• Nature of Topological Phase Transition of Kitaev Quantum Spin Liquids,
    Huanzhi Hu and Frank Krüger,
    Under review, arXiv:2402.12610.

    We investigate the nature of the topological quantum phase transition between the gapless and gapped Kitaev quantum spin liquid
    phases away from the exactly solvable point. The transition is driven by anisotropy of the Kitaev couplings. At the critical point the two
    Dirac points of the gapless Majorana modes merge, resulting in the formation of a semi-Dirac point with quadratic and linear band
    touching directions. We derive an effective Gross-Neveu-Yukawa type field theory that describes the topological phase transition
    in the presence of additional magnetic interactions. We obtain the infrared scaling form of the propagator of the dynamical Ising
    order parameter field and perform a renormalization-group analysis. The universality of the transition is found to be different to that
    of symmetry-breaking phase transitions of semi-Dirac electrons. However, as in the electronic case, the Majorana fermions acquire an
    anomalous dimension, indicative of the breakdown of the fractionalized quasiparticle description.

  • Majorana Fermion Mean-Field Theories of Kitaev Quantum Spin Liquids,
    Shahnam Ghanbari Saheli, Jennifer Lin, Huanzhi Hu and Frank Krüger,
    Phys. Rev. B 109, 014407 (2024).

    We determine the phase diagrams of anisotropic Kitaev-Heisenberg models on the honeycomb lattice using parton mean-field theories
    based on different Majorana fermion representations of the S=1/2 spin operators. Firstly, we use a two-dimensional Jordan-Wigner
    transformation (JWT) involving a semi-infinite snake string operator. In order to ensure that the fermionized Hamiltonian remains local
    we consider the limit of extreme Ising exchange anisotropy in the Heisenberg sector. Secondly, we use the conventional Kitaev
    representation in terms of four Majorana fermions subject to local constraints, which we enforce through Lagrange multipliers. For both
    representations we self-consistently decouple the interaction terms in the bond and magnetization channels and determine the phase
    diagrams as a function of the anisotropy of the Kitaev couplings and the relative strength of the Ising exchange. While both mean-field
    theories produce identical phase boundaries for the topological phase transition between the gapless and gapped Kitaev quantum spin
    liquids, the JWT fails to correctly describe the the magnetic instability and finite-temperature behavior. Our results show that the
    magnetic phase transition is first order at low temperatures but becomes continuous above a certain temperature. At this energy scale,
    we also observe a finite-temperature crossover on the quantum-spin-liquid side, from a fractionalized paramagnet at low temperatures,
    in which gapped flux excitations are frozen out, to a conventional paramagnet at high temperatures.

  • Strain control of a bandwidth-driven spin reorientation in Ca3Ru2O7,
    C.D. Dashwood, A.H. Walker, M.P. Kwasigroch, L.S.I. Veiga, Q. Faure, J.G. Vale, D.G. Porter, P. Manuel,
    D.D. Khalyavin, F. Orlandi, C.V. Colin, O. Fabelo, F. Krüger, R.S. Perry, R.D. Johnson, A.G. Green, D.F. McMorrow,
    Nature Communications 14, 6197 (2023).

    The layered ruthenate family of materials possess an intricate interplay of structural, electronic and magnetic degrees of freedom
    that yields a plethora of delicately balanced ground states. This is exemplified by Ca3Ru2O7, which hosts a complex transition at
    which the lattice parameters jump, the Fermi surface partially gaps and the spins undergo a 90deg in-plane reorientation. Here,
    we show how the transition is driven by a lattice strain that tunes the electronic bandwidth. We apply uniaxial stress to single
    crystals of Ca3Ru2O7, using neutron and resonant x-ray scattering to simultaneously probe the structural and magnetic responses.
    These measurements demonstrate that the transition can be driven by externally induced strain, stimulating the development of a
    theoretical model in which an internal strain is generated self-consistently to lower the electronic energy. We understand the strain
    to act by modifying tilts and rotations of the RuO6 octahedra, which directly influences the nearest-neighbour hopping. Our results
    offer a blueprint for uncovering the driving force behind coupled phase transitions, as well as a novel route to controlling them.

  • Stability of the Néel quantum critical point in the presence of Dirac fermions,
    Huanzhi Hu, Jennifer Lin, Mikolaj D. Uryszek, and Frank Krüger,
    Phys. Rev. B 107, 085113 (2023).

    We investigate the stability of the Néel quantum critical point of two-dimensional quantum antiferromagnets, described by a
    non-linear sigma model (NLSM), in the presence of a Kondo coupling to N flavours of two-component Dirac fermion fields.
    The long-wavelength order parameter fluctuations are subject to Landau damping by electronic particle-hole fluctuations.
    Using momentum-shell RG, we demonstrate that the Landau damping is weakly irrelevant at the Néel quantum critical point,
    despite the fact that the corresponding self-energy correction dominates over the quadratic gradient terms in the IR limit.
    In the ordered phase, the Landau damping increases under the RG, indicative of damped spin-wave excitations. Although the
    Kondo coupling is weakly relevant, sufficiently strong Landau damping renders the Néel quantum critical point quasi-stable
    for N>=4 and thermodynamically stable for N< 4. In the latter case, we identify a new multi-critical point which describes the
    transition between the Néel critical and Kondo run-away regimes. The symmetry breaking at this fixed point results in the
    opening of a gap in the Dirac fermion spectrum. Approaching the multi-critical point from the disordered phase, the fermionic
    quasiparticle residue vanishes, giving rise to non-Fermi-liquid behavior.

  • Magnetic hard-direction ordering in anisotropic Kondo systems,
    M. P. Kwasigroch, Huanzhi Hu, and Frank Krüger, and A. G. Green
    Phys. Rev. B 105, 224418 (2022).

    We present a generic mechanism that explains why many Kondo materials show magnetic ordering along directions that are not
    favoured by the crystal-field anisotropy. Using a renormalization-group (RG) analysis of single impurity Kondo models with
    single-ion anisotropy, we demonstrate that strong fluctuations above the Kondo temperature drive a moment re-orientation over
    a wide range of parameters, e.g. for different spin values S and number of Kondo channels N. In tetragonal systems this can
    happen for both easy-plane or easy axis anisotropy. The characteristic crossing of magnetic susceptibilities is not an artefact
    of the weak-coupling RG treatment but can be reproduced in brute-force perturbation theory. Employing numerical renormalization
    group (NRG), we show that for an under-screened moment (S=1, N=1) with easy-plane anisotropy, a crossing of magnetic
    susceptibilities can also occur in the strong-coupling regime (below the Kondo temperature). This suggests that collective
    magnetic ordering of such under-screened moments would develop along the magnetic hard axis.

    Our work won the first prize in the poster competition at the International Conference on Strongly Correlated Electron Systems
    (SCES2022) in Amsterdam (25-29 July 2022).

  • Interplay of interactions and disorder at the charge-density wave transition of two-dimensional Dirac semimetals,
    Mikolaj D. Uryszek and Frank Krüger,
    Phys. Rev. B 105, 075143 (2022).

    We consider the effects of weak quenched fermionic disorder on the quantum-phase transition between the Dirac semimetal and
    charge density wave (CDW) insulator in two spatial dimensions. The symmetry breaking transition is described by the
    Gross-Neveu-Yukawa (GNY) theory of Dirac fermions coupled to an Ising order parameter field. Treating the disorder using the
    replica method, we consider chemical potential, vector potential (gauge), and random mass disorders, which all arise from
    non-magnetic charge impurities. We self-consistently account for the Landau damping of long-wavelength order-parameter
    fluctuations by using the non-perturbative RPA re-summation of fermion loops, and compute the renormalization-group (RG)
    flow to leading order in the disorder strength and 1/N (N the number of Dirac fermion flavors). We find two fixed points,
    the clean GNY critical point which is stable against weak disorder, and a dirty GNY multi-critical point, at which the chemical
    potential disorder is finite and the other forms of disorder are irrelevant. We investigate the scaling of physical observables at
    this finite-disorder multi-critical point which breaks Lorentz invariance and gives rise to distinct non-Fermi liquid behavior.

  • Non-Fermi liquid behavior below the Néel temperature in the frustrated heavy fermion magnet UAu2,
    Christopher D. O'Neill, Julian L. Schmehr, Harry D. J. Keen, Luke Pritchard Cairns, Dmitry A. Sokolov,
    Andreas Hermann, Didier Wermeille, Pascal Manuel, Frank Krüger, and Andrew D. Huxley
    Proceedings of the National Academy of Sciences, Dec 2021, 118 (49) e2102687118.

    The term Fermi liquid is almost synonymous with the metallic state. The association is known to break down at quantum critical
    points (QCPs), but these require precise values of tuning parameters, such as pressure and applied magnetic field, to exactly
    suppress a continuous phase transition temperature to the absolute zero. Three-dimensional non-Fermi liquid states, apart from
    superconductivity, that are unshackled from a QCP are much rarer and are not currently well understood. Here, we report that the
    triangular lattice system uranium diauride (UAu2) forms such a state with a non-Fermi liquid low-temperature heat capacity
    C/T~ log(1/T) and electrical resistivity ρ(T) - ρ(0) ~ T1.35 far below its Néel temperature. The magnetic order itself has a novel
    structure and is accompanied by weak charge modulation that is not simply due to magnetostriction. The charge modulation
    continues to grow in amplitude with decreasing temperature, suggesting that charge degrees of freedom play an important role
    in the non-Fermi liquid behavior. In contrast with QCPs, the heat capacity and resistivity we find are unusually resilient in
    magnetic field. Our results suggest that a combination of magnetic frustration and Kondo physics may result in the emergence
    of this novel state.

  • Field-Induced Modulated State in the Ferromagnet PrPtAl,
    Christopher D. O'Neill, Gino Abdul-Jabbar, Didier Wermeille, Philippe Bourges,
    Frank Krüger, and Andrew D. Huxley
    Phys. Rev. Lett. 126, 197203 (2021).

    The theory of quantum order-by-disorder (QOBD) explains the formation of modulated magnetic states at the boundary
    between ferromagnetism and paramagnetism in zero field. PrPtAl has been argued to provide an archetype for this.
    Here, we report the phase diagram in magnetic field, applied along both the easy a-axis and hard b-axis. For field aligned
    to the b-axis, we find that the magnetic transition temperatures are suppressed and at low temperature there is a single
    modulated fan state, separating an easy a-axis ferromagnetic state from a field polarized state. This fan state is well
    explained with the QOBD theory in the presence of anisotropy and field. Experimental evidence supporting the QOBD
    explanation is provided by the large increase in the T2 coefficient of the resistivity and direct detection of enhanced
    magnetic fluctuations with inelastic neutron scattering, across the field range spanned by the fan state. This shows that
    the QOBD mechanism can explain field induced modulated states that persist to very low temperature.

  • Fermionic criticality of anisotropic nodal point semimetals away from the upper critical dimension:
    exact 1/Nf exponents
    Mikolaj D. Uryszek, Frank Krüger and Elliot Christou
    Phys. Rev. Research 2, 043265 (2020).

    We consider the fermionic quantum criticality of anisotropic nodal point semimetals in d = dL + dQ spatial dimensions that
    disperse linearly in dL dimensions, and quadratically in the remaining dQ dimensions. When subject to strong interactions,
    these systems are susceptible to semimetal-insulator transitions concurrent with spontaneous symmetry breaking. Such quantum
    critical points are described by effective field theories of anisotropic nodal fermions coupled to dynamical order parameter fields.
    We analyze the universal scaling in the physically relevant spatial dimensions, generalizing to a large number Nf of fermion
    flavors for analytic control. Landau damping by gapless fermionic excitations gives rise to non-analytic self-energy corrections
    to the bosonic order-parameter propagator that dominate the long-wavelength behavior. We show that perturbative momentum
    shell RG leads to non-universal, cutoff dependent results, as it does not correctly account for this non-analytic structure. In turn,
    using a completely general soft cutoff formulation, we demonstrate that the correct IR scaling of the dressed bosonic propagator
    can be deduced by enforcing that results are independent of the cutoff scheme. Using this soft cutoff approach, we compute the
    exact critical exponents for anisotropic semi-Dirac fermions (dL=1, dQ=1) to leading order in 1/Nf, and to all loop orders.
    Applying the same method to relativistic Dirac fermions, we reproduce the critical exponents obtained by other methods,
    such as conformal bootstrap. Unlike in the relativistic case, where the UV-IR connection is re-established at the upper critical
    dimension, non-analytic IR contributions persist near the upper critical line 2dL + dQ=4 of anisotropic nodal fermions.
    We present ε-expansions in both the number of linear and quadratic dimensions. The corrections to critical exponents are
    non-analytic in ε, with a functional form that depends on the starting point on the upper critical line.

  • Non-linear soliton confinement in weakly coupled antiferromagnetic spin chains,
    H. Lane, C. Stock, S.-W. Cheong, F. Demmel, R. A. Ewings, and F. Krüger,
    Phys. Rev. B 102, 024437 (2020).

    We analyze the low-energy dynamics of quasi one dimensional, large-S quantum antiferromagnets with easy-axis anisotropy,
    using a semi-classical non-linear sigma model. The saddle point approximation leads to a sine-Gordon equation which supports
    soliton solutions. These correspond to the movement of spatially extended domain walls. Long-range magnetic order is a
    consequence of a weak inter-chain coupling. Below the ordering temperature, the coupling to nearby chains leads to an energy
    cost associated with the separation of two domain walls. From the kink-antikink two-soliton solution, we compute the effective
    confinement potential. At distances large compared to the size of the solitons the potential is linear, as expected for point-like
    domain walls. At small distances the gradual annihilation of the solitons weakens the effective attraction and renders the potential
    quadratic. From numerically solving the effective one dimensional Schrödinger equation with this non-linear confinement potential
    we compute the soliton bound state spectrum. We apply the theory to CaFe2O4, an anisotropic S=5/2 magnet based upon
    antiferromagnetic zig-zag chains. Using inelastic neutron scattering, we are able to resolve seven discrete energy levels for spectra
    recorded slightly below the Néel temperature TN≈ 200K. These modes are well described by our non-linear confinement model
    in the regime of large spatially extended solitons.

  • Criticality of Dirac fermions in the presence of emergent gauge fields,
    Elliot Christou, Fernando de Juan, and Frank Krüger,
    Phys. Rev. B 101, 155121 (2020).

    We consider spontaneous symmetry-breaking transitions of strongly interacting two-dimensional Dirac fermions minimally coupled
    to mass and emergent gauge field order parameters. Using a renormalization group analysis, we show that the presence of gauge
    fields leads to fermion induced quantum (multi-)criticality where the putative emergent Lorentz invariance is violated. We
    illustrate this with the example of translational symmetry breaking due to charge-density wave order on the honeycomb lattice.
    Finally, we identify that topological phase transitions are well described by this effective field theory.

  • Quantum Criticality of Semi-Dirac Fermions in 2+1 Dimensions,
    Mikolaj D. Uryszek, Elliot Christou, Akbar Jaefari, Frank Krüger, and Bruno Uchoa
    Phys. Rev. B 100, 155101 (2019).

    Two-dimensional semi-Dirac fermions are quasiparticles that disperse linearly in one direction and quadratically in the other.
    We investigate instabilities of semi-Dirac fermions towards charge- and spin-density wave and superconducting order, driven by
    short-range interactions. We analyse the critical behaviour of the Yukawa theories for the different order parameters using Wilson
    momentum shell RG. We generalise to a large number Nf of fermion flavours to achieve analytic control in 2+1 dimensions and
    calculate critical exponents at one-loop order, systematically including 1/Nf corrections . The latter depend on the specific form
    of the bosonic IR propagator in 2+1 dimensions, which needs to be included to regularise divergencies. The 1/Nf corrections
    are surprisingly small, suggesting that the expansion is well controlled in the physical dimension. The order parameter correlations
    inherit the electronic anisotropy of the semi-Dirac fermions, leading to correlation lengths that diverge along the spatial directions
    with distinct exponents, even at the mean-field level. We conjecture that the proximity to the critical point may stabilize novel
    modulated order phases.

  • Hidden Charge Order of Interacting Dirac fermions on the Honeycomb Lattices,
    Elliot Christou, Bruno Uchoa, and Frank Krüger,
    Phys. Rev. B 98, 161120(R) (2018).

    We consider the extended half-filled Hubbard model on the honeycomb lattice for second nearest neighbors interactions.
    Using a functional integral approach, we find that collective fluctuations suppress topological states and instead favor charge ordering,
    in agreement with previous numerical studies. However, we show that the critical point is not of the putative semimetal-Mott insulator
    variety. Due to the frustrated nature of the interactions, the ground state is described by a hidden metallic charge order with semi-Dirac
    excitations. We conjecture that this transition is not in the Gross-Neveu universality class.

  • Quantum Order-by-Disorder in Strongly Correlated Metals,
    A. G. Green, G. Conduit, and F. Krüger,
    Annual Reviews of Condensed Matter Physics 9, pp. 59-77 (2018).

    Entropic forces in classical many-body systems, e.g. colloidal suspensions, can lead to the formation of new phases. Quantum
    fluctuations can have similar effects: spin fluctuations drive the superfluidity of Helium-3 and a similar mechanism operating in metals
    can give rise to superconductivity. It is conventional to discuss the latter in terms of the forces induced by the quantum fluctuations.
    However, focusing directly upon the free energy provides a useful alternative perspective in the classical case and can also be applied
    to study quantum fluctuations. Villain first developed this approach for insulating magnets and coined the term order-by-disorder to
    describe the observed effect. We discuss the application of this idea to metallic systems, recent progress made in doing so, and the
    broader prospects for the future.

  • Topological Triplon Modes and Bound States in a Shastry-Sutherland Magnet,
    P. A. McClarty, F. Krüger, T. Guidi, S. F. Parker, K. Refson, A. W. Parker, D. Prabakharan, and R. Coldea
    Nature Physics 13, 736 (2017).

    The twin discoveries of the quantum Hall effect, in the 1980's, and of topological band insulators, in the 2000's, were landmarks in
    physics that enriched our view of the electronic properties of solids. In a nutshell, these discoveries have taught us that quantum
    mechanical wavefunctions in crystalline solids may carry nontrivial topological invariants which have ramifications for the observable
    physics. One of the side effects of the recent topological insulator revolution has been that such physics is much more widespread
    than was appreciated ten years ago. For example, while topological insulators were originally studied in the context of electron
    wavefunctions, recent work has led to proposals of topological insulators in bosonic systems: in photonic crystals, in the vibrational
    modes of crystals, and in the excitations of ordered magnets. Here we confirm the recent proposal that, in a weak magnetic field, the
    dimerized quantum magnet SrCu2(BO3)2 is a bosonic topological insulator with nonzero Chern number in the triplon bands and
    topologically protected chiral edge excitations.

  • Entanglement entropies and fermion signs of critical metals,
    N. Kaplis, F. Krüger, and J. Zaanen
    Phys. Rev. B 95, 155102 (2017).

    The fermion sign problem is often viewed as a sheer inconvenience that plagues numerical studies of strongly interacting electron
    systems. Only recently, it has been suggested that fermion signs are fundamental for the universal behavior of critical metallic
    systems and crucially enhance their degree of quantum entanglement.In this work we explore potential connections between
    emergent scale invariance of fermion sign structures and scaling properties of bipartite entanglement entropies. Our analysis
    is based on a wavefunction ansatz that incorporates collective, long-range backflow correlations into fermionic Slater determinants.
    Such wavefunctions mimic the collapse of a Fermi liquid at a quantum critical point. Their nodal surfaces -- a representation of the
    fermion sign structure in many-particle configurations space -- show fractal behavior up to a length scale ξ that diverges at a critical
    backflow strength. We show that the Hausdorff dimension of the fractal nodal surface depends on ξ, the number of fermions,
    and the exponent of the backflow. For the same wavefunctions we numerically calculate the second Rényi entanglement entropy S2.
    Our results show a cross-over from volume scaling, S2 ~ℓ^θ (θ=2 in d=2 dimensions), to the characteristic Fermi-liquid behavior
    S2 ~ ℓ ln ℓ on scales larger than ξ. We find that volume scaling of the entanglement entropy is a robust feature of critical backflow
    fermions, independent of the backflow exponent and hence the fractal dimension of the scale invariant sign structure.

  • Stability of a spin-triplet nematic state near to a quantum critical point,
    G. Hannappel, C. J. Pedder, F. Krüger, and A. G. Green
    Phys. Rev. B 93, 235105 (2016).

    We analyze a model of itinerant electrons interacting through a quadrupole density-density repulsion in three dimensions. At the
    mean field level, the interaction drives a continuous Pomeranchuk instability towards d-wave, spin-triplet nematic order, which
    simultaneously breaks the SU(2) spin-rotation and spatial rotational symmetries. This order results in spin antisymmetric, elliptical
    deformations of the Fermi surfaces of up and down spins. We show that the effects of quantum fluctuations are similar to those in
    metallic ferromagnets, rendering the nematic transition first-order at low temperatures. Using the fermionic quantum order-by-disorder
    approach to self-consistently calculate fluctuations around possible modulated states, we show that the first-order transition is
    pre-empted by the formation of a nematic state that is intertwined with a helical modulation in spin space. Such a state is closely
    related to d-wave bond density wave order in square-lattice systems. Moreover, we show that it may coexist with a modulated,
    p-wave superconducting state.

  • Incommensurate spin-density-wave antiferromagnetism in CeRu2Al2B,
    A. Bhattacharyya, D. D. Khalyavin, F. Krüger, D. T. Adroja, A. M. Strydom, W. A. Kockelmann, and A. D. Hillier,
    Phys. Rev. B 93, 060410(R) (2016).

    The newly discovered Ising-type ferromagnet CeRu2Al2B exhibits an additional phase transition at TN= 14.2 K before going to the
    ferromagnetic ground state at TC = 12.8 K. We clarify the nature of this transition through high resolution neutron diffraction
    measurements. The data reveals the presence of a longitudinal incommensurate spin-density wave (SDW) in the temperature range
    between TC and TN. The propagation vector q~(0,0,0.148) is nearly temperature independent in this region and discontinuously
    locks into q=0 at TC. Mean-field calculations of an effective Ising model indicate that the modulated SDW phase is stabilized by
    a strong competition between ferromagnetic and antiferromagnetic exchange interactions. This makes CeRu2Al2B a particularly
    attractive model system to study the global phase diagram of ferromagnetic heavy-fermion metals under influence of magnetic

  • Bose and Mott Glass Phases in Dimerized Quantum Antiferromagnets,
    S. J. Thomson and F. Krüger,
    Phys. Rev. B 92, 180201(R) (2015).

    We examine the effects of disorder on dimerized quantum antiferromagnets in a magnetic field, using the mapping to a lattice gas
    of hard-core bosons with finite-range interactions. Combining a strong-coupling expansion, the replica method, and a one-loop
    renormalization group analysis, we investigate the nature of the glass phases formed. We find that away from the tips of the Mott lobes,
    the transition is from a Mott insulator to a compressible Bose glass, however the compressibility at the tips is strongly suppressed. We
    identify this finding with the presence of a rare Mott glass phase not previously described by any analytic theory for this model and
    demonstrate that the inclusion of replica symmetry breaking is vital to correctly describe the glassy phases. This result suggests that
    the formation of Bose and Mott glass phases is not simply a weak localization phenomenon but is indicative of much richer physics.
    We discuss our results in the context of both ultracold atomic gases and spin-dimer materials.

  • Modulated magnetism in PrPtAl,
    G. Abdul-Jabbar, D. A. Sokolov, C. O'Neill, C. Stock, D. Wermeille, F. Demmel, F. Krüger, A. G. Green,
    F. Levy-Bertrand, B. Grenier, and A. Huxley,
    Nature Physics 11, 321 (2015).

    The transition between paramagnetism and ferromagnetism is the paradigm for a continuous phase transition at finite temperature.
    When such a transition is tuned to zero temperature in clean materials, the growth of low energy zero-point fluctuations potentially
    drives an array of phenomena, including the formation of novel states such as non-conventional superconductivity. Experimentally,
    the growth of the fluctuations is however curtailed and the transition becomes discontinuous as its temperature is reduced. This
    is understood to arise from non-analytic corrections to the free energy that always occur. In a recent theory changes of the excitation
    spectrum are self-consistently considered alongside the ground state. This analysis reveals that a transition to a new state may be
    an alternative outcome. Since the excitation spectrum (the "disorder") is pivotal to promoting the new "order" this mechanism is
    referred to as "order by disorder". Here, we report the discovery of modulated order in PrPtAl, consistent with complex spirals, at
    the boundary between paramagnetism and ferromagnetism that is the first clear experimental realisation of such a state.

  • Symmetry of re-entrant tetragonal phase in Ba(1-x)Na(x)Fe(2)As(2): magnetic versus orbital ordering mechanism,
    D.D. Khalyavin, S.W. Lovesey, P. Manuel, F. Krüger, S. Rosenkranz, J.M. Allred, O. Chmaissem, and R. Osborn,
    Phys. Rev. B 90, 174511 (2014).

    Magneto-structural phase transitions in Ba(1-x)A(x)xFe(2)As(2) (A = K, Na) materials are discussed for both magnetically and orbitally
    driven mechanisms, using a symmetry analysis formulated within the Landau theory of phase transitions. Both mechanisms predict
    identical orthorhombic space-group symmetries for the nematic and magnetic phases observed over much of the phase diagram, but
    they predict different tetragonal space-group symmetries for the newly discovered re-entrant tetragonal phase in Ba(1-x)Na(x)xFe(2)As(2)
    (x ~ 0.24-0.28). In a magnetic scenario, magnetic order with moments along the c-axis, as found experimentally, does not allow any type
    of orbital order, but in an orbital scenario, we have determined two possible orbital patterns, specified by P4/mnc1' and I4221' space
    groups, which do not require atomic displacements relative to the parent I4/mmm1' symmetry and, in consequence, are indistinguishable
    in conventional diffraction experiments. We demonstrate that the three possible space groups are however, distinct in resonant X-ray
    Bragg diffraction patterns created by Templeton-Templeton scattering. This provides an experimental method of distinguishing between
    magnetic and orbital models.

  • Fluctuation driven magnetic hard-axis ordering in metallic ferromagnets,
    F. Krüger, C. J. Pedder, and A. G. Green,
    Phys. Rev. Lett. 113, 147001 (2014).

    We demonstrate that the interplay between soft electronic particle-hole fluctuations and magnetic anisotropies can drive ferromagnetic
    moments to point along a magnetic hard axis. As a proof of concept, we show this behavior explicitly for a generic two-band model with
    local Coulomb and Hund's interactions, and a spin-orbit-induced easy plane anisotropy. The phase diagram is calculated within the
    fermionic quantum order-by-disorder approach, which is based on a self-consistent free-energy expansion around a magnetically ordered
    state with unspecified orientation. Quantum fluctuations render the transition of the easy-plane ferromagnet first-order below a tricritical
    point. At even lower temperatures, directionally dependent transverse fluctuations dominate the magnetic anisotropy and the moments
    flip to lie along the magnetic hard axis. We discuss our findings in the context of recent experiments that show this unusual ordering
    along the magnetic hard direction.

  • Replica symmetry breaking in the Bose glass,
    S. J. Thomson and F. Krüger,
    EPL (Europhysics Letters) 108, 30002 (2014).

    We investigate the nature of the Bose glass phase of the disordered Bose-Hubbard model in d>2 and demonstrate the existence of a
    glass-like replica symmetry breaking (RSB) order parameter in terms of particle number fluctuations. Starting from a strong-coupling
    expansion around the atomic limit, we study the instability of the Mott insulator towards the formation of a Bose glass. We add some
    infinitesimal RSB, following the Parisi hierarchical approach in the most general form, and observe its flow under the momentum-shell
    renormalization group scheme. We find a new fixed point with one-step RSB, corresponding to the transition between the Mott insulator
    and a Bose glass phase with hitherto unseen RSB. The susceptibility associated to infinitesimal RSB perturbation in the Mott insulator
    is found to diverge at the transition with an exponent of gamma=1/d. Our findings are consistent with the expectation of glassy behavior
    and the established breakdown of self-averaging. We discuss the possibility of measuring the glass-like order parameter in optical
    lattice experiments as well as in certain spin systems that are in the same universality class as the Bose-Hubbard model.

  • Resummation of fluctuations near ferromagnetic quantum critical points,
    C. J. Pedder, F. Krüger, and A. G. Green,
    Phys. Rev. B 88, 165109 (2013).

    We present a detailed analysis of the non-analytic structure of the free energy for the itinerant ferromagnet near the quantum
    critical point in two and three dimensions. We analyze a model of electrons with an isotropic dispersion interacting through a contact
    repulsion. A fermionic version of the quantum order-by-disorder mechanism allows us to calculate the free energy as a functional of
    the dispersion in the presence of homogeneous and spiralling magnetic order. We re-sum the leading divergent contributions, to derive
    an algebraic expression for the non-analytic contribution to free energy from quantum fluctuations. Using a recursion which relates
    sub-leading divergences to the leading term, we calculate the full T=0 contribution in d=3. We propose an interpolating functional form,
    which allows us to track phase transition lines at temperatures far below the tricritical point and down to T=0. In d=2, quantum
    fluctuations are stronger and non-analyticities more severe. Using a similar re-summation approach, we find that despite the different
    non-analytic structures, the phase diagrams in two and three dimensions are remarkably similar, exhibiting an incommensurate spiral
    phase near to the avoided quantum critical point.

  • Breakdown of self-averaging in the Bose glass,
    A. Hegg, F. Krüger, and P. Phillips,
    Phys. Rev. B 88, 134206 (2013).

    We study the square-lattice Bose-Hubbard model with bounded random on-site energies at zero temperature. Starting from a dual
    representation obtained from a strong-coupling expansion around the atomic limit, we employ a real-space block decimation scheme.
    This approach is non-perturbative in the disorder and enables us to study the renormalization-group flow of the induced random-mass
    distribution. In both insulating phases, the Mott insulator and the Bose glass, the average mass diverges signaling short range
    superfluid correlations. The relative variance of the mass distribution distinguishes the two phases, renormalizing to zero in the Mott
    insulator and diverging in the Bose glass. Negative mass values in the tail of the distribution signal the presence of rare superfluid
    regions in the Bose glass. The breakdown of self-averaging is evidenced by the divergent relative variance and increasingly
    non-Gaussian distributions. We determine an explicit phase boundary between the Mott insulator and Bose glass.

  • Helical glasses near ferromagnetic quantum criticality,
    S. J. Thomson, F. Krüger, and A. G. Green,
    Phys. Rev. B 87, 224203 (2013).

    We study the effects of quenched charge disorder on the phase reconstruction near itinerant ferromagnetic quantum critical points in
    three spatial dimensions. Combining a replica disorder average with a fermionic version of the quantum-order-by disorder mechanism,
    we show that weak disorder destabilizes the ferromagnetic state and enhances the susceptibility towards incommensurate, spiral
    magnetic ordering. The Goldstone modes of the spiral phase are governed by a 3d-XY model. The induced disorder in the pitch of the
    spiral generates a random anisotropy for the Goldstone modes, inducing vortex lines in the phase of the helical order and rendering
    the magnetic correlations short ranged with a strongly anisotropic correlation length.

  • Disordered driven coupled cavity arrays: Non-equilibrium stochastic mean-field theory,
    G. Kulaitis, F. Krüger, F. Nissen, and J. Keeling,
    Phys. Rev. A 87, 013840 (2013).

    We study the interplay of disorder with pumping and decay in coupled qubit-cavity arrays, the Jaynes-Cummings-Hubbard model. We
    find that relatively weak disorder can wash out the bistability present in the clean pumped system, and that moreover, the combination
    of disorder in on-site energies and decay can lead to effective phase disorder. To explore these questions, we present a non-equilibrium
    generalisation of Stochastic-Mean-Field theory, providing a simple tool to address such questions. This technique is developed for
    rather general forms of light-matter coupling, driving, dissipation, and on-site disorder, making it applicable to a wide range of systems.

  • Quantum order-by-disorder driven phase reconstruction in the vicinity of ferromagnetic quantum critical points,
    U. Karahasanovic, F. Krüger, and A. G. Green,
    Phys. Rev. B 85, 165111 (2012).

    The formation of new phases close to itinerant electron quantum critical points has been observed experimentally in many
    compounds. We present a unified analytical model that explains the emergence of new types of order around itinerant ferromagnetic
    quantum critical points. The central idea of our analysis is that certain Fermi-surface deformations associated with the onset
    of the competing order enhance the phase-space available for low-energy quantum fluctuations and so self-consistently lower
    the free energy. We demonstrate that this quantum order-by-disorder mechanism leads to instabilities towards the formation
    of spiral and d-wave spin nematic phases close to itinerant ferromagnetic quantum critical points in three spatial dimensions.

  • Quantum order-by-disorder near criticality and the secret of partial order in MnSi,
    F. Krüger, U. Karahasanovic, and A. G. Green,
    Phys. Rev. Lett. 108, 067003 (2012).

    The vicinity of quantum phase transitions has proven fertile ground in the search for new quantum phases. We propose a physically
    motivated and unifying description of phase reconstruction near metallic quantum-critical points using the idea of quantum
    order-by-disorder. Certain deformations of the Fermi surface associated with the onset of competing order enhance the phase
    space available for low-energy, particle-hole fluctuations and self-consistently lower the free energy. Applying the notion of quantum
    order-by-disorder to the itinerant helimagnet MnSi, we show that near to the quantum critical point, fluctuations lead to an increase
    of the spiral ordering wave vector and a reorientation away from the lattice favored directions. The magnetic ordering pattern in this
    fluctuation-driven phase is found to be in excellent agreement with the neutron scattering data in the partially ordered phase of MnSi.

  • Spin-wave excitations in the ferromagnetic-metallic and in the charge, orbital and spin ordered states in
    Nd(1-x)Sr(x)MnO(3)$ with x~0.5
    H. Ulbrich, F. Krüger, A. A. Nugroho, D. Lamago, Y. Sidis, M. Braden,
    Phys. Rev. B 84, 094453 (2011).

    Inelastic neutron scattering experiments have been performed on single crystals of Nd(1-x)Sr(x)MnO(3) with x~0.5. Colossal
    magnetoresistance (CMR) in the manganites arises from the interplay between a ferromagnetic metallic and antiferromagnetic charge
    and orbital ordered insulating state. Therefore, it appears important to compare these phases concerning their underlying magnetic
    interaction parameters. Our investigations of the spin-wave disperion in the AFM ordered state of Nd(0.5)Sr(0.5)MnO(3) exhibits a
    strongly anisotropic stiffness. The sign of the anisotropy is characteristic for the site-centered model for charge and orbital ordering in
    half-doped manganites. Within this model, linear spin-wave theory yields a perfect description of the experimental dispersion.
    Furthermore, magnetic excitations in the ferromagnetic metallic state of Nd(1-x)Sr(x)MnO(3) with x=0.49 and x=0.50 exhibit
    nearly the same magnon dispersion which can be described with a Heisenberg model including nearest-neighbor interactions.

  • Two distinct Mott-Insulator to Bose-glass transitions and breakdown of self averaging in the disordered
    Bose-Hubbard model
    F. Krüger, Seungmin Hong, and P. Phillips,
    Phys. Rev. B 84, 115118 (2011).

    We investigate the instabilities of the Mott-insulating phase of the weakly disordered Bose-Hubbard model within a renormalization group
    analysis of the replica field theory obtained by a strong-coupling expansion around the atomic limit. We identify a new order parameter
    and associated correlation length scale that is capable of capturing the transition from a state with zero compressibility, the Mott insulator,
    to one in which the compressibility is finite, the Bose glass. The order parameter is the relative variance of the disorder-induced mass
    distribution. In the Mott insulator, the relative variance renormalizes to zero, whereas it diverges in the Bose glass. The divergence of
    the relative variance signals the breakdown of self-averaging. The length scale governing the breakdown of self-averaging is the distance
    between rare regions. This length scale is finite in the Bose glass but diverges at the transition to the Mott insulator with an exponent of
    $\nu=1/D$ for incommensurate fillings. Likewise, the compressibility vanishes with an exponent of $\gamma=4/D-1$ at the transition. At
    commensurate fillings, the transition is controlled by a different fixed point at which both the disorder and interaction vertices are relevant.

  • Orbital Ordering and Unfrustrated (pi,0) Magnetism from Degenerate Double Exchange in the Pnictides,
    Weicheng Lv, F. Krüger, and P. Phillips,
    Phys. Rev. B 82, 045125 (2010).

    The magnetic excitations of the iron pnictides are explained within a degenerate double-exchange model. The local-moment spins are
    coupled by superexchanges J1 and J2 between nearest and next-nearest neighbors, respectively, and interact with the itinerant electrons
    of the degenerate d(xz) and d(yz) orbitals via a ferromagnetic Hund exchange. The latter stabilizes (pi,0) stripe antiferromagnetism due to
    emergent ferro-orbital order and the resulting kinetic energy gain by hopping preferably along the ferromagnetic spin direction. Taking the
    quantum nature of the spins into account, we calculate the magnetic excitation spectra in the presence of both, super- and double-
    exchange. A dramatic increase of the spin-wave energies at the competing Néel ordering wave vector is found, in agreement with recent
    neutron scattering data. The spectra are fitted to a spin-only model with a strong spatial anisotropy and additional longer ranged couplings
    along the ferromagnetic chains. Over a realistic parameter range, the effective couplings along the chains are negative corresponding to
    unfrustrated stripe antiferromagnetism.

  • Anomalous suppression of the Bose glass at commensurate fillings in the disordered Bose-Hubbard model,
    F. Krüger, Jiansheng Wu, and P. Phillips,
    Phys. Rev. B 80, 094526 (2009),
    Virtual Journal of Atomic Quantum Fluids 1, (4) (2009).

    We study the weakly disordered Bose-Hubbard model on a cubic lattice through a one-loop renormalization group analysis of the
    corresponding effective field theory which is explicitly derived by combining a strong-coupling expansion with a replica average over
    the disorder. The method is applied not only to generic uncorrelated on-site disorder but also to simultaneous hopping disorder
    correlated with the differences of adjacent disorder potentials. Such correlations are inherent in fine-grained optical speckle potentials
    used as a source of disorder in optical lattice experiments. As a result of strong coupling, the strength of the replica mixing disorder
    vertex, responsible for the emergence of a Bose glass, crucially depends on the chemical potential and the Hubbard repulsion and
    vanishes to leading order in the disorder at commensurate boson fillings. As a consequence, at such fillings a direct transition
    between the Mott-insulator and the superfluid in the presence of disorder cannot be excluded on the basis of a one-loop
    calculation. At incommensurate fillings, at a certain length scale, the Mott insulator will eventually become unstable towards
    the formation of a Bose glass. Phase diagrams as a function of the microscopic parameters are presented and the finite-size
    crossover between the Mott-insulating state and the Bose glass is analyzed.

  • Phase diagram of the frustrated, spatially anisotropic S=1 antiferromagnet on a square lattice,
    H. C. Jiang, F. Krüger, J. E. Moore, D. N. Sheng, J. Zaanen, and Z. Y. Weng,
    Phys. Rev. B 79, 174409 (2009).

    We study the S=1 square lattice Heisenberg antiferromagnet with spatially anisotropic nearest neighbor couplings J1x, J1y
    frustrated by a next-nearest neighbor coupling J2 numerically using the density-matrix renormalization group (DMRG) method and
    analytically employing the Schwinger-Boson mean-field theory (SBMFT). Up to relatively strong values of the anisotropy, within both
    methods we find quantum fluctuations to stabilize the Neel ordered state above the classically stable region. Whereas SBMFT
    suggests a fluctuation-induced first order transition between the Néel state and a stripe antiferromagnet for 1/3<J1x/J1y<1
    and an intermediate paramagnetic region opening only for very strong anisotropy, the DMRG results clearly demonstrate that the two
    magnetically ordered phases are separated by a quantum disordered region for all values of the anisotropy with the remarkable implication
    that the quantum paramagnetic phase of the spatially isotropic J1-J2 model is continuously connected to the limit of decoupled
    Haldane spin chains. Our findings indicate that for S=1 quantum fluctuations in strongly frustrated antiferromagnets are crucial and
    not correctly treated on the semiclassical level.

  • Spin-orbital frustrations and anomalous metallic state in iron-pnictide superconductors,
    F. Krüger, S. Kumar, J. Zaanen, and J. van den Brink,
    Phys. Rev. B 79, 054504 (2009),
    Virtual Journal of Applications of Superconductivity 16, (4) (2009).

    We develop an understanding of the anomalous metal state of the parent compounds of recently discovered iron based superconductors
    starting from a strong coupling viewpoint, including orbital degrees of freedom. On the basis of an intermediate-spin (S=1) state for the
    Fe(2+) ions, we derive a Kugel-Khomskii spin-orbital Hamiltonian for the active t(2g) orbitals. It turns out to be a highly complex
    model with frustrated spin and orbital interactions. We compute its classical phase diagrams and provide an understanding for the stability
    of the various phases by investigating its spin-only and orbital-only limits. The experimentally observed spin-stripe state is found to be
    stable over a wide regime of physical parameters and can be accompanied by three different types of orbital orders. Of these the orbital-
    ferro and orbital-stripe orders are particularly interesting since they break the in-plane lattice symmetry --a robust feature of the undoped
    compounds. We compute the magnetic excitation spectra for the effective spin Hamiltonian, observing a strong reduction of the ordered
    moment, and point out that the proposed orbital ordering pattern can be measured in resonant X-ray diffraction.

  • Fermionic quantum criticality and the fractal nodal surface,
    F. Krüger and J. Zaanen,
    Phys. Rev. B 78, 035104 (2008).

    The complete lack of theoretical understanding of the quantum critical states found in the heavy fermion metals and the normal states
    of the high-Tc superconductors is routed in deep fundamental problem of condensed matter physics: the infamous minus signs
    associated with Fermi-Dirac statistics render the path integral non-probabilistic and do not allow to establish a connection with critical
    phenomena in classical systems. Using Ceperley's constrained path-integral formalism we demonstrate that the workings of scale
    invariance and Fermi-Dirac statistics can be reconciled. The latter is self-consistently translated into a geometrical constraint structure.
    We prove that this "nodal hypersurface" encodes the scales of the Fermi liquid and turns fractal when the system becomes quantum
    critical. To illustrate this we calculate nodal surfaces and electron momentum distributions of Feynman backflow wave functions and indeed
    find that with increasing backflow strength the quasiparticle mass gradually increases, to diverge when the nodal structure becomes fractal.
    Such a collapse of a Fermi liquid at a critical point has been observed in the heavy-fermion intermetallics in a spectacular fashion.

    Triggered by our paper, the visualization of nodal surfaces of backflow
    wavefunctions has been implemented as a
    MATHEMATICA demonstration project.

    Synopsis: "Fractals and quantum criticality" by Sarma Kancharla

  • Pacifying the Fermi-liquid: battling the devious fermion signs,
    J. Zaanen, F. Krüger, J.-H. She, D. Sadri, S. I. Mukhin,
    Iranian Journal of Physics Research, Vol. 8, No. 2, 39 (2008).

    The fermion sign problem is studied in the path integral formalism. The standard picture of Fermi liquids is first critically analyzed, pointing
    out some of its rather peculiar properties. The insightful work of Ceperley in constructing fermionic path integrals in terms of constrained
    worldlines is then reviewed. In this representation, the minus signs associated with Fermi-Dirac statistics are self consistently translated
    into a geometrical constraint structure (the nodal hypersurface) acting on an effective bosonic dynamics. As an illustrative example we use
    this formalism to study 1+1-dimensional systems, where statistics are irrelevant, and hence the sign problem can be circumvented. In this
    low-dimensional example, the structure of the nodal constraints leads to a lucid picture of the entropic interaction essential to one-
    dimensional physics. Working with the path integral in momentum space, we then show that the Fermi gas can be understood by analogy
    to a Mott insulator in a harmonic trap. Going back to real space, we discuss the topological properties of the nodal cells, and suggest
    a new holographic conjecture relating Fermi liquids in higher dimensions to soft-core bosons in one dimension. We also discuss some
    possible connections between mixed Bose/Fermi systems and supersymmetry.

  • Magnetic fluctuations in n-type high-Tc superconductors reveal breakdown of fermiology: Experiments and Fermi-liquid/
    RPA calculations
    F. Krüger, S. D. Wilson, L. Shan, S. Li, Y. Huang, H. H. Wen, S. C. Zhang, Pengcheng Dai, J. Zaanen,
    Phys. Rev. B 76, 094506 (2007).

    By combining experimental measurements of the quasiparticle and dynamical magnetic properties of optimally electron-doped
    Pr(0.88)LaCe(0.12)CuO4 with theoretical calculations we demonstrate that the conventional fermiology approach cannot possibly account
    for the magnetic fluctuations in these materials. In particular, we perform tunneling experiments on the very same sample for which a
    dynamical magnetic resonance has been reported recently and use photoemission data by others on a similar sample to characterize the
    fermionic quasi-particle excitations in great detail. We subsequently use this information to calculate the magnetic response within the
    conventional fermiology framework as applied in a large body of work for the hole-doped superconductors to find a profound disagreement
    between the theoretical expectations and the measurements: this approach predicts a step-like feature rather than a sharp resonance
    peak, it underestimates the intensity of the resonance by an order of magnitude, it suggests an unreasonable temperature dependence
    of the resonance, and most severely, it predicts that most of the spectral weight resides in incommensurate wings which are a key
    feature of the hole-doped cuprates but have never been observed in the electron-doped counterparts. Our findings strongly suggest that
    the magnetic fluctuations reflect the quantum-mechanical competition between antiferromagnetic and superconducting orders.

  • Is deconfined quantum criticality in frustrated antiferromagnets ruled out by generic fluctuation induced
    first-order behavior?
    F. Krüger,
    J. Supercond. Nov. Magn. 20, 575 (2007).

    Proceeding for International Conference Stripes2006, Rome, Italy.

  • Frustrated Heisenberg antiferromagnets: fluctuation induced first order vs deconfined quantum criticality,
    F. Krüger and S. Scheidl,
    Europhys. Lett. 74, 896 (2006).

    Recently it was argued that quantum phase transitions can be radically different from classical phase transitions with as a highlight the
    'deconfined critical points' exhibiting fractionalization of quantum numbers due to Berry phase effects. Such transitions are supposed to
    occur in frustrated ('J1-J2') quantum magnets. We have developed a novel renormalization approach for such systems which is
    fully respecting the underlying lattice structure. According to our findings, another profound phenomenon is around the corner:
    a fluctuation induced (order-out-of-disorder) first order transition. This has to occur for large spin and we conjecture that it is responsible
    for the weakly first order behavior recently observed in numerical simulations for frustrated S=1/2 systems.

  • Spin-wave dispersion in orbitally ordered La$_{1/2}$Sr$_{3/2}$MnO$_4$,
    D. Senff, F. Krüger, S. Scheidl, M. Benomar, Y. Sidis, S. Demmel, and M. Braden,
    Phys. Rev. Lett. 96, 257201 (2006).

    The magnon dispersion in the charge, orbital and spin ordered phase in La(1/2)Sr(3/2)MnO4 has been studied by means of
    inelastic neutron scattering. We find an excellent agreement with a magnetic interaction model basing on the CE-type superstructure.
    The magnetic excitations are dominated by ferromagnetic exchange parameters revealing a nearly-one dimensional character at high
    energies. The nearest neighbor ferromagnetic interaction in La(1/2)Sr(3/2)MnO4 is significantly larger than the one in the
    metallic ferromagnetically ordered manganites. The large ferromagneticinteraction in the charge/orbital ordered phase appears to be
    essential for the capability of manganites to switch between metallic and insulating phases.

  • The spin excitation spectrum in striped bilayer compounds,
    F. Krüger and S. Scheidl,
    Phys. Rev. B 70, 064421 (2004),
    Virtual Journal of Applications of Superconductivity 7, (5) (2004).

    The spin dynamics of bilayer cuprate compounds are studied in a basic model. The magnetic spectral properties are calculated in linear
    spin-wave theory for several stripe configurations which differ by the relative location of the stripes in the layers. We focus on the bilayer
    splitting of the magnon bands near the incommensurate low energy peaks as well as near the pi resonance, distinguishing between the
    odd and even channel. We find that a x-shaped dispersion near the pi resonance is generic for stripes. By comparison of our results to
    neutron scattering data for YBa(2)Cu(3)O(6+x) we conclude that the stripe model is consistent with characteristic features of
    bilayer high-Tc compounds.

  • Spin dynamics of stripes,
    F. Krüger and S. Scheidl,
    Phys. Rev. B. 67, 134512 (2003).

    The spin dynamics of stripes in high-temperature superconductors and related compounds is studied in the framework of a spin-wave
    theory for a simple spin-only model. The magnon dispersion relation and the magnetic structure factor are calculated for diagonal and
    vertical stripes. Acoustical as well as optical bands are included in the analysis. The incommensurability and the pi resonance
    appear as complementary features of the band structure at different energy scales. The dependence of spin-wave velocities and
    resonance frequencies on the stripe spacing and coupling is calculated. At low doping, the resonance frequency is found to scale
    roughly inversely proportional to the stripe spacing. The favorable comparison of the results with experimental data suggests
    that the spin-only model provides a suitable and simple basis for calculating and understanding the spin dynamics of stripes.

  • Spin and charge ordering transitions in stripes,
    F. Krüger and S. Scheidl,
    J. Phys. IV France 12, Pr9-259 (2002).

    Proceeding for International Workshop on Electronic Crystal (ECRYS), St. Flour, France.

  • Non-universal ordering of spin and charge in stripe phases,
    F. Krüger and S. Scheidl,
    Phys. Rev. Lett. 89, 095701 (2002).

    We study the interplay of topological excitations in stripe phases: charge dislocations, charge loops, and spin vortices. In two dimensions
    these defects interact logarithmically on large distances. Using a renormalization-group analysis in the Coulomb gas representation of
    these defects, we calculate the phase diagram and the critical properties of the transitions. Depending on the interaction parameters,
    spin and charge order can disappear at a single transition or in a sequence of two transitions (spin-charge separation). These transitions
    are non-universal with continuously varying critical exponents. We also determine the nature of the points where three phases coexist.

  • Bond-disordered spin systems: Theory and application to doped high-$T_\textrm{c}$ compounds,
    F. Krüger and S. Scheidl,
    Phys. Rev. B 65, 224502 (2002).

    We examine the stability of magnetic order in a classical Heisenberg model with quenched random exchange couplings. This system
    represents the spin degrees of freedom in high-Tc compounds with immobile dopants. Starting from a replica representation
    of the nonlinear sigma model, we perform a renormalization-group analysis. The importance of cumulants of the disorder distribution
    to arbitrarily high orders necessitates a functional renormalization scheme. From the renormalization flow equations we determine the
    magnetic correlation length numerically as a function of the impurity concentration and of temperature. From our analysis follows that
    two-dimensional layers can be magnetically ordered for arbitrarily strong but sufficiently diluted defects. We further consider the
    dimensional crossover in a stack of weakly coupled layers. The resulting phase diagram is compared with experimental data for