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One of the key features of hole-doped cuprates is the presence of an extended pseudogap phase, whose microscopic origin has been the subject of intense investigation since its discovery and is believed to be crucial for understanding high-temperature superconductivity. Various explanations have been proposed for the pseudogap, including links to symmetry-breaking orders such as stripes or pairing, and the emergence of novel fractionalized Fermi liquid (FL*) phases. The topological nature of the FL* phase has been identified as a scenario compatible with a small Fermi surface without symmetry breaking, as suggested experimentally. With recent experimental and numerical studies supporting an intricate relationship between stripe order and the pseudogap phase, we here propose an alternative FL* scenario: a fractionalized Fermi liquid with a geometric origin (GFL*) driven by fluctuating domain walls. The essential mechanism behind our proposal is hidden order, where the proliferation of domain walls stabilized by charge fluctuations obscures the underlying long-range antiferromagnetic order in real-space, but order is preserved in the reference frame of the background spins. As a result, well-defined fermionic quasiparticles in the form of magnetic polarons exist, which couple to $\mathbb{Z}_2$ topological excitations of the domain wall string-net condensate in the ground state and constitute a small Fermi surface. At a critical doping value, we argue that hidden order is lost, driving a transition to a regular Fermi liquid at a hidden quantum critical point (hQCP) featuring quantum critical transport properties. Our GFL* framework provides a compelling connection between the antiferromagnetic, stripe, and pseudogap phases, and suggests a possible unification of superconductivity in (electron and hole) doped cuprates and heavy fermion compounds., Comment: 11 pages, 4 figures

Quantum annealing is a computational approach designed to leverage quantum fluctuations for solving large-scale classical optimization problems. Although incorporating standard transverse field (TF) terms in the annealing process can help navigate sharp minima, the potential for achieving a scalable quantum advantage for general optimization problems remains uncertain. Here, we examine the effectiveness of including chaotic quantum driver Hamiltonians in the annealing dynamics. Specifically, we investigate driver Hamiltonians based on a bosonic spin version of the Sachdev-Ye-Kitaev (SYK) model, which features a high degree of non-locality and non-commutativity. Focusing on MaxCut instances on regular graphs, we find that a considerable proportion of SYK model instances demonstrate significant speedups, especially for challenging graph configurations. Additionally, our analysis of time-to-solution scalings for the low autocorrelation binary sequence (LABS) problem suggests that SYK-type fluctuations can outperform traditional transverse field annealing schedules in large-scale optimization tasks., Comment: 9+2 pages

Due to their inherent capabilities of capturing non-local dependencies, Transformer neural networks have quickly been established as the paradigmatic architecture for large language models and image processing. Next to these traditional applications, machine learning methods have also been demonstrated to be versatile tools in the analysis of image-like data of quantum phases of matter, e.g. given snapshots of many-body wave functions obtained in ultracold atom experiments. While local correlation structures in image-like data of physical systems can reliably be detected, identifying phases of matter characterized by global, non-local structures with interpretable machine learning methods remains a challenge. Here, we introduce the correlator Transformer (CoTra), which classifies different phases of matter while at the same time yielding full interpretability in terms of physical correlation functions. The network's underlying structure is a tailored attention mechanism, which learns efficient ways to weigh local and non-local correlations for a successful classification. We demonstrate the versatility of the CoTra by detecting local order in the Heisenberg antiferromagnet, and show that local gauge constraints in one- and two-dimensional lattice gauge theories can be identified. Furthermore, we establish that the CoTra reliably detects non-local structures in images of correlated fermions in momentum space (Cooper pairs) and that it can distinguish percolating from non-percolating images.

Fermi surface reconstruction in cuprates can lead to an abrupt change in the Fermi momentum $k_F$ between different phases. This phenomenon remains subject of debate and is at the heart of an ongoing discussion about the nature of the metallic state in the pseudogap regime. Here we study a minimal model of a $k_F$ changing crossover in the one-dimensional Fermi-Hubbard model, where a tuning of the onsite interaction leads to a crossover between a spin-$1/2$ Luttinger liquid with small Fermi momentum and a spinless chargon liquid with large Fermi momentum. We attribute this to an emergent $U(1)$ symmetry in the strongly correlated limit, which can be used to derive a modified Luttinger sum rule recovering the large Fermi momentum. We analyse Friedel oscillations at the edge of a system to directly probe the change of Fermi momentum at zero and non-zero temperature. This paves the way for a direct experimental observation of changes of the Fermi momentum using ultracold fermions in a quantum gas microscope, with possible extensions to higher dimensional systems., Comment: 8 pages, 7 figures

The simulation of high-temperature superconducting materials by implementing strongly correlated fermionic models in optical lattices is one of the major objectives in the field of analog quantum simulation. Here we show that local control and optical bilayer capabilities combined with spatially resolved measurements create a versatile toolbox to study fundamental properties of both nickelate and cuprate high-temperature superconductors. On the one hand, we present a scheme to implement a mixed-dimensional (mixD) bilayer model that has been proposed to capture the essential pairing physics of pressurized bilayer nickelates. This allows for the long-sought realization of a state with long-range superconducting order in current lattice quantum simulation machines. In particular, we show how coherent pairing correlations can be accessed in a partially particle-hole transformed and rotated basis. On the other hand, we demonstrate that control of local gates enables the observation of $d$-wave pairing order in the two-dimensional (single-layer) repulsive Fermi-Hubbard model through the simulation of a system with attractive interactions. Lastly, we introduce a scheme to measure momentum-resolved dopant densities, providing access to observables complementary to solid-state experiments -- which is of particular interest for future studies of the enigmatic pseudogap phase appearing in cuprates., Comment: 15+5 pages

Moir\'e materials provide a unique platform for studies of correlated many-body physics of the Fermi-Hubbard model on triangular spin-charge lattices. Bilayer Hubbard models are of particular significance with regard to the physics of Mott insulating states and their relation to unconventional superconductivity, yet their experimental implementation in moir\'e systems has so far remained elusive. Here, we demonstrate the realization of a staggered bilayer triangular lattice of electrons in an antiparallel MoSe$_{2}$/WS$_{2}$ heterostructure. The bilayer lattice emerges due to strong electron confinement in the moir\'e potential minima and the near-resonant alignment of conduction band edges in MoSe$_{2}$ and WS$_{2}$. As a result, charge filling proceeds layer-by-layer, with the first and second electron per moir\'e cell consecutively occupying first the MoSe$_{2}$ and then the WS$_{2}$ layer. We describe the observed charging sequence by an electrostatic model and provide experimental evidence of spin correlations on the vertically offset and laterally staggered bilayer lattice, yielding absolute exciton Land\'e factors as high as $600$ at lowest temperatures. The bilayer character of the implemented spin-charge lattice allows for electrostatic tunability of Ruderman-Kittel-Kasuya-Yosida magnetism, and establishes antiparallel MoSe$_{2}$/WS$_{2}$ heterostructures as a viable platform for studies of bilayer Hubbard model physics with exotic magnetic phases on frustrated lattices.

The relation between d-wave superconductivity and stripes is fundamental to the understanding of ordered phases in cuprates. While experimentally both phases are found in close proximity, numerical studies on the related Fermi-Hubbard model have long been investigating whether stripes precede, compete or coexist with superconductivity. Such stripes are characterised by interleaved charge and spin density wave ordering where fluctuating lines of dopants separate domains of opposite antiferromagnetic order. Here we show first signatures of stripes in a cold-atom Fermi-Hubbard quantum simulator. By engineering a mixed-dimensional system, we increase their typical energy scales to the spin exchange energy, enabling us to access the interesting crossover temperature regime where stripes begin to form. We observe extended, attractive correlations between hole dopants and find an increased probability to form larger structures akin to stripes. In the spin sector, we study correlation functions up to third order and find results consistent with stripe formation. These higher-order correlation measurements pave the way towards an improved microscopic understanding of the emergent properties of stripes and their relation to other competing phases. More generally, our approach has direct relevance for newly discovered high-temperature superconducting materials in which mixed dimensions play an essential role., Comment: 16 pages, 21 figures

The physics of doped Mott insulators is at the heart of strongly correlated materials and is believed to constitute an essential ingredient for high-temperature superconductivity. In systems with higher SU(N) spin symmetries, even richer magnetic ground states appear at a filling of one particle per site compared to the case of SU(2) spins, but their fate upon doping remains largely unexplored. Here we address this question by studying a single hole in the SU(3) $t$-$J$ model, whose undoped ground state features long-range, diagonal spin stripes. By analyzing both ground state and dynamical properties utilizing the density matrix renormalization group, we establish the appearence of magnetic polarons consisting of chargons and flavor defects, whose dynamics is constrained to a single effective dimension along the ordered diagonal. We semi-analytically describe the system using geometric string theory, where paths of hole motion are the fundamental degrees of freedom. With recent advances in the realization and control of SU(N) Fermi-Hubbard models with ultracold atoms in optical lattices, our results can directly be observed in quantum gas microscopes with single-site resolution. Our work suggests the appearance of intricate ground states at finite doping constituted by emergent, coupled Luttinger liquids along diagonals, and is a first step towards exploring a wealth of physics in doped SU(N) Fermi-Hubbard models on various geometries., Comment: 5 + 5 pages

Ever since the discovery of high-temperature superconductivity in cuprates, gaining microscopic insights into the nature of pairing in strongly correlated, repulsively interacting fermionic systems has remained one of the greatest challenges in modern condensed matter physics. Following recent experiments reporting superconductivity in the bilayer nickelate La$_3$Ni$_2$O$_7$ (LNO) with remarkably high critical temperatures of $T_c = 80$ K, it has been argued that the low-energy physics of LNO can be described by the strongly correlated, mixed-dimensional bilayer $t-J$ model. Here we investigate this bilayer system and utilize density matrix renormalization group techniques to establish a thorough understanding of the model and the magnetically induced pairing through comparison to the perturbative limit of dominating inter-layer spin couplings. In particular, this allows us to explain appearing finite-size effects, firmly establishing the existence of long-range superconducting order in the thermodynamic limit that is described by the XXZ universality class of hard-core bosons constituted by $s$-wave singlet pairs. As the effective model in the perturbative limit is known to show linear resistivity above the superconducting transition temperature, we propose a pair-based interpretation of the extended strange metal phase observed in LNO. By analyzing binding energies, we predict a BEC-BCS crossover as a function of the Hamiltonian parameters, whereas LNO is anticipated to lie on the BCS side in vicinity of the transition. We find large binding energies of the order of the inter-layer coupling that suggest strikingly high critical temperatures of the Berezinskii-Kosterlitz-Thouless transition, raising the question whether (mixD) bilayer superconductors possibly facilitate critical temperatures above room temperature., Comment: 8 + 5 pages

Microscopically understanding competing orders in strongly correlated systems is a key challenge in modern quantum many-body physics. For example, the study of magnetic polarons and their relation to pairing in the Fermi-Hubbard model in different geometries remains one of the central questions, and may help to understand the mechanism underlying unconventional superconductivity in cuprates or transition metal dichalcogenides. With recent advances in analog quantum simulation of the Fermi-Hubbard model on non-bipartite lattices, frustrated physics can now be explored experimentally in systems lacking particle-hole symmetry. Here, we study the singly doped $t-J$ model on the triangular lattice, focusing on the competition between kinetic and magnetic frustration as a function of temperature. In doublon doped systems, we uncover a crossover between Nagaoka-type ferromagnetic (FM) correlations at high temperature and exchange mediated antiferromagnetic (AFM) order around the doublon at low temperature. For hole doped systems, kinetic Haerter-Shastry-type AFM at high temperature as well as exchange interactions at low temperature favor $120^{\circ}$ order, strengthening magnetic correlations compared to the undoped system. In the ground state, the presence of AFM correlations throughout a wide range of interactions indicates confinement of both types of dopants. In this regime we firmly establish the presence of linear confining potentials via energy scaling arguments, supporting the picture of geometric strings in the frustrated triangular $t-J$ model., Comment: 5 + 5 pages

Microscopically understanding and classifying phases of matter is at the heart of strongly-correlated quantum physics. With quantum simulations, genuine projective measurements (snapshots) of the many-body state can be taken, which include the full information of correlations in the system. The rise of deep neural networks has made it possible to routinely solve abstract processing and classification tasks of large datasets, which can act as a guiding hand for quantum data analysis. However, though proven to be successful in differentiating between different phases of matter, conventional neural networks mostly lack interpretability on a physical footing. Here, we combine confusion learning with correlation convolutional neural networks, which yields fully interpretable phase detection in terms of correlation functions. In particular, we study thermodynamic properties of the 2D Heisenberg model, whereby the trained network is shown to pick up qualitative changes in the snapshots above and below a characteristic temperature where magnetic correlations become significantly long-range. We identify the full counting statistics of nearest neighbor spin correlations as the most important quantity for the decision process of the neural network, which go beyond averages of local observables. With access to the fluctuations of second-order correlations -- which indirectly include contributions from higher order, long-range correlations -- the network is able to detect changes of the specific heat and spin susceptibility, the latter being in analogy to magnetic properties of the pseudogap phase in high-temperature superconductors. By combining the confusion learning scheme with transformer neural networks, our work opens new directions in interpretable quantum image processing being sensible to long-range order., Comment: 15+3 pages

Unveiling the microscopic origins of quantum phases dominated by the interplay of spin and motional degrees of freedom constitutes one of the central challenges in strongly correlated many-body physics. When holes move through an antiferromagnetic spin background, they displace the positions of spins, which induces effective frustration in the magnetic environment. However, a concrete characterization of this effect in a quantum many-body system is still an unsolved problem. Here we present a Hamiltonian reconstruction scheme that allows for a precise quantification of hole-motion-induced frustration. We access non-local correlation functions through projective measurements of the many-body state, from which effective spin-Hamiltonians can be recovered after detaching the magnetic background from dominant charge fluctuations. The scheme is applied to systems of mixed dimensionality, where holes are restricted to move in one dimension, but SU(2) superexchange is two-dimensional. We demonstrate that hole motion drives the spin background into a highly frustrated regime, which can quantitatively be described by an effective $J_1-J_2-$type spin model. We exemplify the applicability of the reconstruction scheme to ultracold atom experiments by recovering effective spin-Hamiltonians of experimentally obtained 1D Fermi-Hubbard snapshots. Our method can be generalized to fully 2D systems, enabling promising microscopic perspectives on the doped Hubbard model., Comment: 8 + 6 pages

Microscopically understanding competing orders in strongly correlated systems is a key challenge in modern quantum many-body physics. For example, the origin of stripe order and its relation to pairing in the Fermi-Hubbard model remains one of the central questions, and may help to understand the origin of high-temperature superconductivity in cuprates. Here, we analyze stripe formation in the doped mixed-dimensional (mixD) variant of the $t-J$ model, where charge carriers are restricted to move only in one direction, whereas magnetic SU(2) interactions are two-dimensional. Using the density matrix renormalization group at finite temperature, we find a stable vertical stripe phase in the absence of pairing, featuring incommensurate magnetic order and long-range charge density wave profiles over a wide range of dopings. We find high critical temperatures on the order of the magnetic coupling $\sim J/2$, hence being within reach of current quantum simulators. Snapshots of the many-body state, accessible to quantum simulators, reveal hidden spin correlations in the mixD setting, whereby antiferromagnetic correlations are enhanced when considering purely the magnetic background. The proposed model can be viewed as realizing a parent Hamiltonian of the stripe phase, whose hidden spin correlations lead to the predicted resilience against quantum and thermal fluctuations., Comment: 5 + 6 pages

In the presence of boundaries, the entanglement entropy in lattice models is known to exhibit oscillations with the (parity of the) length of the subsystem, which however decay to zero with increasing distance from the edge. We point out in this article that, when the subsystem starts at the boundary and ends at an impurity, oscillations of the entanglement (as well as of charge fluctuations) appear which do not decay with distance, and which exhibit universal features. We study these oscillations in detail for the case of the XX chain with one modified link (a conformal defect) or two successive modified links (a relevant defect), both numerically and analytically. We then generalize our analysis to the case of extended (conformal) impurities, which we interpret as SSH models coupled to metallic leads. In this context, the parity effects can be interpreted in terms of the existence of non-trivial topological phases.

Two-photon photopolymerization delivers prints without support structures and minimizes layering artifacts in a broad range of materials. This volumetric printing approach scans a focused light source throughout the entire volume of a resin vat and takes advantage of the quadratic power dependence of two photon absorption to produce photopolymerization exclusively at the focal point. While this approach has advantages, the widespread adoption of two photon photopolymerization is hindered by the need for expensive ultrafast lasers and extremely slow print speeds. Here we present an analogous quadratic process, triplet-triplet-annihilation-driven 3D printing, that enables volumetric printing at a focal point driven by <4 milliwatt-power continuous wave excitation. To induce photopolymerization deep within a vat, the key advance is the nanoencapsulation of photon upconversion solution within a silica shell decorated with solubilizing polymer ligands. This scalable self-assembly approach allows for scatter-free nanocapsule dispersal in a variety of organic media without leaking the capsule contents. We further introduce an excitonic strategy to systematically control the upconversion threshold to support either monovoxel or parallelized printing schemes, printing at power densities multiple orders of magnitude lower than power densities required for two-photon-based 3D printing. The application of upconversion nanocapsules to volumetric 3D printing provides access to the benefits of volumetric printing without the current cost, power, and speed drawbacks. The materials demonstrated here open opportunities for other triplet fusion upconversion-controlled applications.

We show how measuring real space properties such as the charge density in a quasiperiodic system can be used to gain insight into their topological properties. In particular, for the Fibonacci chain, we show that the total onsite charge oscillates when plotted in the appropriate coordinates, and the number of oscillations is given by the Chern number of the gap in which the Fermi level lies. We show that these oscillations have two distinct interpretations, obtained by extrapolating results from the two extreme limits of the Fibonacci chain -- the valence bond picture in the strong modulation limit, and perturbation around the periodic chain in the weak modulation limit. This effect is found to remain robust at moderate interactions, as well as in the presence of disorder. We conclude that experimental measurement of the real space charge distribution can yield information on topological properties in a straightforward way., Comment: 8 pages, 6 figures

Collective excitations in topologically non-trivial systems have attracted considerable attention in recent years. Here we study plasmons in the Su-Schrieffer-Heeger model whose low-energy electronic band is only partially filled, such that the system is metallic. Using the random phase approximation, we calculate the intra- and inter-band polarization functions and determine the bulk plasmonic dispersion from the dielectric function within the random phase approximation. We find that the sub-lattice basis states strongly affect the polarization functions and therefore control the system's plasmonic excitations. By varying the real-space separation of these local orbitals, one can thus selectively enhance or suppress the plasmonic energies via a tunable trade-off between intra-band and inter-band screening processes. Specifically, this mechanism can be used to stabilize undamped high energy plasmons that have already been reported in related models. We propose scenarios on how to control and observe these effects in experiments.

We consider topological protection mechanisms in dissipative quantum systems in the presence of quenched disorder, with the intent to prolong coherence times of qubits. The physical setting is a network of qubits and dissipative cavities whose coupling parameters are tunable, such that topological edge states can be stabilized. The evolution of a fiducial qubit is entirely determined by a non-Hermitian Hamiltonian which thus emerges from a bona-fide physical process. It is shown how even in the presence of disorder winding numbers can be defined and evaluated in real space, as long as certain symmetries are preserved. Hence we can construct the topological phase diagrams of noisy open quantum models, such as the non-Hermitian disordered Su-Schrieffer- Heeger dimer model and a trimer model that includes longer-range couplings. In the presence of competing disorder parameters, interesting re-entrance phenomena of topologically non-trivial sectors are observed. This means that in certain parameter regions, increasing disorder drastically increases the coherence time of the fiducial qubit.

We analyze collective excitations in models of two-dimensional topological insulators using the random phase approximation. In a two-dimensional extension of the Su-Schrieffer-Heeger model, edge plasmonic excitations with induced charge-density distributions localized at the boundaries of the system are found in the topologically non-trivial phase, dispersing similarly as one-dimensional bulk plasmons in the conventional Su-Schrieffer-Heeger chain. For two-dimensional bulk collective modes, we reveal regimes of enhanced inter-band wave function correlations, leading to characteristic hardening and softening of inter- and intra-band bulk plasmonic branches, respectively. In the two-dimensional Haldane Chern insulator model, chiral, uni-directional edge plasmons in nano-ribbon architectures are observed, which can be characterized by an effective Coulomb interaction cross section. Bulk collective excitations in the two-dimensional Haldane model are shown to be originated by single-particle band structure details in different topological phases.

For developing a detailed network model of the brain based on image reconstructions, it is necessary to spatially resolve crossing nerve fibers. The accuracy hereby depends on many factors, including the spatial resolution of the imaging technique. 3D Polarized Light Imaging (3D-PLI) allows the three-dimensional reconstruction of nerve fiber tracts in whole brain sections with micrometer in-plane resolution, but leaves uncertainties in pixels containing crossing fibers. Here we introduce Scattered Light Imaging (SLI) to resolve the substructure of nerve fiber crossings. The measurement is performed on the same unstained histological brain sections as in 3D-PLI. By illuminating the brain sections from different angles and measuring the transmitted (scattered) light under normal incidence, SLI provides information about the underlying nerve fiber structure. A fully automated evaluation of the resulting light intensity profiles has been developed, allowing the user to extract various characteristics, like the individual directions of in-plane crossing nerve fibers, for each image pixel at once. We validate the reconstructed nerve fiber directions against results from previous simulation studies, scatterometry measurements, and fiber directions obtained from 3D-PLI. We demonstrate in different brain samples (human optic tracts, vervet monkey brain, rat brain) that the 2D fiber directions can be reliably reconstructed for up to three crossing nerve fiber bundles in each image pixel with an in-plane resolution of up to 6.5 $\mu$m. We show that SLI also yields reliable fiber directions in brain regions with low 3D-PLI signals coming from regions with a low density of myelinated nerve fibers or out-of-plane fibers. In combination with 3D-PLI, the technique can be used for a full reconstruction of the three-dimensional nerve fiber architecture in the brain., Comment: 30 pages, 16 figures

This article revisits Kobayasi's and Yosiki's algorithm for conversion of OSA-UCS into XYZ cooordinates. It corrects some mistakes on the involved functions and initial guesses and shows that that hundreds of thousands of coordinates can be converted in less than a second with full accuracy.

This note is concerned with the CIECAM02 color appearance model and its successor, the CAM16 color appearance model. Several algorithmic flaws are pointed out and remedies are suggested. The resulting color model is algebraically equivalent to CIECAM02/CAM16, but shorter, more efficient, and works correctly for all edge cases.

The stable and unstable solutions of a square 2D extreme type-II superconductor are studied in the field of a magnetic disc. We use a preconditioned Newton-Krylov solver to find the solutions and use numerical continuation to track the solutions as the field strength varies. For a disc with a small radius, we have identified generic scenarios through which the state loses and regains its stability.

We describe a systematic approach for the efficient numerical solution of nonlinear Schr\"odinger-type partial differential equations of the form $(K +V + g|\psi|^2)\psi=0$, with an energy operator $K$, a scalar potential $V$, and a scalar parameter $g$. Instrumental to the approach are developments in numerical linear and nonlinear algebra, specifically numerical parameter continuation. We demonstrate how a continuous sequence of solutions can be obtained regardless of their stability, so that finally the spectrum of stable and unstable solutions in the specified parameter range is fully revealed. The method is demonstrated for the GL equation in a three-dimensional superconducting domain with an inhomogeneous magnetic field, a numerically demanding problem known to have an involved solution landscape.

The authors propose a recycling Krylov subspace method for the solution of a sequence of self-adjoint linear systems. Such problems appear, for example, in the Newton process for solving nonlinear equations. Ritz vectors are automatically extracted from one MINRES run and then used for self-adjoint deflation in the next. The method is designed to work with arbitrary inner products and arbitrary self-adjoint positive-definite preconditioners whose inverse can be computed with high accuracy. Numerical experiments with nonlinear Schr\"odinger equations indicate a substantial decrease in computation time when recycling is used.

This paper considers the extreme type-II Ginzburg--Landau equations, a nonlinear PDE model for describing the states of a wide range of superconductors. Based on properties of the Jacobian operator and an AMG strategy, a preconditioned Newton--Krylov method is constructed. After a finite-volume-type discretization, numerical experiments are done for representative two- and three-dimensional domains. Strong numerical evidence is provided that the number of Krylov iterations is independent of the dimension $n$ of the solution space, yielding an overall solver complexity of O(n).

This paper considers the extreme type-II Ginzburg-Landau equations that model vortex patterns in superconductors. The nonlinear PDEs are solved using Newton's method, and properties of the Jacobian operator are highlighted. Specifically, it is illustrated how the operator can be regularized using an appropriate phase condition. For a two-dimensional square sample, the numerical results are based on a finite-difference discretization with link variables that preserves the gauge invariance. For two exemplary sample sizes, a thorough bifurcation analysis is performed using the strength of the applied magnetic field as a bifurcation parameter and focusing on the symmetries of this system. The analysis gives new insight in the transitions between stable and unstable states, as well as the connections between stable solution branches., Comment: 31 pages