Coulomb Phase Research Articles

Cooperative magnetic phenomena in artificial spin systems: spin liquids, Coulomb phase and fragmentation of magnetism – a colloquium

Two-dimensional arrays of interacting magnetic nanostructures offer a remarkable playground for simulating, experimentally, lattice spin models. Initially designed to capture the low-energy physics of highly frustrated magnets, they quickly became a lab-on-chip platform to investigate cooperative magnetic phenomena often associated with classical frustrated magnetism. This article reviews the many-body physics which can be visualized, directly in real space, through the magnetic imaging of artificial arrays of magnetic nanostructures. Particular attention is paid to classical spin liquid states, magnetic Coulomb phases and magnetic moment fragmentation. Other phenomena, such as complex magnetic ordering, charge crystallization and monopole-like excitations, are also described in light of the recent advances in the field.

Gauge theory for the cuprates near optimal doping

We describe the phase diagram of a 2+1 dimensional SU(2) gauge theory of fluctuating incommensurate spin density waves for the hole-doped cuprates. Our primary assumption is that all low energy fermionic excitations are gauge neutral and electron-like, while the spin density wave order is fractionalized into Higgs fields transforming as adjoints of the gauge SU(2). The confining phase of the gauge theory is a conventional Fermi liquid with a large Fermi surface (and its associated $d$-wave superconductor). There is a quantum phase transition to a Higgs phase describing the `pseudogap' at lower doping. Depending upon the quartic terms in the Higgs potential, the Higgs phase exhibits one or more of charge density wave, Ising-nematic, time-reversal odd scalar spin chirality, and $\mathbb{Z}_2$ topological orders. It is notable that the emergent broken symmetries in our theory of fluctuating spin density waves co-incide with those observed in diverse experiments. For the electron-doped cuprates, the spin density wave fluctuations are at wavevector $(\pi,\pi)$, and then the corresponding SU(2) gauge theory only has a crossover between the confining and Higgs regimes, with an exponentially large confinement scale deep in the Higgs regime. On the Higgs side, for both the electron- and hole-doped cases, and at scales shorter than the confinement scale (which can be infinite when $\mathbb{Z}_2$ topological order is present), the electron spectral function has a `fractionalized Fermi liquid (FL*)' form with small Fermi surfaces. We also describe the deconfined quantum criticality of the Higgs transition in the limit of a large number of Higgs flavors, and perturbatively discuss its coupling to fermionic excitations.

The interplay of hadronic amplitudes and Coulomb phase in LHC measurements at 13 TeV

Detailed analysis of the measurements of differential cross sections in the forward region of pp elastic scattering at 13 TeV in LHC is performed. The structures of the real and imaginary parts of the scattering amplitude are investigated, both requiring exponential and linear factors. Representations of the data in different conditions are compared, investigating the role of the Coulomb-nuclear interference and satisfying predictions from dispersion relations. The additivity of nuclear and Coulomb eikonal phases to determine the interference phase in pp scattering is submitted to a comparison with assumption of phase equal to zero. The structures of the real part of the scattering amplitude under the two assumptions are examined, and direct comparison is performed with the information extracted from the data. The alternative criteria for the treatment of the interference phase lead to different values for the ρ parameter between 0.1 and 0.13 while σ differ by 0.18 mb, and it is shown that the available data cannot discriminate the two choices. As an alternative study, we also present the results for the amplitude parameters and χ2 values when constant values of 2 and 3 radians are subtracted from the analytical expression for the interference phase.

Dynamical structure factor of the triangular antiferromagnet: Schwinger boson theory beyond mean field

We compute the zero temperature dynamical structure factor $S({\bf q},\omega)$ of the triangular lattice Heisenberg model (TLHM) using a Schwinger boson approach that includes the Gaussian fluctuations ($1/N$ corrections) of the saddle point solution. While the ground state of this model exhibits a well-known 120$^{\circ}$ magnetic ordering, experimental observations have revealed a strong quantum character of the excitation spectrum. We conjecture that this phenomenon arises from the proximity of the ground state of the TLHM to the quantum melting point separating the magnetically ordered and spin liquid states. Within this scenario, magnons are described as collective modes (two spinon-bound states) of a spinon condensate (Higgs phase) that spontaneously breaks the SU(2) symmetry of the TLHM. Crucial to our results is the proper account of this spontaneous symmetry breaking. The main qualitative difference relative to semi-classical treatments ($1/S$ expansion) is the presence of a high-energy spinon continuum extending up to about three times the single-magnon bandwidth. In addition, the magnitude of the ordered moment ($m=0.224$) agrees very well with numerical results and the low energy part of the single-magnon dispersion is in very good agreement with series expansions. Our results indicate that the Schwinger boson approach is an adequate starting point for describing the excitation spectrum of some magnetically ordered compounds that are near the quantum melting point separating this Higgs phase from the {\it deconfined} spin liquid state.

Duality and an exact Landau-Ginzburg potential for quasi-bosonic Chern-Simons-Matter theories

It has been conjectured that Chern-Simons (CS) gauged ‘regular’ bosons in the fundamental representation are ‘level-rank’ dual to CS gauged critical fermions also in the fundamental representation. Generic relevant deformations of these conformal field theories lead to one of two distinct massive phases. In previous work, the large N thermal free energy for the bosonic theory in the unHiggsed phase has been demonstrated to match the corresponding fermionic results under duality. In this note we evaluate the large N thermal free energy of the bosonic theory in the Higgsed phase and demonstrate that our results, again, perfectly match the predictions of duality. Our computation is performed in a unitary gauge by integrating out the physical excitations of the theory — i.e. W bosons — at all orders in the ’t Hooft coupling. Our results allow us to construct an exact quantum effective potential for overline{phi}phi , the lightest gauge invariant scalar operator in the theory. In the zero temperature limit this exact Landau-Ginzburg potential is non-analytic at overline{phi}phi = 0. The extrema of this effective potential at positive overline{phi}phi solve the gap equations in the Higgsed phase while the extrema at negative overline{phi}phi solve the gap equations in the unHiggsed phase. Our effective potential is bounded from below only for a certain range of x6 (the parameter that governs sextic interactions of ϕ). This observation suggests that the regular boson theory has a stable vacuum only when x6 lies in this range.

Magnetic clustering, half-moons, and shadow pinch points as signals of a proximate Coulomb phase in frustrated Heisenberg magnets

We study the formation of magnetic clusters in frustrated magnets in their cooperative paramagnetic regime. For this purpose, we consider the $J_1$-$J_2$-$J_3$ classical Heisenberg model on kagome and pyrochlore lattices with $J_2 = J_3=J$. In the absence of farther-neighbor couplings, $J=0$, the system is in the Coulomb phase with magnetic correlations well characterized by pinch-point singularities. Farther-neighbor couplings lead to the formation of magnetic clusters, which can be interpreted as a counterpart of topological-charge clusters in Ising frustrated magnets [T. Mizoguchi, L. D. C. Jaubert and M. Udagawa, Phys. Rev. Lett. {\bf 119}, 077207 (2017)]. The concomitant static and dynamical magnetic structure factors, respectively $\mathcal{S}({\bm{q}})$ and $\mathcal{S}({\bm{q}},\omega)$, develop half-moon patterns. As $J$ increases, the continuous nature of the Heisenberg spins enables the half-moons to coalesce into connected `star' structures spreading across multiple Brillouin zones. These characteristic patterns are a dispersive complement of the pinch point singularities, and signal the proximity to a Coulomb phase. Shadows of the pinch points remain visible at finite energy, $\omega$. This opens the way to observe these clusters through (in)elastic neutron scattering experiments. The origin of these features are clarified by complementary methods: large-$N$ calculations, semi-classical dynamics of the Landau-Lifshitz equation, and Monte Carlo simulations. As promising candidates to observe the clustering states, we revisit the origin of "spin molecules" observed in a family of spinel oxides $AB_2$O$_4$ ($A=$ Zn, Hg, Mg, $B=$ Cr, Fe).

Residual entropies for three-dimensional frustrated spin systems with tensor networks

We develop a technique for calculating three-dimensional classical partition functions using projected entangled-pair states (PEPS). Our method is based on variational PEPS optimization algorithms for two-dimensional quantum spin systems, and allows us to compute free energies directly in the thermodynamic limit. The main focus of this work is classical frustration in three-dimensional many-body systems leading to an extensive ground-state degeneracy. We provide high-accuracy results for the residual entropy of the dimer model on the cubic lattice, water-ice $I_h$ and water-ice $I_c$. In addition, we show that these systems are in a Coulomb phase by computing the dipolar form of the correlation functions. As a further benchmark of our methods, we calculate the critical temperature and exponents of the Ising model on the cubic lattice.

Pinch point singularities of tensor spin liquids

Recently, a new class of three-dimensional spin liquid models have been theoretically discovered, which feature generalized Coulomb phases of emergent symmetric tensor $U(1)$ gauge theories. These ``higher rank'' tensor models are particularly intriguing due to the presence of quasiparticles with restricted mobility, such as fractons. We investigate universal experimental signatures of tensor Coulomb phases. Most notably, we show that tensor Coulomb spin liquids (both quantum and classical) feature characteristic pinch point singularities in their spin-spin correlation functions, accessible via neutron scattering, which can be readily distinguished from pinch points in conventional $U(1)$ spin liquids. These pinch points can thus serve as a crisp experimental diagnostic for such phases. We also tabulate the low-temperature heat capacity of various tensor Coulomb phases, which serves as a useful additional diagnostic in certain cases.

Multiple Coulomb phase in the fluoride pyrochlore CsNiCrF6

The Coulomb phase is an idealized state of matter whose properties are determined by factors beyond conventional considerations of symmetry, including global topology, conservation laws and emergent order. Theoretically, Coulomb phases occur in ice-type systems such as water ice and spin ice; in dimer models; and in certain spin liquids. However, apart from ice-type systems, more general experimental examples are very scarce. Here we study the partly disordered material CsNiCrF6 and show that this material is a multiple Coulomb phase with signature correlations in three degrees of freedom: charge configurations, atom displacements and spin configurations. We use neutron and X-ray scattering to separate these correlations and to determine the magnetic excitation spectrum. Our results show how the structural and magnetic properties of apparently disordered materials may inherit, and be dictated by, a hidden symmetry—the local gauge symmetry of an underlying Coulomb phase. Neutron and X-ray scattering experiments show that the partly disordered material CsNiCrF6 supports multiple Coulomb phases with structural and magnetic properties dictated by underlying local gauge symmetry.

What symmetry is actually broken in the Higgs phase of a gauge-Higgs theory?

In SU($N$) gauge-Higgs theories, with a single Higgs field in the fundamental representation, there exists in addition to the local gauge symmetry a global SU(2) symmetry, at $N=2$, and a global U(1) symmetry, for $N \ne 2$. We construct a gauge-invariant order parameter for the breaking of these global symmetries in the Higgs sector, and calculate numerically the transition lines, in coupling-constant space, for SU(2) and SU(3) gauge theories with unimodular Higgs fields. The order parameter is non-local, and therefore its non-analyticity does not violate the theorem proved by Osterwalder and Seiler. We then show that there exists a transition, in gauge-Higgs theories, between two types of confinement: ordinary color neutrality in the Higgs region, and a stronger condition, which we have called "separation-of-charge confinement," in the confinement region. We conjecture that the symmetry-breaking transition coincides with the transition between these two physically different types of confinement.

Confining solitons in the Higgs phase of \u2102PN \u22121 model: self-consistent exact solutions in large-N limit

The quantum ℂPN −1 model is in the confining (or unbroken) phase with a full mass gap in an infinite space, while it is in the Higgs (broken or deconfinement) phase accompanied with Nambu-Goldstone modes in a finite space such as a ring or finite interval smaller than a certain critical size. We find a new self-consistent exact solution describing a soliton in the Higgs phase of the ℂPN −1 model in the large-N limit on a ring. We call it a confining soliton. We show that all eigenmodes have real and positive energy and thus it is stable.

Vector particle scattering on the lattice

In this work, we present an explicit form of the Luescher equation and consider the construction of the operators in different irreducible representations for the case of scattering of two vector particles. The formalism is applied to scalar QED in the Higgs Phase, where the $U(1)$ gauge boson acquires mass.

Coulomb phase in high harmonic generation

We derive a regularized expression for the Coulomb term in the phase of high order harmonics emitted in the interaction of intense laser pulses with atoms. The calculation is based on the formalism of quantum orbits and on the imaginary time method and allows one to match the initially divergent Coulomb integral with the phases of the atomic wave function at the instants of ionization and recombination, leading to a finite and closed analytic formula. This complex-valued Coulomb phase can considerably modify both the yield and the interference structure of high harmonic spectra. Its possible implications for numerical calculations of the high harmonic signal are discussed.

Higgs mechanism in higher-rank symmetric U(1) gauge theories

We use the Higgs mechanism to investigate connections between higher-rank symmetric $U(1)$ gauge theories and gapped fracton phases. We define two classes of rank-2 symmetric $U(1)$ gauge theories: the $(m,n)$ scalar and vector charge theories, for integer $m$ and $n$, which respect the symmetry of the square (cubic) lattice in two (three) spatial dimensions. We further provide local lattice rotor models whose low energy dynamics are described by these theories. We then describe in detail the Higgs phases obtained when the $U(1)$ gauge symmetry is spontaneously broken to a discrete subgroup. A subset of the scalar charge theories indeed have X-cube fracton order as their Higgs phase, although we find that this can only occur if the continuum higher rank gauge theory breaks continuous spatial rotational symmetry. However, not all higher rank gauge theories have fractonic Higgs phases; other Higgs phases possess conventional topological order. Nevertheless, they yield interesting novel exactly solvable models of conventional topological order, somewhat reminiscent of the color code models in both two and three spatial dimensions. We also investigate phase transitions in these models and find a possible direct phase transition between four copies of $\mathbb{Z}_2$ gauge theory in three spatial dimensions and X-cube fracton order.

Spin-Ice Thin Films: Large- N Theory and Monte Carlo Simulations

We explore the physics of highly frustrated magnets in confined geometries, focusing on the Coulomb phase of pyrochlore spin ices. As a specific example, we investigate thin films of nearest-neighbor spin ice, using a combination of analytic large-N techniques and Monte Carlo simulations. In the simplest film geometry, with surfaces perpendicular to the [001] crystallographic direction, we observe pinch points in the spin-spin correlations characteristic of a two-dimensional Coulomb phase. We then consider the consequences of crystal symmetry breaking on the surfaces of the film through the inclusion of orphan bonds. We find that when these bonds are ferromagnetic, the Coulomb phase is destroyed by the presence of fluctuating surface magnetic charges, leading to a classical Z_2 spin liquid. Building on this understanding, we discuss other film geometries with surfaces perpendicular to the [110] or the [111] direction. We generically predict the appearance of surface magnetic charges and discuss their implications for the physics of such films, including the possibility of an unusual Z_3 classical spin liquid. Finally, we comment on open questions and promising avenues for future research.

Charge and spin correlations in the monopole liquid

A \emph{monopole liquid} is a magnetic charge-disordered spin system defined over an Ising pyrochlore lattice, with one single topological charge or \emph{monopole} in each tetrahedron. We define a simple model Hamiltonian for this system and compare its thermodynamics at zero magnetic field with that of spin ice ---a phase free of these charges. In spite of the liquid-like correlations between charges, we find that spins in the charged phase are uncorrelated at all temperatures, like in a perfect paramagnet. The addition of nearest neighbors interactions favoring neutral `2in-2out' excitations as a perturbation has a peculiar effect. While they decrease charge-charge correlations, new spin correlations resembling those in spin ice appear on increasing temperature. This helps us understand why dipolar correlations are observed in spin ices at unexpectedly high temperatures, and the major role of double excitations in erasing the Coulomb phase correlations. Ferromagnetic interactions strengthen the charges short range order and its associated spin correlations. Finally, we discuss how the monopole liquid can be related to other systems and materials where different phases of \emph{monopole matter} have been observed.

Topological order in the pseudogap metal

We compute the electronic Green's function of the topologically ordered Higgs phase of a SU(2) gauge theory of fluctuating antiferromagnetism on the square lattice. The results are compared with cluster extensions of dynamical mean field theory, and quantum Monte Carlo calculations, on the pseudogap phase of the strongly interacting hole-doped Hubbard model. Good agreement is found in the momentum, frequency, hopping, and doping dependencies of the spectral function and electronic self-energy. We show that lines of (approximate) zeros of the zero-frequency electronic Green's function are signs of the underlying topological order of the gauge theory and describe how these lines of zeros appear in our theory of the Hubbard model. We also derive a modified, nonperturbative version of the Luttinger theorem that holds in the Higgs phase.

Generation of the relic neutrino asymmetry in a hot plasma of the early universe

The neutrino asymmetry in the early universe plasma, [Formula: see text], is calculated both before and after the electroweak phase transition (EWPT). In the Standard Model, before EWPT, the leptogenesis is well known to be driven by the abelian anomaly in a massless hypercharge field. The generation of the neutrino asymmetry in the Higgs phase after EWPT, in its turn, has not been considered previously because of the absence of any quantum anomaly in an external electromagnetic field for such electroneutral particles as neutrino, unlike the Adler–Bell–Jackiw anomaly for charged left and right polarized massless electrons in the same electromagnetic field. Using the neutrino Boltzmann equation, modified by the Berry curvature term in the momentum space, we establish the violation of the macroscopic neutrino current in plasma after EWPT and exactly reproduce the nonconservation of the lepton current in the symmetric phase before EWPT arising in quantum field theory due to the nonzero lepton hypercharge and corresponding triangle anomaly in an external hypercharge field. In the last case, the nonconservation of the lepton current is derived through the kinetic approach without a computation of corresponding Feynman diagrams. Then, the new kinetic equation is applied for the calculation of the neutrino asymmetry accounting for the Berry curvature and the electroweak interaction with background fermions in the Higgs phase. Such an interaction generates a neutrino asymmetry through the electroweak coupling of neutrino currents with electromagnetic fields in plasma, which is [Formula: see text]. It turns out that this effect is especially efficient for maximally helical magnetic fields.

Duality in a supersymmetric gauge theory from a perturbative viewpoint

We study duality in $\mathcal{N}=1$ supersymmetric QCD in the non-Abelian Coulomb phase, order-by-order in scheme-independent series expansions. Using exact results, we show how the dimensions of various fundamental and composite chiral superfields, and the quantities $a$, $a/c$, and $b$ at superconformal fixed points of the renormalization group emerge in scheme-independent series expansions in the electric and magnetic theories. We further demonstrate that truncations of these series expansions to modest order yield very accurate approximations to these quantities.

DRGT theory of massive gravity from spontaneous symmetry breaking

In this note we propose a topological action for a Poincare times diffeomorphism invariant gauge theory. We show that there is Higgs phase where the gauge symmetry is spontaneous broken to a diagonal Lorentz subgroup and gives the Einstein–Hilbert action plus the dRGT potential terms. In this vacuum, there are five (three from Goldstone modes) propagating degrees of freedom which form polarizations of a massive spin 2 particle, an extra healthy heavy scalar (Higgs) mode and no Boulware–Deser ghost mode. We further show that the action can be derived in a limit from a topological de Sitter invariant gauge theory in 4 dimensions.