP invariance due to the θ -vacuum angle in non-Abelian gauge theories, quantum effects due to two self-adjoint extension parameters θ x and θ y explicitly break the continuous translation invariance of the classical theory. This reduces the symmetry to a discrete magnetic translation group and leads to finite degeneracy. Similar to a particle moving on a cone, a particle in a constant magnetic field shows a very peculiar realization of accidental symmetry in quantum mechanics.

The magnetic translation group for a finite system and the Born–Karman boundary condition

The symmetry of the electron on the square planar lattice and in the magnetic field perpendicular to it is described by the magnetic translation group. The boundary condition applied to the system provides, as a consequence of the boundary equivalence, the additional equations for the possible value of magnetic fluxes penetrating the elementary cell of the lattice. These restrictions depend on the chosen gauge.

Magnetic translation group for graphene

The group-theoretical study of an electron in a two-dimensional lattice in the presence of a magnetic field yields lattices of lines of magnetic flux. The symmetry of the system is described by the magnetic translation group. The structure of the group has been determined for a selected gauge. The calculation was performed for a finite lattice corresponding to nano-scale carbon structures.

Construction of superconducting pairs in the two‐dimensional electron gas for the model of itinerant electrons

AbstractWe consider a 3 × 3 finite plane lattice with Born–von Karman boundary conditions, threaded by magnetic field perpendicular to it, such that two quanta of flux cover exactly three Bravais cells. The corresponding magnetic translation group is determined, and the single‐electron energy levels of the system are labelled by three‐dimensional irreps of this group. We perform also the Clebsch–Gordan decomposition, and interpret selected irreps as the relevant superconducting pairs. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)