Nonrelativistic Scalar Field Research Articles

Self-similar solutions for fuzzy dark matter

Self-similar solutions for fuzzy dark matter are very different from their counterparts in the standard cold dark matter model. In contrast to the familiar hierarchical collapse of the current model for structure formation, they correspond to an inverse hierarchical blow-up. This fact highlights the gravitational cooling process, which in the absence of dissipation, allows the system to eject excess energy through the intermittent expulsion of clumps of matter. These surprising behaviours are due to the wave properties of the Schrödinger equation and to the quantum pressure. These features are printed in the Eulerian density-velocity representation of the nonrelativistic scalar field, or in the Lagrangian representation in the mass-shell trajectories.

Self-similar solutions for fuzzy dark matter

Fuzzy dark matter (FDM) models admit self-similar solutions which are very different from the standard cold dark matter (CDM) self-similar solutions and do not converge to the latter in the semiclassical limit. In contrast with the familiar CDM hierarchical collapse, they correspond to an inverse-hierarchy blowup. Constant-mass shells start in the nonlinear regime, at early times, with small radii and high densities, and expand to reach at late times the Hubble flow, up to small linear perturbations. Thus, larger masses become linear first. This blowup approximately follows the Hubble expansion, so that the central density contrast remains constant with time, although the width of the self-similar profile shrinks in comoving coordinates. As in a gravitational cooling process, matter is ejected from the central peaks through successive clumps. As in wave systems, the velocities of the geometrical structures and of the matter do not coincide, and matter slowly moves from one clump to the next, with intermittent velocity bursts at the transitions. These features are best observed using the density-velocity representation of the nonrelativistic scalar field, or the mass-shell trajectories, than with the Husimi phase-space distribution, where an analog of the Heisenberg uncertainty principle blurs the resolution in the position or velocity direction. These behaviors are due to the quantum pressure and the wavelike properties of the Schr\"odinger equation. Although the latter has been used as an alternative to $N$-body simulations for CDM, these self-similar solutions show that the semiclassical limit needs to be handled with care.

Real scalar field, the nonrelativistic limit, and the cosmological expansion

The existing transformation from a relativistic real scalar field to a complex non-relativistic scalar field by Namjoo, Guth, and Kaiser is generalized from Minkowski space to a more general background metric. In that case the transformation is not purely algebraic any more but determined by a differential equation. We apply the generalized transformation to a real scalar with $\phi^4$ interaction on an Friedmann-Lema\^itre-Robertson-Walker cosmologically expanding background and calculate the resulting non-relativistic action up to second order in small parameters. We also show that the transformation can be interpreted as a Bogoliubov transformation between relativistic and non-relativistic creation and annihilation operators and comment on emerging symmetries in the non-relativistic theory.

Effective theories for a nonrelativistic field in an expanding universe: Induced self-interaction, pressure, sound speed, and viscosity

A massive, nonrelativistic scalar field in an expanding spacetime is usually approximated by a pressureless perfect fluid, which leads to the standard conclusion that such a field can play the role of cold dark matter. In this paper, we systematically study these approximations, incorporating subleading corrections. We provide two equivalent effective descriptions of the system, each of which offers its own advantages and insights: (i) A nonrelativistic effective field theory (EFT) with which we show that the relativistic corrections induce an effective self-interaction for the nonrelativistic field. As a byproduct, our EFT also allows one to construct the exact solution, including oscillatory behavior, which is often difficult to achieve from the exact equations. (ii) An effective (imperfect) fluid description, with which we demonstrate that, for a perturbed Friedmann-Lemaître- Robertson-Walker (FLRW) universe: (a) The pressure is small but nonzero (and positive), even for a free theory with no tree-level self-interactions. (b) The sound speed of small fluctuations is also nonzero (and positive), reproducing already known leading-order results, correcting a subdominant term, and identifying a new contribution that had been omitted in previous analyses. (c) The fluctuations experience a negative effective bulk viscosity. The positive sound speed and the negative bulk viscosity act in favor of and against the growth of overdensities, respectively. The net effect may be considered a smoking gun for ultra-light dark matter.

Formation, gravitational clustering, and interactions of nonrelativistic solitons in an expanding universe

We investigate the formation, gravitational clustering, and interactions of solitons in a self-interacting, nonrelativistic scalar field in an expanding universe. Rapid formation of a large number of solitons is driven by attractive self-interactions of the field, whereas the slower clustering of solitons is driven by gravitational forces. Driven closer together by gravity, we see a rich plethora of dynamics in the soliton ``gas'' including mergers, scatterings, and formation of soliton binaries. The numerical simulations are complemented by analytic calculations and estimates of (i) the relevant instability length scales and timescales, (ii) individual soliton profiles and their stability, (iii) number density of produced solitons, and (iv) the two-point correlation function of soliton positions as evidence for gravitational clustering.

Klein–Gordon field from the XXZ Heisenberg model

We examine the recently introduced idea of Spin-Field Correspondence (SFC) focusing on the example of the spin system described by the XXZ Heisenberg model with external magnetic field. The Hamiltonian of the resulting nonlinear scalar field theory is derived for arbitrary value of the anisotropy parameter [Formula: see text]. We show that the linear scalar field theory is reconstructed in the large spin limit. For [Formula: see text], a nonrelativistic scalar field theory satisfying the Born reciprocity principle is recovered. As expected, for the vanishing anisotropy parameter [Formula: see text], the standard relativistic Klein–Gordon field is obtained. Various aspects of the obtained class of the scalar fields are studied, including the fate of the relativistic symmetries and the properties of the emerging interaction terms. We show that, in a certain limit, the so-called polymer quantization of the field variables is recovered. This and other discussed properties suggest a possible relevance of the considered framework in the context of quantum gravity.

Universal horizons and Hawking radiation in nonprojectable 2d Hořava gravity coupled to a nonrelativistic scalar field

In this paper, we study the non-projectable 2d Ho\v{r}ava gravity coupled with a non-relativistic scalar field, where the coupling is in general non-minimal and of the form $f(\phi)R$, where $f(\phi)$ is an arbitrary function of the scalar field $\phi$, and $R$ denotes the 2d Ricci scalar. In particular, we first investigate the Hamiltonian structure, and show that there are two-first and two-second class constraints, similar to the pure gravity case, but now the local degree of freedom is one, due to the presence of the scalar field. Then, we present various exact stationary solutions of this coupled system, and find that some of them represent black holes but now with universal horizons as their boundaries. At these horizons, the Hawking radiations are thermal with temperatures proportional to their surface gravities, which normally depend on the non-linear dispersion relations of the particles radiated, similar to the (3+1)-dimensional case.

Spin-Field Correspondence

In the recent article Phys. Lett. B 2016, 759, 424–429, a new class of field theories called Nonlinear Field Space Theory was proposed. In this approach, the standard field theories are considered as linear approximations to some more general theories characterized by nonlinear field phase spaces. The case of spherical geometry is especially interesting due to its relation with the spin physics. Here, we explore this possibility, showing that classical scalar field theory with such a field space can be viewed as a perturbation of a continuous spin system. In this picture, the spin precession and the scalar field excitations are dual descriptions of the same physics. The duality is studied in the example of the Heisenberg model. It is shown that the Heisenberg model coupled to a magnetic field leads to a non-relativistic scalar field theory, characterized by quadratic dispersion relation. Finally, on the basis of analysis of the relation between the spin phase space and the scalar field theory, we propose the Spin-Field correspondence between the known types of fields and the corresponding spin systems.

Torsional Newton–Cartan geometry from Galilean gauge theory

Using the recently advanced Galilean gauge theory (GGT) we give a comprehensive construction of torsional Newton–Cartan (NC) geometry. The coupling of a Galilean symmetric model with background NC geometry following GGT is illustrated by a free nonrelativistic scalar field theory. The issue of spatial diffeomorphism (Son and Wingate 2006 Ann. Phys. 321 197–224; Banerjee et al 2015 Phys. Rev. D 91 084021) is focussed from a new angle. The expression of the torsionful connection is worked out, which is in complete parallel with the relativistic theory. Also, smooth transition of the connection to its well known torsionless expression is demonstrated. A complete (implicit) expression of the torsion tensor for the NC spacetime is provided where the first-order variables occur in a suggestive way. The well known result for the temporal part of torsion is reproduced from our expression.

Holographic superconductors in Hořava–Lifshitz gravity

We consider holographic superconductors related to the Schwarzschild black hole in the low energy limit of Hořava–Lifshitz spacetime. The nonrelativistic electromagnetic and scalar fields are introduced to construct a holographic superconductor model in Hořava–Lifshitz gravity and the results show that the α2 term plays an important role, modifying the conductivity curve line by means of an attenuation of the conductivity.

Spontaneous generation of a crystalline ground state in a higher derivative theory

The possibility of Spontaneous Symmetry Breaking in momentum space in a generic Lifshitz scalar model–a non-relativistic scalar field theory with higher spatial derivative terms–has been studied. We show that the minimum energy state, the ground state, has a lattice structure, where the translation invariance of the continuum theory is reduced to a discrete translation symmetry. The scale of translation symmetry breaking (or induced lattice spacing) is proportional to the inverse of the momentum of the condensate particle. The crystalline ground state is stable under excitations below a certain critical velocity. The small fluctuations above the ground state can have a phonon like dispersion under suitable choice of parameters.At the beginning we have discussed the effects of next to nearest neighbor interaction terms in a model of linear triatomic molecule depicted by a linear system of three particles of same mass connected by identical springs. This model is relevant since in the continuum limit the next to nearest neighbor interaction terms generate higher (spatial) derivative wave equation, the main topic of this paper.

Method for simulatingO(N)lattice models at finite density

We present a method for simulating relativistic and nonrelativistic scalar field theories at finite density, with matter transforming in the fundamental representation of the global symmetry group O(N). The method avoids the problem of complex probability weights which is present in conventional path integral Monte Carlo algorithms. To verify our approach, we simulate the free and interacting relativistic U(1){approx_equal}O(2) theory in 2+1 dimensions. We compute the two-point correlation function and charge density as a function of chemical potential in the free theory. At weak |{phi}|{sup 4} coupling and zero temperature we map the m{sup 2}-{mu} phase diagram and compare our numerical results with perturbative calculations. Finally, we compute properties of the T-{mu} phase diagram in the vicinity of the phase transition and at bare self-couplings large compared to the temperature and chemical potential.

Non-relativistic scalar field on the quantum plane

We apply the coherent state approach to the non-commutative plane to check the one-loop finiteness of the two-point and four-point functions of a non-relativistic scalar field theory in 2+1 dimensions. We show that the two-point and four-point functions of the model are finite at one-loop level and one recovers the divergent behavior of the model in the limit θ→0+ by appropriate redefinition of the non-commutativity parameter.

Three-body system with short-range interactions

Within the framework of non-relativistic scalar effective field theory it is shown that the problem of the cutoff dependence of the leading order amplitude for a particle scattering off a two-body bound state can be solved without introducing three-body forces.

Nonrelativistic noncommutative field theory and UV-IR mixing

We study a non-commutative non-relativistic scalar field theory in 2+1 dimensions. The theory shows the UV/IR mixing typical of QFT on non-commutative spaces. The one-loop correction to the two-point function turns out to be given by a delta-function in momentum space. The one-loop correction to the four-point function is of logarithmic type. We also evaluate the thermodynamic potential at two-loop order. To avoid an IR-singularity we have to introduce a chemical potential. The suppression of the non-planar contribution with respect to the planar one turns out to depend crucially on the value of the chemical potential.

Vortex solutions of nonrelativistic Fermion and scalar field theories coupled to Maxwell-Chern-Simons gauge fields

We have constructed nonrelativistic fermion and scalar field theories coupled to a Maxwell-Chern-Simons gauge field which admit static multi-vortex solutions. This is achieved by introducing a magnetic coupling term in addition to the usual minimal coupling.

Static properties and motion of vortices in charged fluids

It is shown that a non-relativistic scalar field minimally coupled to electromagnetism supports in the presence of a homogeneous background electric charge density the existence of smooth, finite-energy topologically stable flux vortices. The static properties of such vortices and vortex pairs are studied in detail in the context of a two-parameter model describing this system as a special case. The interaction potential of two minimal vortices is obtained for various values of the parameters. It is proven analytically that a free vortex is spontaneously pinned, while under the action of an external force it moves with a calculable speed perpendicular to it. In a homogeneous external current J the vortex velocity is V = − J , reminiscent of the opposite-sign Hall effect in superconductors. Other theories with the same vortex behaviour are briefly discussed.

Nonrelativistic field-theoretic scale anomaly.

We construct a nonrelativistic scalar field theory with a quartic self-interaction in 2+1 dimensions as an infinite mass limit of a relativistic theory, and calculate the two-particle scattering amplitude and two-particle bound-state energy. We show that the results are the same as for quantum mechanics of two scalar particles interacting via a $\ensuremath{\delta}$-function potential. Renormalization of the theory reveals an anomalous breaking of scale symmetry. The renormalization group structure is presented and the anomaly is expressed in terms of the trace of the energy-momentum tensor.

Domain walls and the Holstein-Primakoff transformations

The problem of determining the interaction of spin waves in the presence of a domain wall is examined. The Hamiltonian for a ferromagnet with a uniaxial anisotropy is rewritten in terms of harmonic oscillators with the use of the Holstein-Primakoff transformations. In the continuum limit a nonrelativistic scalar field theory is obtained. The equation of motion in three spatial dimensions admits domain-wall solutions. The Hamiltonian is expanded about the classical solution and the quadratic terms are used to generate the spin-wave spectrum. The results are shown to be consistent with other methods. The space of states and propagators are exhibited, and the extension to the higher-order terms governing spin-wave interactions is discussed.