Complex Order Parameter Research Articles

Why hyperdensity functionals describe any equilibrium observable.

We give an introductory account of the recent hyperdensity functional theory for the equilibrium statistical mechanics of soft matter systems [F. Sammüller et al., Phys. Rev. Lett. 133, 098201 (2024)]. Hyperdensity functionals give access to the behaviour of arbitrary thermal observables in spatially inhomogeneous equilibrium many-body systems. The approach is based on classical density functional theory applied to an extended ensemble using standard functional techniques. The associated formally exact generalized Mermin-Evans functional relationships can be represented efficiently by neural functionals. These neural networks are trained via simulation-based supervised machine learning and they allow one to carry out efficient functional calculus using automatic differentiation and numerical functional line integration. Exact sum rules, including hard wall contact theorems and hyperfluctuation Ornstein-Zernike equations, interrelate the different correlation functions. We lay out close connections to hyperforce correlation sum rules [S. Robitschko et al., Commun. Phys. 7, 103 (2024)] that arise from statistical mechanical gauge invariance [J. Müller et al., Phys. Rev. Lett. (to appear); arXiv:2406.19235]. Further quantitative measures of collective self-organization are provided by hyperdirect correlation functionals and spatially resolved hyperfluctuation profiles. The theory facilitates to gain deep insight into the inherent structuring mechanisms that govern the behaviour of both simple and complex order parameters in coupled many-body systems.&#xD.

Finite-size effect in Kuramoto oscillators with higher-order interactions.

Finite-size systems of a Kuramoto model display intricate dynamics, especially in the presence of multi-stability where both coherent and incoherent states coexist. We investigate such a scenario in globally coupled populations of Kuramoto phase oscillators with higher-order interactions and observe that fluctuations inherent to finite-size systems drive the transition to the synchronized state before the critical point in the thermodynamic limit. Using numerical methods, we plot the first exit-time distribution of the magnitude of a complex order parameter and obtain numerical transition probabilities across various system sizes. Furthermore, we extend this study to a two-population oscillator system, and, using the velocity field of the associated order parameters, show the emergence of a new fixed point corresponding to a partially synchronized state arising due to the finite-size effect, which is absent in the thermodynamics limit.

Discrete Z4 Symmetry in Quantum Gravity

We consider the discrete Z4 symmetry i^, which takes place in the scenario of quantum gravity where the gravitational tetrads emerge as the order parameter—the vacuum expectation value of the bilinear combination of fermionic operators. Under this symmetry operation, i^, the emerging tetrads are multiplied by the imaginary unit, i^eμa=−ieμa. The existence of such symmetry and the spontaneous breaking of this symmetry are also supported by the consideration of the symmetry breaking scheme in the topological superfluid 3He-B. The order parameter in 3He-B is also the bilinear combination of the fermionic operators. This order parameter is the analog of the tetrad field, but it has complex values. The i^-symmetry operation changes the phase of the complex order parameter by π/2, which corresponds to the Z4 discrete symmetry in quantum gravity. We also considered the alternative scenario of the breaking of this Z4 symmetry, in which the i^-operation changes sign of the scalar curvature, i^R=−R, and thus the Einstein–Hilbert action violates the i^-symmetry. In the alternative scenario of symmetry breaking, the gravitational coupling K=1/16πG plays the role of the order parameter, which changes sign under i^-transformation.

Hyperdensity Functional Theory of Soft Matter.

We present a scheme for investigating arbitrary thermal observables in spatially inhomogeneous equilibrium many-body systems. Extending the grand canonical ensemble yields any given observable as an explicit hyperdensity functional. Associated local fluctuation profiles follow from an exact hyper-Ornstein-Zernike equation. While the local compressibility and simple observables permit analytic treatment, complex order parameters are accessible via simulation-based supervised machine learning of neural hyperdirect correlation functionals. We exemplify efficient and accurate neural predictions for the cluster statistics of hard rods, square-well rods, and hard spheres. The theory allows one to treat complex observables, as is impossible in standard density functional theory.

Multimodal Approach Reveals the Symmetry-Breaking Pathway to the Broken Helix in EuIn2As2

Understanding and manipulating emergent phases, which are themes at the forefront of quantum-materials research, rely on identifying their underlying symmetries. This general principle has been particularly prominent in materials with coupled electronic and magnetic degrees of freedom, in which magnetic order influences the electronic band structure and can lead to exotic topological effects. However, identifying symmetry of a magnetically ordered phase can pose a challenge, particularly in the presence of small domains. Here we introduce a multimodal approach for determining magnetic structures, which combines symmetry-sensitive optical probes, scattering, and group-theoretical analysis. We apply it to EuIn2As2, a material that has received attention as a candidate axion insulator. While first-principles calculations predict this state on the assumption of a simple collinear antiferromagnetic structure, subsequent neutron-scattering measurements reveal a much more intricate magnetic ground state characterized by two coexisting magnetic wave vectors reached by successive thermal phase transitions. The proposed high- and low-temperature phases are a spin helix and a state with interpenetrating helical and Néel antiferromagnetic order termed a “broken helix,” respectively. Employing a multimodal approach, we identify the magnetic structure associated with these two phases of EuIn2As2. We find that the higher-temperature phase is characterized by a variation of the magnetic moment amplitude from layer to layer, with the moment vanishing entirely in every third Eu layer. The lower-temperature structure is similar to the broken helix, with one important difference: Because of local strain, the relative orientation of the magnetic structure and the lattice is not fixed. Consequently, the symmetry required to protect the axion phase is not generically protected in EuIn2As2, but we show that it can be restored if the magnetic structure is tuned with uniaxial strain. Finally, we present a spin Hamiltonian that identifies the spin interactions that account for the complex magnetic order in EuIn2As2. Our work highlights the importance of a multimodal approach in determining the symmetry of complex order parameters. Published by the American Physical Society 2024

Nearly critical superfluids in Keldysh-Schwinger formalism

We examine the effective theory of critical dynamics near superfluid phase transitions in the framework of the Keldysh-Schwinger formalism. We focus on the sector capturing the dynamics of the complex order parameter and the conserved current corresponding to the broken global symmetry. After constructing the theory up to quadratic order in the a-fields, we compare the resulting stochastic system with Model F as well as with holography. We highlight the role of a time independent gauge symmetry of the effective theory also known as “chemical shift”. Finally, we consider the limiting behaviour at energies much lower than the gap of the amplitude mode by integrating out the high energy degrees of freedom to reproduce the effective theory of superfluids.

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All superconductors in a magnetic field are characterized by three critical magnetic fields: lower critical Hc1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{c1}$$\\end{document}, upper critical Hc2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{c2}$$\\end{document} and thermodynamic critical field Hc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{c}$$\\end{document}. Only two sets of inequalities Hc2>Hc>Hc1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{c2}>H_c>H_{c1}$$\\end{document} or Hc1>Hc>Hc2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{c1}>H_c>H_{c2}$$\\end{document} are possible in a single-component superconductor. Here, we report our study of the critical fields in multicomponent superconductors with two superconducting components in the framework of the Ginzburg-Landau functional. We derive the relationship between the phases of the components of the superconducting complex order parameter from the charge conservation law in explicit form and insert it into the Ginzburg-Landau functional. Using the modified Ginzburg-Landau equation, we acquire the single vortex state including the analytical expression for asymptotics. Also, we obtain the analytical form for the state in the upper critical field. We find that in some cases an unusual sequence of critical fields Hc1,Hc2>Hc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{c1},H_{c2}>H_c$$\\end{document} can be realized in multicomponent superconductors.

Superconducting gap and critical behavior in the Iron-Pnictides

In the phase diagram of iron pnictides, superconductivity arises at the border of antiferromagnetism, which raises the question of the role of symmetry of the gap and quantum criticality. Although more than 15-years of extensive research, the microscopic origin of the pairing symmetry inside the superconducting (SC) dome and its link to quantum criticality still remains elusive. Here, we report two new findings on BaFe2−xNixAs2: (1) A sharp peak in the x-dependence of the lower and upper critical fields, the SC critical current density Jc, the size of the jump in the specific heat ΔCel/T and the Sommerfeld coefficient (γ) at the optimum composition x = 0.10, where the SC transition temperature Tc reaches a maximum. Our obtained reliable values as a function of doping of the normal-state Sommerfeld coefficient increase with doping, illustrating the strong competition between magnetism and superconductivity and attributed to closing of spin density wave gap with Ni doping. (2) We show that doping induced a sudden change of the gap structure from nodeless to nodal. Our results imply that the superconductivity in BaFe2−xNixAs2 is closely linked to the quantum criticality and is characterized by a complex order parameter.

Transferring orbital angular momentum to an electron beam reveals toroidal and chiral order

Orbital angular momentum (OAM) and torque transfer play central roles in a wide range of magnetic textures and devices including skyrmions and spin-torque electronics. Analogous topological structures are now also being explored in ferroelectrics, including polarization vortex arrays in ferroelectric/dielectric superlattices. Unlike magnetic toroidal order, electric toroidal order does not couple directly to linear external fields. Instead, we find that the presence of an electric toroidal moment in a ferrorotational phase transfers measurable torque and OAM to a localized electron beam in the ballistic limit. We record these torque transfers from a high-energy electron beam using a momentum-resolved detector. This approach provides a high-sensitivity method to detect polarization fields and their more complex order parameters and topologies. In addition to toroidal order, we also demonstrate high-precision measurements of vorticity and chirality for polar vortexlike phases.

Simulations of Dynamical Electronic Vortices in Charge and Spin Density Waves

Charge and spin density waves are typical symmetry broken states of quasi one-dimensional electronic systems. They demonstrate such common features of all incommensurate electronic crystals as a spectacular non-linear conduction by means of the collective sliding and susceptibility to the electric field. These phenomena ultimately require for emergence of static and transient topological defects: there are dislocations as space vortices and space-time vortices known as phase slip centers, i.e., a kind of instantons. Dislocations are statically built-in under a transverse electric field; their sweeping provides a conversion among the normal carriers and condensate which ensures the onset of the collective sliding. A special realization in a high magnetic field, when the density wave is driven by the Hall voltage, originated by quantized normal carriers, reveals the dynamic vorticity serving to annihilate compensating normal and collective currents. Spin density waves, with their rich multiplicative order parameter, bring to life complex objects with half-integer topologically bound vorticities in charge and spin degrees of freedom. We present the basic concepts and modelling results of the stationary states and their transient dynamics involving vorticity. The models take into account multiple fields in their mutual non-linear interactions: the complex order parameter, the self-consistent electric field, and the reaction of normal carriers. We explore the traditional time-dependent Ginzburg–Landau approach and introduce its generalization allowing the treatment of intrinsic normal carriers. The main insights and illustrations come from numerical solutions to partial differential equations for the dissipative dynamics of one and two space dimensions.

Quantum quenches of an SO(5) pseudospin reveal Higgs bosons

A controlled dynamical probe measurement of complex order parameter fluctuations of a system may reveal its massive collective excitations. Here, we design dynamical quench protocols to excite independently all ten midgap Higgs bosons in the isotropic Balian-Werthamer state of a spinful $p$-wave superfluid or superconductor. The analysis is based on microscopic equations of motion of an SO(5) pseudospin, an extension of the usual Bloch equation to a five-dimensional space. Key to these protocols is the realization of quenches that break the rotational symmetry of the kinetic energy and exploit the irreducible representation of the angular momentum $J=2$. For perturbative quenches, we find (nondecaying) periodic oscillations in time of these Higgs modes. We present experimental protocols for superconductors (superfluids), with the intention of unveiling the nature of their order parameters.

Coherent detection of hidden spin-lattice coupling in a van der Waals antiferromagnet.

Strong interactions between different degrees of freedom lead to exotic phases of matter with complex order parameters and emergent collective excitations. Conventional techniques, such as scattering and transport, probe the amplitudes of these excitations, but they are typically insensitive to phase. Therefore, novel methods with phase sensitivity are required to understand ground states with phase modulations and interactions that couple to the phase of collective modes. Here, by performing phase-resolved coherent phonon spectroscopy (CPS), we reveal a hidden spin-lattice coupling in a vdW antiferromagnet FePS3 that eluded other phase-insensitive conventional probes, such as Raman and X-ray scattering. With comparative analysis and analytical calculations, we directly show that the magnetic order in FePS3 selectively couples to the trigonal distortions through partially filled t2g orbitals. This magnetoelastic coupling is linear in magnetic order and lattice parameters, rendering these distortions inaccessible to inelastic scattering techniques. Our results not only capture the elusive spin-lattice coupling in FePS3 but also establish phase-resolved CPS as a tool to investigate hidden interactions.

Behavior of circular chains of nonlinear oscillators with Kuramoto-like local coupling

The conditions under which synchronization is achieved for a one-dimensional ring of identical phase oscillators with Kuramoto-like local coupling are studied. The system is approached in the weakly coupled approximation as phase units. Instead of global couplings, the nearest-neighbor interaction is assumed. Units are pairwise coupled by a Kuramoto term driven by their phase differences. The system exhibits a rich set of behaviors depending on the balance between the natural frequency of isolated units and the self-feedback. The case of two oscillators is solved analytically, while a numerical approach is used for N > 2. Building from Kuramoto, the approach to synchronization, when possible, is studied through a local complex order parameter. The system can eventually evolve as a set of coupled local communities toward a given phase value. However, the approach to the stationary state shows a non-monotonous non-trivial dynamic.

Local Dirac Synchronization on networks.

We propose Local Dirac Synchronization that uses the Dirac operator to capture the dynamics of coupled nodes and link signals on an arbitrary network. In Local Dirac Synchronization, the harmonic modes of the dynamics oscillate freely while the other modes are interacting non-linearly, leading to a collectively synchronized state when the coupling constant of the model is increased. Local Dirac Synchronization is characterized by discontinuous transitions and the emergence of a rhythmic coherent phase. In this rhythmic phase, one of the two complex order parameters oscillates in the complex plane at a slow frequency (called emergent frequency) in the frame in which the intrinsic frequencies have zero average. Our theoretical results obtained within the annealed approximation are validated by extensive numerical results on fully connected networks and sparse Poisson and scale-free networks. Local Dirac Synchronization on both random and real networks, such as the connectome of Caenorhabditis Elegans, reveals the interplay between topology (Betti numbers and harmonic modes) and non-linear dynamics. This unveils how topology might play a role in the onset of brain rhythms.

Complex-tensor theory of simple smectics

Matter self-assembling into layers generates unique properties, including structures of stacked surfaces, directed transport, and compact area maximization that can be highly functionalized in biology and technology. Smectics represent the paradigm of such lamellar materials — they are a state between fluids and solids, characterized by both orientational and partial positional ordering in one layering direction, making them notoriously difficult to model, particularly in confining geometries. We propose a complex tensor order parameter to describe the local degree of lamellar ordering, layer displacement and orientation of the layers for simple, lamellar smectics. The theory accounts for both dislocations and disclinations, by regularizing singularities within defect cores and so remaining continuous everywhere. The ability to describe disclinations and dislocation allows this theory to simulate arrested configurations and inclusion-induced local ordering. This tensorial theory for simple smectics considerably simplifies numerics, facilitating studies on the mesoscopic structure of topologically complex systems.

Nonabelian Ginzburg–Landau theory for ferroelectrics

The Ginzburg–Landau theory, which was introduced to phenomenologically describe the destruction of superconductivity by a magnetic field at the beginning, has brought up much more knowledge beyond the original one as a mean-field theory of thermodynamics states. There the complex order parameter plays an important role. Here we propose a macroscopic theory to describe the features of ferroelectrics by a two-component complex order parameter coupled to nonabelian gauge potentials that provide more freedom to reflect interplays between different measurables. Within this theoretical framework, some recently discovered empirical static and time-independent phenomena, such as vortex, anti-vortex, spiral orders can be obtained as solutions for different gauge potentials. It is expected to bring in a new angle of view with more elucidation than the traditional one that takes the polarization as order parameter.

Generalization of the Kuramoto model to the Winfree model by a symmetry breaking coupling

We construct a nontrivial generalization of the paradigmatic Kuramoto model by using an additional coupling term that explicitly breaks its rotational symmetry resulting in a variant of the Winfree model. Consequently, we observe the characteristic features of the phase diagrams of both the Kuramoto model and the Winfree model depending on the degree of the symmetry breaking coupling strength for unimodal frequency distribution. The phase diagrams of both the Kuramoto and the Winfree models resemble each other for symmetric bimodal frequency distribution for a range of the symmetry breaking coupling strength except for region shift and difference in the degree of spread of the macroscopic dynamical states and bistable regions. The dynamical transitions in the bistable states are characterized by an abrupt (first-order) transition in both the forward and reverse traces. For asymmetric bimodal frequency distribution, the onset of bistable regions depends on the degree of the asymmetry. Large degree of the symmetry breaking coupling strength promotes the synchronized stationary state, while a large degree of heterogeneity, proportional to the separation between the two central frequencies, facilitates the spread of the incoherent and standing wave states in the phase diagram for a low strength of the symmetry breaking coupling. We deduce the low-dimensional equations of motion for the complex order parameters using the Ott-Antonsen ansatz for both unimodal and bimodal frequency distributions. We also deduce the Hopf, pitchfork, and saddle-node bifurcation curves from the evolution equations for the complex order parameters mediating the dynamical transitions. Simulation results of the original discrete set of equations of the generalized Kuramoto model agree well with the analytical bifurcation curves.

Supercooling of the A phase of 3He

Because of the extreme purity, lack of disorder, and complex order parameter, the first-order superfluid 3He A–B transition is the leading model system for first order transitions in the early universe. Here we report on the path dependence of the supercooling of the A phase over a wide range of pressures below 29.3 bar at nearly zero magnetic field. The A phase can be cooled significantly below the thermodynamic A–B transition temperature. While the extent of supercooling is highly reproducible, it depends strongly upon the cooling trajectory: The metastability of the A phase is enhanced by transiting through regions where the A phase is more stable. We provide evidence that some of the additional supercooling is due to the elimination of B phase nucleation precursors formed upon passage through the superfluid transition. A greater understanding of the physics is essential before 3He can be exploited to model transitions in the early universe.

Eulerian fractions and Stirling, Bernoulli and Euler functions with complex order parameters and their impact on the polylogarithm function

We first study some generalizations of Eulerian fractions with complex order parameter and investigate their interrelationship with likewise generalized Eulerian functions as well as Stirling functions. We apply the new approach to polylogarithms of non-integral order, for which only a few values are known in closed form. In particular, we present a structural solution of the counterpart of an old conjecture of Mengoli and Euler in the polylogarithm case with the aid of Riemann?s zeta function and the Dirichlet eta and beta functions.

Nearly critical holographic superfluids

We study the nearly critical behaviour of holographic superfluids at finite temperature and chemical potential in their probe limit. This allows us to examine the coupled dynamics of the full complex order parameter with the charge density of the system. We derive an effective theory for the long wavelength limit of the gapless and pseudo-gapped modes by using analytic techniques in the bulk. We match our construction with Model F in the classification of Hohenberg and Halperin and compute the complex dissipative kinetic transport coefficient in terms of thermodynamics and black hole horizon data. We carry out an analysis of the corresponding modes and argue that at finite density the dispersion relations are discontinuous between the normal and the broken phase. We compare and contrast our results with earlier numerical work.