Disorder-induced quantum-Griffiths-like features from a non-conventional renormalization group analysis near four dimensions

We investigate the role played by symmetry conserving quenched disorder on quantum criticality of a variety of d-dimensional systems with a continuous symmetry order parameter. We employ a non-standard procedure which combines a preliminary reduction to an effective classical random problem and a successive conventional renormalization group treatment. Solving the effective flow equations to first order in ε=4−d and then restoring the original coupling parameters, for d<4 we find a quantum critical point scenario exhibiting unusual features, which remind us of some predictions of the quantum Griffiths phase model.     

Quantum critical behavior of clean itinerant ferromagnets

We consider the quantum ferromagnetic transition at zero temperature in clean itinerant electron systems. We find that the Landau-Ginzburg-Wilson order parameter field theory breaks down since the electron-electron interaction leads to singular coupling constants in the Landau- Ginzburg-Wilson functional. These couplings generate an effective long-range interaction between the spin or order parameter fluctuations of the form 1 <r ² d?1, with d the spatial dimension. This leads to unusual scaling behavior at the quantum critical point in 1 < d ≤ 3, which we determine exactly. We also discuss the quantum-to-classical crossover at small but finite temperatures, which is characterized by the appearance of multiple temperature scales. A comparison with recent results on disordered itinerant ferromagnets is given.     

Quantum effects in the critical properties of the transverse Ising model for 2 < d ⩽ 3

The crossover behaviour of a d-dimensional (2<d?3) Ising model in a transverse field Г is investigated near the multicritical point $\text{[Г, T] = [Г}{}_{\mathrm{c}}\text{(0), 0].}$ A renormalization scheme which removes divergences in the zero-temperature limit is presented. The crossover exponent and scaling function for the longitudinal susceptibility are found.     

Finite-size crossover in systems with slab geometry

A renormalization group study of the finite-size (dimensional) crossover is carried out with the help pf ε = 4 − d and ε₀ = 3 − d expansion techniques. The finite-size crossover and the invariance relation for the length scale transformation are proven up to the two-loop approximation. The formal equivalence between the finite-size crossover in classical systems and the quantum-to-classical dimensional crossover in certain quantum statistical models is emphasized and exploited. The finite-size corrections to the fluctuation shift of the critical temperature and the width of the critical region are investigated. It is shown that the shift exponent λ describing the fractional rounding of the critical temperature obeys the relation λ = D − 2, where D is the dimensionality of the system.     

A fixed point for quantum gravity

Using the 1/N expansion a fixed point of the renormalization group is found for quantized gravitational theories which is non-trivial in all dimensions, d, including four. Using the fixed point it is shown how Einstein's theory can be renormalized for 3<d<4. In four dimensions the pure Einstein theory does not exist, but the R + C_μναβ² theory does. It is shown how gravitational theories whose quantum lagrangians are scale invariant may be renormalized such that the scale invariance is broken only by the choice of the critical renormalization group trajectory. A comparison is made with the renormalization of four-fermion and Yukawa theories in 4?? dimensions which suggests that quantum gravity might exist in four dimensions even if those theories do not.     

A study of the critical behavior of the O(N) linear and nonlinear σ models

A study of the large N behavior of both the O(N) linear and nonlinear σ models is presented. The purpose is to investigate the relationship between the disordered (ordered) phase of the linear and nonlinear sigma models. Utilizing operator product expansions and stability analyses, it is shown that for 2 ≤ d < 4, it is the dimensionless renormalized quartic coupling and $\text{λ}{}^1$ is the IR fixed point) limit of the linear σ model which yields the nonlinear σ model. It is also shown that stable large N linear σ models with λ < 0 (σ is the bare quartic coupling) can exist (at least in the context of no tachyonic states being present). A criteria valid for all dimensionalities d, less than four, is derived which determines when λ < 0 models are tachyonic free. Arguments are given showing that the d = 4 large N linear (for λ > 0) and nonlinear models are trivial. This result (i.e., triviality) is well known but only for one and two component models. Interestingly enough, the λ < 0, d = 4 linear σ model remains nontrivial and tachyonic free.     

Quantum tricriticality in transverse Ising-like systems

The quantum tricriticality of d-dimensional transverse Ising-like systems is studied by means of a perturbative renormalization group approach focusing on static susceptibility. This allows us to obtain the phase diagram for 3 ≤ d < 4, with a clear location of the critical lines ending in the conventional quantum critical points and in the quantum tricritical one, and of the tricritical line for temperature T ≥ 0. We determine also the critical and the tricritical shift exponents close to the corresponding ground state instabilities. Remarkably, we find a tricritical shift exponent identical to that found in the conventional quantum criticality and, by approaching the quantum tricritical point increasing the non-thermal control parameter r, a crossover of the quantum critical shift exponents from the conventional value φ = 1/(d − 1) to the new one φ = 1/2(d − 1). Besides, the projection in the (r,T)-plane of the phase boundary ending in the quantum tricritical point and crossovers in the quantum tricritical region appear quite similar to those found close to an usual quantum critical point. Another feature of experimental interest is that the amplitude of the Wilsonian classical critical region around this peculiar critical line is sensibly smaller than that expected in the quantum critical scenario. This suggests that the quantum tricriticality is essentially governed by mean-field critical exponents, renormalized by the shift exponent φ = 1/2(d − 1) in the quantum tricritical region.     

Mutual Composite Fermion and Composite Boson approaches to balanced and imbalanced bi-layer quantum Hall system: An electronic analogy of the Helium 4 system

We use both Mutual Composite Fermion (MCF) and Composite Boson (CB) approach to study balanced and imbalanced Bi-layer Quantum Hall systems (BLQH) and make critical comparisons between the two approaches. We find the CB approach is superior to the MCF approach in studying ground states with different kinds of broken symmetries. In the phase representation of the CB theory, we first study the Excitonic superfluid (ESF) state. The theory puts spin and charge degree freedoms in the same footing, explicitly bring out the spin-charge connection and classify all the possible excitations in a systematic way. Then in the dual density representation of the CB theory, we study possible intermediate phases as the distance increases. We propose there are two critical distances d_c1 < d_c2 and three phases as the distance increases. When 0 < d < d_c1, the system is in the ESF state which breaks the internal U(1) symmetry, when d_c1 < d < d_c2, the system is in an pseudo-spin density wave (PSDW) state which breaks the translational symmetry, there is a first-order transition at d_c1 driven by the collapsing of magneto-roton minimum at a finite wavevector in the pseudo-spin channel. When d_c2 < d < ∞, the system becomes two weakly coupled ν = 1/2 Composite Fermion Fermi Liquid (FL) state. There is also a first-order transition at d = d_c2. We construct a quantum Ginzburg Landau action to describe the transition from ESF to PSDW which break the two completely different symmetries. By using the QGL action, we explicitly show that the PSDW takes a square lattice and analyze in detail the properties of the PSDW at zero and finite temperature. We also suggest that the correlated hopping of vacancies in the active and passive layers in the PSDW state leads to very large and temperature-dependent drag consistent with the experimental data. Then we study the effects of imbalance on both ESF and PSDW. In the ESF side, the system supports continuously changing fractional charges as the imbalance changes. In the PSDW side, there are two quantum phase transitions from the commensurate excitonic solid to an incommensurate excitonic solid and then to the excitonic superfluid state. We also comment on the effects of disorders and compare our results with the previous work. The very rich and interesting phases and phase transitions in the pseudo-spin channel in the BLQH is quite similar to those in ⁴He system with the distance playing the role of the pressure. A BLQH system in a periodic potential is also discussed. The Quantum Hall state to Wigner crystal transition in single layer Quantum Hall system is studied.     

Classical and quantum turbulence

The reconnection of two singularities in 2D, 3D, and 4D classical and quantum turbulence is examined. Singularity reconnection plays an essential role in the dissipation of the incompressible part of kinetic energy. A reconnection condition 2(d_s+1)≥d+1 is derived, which crucially depends on the dimension d_s of the singular structure in relation to the spatial dimension d of the system. The feasibility of this condition is examined using direct numerical simulations of the Navier-Stokes and Gross-Pitaevskii equations for the classical and quantum turbulence, respectively. We observed that the condition was satisfied for d=3 and 4, in agreement with the occurrence of energy cascades in both classical and quantum turbulence in those dimensions.     

Quantum field theory with an interaction on the boundary

We consider quantum theory of fields ? defined on a D dimensional manifold (bulk) with an interaction V (?) concentrated on a d < D dimensional surface (brane). Such a quantum field theory can be less singular than the one in d dimensions with an interaction V (?). It is shown that scaling properties of fields on the brane are different from the ones in the bulk. We discuss as an example fields on de Sitter space.     

Renormalization group in the large N limit

The normalization group in the large N limit is described and its fixed point and eigenvalues determined for 2 < d < 4. N is the number of components of the order parameter and d is the dimension.     

Critical velocities and two mechanisms of transition in superfluid liquid He

Two mechanisms of transition of the superfluid liquid ⁴He to quantum turbulence regimes are proposed for the case when the influence of the normal fluid on superfluid flow is suppressed by introducing superleaks at the ends of the capillary. Using dimensional analysis it is found that in the roton mechanism the critical velocity depends on channel size as vc∝d−1/4, matching the experiments. For the second, super-flow mechanism, the analysis of independent parameters relevant for this phenomena leads to critical velocity depending on d as vc∝d−1. It is shown that turbulence for super-flow mechanism arises when a “quantum Reynolds number” exceeds some critical value which is about 10³ for 1D geometry. The dimensional analysis of the equation for critical velocity of superfluid flow without superleaks at the ends of the capillary is also presented.     

Finite-size scaling of O(n) models in higher dimensions

We propose a finite-size scaling hypothesis for O(n) models, with n ⩾ 2, in geometry L^d−d′ × ∞^d′, with d > 4 and d′ ⩽ 2, subject to periodic boundary conditions. Several predictions, for T < T_c as well as T ≈ T_c, are made and are verified analytically for the special case of the spherical model (n = ∞).     

On the dynamics of the Ising model in a transverse field

Including mode coupling terms we give the equation of motion for the classical version of the transverse Ising model. The critical behaviour for d ? 2 is dominated by the mode coupling terms. The corresponding kinetic coefficient diverges with $\text{z =}\text{1}\text{2}\text{(d ? 2)}$. For 2 < d ? 4 the model behaves like a relaxator model.     

Critical behaviour at the displacive limit of structural phase transitions

For a special critical point at zero temperature,T _c =0, which is called the displacive limit of a classical or of a quantum-mechanical model showing displacive phase transitions, we derive a set of static critical exponents in the large-n limit. Due to zero-point motions and quantum fluctuations at low temperatures, the exponents of the quantum model are different from those of the classical model. Moreover, we report results on scaling functions, corrections to scaling, and logarithmic factors which appear ford=2 in the classical case and ford=3 in the quantum-mechanical case. Zero-point motions cause a decrease of the critical temperature of the quantum model with respect to the classicalT _c , which implies a difference between the classical and the quantum displacive limit. However, finite critical temperatures are found in both cases ford>2, while critical fluctuations still occur atT _c =0 for 0<d≦2 in the classical case and for 1 <d≦2 in the quantum model. Further we derive the slope of the critical curve at the classical displacive limit exactly. The absence of 1/n-corrections to the exponents of the classical model is also discussed.     

Nontrivial critical behaviour in a lattice model of random surfaces

A model of “planar random surfaces without spikes” on hypercubical lattices was introduced some years ago as a discretization of quantum string theory. We review some general properties of this model and present results from a Monte Carlo study of its critical behaviour in d = 4, 8 and 10 dimensions. In d = 4 dimensions we find a Hausdorff dimension d_H ≈ 4 and an anomalous dimensions η ≈ 1. These critical exponents imply a deviation from mean field theory in contrast to other lattice random surface models. Furthermore, we find evidence for mean field behaviour in 8 and 10 dimensions, indicating an upper critical dimension d_c^u ⩽ 8.     

Hot́ hole anisotropic effect in silicon and germanium

Anisotropic effect of the hole drift velocity in silicon and germanium has been investigated with the time of flight technique by applying the electric field parallel to the <100 and <111 crystallographic axis. The measurements were performed for electric fields ranging from 10 to 3 × 10⁴V/cm and temperatures from 40 to 200°K. The results indicate that the anisotropic effect v_d<100/v_d<111 increases with decreasing temperature and increasing electric field, and reaches a saturation value at high electric fields (? 10⁴V/cm). The maximum anisotropic effect for Ge is 1.25 at 40°K and for Si is 1.2 at 45°K. A qualitative analysis of the experimental data indicates that the anisotropic effect is due to the warped heavy-valence-band shape.     

Quantum Tunneling and Caustics under Inverse Square Potential

We examine the quantization of a harmonic oscillator with inverse square potential V(x)=(mω²/2) x²+g/x² on the line −∞<x<∞. We find that, for 0<g<3?²/(8m), the system admits a U(2) family of inequivalent quantizations allowing for quantum tunneling through the infinite potential barrier at x=0. These are a generalization of the conventional quantization applied to the Calogero model in which no quantum tunneling is allowed. The tunneling renders the classical caustics which arise under the potential anomalous at the quantum level, leading to the possibility of copying the profile of an arbitrary state from one side x>0, say, to the other x<0.     

Conformal field theories near a boundary in general dimensions

The implications of restricted conformal invariance under conformal transformations preserving a plane boundary are discussed for general dimensions d. Calculations of the universal function of a conformal invariant ξ which appears in the two-point function of scalar operators in conformally invariant theories with a plane boundary are undertaken to first order in the ge = 4 − d expansion for the operator φ² in φ⁴ theory. The form for the associated functions of ξ for the two-point functions for the basic field φ^α and the auxiliary field λ in the N → ∞ limit of the O(N) nonlinear sigma model for any d in the range 2 < d < 4 are also rederived. These results are obtained by integrating the two-point functions over planes parallel to the boundary, defining a restricted two-point function which may be obtained more simply. Assuming conformal invariance this transformation can be inverted to recover the full two-point function. Consistency of the results is checked by considering the limit d → 4 and also by analysis of the operator product expansions for φ^αφ^β and λλ. Using this method the form of the two-point function for the energy-momentum tensor in the conformal O(N) model with a plane boundary is also found. General results for the sum of the contributions of all derivative operators appearing in the operator product expansion, and also in a corresponding boundary operator expansion, to the two-point functions are also derived making essential use of conformal invariance.