Anisotropic Scale Invariance Research Articles

Exact four-point function and OPE for an interacting quantum field theory with space/time anisotropic scale invariance

We identify a nontrivial yet tractable quantum field theory model with space/time anisotropic scale invariance, for which one can exactly compute certain four-point correlation functions and their decompositions via the operator-product expansion(OPE). The model is the Calogero model, non-relativistic particles interacting with a pair potential frac{g}{{left|x-yright|}^2} in one dimension, considered as a quantum field theory in one space and one time dimension via the second quantisation. This model has the anisotropic scale symmetry with the anisotropy exponent z = 2. The symmetry is also enhanced to the Schrödinger symmetry. The model has one coupling constant g and thus provides an example of a fixed line in the renormalisation group flow of anisotropic theories.We exactly compute a nontrivial four-point function of the fundamental fields of the theory. We decompose the four-point function via OPE in two different ways, thereby explicitly verifying the associativity of OPE for the first time for an interacting quantum field theory with anisotropic scale invariance. From the decompositions, one can read off the OPE coefficients and the scaling dimensions of the operators appearing in the intermediate channels. One of the decompositions is given by a convergent series, and only one primary operator and its descendants appear in the OPE. The scaling dimension of the primary operator we computed depends on the coupling constant. The dimension correctly reproduces the value expected from the well-known spectrum of the Calogero model combined with the so-called state-operator map which is valid for theories with the Schrödinger symmetry. The other decomposition is given by an asymptotic series. The asymptotic series comes with exponentially small correction terms, which also have a natural interpretation in terms of OPE.

Dimensional crossovers and Casimir forces for the Bose gas in anisotropic optical lattices

We consider the Bose gas on a $d$-dimensional anisotropic lattice employing the imperfect (mean-field) gas as a prototype example. We study the dimensional crossover arising as a result of varying the dispersion relation at finite temperature $T$. We analyze in particular situations where one of the relevant effective dimensionalities is located at or below the lower critical dimension, so that the Bose-Einstein condensate becomes expelled from the system by anisotropically modifying the lattice parameters controlling the kinetic term in the Hamiltonian. We clarify the mechanism governing this phenomenon. Subsequently we study the thermodynamic Casimir effect occurring in this system. We compute the exact profile of the scaling function for the Casimir energy. As an effect of strongly anisotropic scale invariance, the Casimir force below or at the critical temperature $T_c$ may be repulsive even for periodic boundary conditions. The corresponding Casimir amplitude is universal only in a restricted sense, and the power law governing the decay of the Casimir interaction becomes modified. We also demonstrate that, under certain circumstances, the scaling function is constant for suffciently large values of the scaling variable, and in consequence is not an analytical function. At $T > T_c$ the Casimir-like interactions reflect the structure of the correlation function, and, for certain orientations of the confining walls, show exponentially damped oscillatory behavior so that the corresponding force is attractive or repulsive depending on the distance.

Lifshitz scaling, microstate counting from number theory and black hole entropy

Non-relativistic field theories with anisotropic scale invariance in (1+1)-d are typically characterized by a dispersion relation E ∼ kz and dynamical exponent z > 1. The asymptotic growth of the number of states of these theories can be described by an extension of Cardy formula that depends on z. We show that this result can be recovered by counting the partitions of an integer into z-th powers, as proposed by Hardy and Ramanujan a century ago. This gives a novel duality relationship between the characteristic energy of the dispersion relation with the cylinder radius and the ground state energy. For free bosons with Lifshitz scaling, this relationship is shown to be identically fulfilled by virtue of the reflection property of the Riemann ζ-function. The quantum Benjamin-Ono2 (BO2) integrable system, relevant in the AGT correspondence, is also analyzed. As a holographic realization, we provide a special set of boundary conditions for which the reduced phase space of Einstein gravity with a couple of U (1) fields on AdS3 is described by the BO2 equations. This suggests that the phase space can be quantized in terms of quantum BO2 states. Indeed, in the semiclassical limit, the ground state energy of BO2 coincides with the energy of global AdS3, and the Bekenstein-Hawking entropy for BTZ black holes is recovered from the anisotropic extension of Cardy formula.

Unconventional scaling theory in disorder-driven quantum phase transition

We clarify novel forms of scaling functions of conductance, critical conductance distribution and localization length in a disorder-driven quantum phase transition between band insulator and Weyl semimetal phases. Quantum criticality of the phase transition is controlled by a clean-limit fixed point with spatially anisotropic scale invariance. We argue that the anisotropic scale invariance is reflected on unconventional scaling function forms in the quantum phase transition. We verify the proposed scaling function forms in terms of transfer-matrix calculations of conductance and localization length in a tight-binding model.

Anisotropic Weyl invariance

We consider a non-relativistic free scalar field theory with a type of anisotropic scale invariance in which the number of coordinates “scaling like time” is generically greater than one. We propose the Cartesian product of two curved spaces, the metric of each space being parameterized by the other space, as a notion of curved background to which the theory can be extended. We study this type of geometries, and find a family of extensions of the theory to curved backgrounds in which the anisotropic scale invariance is promoted to a local, Weyl-type symmetry.

Numerically efficient algorithms for anisotropic scale and translation Tchebichef moment invariants

Anisotropic scale and translation invariants (ASTI) for Tchebichef moments have been proposed by Zhu et al. [27]. Since these invariants are derived via the decomposition of Tchebichef polynomials, it is unavoidable that the invariant algorithms inherit the complexities from the Tchebichef polynomials defined in terms of hypergeometric functions. Furthermore, in order to achieve anisotropic scale and translation invariance, the computation of translation invariants and scale invariants need to be performed sequentially. These have turned out to be the bottleneck for the invariant algorithms. Experimental results show that some of the computed ASTI features from symmetric patterns are less accurate. Thus, we would like to extend the work of Zhu et al.[27] to simplify the complexity of the algorithms and further improve the accuracy of the computed features. The three terms recurrence relation of the Tchebichef polynomials has been used to simplify and improve the computational efficiency of the invariant algorithms. Skew transformations are deployed to enhance the numerical accuracy of ASTI for Tchebhcief moments. Our studies show that the skewed features are less sensitive to noise and significantly enhance the accuracy of pattern recognition systems. This has been verified by the experiments on recognition of printed English letters and leaf patterns corrupted by noise and scaled and translated deformations. The simplification of the algorithms using the orthogonal property of basis functions also can be used to simplify more complex invariants like affine invariants of discrete Tchebichief moments. It can also be extended to derive invariants for other orthogonal based moments like Legendre moments, Krawtchouk moments, Hahn moments, etc.

D3–D5 theories with unquenched flavors

We construct the string duals of the defect theories generated when Nf flavor D5-branes intersect Nc color D3-branes along a 2+1 dimensional subspace. We work in the Veneziano limit in which Nc and Nf are large and Nf/Nc is fixed. By smearing the D5-branes, we find supergravity solutions that take into account the backreaction of the flavor branes and preserve two supercharges. When the flavors are massless the resulting metric displays an anisotropic Lifshitz-like scale invariance. The case of massive quarks is also considered.

Anisotropic scale invariant spacetimes and black holes in Zwei-Dreibein Gravity

We show that Zwei-Dreibein Gravity (ZDG), a bigravity theory recently proposed by Bergshoeff, de Haan, Hohm, Merbis, and Townsend in Phys.Rev.Lett. 111 (2013) 111102, admits exact solutions with anisotropic scale invariance. These type of geometries are the three-dimensional analogues of the spacetimes which were proposed as gravity duals for condensed matter systems. In particular, we find Schr\"odinger invariant spaces as well as Lifshitz spaces with arbitrary dynamical exponent $z$. We also find black holes that are asymptotically Lifshitz with $z=3$, showing that these (non-constant curvature) solutions of New Massive Gravity (NMG) are persistent after the introduction of the infinite tower of higher-curvature terms of ZDG, provided a renormalization of the parameters. Black holes in asymptotically warped Anti-de Sitter spaces are also found. Interestingly, in almost all the geometries studied in this work, the metric associated with the second dreibein turns out to be equivalent, up to a constant global factor, to the first one. This phenomenon has been previously observed in other bigravity theories in asymptotically flat and asymptotically Anti-de Sitter backgrounds. However, for the particular case of the $z=3$ Lifshitz black hole, here we found that the second metric corresponds to a different black hole that coincides with the former only in the asymptotic region. In fact, we find a new family of $z=3$ black holes that corresponds to a one-parameter deformation of the NMG solution.

Quantification of anisotropic scale invariance of geochemical anomalies associated with Sn-Cu mineralization in Gejiu, Yunan Province, China

In order to understand various anisotropic scale invariant systems, two different models are combined into a single method: the scale invariant generator (SIG) model quantifies anisotropies by estimating the generalized scale invariance (GSI) in frequency domain, which is a form of scale transformation in GSI representing how a scaling field is stratified and how it rotates; and the spectrum–area (S–A) model quantifies anisotropies by estimating the anisotropic scaling exponent in 2D frequency domain and decomposes multi-component data into separate patterns based on their distinct anisotropic scaling properties. The application of the combined model of S–A and SIG to stream sediment geochemical data from the Gejiu mineral district has shown that the new method can be used not only for separation of anomaly from background but also for characterization of anisotropic scale invariance properties such as differential compression and rotational scaling. Anomalies obtained by means of the S–A model can be utilized to delineate target areas for mineral exploration. The parameters obtained by means of the SIG model for the decomposed anomalies are useful for characterization of their orientation and compression distributions, which may provide insights to controls on mineralization or anomaly formation.

Type IIB asymptotically Lifshitz black holes

AbstractWe present the first black hole solutions with Lifshitz asymptotics found in string theory by using a consistent truncation of type IIB to four dimensions. These solutions are expected to be dual to models of strongly correlated electrons with anisotropic scale invariance with z = 2 at finite temperature. The Lifshitz black holes found display new interesting features that contrast with previous results in phenomenological models. In particular, they behave similarly to AdS‐Schwarzschild black holes: extremality is never reached and small black holes are thermodymanically unstable, signaling a Hawking‐Page like phase transition at low temperatures.

Holography for chiral scale-invariant models

Deformation of any d-dimensional conformal field theory by a constant null source for a vector operator of dimension (d + z -1) is exactly marginal with respect to anisotropic scale invariance, of dynamical exponent z. The holographic duals to such deformations are AdS plane waves, with z=2 being the Schrodinger geometry. In this paper we explore holography for such chiral scale-invariant models. The special case of z=0 can be realized with gravity coupled to a scalar, and is of particular interest since it is related to a Lifshitz theory with dynamical exponent two upon dimensional reduction. We show however that the corresponding reduction of the dual field theory is along a null circle, and thus the Lifshitz theory arises upon discrete light cone quantization of an anisotropic scale invariant field theory.

On conjectured local generalizations of anisotropic scale invariance and their implications

The theory of generalized local scale invariance of strongly anisotropic scale invariant systems proposed some time ago by Henkel [M. Henkel, Phenomenology of local scale invariance: from conformal invariance to dynamical scaling, Nucl. Phys. B 641 (3) (2002) 405–486] is examined. The case of so-called type-I systems is considered. This was conjectured to be realized by systems at m-axial Lifshitz points; in support of this claim, scaling functions of two-point cumulants at the uniaxial Lifshitz point of the three-dimensional ANNNI model were predicted on the basis of this theory and found to be in excellent agreement with Monte Carlo results [M. Pleimling, M. Henkel, Anisotropic scaling and generalized conformal invariance at Lifshitz points, Phys. Rev. Lett. 87 (12) (2001) 125702]. The consequences of the conjectured invariance equations are investigated, with emphasis put on this case. It is shown that fewer solutions than anticipated by Henkel generally exist and contribute to the scaling functions if these equations are assumed to hold for all (positive and negative) values of the d-dimensional space (or space time) coordinates ( t , r ) ∈ R × R d − 1 . Specifically, a single rather than two independent physically acceptable solutions exists in the case relevant for the mentioned fit of Monte Carlo data for the ANNNI model. Renormalization-group improved perturbation theory in 4 + m / 2 − ϵ dimensions is used to determine the scaling functions of the order-parameter and energy-density two-point cumulants in momentum space to two-loop order. The results are mathematically incompatible with Henkel's predictions except in free-field-theory cases. However, the scaling function of the energy-density cumulant we obtain for m = 1 upon extrapolation of our two-loop RG results to d = 3 differs numerically little from that of an effective free field theory.

An analytic Lifshitz black hole

A Lifshitz point is described by a quantum field theory with anisotropic scale invariance (but not Galilean invariance). In [8], gravity duals were conjectured for such theories. We construct analytically a black hole that asymptotes to a vacuum Lifshitz solution; this black hole solves the equations of motion of some simple (but somewhat strange) extensions of the models of [8]. We study its thermodynamics and scalar response functions. The scalar wave equation turns out to be exactly solvable. Interestingly, the Green's functions do not exhibit the ultralocal behavior seen previously in the free Lifshitz scalar theory.

Holographic stress tensor for non-relativistic theories

We discuss the calculation of the field theory stress tensor from the dual geometry for two recent proposals for gravity duals of non-relativistic conformal field theories. The first of these has a Schrödinger symmetry including Galilean boosts, while the second has just an anisotropic scale invariance (the Lifshitz case). For the Lifshitz case, we construct an appropriate action principle. We propose a definition of the non-relativistic stress tensor complex for the field theory as an appropriate variation of the action in both cases. In the Schrödinger case, we show that this gives physically reasonable results for a simple black hole solution and agrees with an earlier proposal to determine the stress tensor from the familiar AdS prescription. In the Lifshitz case, we solve the linearised equations of motion for a general perturbation around the background, showing that our stress tensor is finite on-shell.

On string theory duals of Lifshitz-like fixed points

We present type IIB supergravity solutions which are expected to be dual to certain Lifshitz-like fixed points with anisotropic scale invariance. They are expected to describe a class of D3-D7 systems and their finite temperature generalizations are straightforward. We show that there exist solutions that interpolate between these anisotropic solutions in the IR and the standard AdS5 solutions in the UV. This predicts anisotropic RG flows from familiar isotropic fixed points to anisotropic ones. In our case, these RG flows are triggered by a non-zero theta-angle in Yang-Mills theories that linearly depends on one of the spatial coordinates. We study the perturbations around these backgrounds and discuss the possibility of instability. We also holographically compute their thermal entropies, viscosities, and entanglement entropies.

Worldsheet analysis of gauge/gravity dualities

Gauge/gravity dualities are investigated from the worldsheet point of view. In [H. Kawai, T. Suyama, AdS/CFT correspondence as a consequence of scale invariance, Nucl. Phys. B 789 (2008) 209, arXiv:0706.1163 [hep-th]; H. Kawai, T. Suyama, Some implications of perturbative approach to AdS/CFT correspondence, Nucl. Phys. B 794 (2008) 1, arXiv:0708.2463 [hep-th]], a duality between 4d SYM and supergravity on AdS 5 × S 5 has been partly explained by using an anisotropic scale invariance of worldsheet theory. In this paper, we refine the argument and generalize it to lower dimensional cases. We show the correspondence between the Wilson loops in ( p + 1 ) -d SYM and the minimal surface in the black p-brane background. Although the scale invariance does not exist in these cases, the generalized scale transformation can be utilized. We also find that the energy density of open strings can be related to the ADM mass of the p-brane without relying on this symmetry.

Scaling turbulent atmospheric stratification. III: Space–time stratification of passive scalars from lidar data

AbstractIn this third and final part of the series, we concentrate on the temporal behaviour of atmospheric passive scalars. We first recall that—although the full (x, y, z, t) turbulent processes respect an anisotropic scale invariance—that due to advection—the generator will generally not be a diagonal matrix. This implies that the scaling of (1‐D) temporal series will generally involve three exponents in real space: 1/3, 1/2, 3/5, for spectra βτ = 5/3, 2, 11/5, with the first and last corresponding to domination by advection (horizontal and vertical respectively), and the second to pure temporal development (no advection). We survey the literature and find that almost all the empirical βτ values are indeed in the range 5/3 to 2. We then use meteorological analyses to argue that, although pure temporal development is unlikely to be dominant for time‐scales less than the eddy turnover time of the largest structures (about 2 weeks), an intermittent vertical velocity could quite easily explain the occasionally observed βτ ≈ 2 spectra.We then use state‐of‐the‐art vertically pointing lidar data of backscatter ratios from both aerosols and cirrus clouds yielding several (z, t) vertical space–time cross‐sections with resolution of 3.75 m in the vertical, 0.5–30 s in time and spanning 3–4 orders of magnitude in temporal scale. We first test the predictions of the anisotropic, multifractal extension of the Corrsin‐Obukhov law in the vertical and in time, separately finding that the cirrus and aerosol backscatters both followed the theoretical (anisotropic) scalings accurately; three of the six cases show dominance by the horizontal wind, the others by the vertical wind. In order to test the theory in arbitrary directions in this (z, t) space, and in order to get more complete information about the underlying physical scale, we develop and apply a new Anisotropic Scaling Analysis Technique (ASAT) which is based on a nonlinear space–time coordinate transformation. This transforms the original differential scaling into standard self‐similar scaling; there remains only a ‘trivial’ anisotropy. This method is used in real space on 2‐D structure functions. It is applied to both the new (z, t) data as well as the (x, z) data discussed in part II. Using ASAT, we verify the theory to within about 10% over more than three orders of magnitude of space–time scales in arbitrary directions in (x, z) and (z, t) spaces. By considering the high‐ (and low‐) order structure functions, we verify the theory for both weak and strong structures; as predicted, their average anisotropies are apparently the same.Putting together the results for (x, z) and (z, t), and assuming that there is no overall stratification in the horizontal (x, y) plane, we find that the overall (x, y, z, t) space is found to have an effective ‘elliptical dimension’ characterizing the overall space–time stratification equal to Deff, st = 3.21 ± 0.05. Copyright © 2008 Royal Meteorological Society

Turbulence of near-inertial waves in the continuously stratified fluid

Abstract By using the normal form of continuously stratified “primitive” equations of geophysical fluid dynamics with density (in the ocean), or potential temperature (in the atmosphere) playing the role of the vertical coordinate, we decouple vortex and wave motions in the system, introduce normal variables, and derive the effective Hamiltonian for waves with frequencies close to the inertial frequency (near-inertial waves, NIW). We then apply the weak turbulence approach to the random-phase ensembles of these waves. We show how the anisotropic scale-invariance of NIW may be exploited in order to obtain the stationary power-law spectra. The non-decay anisotropic scale-invariant dispersion laws of the NIW-type were not studied previously in the weak-turbulence literature.

A New Model for Quantifying Anisotropic Scale Invariance and for Decomposition of Mixing Patterns

A new power–law function has been derived to represent the relationship between area of the set consisting of wave numbers with spectral energy density above S (A(>S)) on the two-dimensional frequency plane and S. The power–law relation holds if the field concerned possessing isotropic scale invariance or generalized scaling invariance involves rotational and ratio-scale changing transforms. The equation is valid for dealing with common exploration geophysical and geochemical fields encountered in mineral exploration and environmental assessment. This power–law function not only provides a new model for characterizing anisotropic scaling invariance for generalized scaling field, for example, estimating the power exponent of power spectrum of generalized scale invariance measure in frequency domain, but also forms a theoretical base for the S–A filtering technique developed for decomposing a mixing field into components on the basis of distinct scaling properties in the frequency domain. It is demonstrated that the method has potential to become a general technique for image processing and pattern recognition.

The dilogarithm function for complex argument

This paper summarizes the basic properties of the Euler dilogarithm function, often referred to as the Spence function. These include integral representations, series expansions, linear and quadrat...