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Small normal-metal rings threaded by a constant magnetic flux have been shown to carry a mesoscopic persistent current at low temperatures. The current is a few-electron effect and its sign and amplitude depend on the microscopic configuration of disorder. Assuming a Gaussian current distribution, we characterize the effect by three quantities, the rms or typical total current ${\mathit{I}}^{\mathrm{typ}}$=〈${\mathit{I}}^{2}$${\mathrm{〉}}_{\mathit{D}}^{1/2}$, the average current ${\mathit{I}}^{\mathrm{av}}$=〈I${\mathrm{〉}}_{\mathit{D}}$, and the typical single-level current ${\mathit{i}}^{\mathrm{typ}}$=〈${\mathit{i}}^{2}$${\mathrm{〉}}_{\mathit{D}}^{1/2}$. Specifically, we review and extend the analytical calculations for the typical total and single-level currents focusing on the case of noninteracting electrons in disordered rings in the regime of diffusive transport. We calculate and discuss those current-current correlation functions that describe the dependences of the persistent current on filling, flux, and disorder configuration. Only the single-electron contribution discussed in this paper is known to contribute to the first, ${\mathrm{\ensuremath{\varphi}}}_{0}$-periodic harmonic of the total current in a single ring. The second harmonic also contains an interaction-induced contribution proposed by Ambegaokar and Eckern that survives the disorder average. The Thouless correlation energy ${\mathit{E}}_{\mathit{c}}$ is the characteristic energy scale for the amplitude of the total current and its dependences on filling, temperature, and inelastic scattering. The persistent current is sensitive to changing the position of a single impurity. We compare our results with the recent single-ring experiment by Chandrasekhar et al.

We study the function-induced persistent current close to ${\mathit{T}}_{\mathit{c}}$ in effectively one-dimensional superconducting rings threaded by a magnetic flux \ensuremath{\varphi}. Using a transfer-operator approach to the Ginzburg-Landau theory we calculate the dependences of the persistent current on temperature, flux, and ${\mathit{T}}_{\mathit{c}}$. In agreement with a simple physical argument we find that the temperature dependence of the current amplitude is similar to that of the order-parameter fluctuations 〈\ensuremath{\Vert}\ensuremath{\Delta}${\mathrm{\ensuremath{\Vert}}}^{2}$〉. We discuss the observability of a flux-periodic response below ${\mathit{T}}_{\mathit{c}}$ in view of the existence of metastable states and phase slips. Our predictions apply to experiments on superconductors whose ${\mathit{T}}_{\mathit{c}}$ is larger than the correlation energy ${\mathit{E}}_{\mathit{c}}$.

We present a microscopic calculation of the disorder-averaged persistent current for a mesoscopic ring with a fixed number of electrons. Using an idea by Imry, we show that the event Fourier components of the average current dominate and are related to the typical single-level current D ∞ D/0 , which can be calculated by a Green's-function-averaging technique. For the small even harmonics p, the result simplifies to D ∼I 0 /2πM, where M is the number of transverse channels. Its relevance to multiring experiments is discussed

We describe the status of our effort to realize a first neutrino factory and the progress made in understanding the problems associated with the collection and cooling of muons towards that end. We summarize the physics that can be done with neutrino factories as well as with intense cold beams of muons. The physics potential of muon colliders is reviewed, both as Higgs Factories and compact high energy lepton colliders. The status and timescale of our research and development effort is reviewed as well as the latest designs in cooling channels including the promise of ring coolers in achieving longitudinal and transverse cooling simultaneously. We detail the efforts being made to mount an international cooling experiment to demonstrate the ionization cooling of muons., Comment: 112 pages, Large report

Semiclassical theory relates mesoscopic persistent currents to classical periodic orbits. This is used to demonstrate that persistent currents are a sensitive probe of quantum chaos, exhibiting different universal dependences on the Fermi velocity for integrable and for completely chaotic ballistic microstructures. A study of the Sinai-billiard geometry reveals that the universality classes do not coincide with those of the spectral spacing distribution. Our conclusions are supported by numerical calculations.

The status of the field is excellent, much is in flux. For the first time we have experimental evidence [1] for persistent currents in small normal metal rings threaded by a magnetic flux. And there are new theoretical developments that indicate that in addition to the single-electron (or quasi-particle) contribution [2, 3] to the persistent current there are also collective ones [4].

We study the average persistent current in billiard systems, which can be realized experimentally in ballistic microstructures. We show that the dependences of the average current on the Fermi velocity and the system size are determined by exponents, which differ for integrable and completely chaotic billiards.

We propose scaling relationships for the disorder-averaged persistent current in mesoscopic normal-metal rings, as suggested by our analytical theory, and show how these allow one to resolve a discrepancy between recent numerical and analytical results for the average current.

The inhomogeneous generalization of the Migdal‐Kadanoff approximate recursion relations is used to study the properties of Ising magnets with spatially random (quenched) couplings. Simple calculations in two dimensions produce qualitatively reasonable phase diagrams and critical exponents for both a dilution problem (nearest‐neighbor couplings 0 and J) and a spin‐glass system (nearest‐neighbor coupling +J and −J. Other related work is briefly reviewed.

A numerical technique, termed the scaling-field method, is developed for solving by successive approximation Wilson's exact renormalization-group equation for critical phenomena in three-dimensional spin systems. The approach uses the scaling-field representation of the Wilson equation derived by Riedel, Golner, and Newman. A procedure is proposed for generating in a nonperturbative and unbiased fashion sequences of successively larger truncations to the infinite hierarchy of scaling-field equations. A "principle of balance" is introduced and used to provide a self-consistency criterion. The approach is then applied to the isotropic $N$-vector model. Truncations to order 13 (10, when $N=1$) scaling-field equations yield the leading critical exponents, $\ensuremath{\nu}$ and $\ensuremath{\eta}$, and several of the correction-to-scaling exponents, ${\ensuremath{\Delta}}_{m}$, to high precision. Results for $N=0, 1, 2, \mathrm{and} 3$ are tabulated. For the Ising case ($N=1$), the estimates $\ensuremath{\nu}=0.626\ifmmode\pm\else\textpm\fi{}0.009$, $\ensuremath{\eta}=0.040\ifmmode\pm\else\textpm\fi{}0.007$, and ${\ensuremath{\Delta}}_{1}\ensuremath{\equiv}{\ensuremath{\Delta}}_{400}=0.54\ifmmode\pm\else\textpm\fi{}0.05$ are in good agreement with recent high-temperature-series results, though exhibiting larger confidence limits at the present level of approximation. For the first time, estimates are obtained for the second and third correction-to-scaling exponents. For example, for the Ising model the second "even" and first "odd" correction-to-scaling exponents are ${\ensuremath{\Delta}}_{422}=1.67\ifmmode\pm\else\textpm\fi{}0.11$ and ${\ensuremath{\Delta}}_{500}=1.5\ifmmode\pm\else\textpm\fi{}0.3$, respectively. Extensions necessary to improve the accuracy of the calculation are discussed, while applications of the approach to anisotropic $N$-vector models are described elsewhere. Finally, the scaling-field method is compared with other techniques for the high-precision calculation of critical phenomena in three dimensions, i.e., high-temperature-series, Monte Carlo renormalization-group, and field-theoretic perturbation expansions.

Critical properties of models defined by continuous-spin Landau Hamiltonians of cubic symmetry are calculated as functions of spatial dimensionality, $2.8\ensuremath{\le}d\ensuremath{\le}4$, and number of spin components, $N$. The investigation employs the scaling-field method developed by Golner and Riedel for Wilson's exact momentum-space renormalization-group equation. Fixed points studied include the isotropic and decoupled Ising ($\ensuremath{-}2\ensuremath{\le}Nl\ensuremath{\infty}$), the face-and corner-ordered cubic ($1\ensuremath{\le}Nl\ensuremath{\infty}$), and, via the replica method for $N\ensuremath{\rightarrow}0$, the quenched random Ising fixed point. Variations of $N$ and $d$ are used to link the results to exact results or results from other calculational methods, such as $\ensuremath{\epsilon}$ expansions near two and four dimensions. This establishes the consistency of the calculation for three dimensions. Specifically, truncated sets involving seven (twelve) scaling-field equations are derived for the cubic $N$-vector model. A stable random Ising fixed point is found and shown to be distinct from the cubic fixed point and to connect, as a function of $d$, with the Khmelnitskii ${\ensuremath{\epsilon}}^{\frac{1}{2}}$ fixed point. At $d=3$, the short truncation yields $\ensuremath{\alpha}\ensuremath{\approx}0.11$ for the pure Ising and $\ensuremath{\alpha}\ensuremath{\approx}\ensuremath{-}0.09$ for the random Ising fixed point. A search for a random tricritical fixed point was inconclusive. For the $N$-component cubic model, the spin dimensionality ${N}_{c}$, at which the isotropic and cubic fixed points change stability, is determined as a function of $d$. The results support ${N}_{c}g3$ for three dimensions.

We have performed analytical calculations and computer simulations to study the persistent current I in small isolated one-dimensional metal rings enclosing a magnetic flux \ensuremath{\varphi}. We have calculated I as a function of flux \ensuremath{\varphi}, ring circumference L, temperature T, chemical potential \ensuremath{\mu}, and randomness W. We find that I decreases exponentially with T, that I decreases as ${W}^{2}$ for weak W and as exp(-L/\ensuremath{\xi}) for strong W, where \ensuremath{\xi} denotes the localization length, and that I is periodic in \ensuremath{\varphi} with period ${\ensuremath{\varphi}}_{0}$=hc/e. For certain averaging procedures the periodicity may change to ${\ensuremath{\varphi}}_{0}$/2.

Relations between two‐dimensional physical systems, continuous‐spin Hamiltonians, and discrete spin or Potts models are established by Landau symmetry and renormalization‐group arguments. Experimental realizations of XY and Heisenberg models with cubic anisotropy as well as of different Potts models are reviewed. Then a unified model encompassing these classes of models is introduced. For the special case of a six‐state discrete model a phase diagram is proposed based on duality and Migdal renormalization‐group transformations.

The cubic lattice gas, a generalization of the $n$-component face-cubic model, is studied in two dimensions by variational renormalization-group techniques for general $n$, and by an exact mapping onto a solid-on-solid model for $n=2$. This approach reveals new interconnections among the class of cubic, Potts, and Ashkin-Teller phase-transition phenomena. In particular, it is shown that the fixed line of the Ashkin-Teller lattice gas serves as the backbone for a hierarchy of pairs of fixed points which branch off from it, as $n$ is decreased from 2, at special points where two operators are marginal. The phase diagrams of the pure and dilute $n$-component cubic models are obtained and the orders of all transitions are determined. The cubic phase transition is found to be first order for all $ng{n}_{c}=2$. The evolution of the results as a function of dimensionality is discussed briefly.

A novel renormalization-group description of the phase transitions of the Potts and cubic models discovered by Nienhuis et al. is reviewed.

Critical properties of the Q-state Potts model for dimensions 3≤d≤6 are calculated by means of Wilson's exact momentum-space renormalization-group equation. The scaling-field method of Golner and Riedel is used to approximate the functional differential equation by a set of 11 ordinary coupled differential equations. For d=4−e, lines of critical and tricritical Potts fixed points are found as functions of Q that annihilate as Q approaches a critical value Qc=2+e2a+O(e3). For Q>Qc, the Potts transition is first order. Along these fixed lines the critical and tricritical exponents (upper and lower sign, respectively) are to leading order: 1ν=2−16[e±(e2−aδ)12], φν=∓(e2−aδ)12, and η=[e±(e2−aδ)12]2216+bδ, where e=4−d, δ=Q−2, and δ≤δc=e2a+O(e3). While the form of the e and δ dependences is exact, the coefficients a and b cannot be obtained systematically by e expansion, since the upper critical dimensionality of the Potts model is six when Q≠2. In our truncation, a=6.52 and b=0.065. The results have been extended to dimensions 3.4≤d≲4 by solving the renormalization-group equations numerically. The percolation limit of the Potts model, Q=1, is also investigated and the critical exponents νP,φP, and ηP determined as functions of dimension for 3≤d≤6.

The randomly bond-dilute two-dimensional nearest-neighbor Ising model on the square lattice is studied by renormalization-group methods based on the Migdal-Kadanoff approximate recursion relations. Calculations give both thermal and magnetic exponents associated with the percolative fixed point. Differential recursion relations yield a phase diagram which is in quantitative agreement with all known results. Curves for the specific heat, percolation probability, and magnetization are displayed. The critical region of the specific heat becomes unobservably narrow well above the percolation threshold ${p}_{c}$. This provides a possible explanation for the apparent specific-heat rounding in certain experiments.

Persistent currents in small nonsuperconducting rings threaded by a magnetic flux are a manifestation of novel quantum effects in submicron systems. We present theoretical results for one-channel and multichannel systems concerning the dependence of the current amplitude on the number of channels and geometry, temperature, and degree of disorder. Inelastic scattering is considered for one-channel loops only. We also discuss the observability of the effect.

The density of vortex excitations and their contribution to the specific heat are discussed for the Villain planar model. Exact results are derived for the low- and high-temperature limits, and a renormalization-group interpolation scheme is developed for intermediate temperatures.

A systematic study of the possibility of weakly incommensurate phases with a honeycomb domain wall network structure in uniaxial systems is presented. Competition between breathing entropy, the energy needed to tilt a domain wall, and the core energy of intersections can stabilize honeycomb phases in uniaxial systems. Two types of honeycomb networks are introduced, O-shaped and X-shaped. Breathing entropy turns out to be much weaker, compared to meander entropy, in uniaxial systems than in isotropic ones. Still, our study suggests that at the chiral melting transition the commensurate solid melts into an incommensurate fluid with a local short ranged O-shaped network order. Moreover, breathing entropy of X-shaped honeycomb networks might change the Pokrovsky-Talapov nature of the commensurate to striped incommensurate solid transition.

Persistent currents in small nonsuperconducting rings threaded by a magnetic flux are a manifestation of novel quantum effects in submicron systems. We present results concerning the dependence of the current amplitude on the number of channels, temperature, and degree of disorder. We also discuss the observability of the effect.

The results of a variational renormalisation-group calculation for the magnetic exponent yH of the two-dimensional q-state Potts model suggest a simple relationship between yH and the exactly known critical exponent yT8v of the eight-vertex model. The relation allows one to predict the critical and tricritical magnetic exponent delta of the q-state Potts model as a function of q.

A novel momentum-space renormalization-group (RG) technique, termed the scaling-field method, is proposed for the investigation of critical phenomena in three-dimensional systems. The approach provides a method for solving Wilson's exact functional differential equation for RG Hamiltonians H l [ σ ] by successive approximation, and allows the determination of both critical exponents and scaling functions. In other papers the method is used to calculate to high precision the critical exponents of the isotropic N -vector model in three dimensions and to investigate the Potts, percolation, cubic, and random Ising models in ranges of dimensions between the upper critical and three dimensions. The purpose of this article is to present the foundations of the method. The scaling-field representation of the Wilson functional RG equation is derived by using the expansion of the RG Hamiltonian H l [ σ ] in terms of the hierarchy of Gaussian operators of appropriate symmetries. The expansion coefficients are termed scaling fields. The algorithm for the computation of the scaling-field coupling coefficients is developed for the isotropic N -vector model. Technical details and examples are given in a series of Appendices. The invariance of the Wilson equation under physically redundant scale changes of the spin variable σ is carefully investigated for both the functional renormalization group and the scaling-field representation. The understanding of this invariance yields the operational definition of the spin-rescaling function that enters the Wilson equation as a parameter. Exact recursion relations for thermodynamic and correlation functions are derived by using the functional-integral representation of the Wilson equation. The recursion relations provide the starting point for the computation of scaling functions. Finally, as an example and a guide for the development of numerical techniques, the scaling-field equations for the isotropic N -vector model are solved by e expansion and in the spherical-model limit.

Two-dimensional anisotropic $N$-vector models are discussed in three contexts. (i) A comprehensive approach to the description of phase transitions in two-dimensional physical systems is outlined. It involves the identification of discrete models for critical phenomena in two-dimensional systems (such as adsorbed thin films) and their investigation by symmetry, duality, and Migdal renormalization-group methods. The identification is based on the Landau-Ginzburg-Wilson Hamiltonian concept and universality arguments. (ii) Relations among anisotropic continuous-spin Hamiltonians and discrete models are established by the Hubbard transformation and the Migdal renormalization-group transformation. Discrete models are conjectured to be equivalent to $N$-component continuous-spin models with local anisotropies. For example, it is shown that the Migdal recursion relations map the continuous-spin, cubic Heisenberg Hamiltonian onto the discrete cubic model. (iii) Many of the anisotropic $N$-vector Hamiltonians can be associated with discrete models that have the form of a generalized Potts model. Such a model, termed (${N}_{\ensuremath{\alpha}}$, ${N}_{\ensuremath{\beta}}$) model, is defined in terms of two interacting Potts-like variables associated with each lattice site, and is analyzed by duality and renormalization-group methods. The (${N}_{\ensuremath{\alpha}}$, ${N}_{\ensuremath{\beta}}$) Hamiltonian provides a unified description for large classes of discrete models. The concepts are exemplified by a detailed discussion of the two-dimensional Heisenberg model with cubic anisotropy, which has applications to the magnetic $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}$ phase transition in overlayers of molecular oxygen on graphite. New experiments for the study of this system are also discussed.

A generalization of the Niemeijer-van Leeuwen renormalization-group transformation treating disordered cells of spins as vacancies is introduced and applied to the two-dimensional $q$-state Potts model. Finite-lattice approximations yield the changeover from continuous to first-order phase transitions predicted by Baxter but not observed in previous renormalization-group calculations. Exact results are conjectured for the tricritical exponents of dilute Potts models.

Scaling laws for multicomponent systems near the tricritical point are derived by relating the "scales" of the competing second-order and tricritical phase transitions. The approach leads to a consistent description of the thermodynamics of the superfluid and phase-separation transitions in ${\mathrm{He}}^{3}$-${\mathrm{He}}^{4}$ mixtures.

The asymptotic critical form of thermodynamic functions is analyzed by means of renormalization-group techniques. If certain exponent relations are satisfied, then the critical behavior is not described by a simple power law, but a power law multiplied by a fractional power of a logarithm. The approach is applied to two special systems whose critical exponents are molecular-field-like. (i) For ordinary critical transitions in four dimensions we find the same logarithmic factors previously computed by Larkin and Khmel'nitskii. (ii) For tricritical transitions in three dimensions we compute the logarithmic corrections to the molecularfield tricritical behavior discussed in an earlier publication.

A semi-microscopic scaling-field theory is developed for crossover phenomena near critical and tricritical points. The theory is based on a renormalization-group description of a model with two competing fixed points (such as a critical and a tricritical fixed point) in terms of scaling fields. The coupled nonlinear differential equations for scaling fields are truncated such as to preserve the physics essential for crossover phenomena. The approach allows the explicit calculation of thermodynamic functions for (i) tricritical systems and (ii) critical systems with an irrelevant scaling field. We obtain, for example, an explicit expression for the scaling function of the susceptibility, which describes the crossover from the tricritical to the critical region. The idea of "flow diagrams" in the scaling-field space is used to characterize crossover phenomena globally in the whole critical region. The concept of asymptotic critical exponents is generalized and effective critical exponents are introduced as logarithmic derivatives of thermodynamic quantities with respect to experimental fields and scaling fields, respectively. By using the method of effective exponents the size of the crossover region between regions of different asymptotic critical behavior is estimated. For the susceptibility, the width of the crossover region in decades of the effective temperature variable is roughly equal to the inverse of the crossover exponent. In the case of a critical system with a slow transient the asymptotic critical exponent is only reached extremely close to the critical point (unless the amplitude of the transient vanishes). It might then be impossible to determine the asymptotic exponent experimentally or by conventional series-expansion techniques, and an analysis of the data in terms of effective exponents is the alternative. The scaling-field approach is applied to three systems with crossover phenomena: (i) the model for tricritical systems with molecular-field tricritical exponents, (ii) the Ashkin-Teller model in three dimensions, and (iii) a model for phase transitions with Fisher exponent renormalization due to a constraint.

The tricritical behavior of a classical three-well-potential model for two-component systems (such as ${\mathrm{He}}^{3}$-${\mathrm{He}}^{4}$ mixtures) is discussed by using renormalization-group techniques. The tricritical exponents and scaling fields are calculated for three dimensions.

Scaling‐field equations for the hypercubic N‐vector model are derived from Wilson’s exact renormalization‐group equation and applied to the determination of NC*, the instability point of the isotropic fixed point as a function of N, and to the estimation of the critical point exponents associated with the isotropic, cubic, and decoupled Ising fixed points in hypercubic systems of three spatial dimensions.

A new method is reviewed for the numerical calculation to high precision of critical phenomena in three-dimensional spin systems. The approach is based on Wilson’s exact functional renormalization-group equation. Results are presented for the critical exponents of the isotropic N-vector model.

The correlations in the energy spectrum and persistent current of electrons in one-dimensional disordered rings threaded by an Aharonov-Bohm magnetic flux are shown to be so strong that the current of any level is sufficient to cancel the sum of the contributions of all previous levels. Hence, the sign of the sum of the n lowest single-level currents alternates as a function of n. A new relation between the real part of the inverse transmission coefficient and the energy spectrum is also derived.

Persistent currents in multichannel normal-metal rings threaded by a magnetic flux are treated by a Green's-function approach. The diffusive region is analyzed in detail and the typical current predicted as a function of disorder, temperature, and number of channels. The amplitude and temperature sensitivity of the effect are found to be governed by correlations in the energy spectrum of the ring, of range ${E}_{c}$ proportional to the Thouless energy. The results are relevant to persistent-current experiments in mesoscopic metal rings.

The $\ensuremath{\epsilon}$-expansion formula for the critical exponent $\ensuremath{\eta}$ derived by Shukla and Green from Wilson's exact renormalization-group equation is found to yield the standard result $\ensuremath{\eta}=\frac{1}{54}{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, independent of the choice of the incomplete-integration function. The result is obtained analytically in the limit $a\ensuremath{\rightarrow}\ensuremath{\infty}$ and numerically for arbitrary $a$, where $a$ denotes the normalization constant of the Gaussian fixed-point functional.