Off-critical Region Research Articles

Emergence of long-range correlations in random networks

We perform an analysis of the long-range degree correlation of the giant component (GC) in an uncorrelated random network by employing generating functions. By introducing a characteristic length, we find that a pair of nodes in the GC is negatively degree-correlated within the characteristic length and uncorrelated otherwise. At the critical point, where the GC becomes fractal, the characteristic length diverges and the negative long-range degree correlation emerges. We further propose a correlation function for degrees of two nodes separated by the shortest path length l, which behaves as an exponentially decreasing function of distance in the off-critical region. The correlation function obeys a power-law with an exponential cutoff near the critical point. The Erdős-Rényi random graph is employed to confirm this critical behavior.

Quons in a quantum dissipative system

String theory proves to be an imperative tool to explore the critical behavior of the quantum dissipative system. We discuss the quantum particles moving in two dimensions, in the presence of a uniform magnetic field, subject to a periodic potential and a dissipative force, which are described by the dissipative Wannier–Azbel–Hofstadter (DWAH) model. Using string theory formulation of the model, we find that the elementary excitations of the system at the generic points of the off-critical regions, in the zero temperature limit are quons, which satisfy [Formula: see text]-deformed statistics.

Theory of temporal fluctuations in isolated quantum systems

When an isolated quantum system is driven out of equilibrium, expectation values of general observables start oscillating in time. This paper reviews the general theory of such temporal fluctuations. We first survey some results on the strength of such temporal fluctuations. For example temporal fluctuations are exponentially small in the system's volume for generic systems whereas they fall-off algebraically in integrable systems. We then concentrate on the so-called quench scenario where the system is driven out-of-equilibrium under the application of a sudden perturbation. For sufficiently small perturbations, temporal fluctuations of physical observables can be characterized in full generality and can be used as an effective tool to probe quantum criticality of the underlying model. In the off-critical region the distribution becomes Gaussian. Close to criticality the distribution becomes a universal function uniquely characterized by a single critical exponent, that we compute explicitly. This contrasts standard equilibrium quantum fluctuations for which the critical distribution depends on a numerable set of critical coefficients and is known only for limited examples. The possibility of using temporal fluctuations to determine pseudo-critical boundaries in optical lattice experiments is further reviewed.

Doubling of entanglement spectrum in tensor renormalization group

We investigate the entanglement spectrum in HOTRG ---tensor renormalization group (RG) method combined with the higher order singular value decomposition--- for two-dimensional (2D) classical vertex models. In the off-critical region, it is explained that the entanglement spectrum associated with the RG transformation is described by `doubling' of the spectrum of a corner transfer matrix. We then demonstrate that the doubling actually occurs for the square-lattice Ising model by HOTRG calculations up to $D = 64$, where $D$ is the cut-off dimension of tensors. At the critical point, we also find that a non-trivial $D$ scaling behavior appears in the entanglement entropy. We mention about the HOTRG for the 1D quantum system as well.

Shear Induced Phase Boundary Shift in the Critical and Off-Critical Regions for a Polybutadiene/Polyisoprene Blend

The rheology of near- and off-critical elastomeric blends of polybutadiene (PB)/ low vinyl content polyisoprene (LPI) has been studied as a function of temperature, heating rate, and shear frequency. Depending on the composition, near or far away from the critical point, blends showed different behaviors in several aspects: temperature ramp curves, shift of the apparent binodal and spinodal points under oscillatory shear at the given frequency and strain amplitude. The composition dependent rheological responses are interpreted by the differences in the amplitude of the critical fluctuation near the phase boundary and in the shear induced mixing mechanisms between near- and off-critical blends. For near-critical blends, the critical fluctuation is large enough to induce a considerable extra stress when the blend is still in the miscible state. As a result, a heating rate independent upturn of G′ can be observed and the apparent spinodal point can be greatly shifted through the strong suppression of the la...

STRING THEORY AND DUALITIES IN THE QUANTUM DISSIPATIVE HOFSTADTER SYSTEM

We study the dualities of the quantum dissipative Hofstadter system which describes particles moving in two dimensions, subject to a uniform magnetic field, a periodic potential and a dissipative force. Using the string theory formulation, we show that the system has two kinds of dualities. The duality, previously known as the exact duality in the literature is shown to correspond to a subgroup of the T-dual symmetry group unbroken by the periodic boundary potential in string theory. The other duality is a particle–kink duality in the noncommutative open string theory which is a generalized Schmid duality in the presence of the uniform magnetic field. The kinks of the dissipative Hofstadter model are found to be noncommutative objects. The particle–kink duality, which is called previously the approximate duality, is shown to be also exact. In contrast to the previous derivation, which is based on the Coulomb gas expansion of the partition function and asserts that the duality holds only approximately in the regime of strong magnetic field, the string theory formulation proves that duality holds always exactly in the off-critical regions where the periodic potential becomes strong, regardless of the strength of the magnetic field. The dualities of the DHM may also be useful for studying the rolling tachyon in string theory in the presence of the Neveu–Schwarz (NS) B-field, since both DHM and rolling tachyon in the presence of NS B-field are described by the same action.

Quantum Brownian motion on a triangular lattice and Fermi-Bose equivalence: an application of boundary state formulation

We discuss the Bose-Fermi equivalence in the quantum Brownian motion (QBM) on a triangular lattice, mapping the action for the QBM into a string theory action with a periodic boundary tachyon potential. We construct new Klein factors which are more appropriate than the conventional ones to deal with the quantum field theories defined on a two dimensioanl space-time with boundaries. Using the Fermi-Bose equivalence with the new Klein factors, we show that the model for the quantum Bownian motion on a triangular lattice is equivalent to the Thirring model with boundary terms, which are quadratic in fermion field operators, in the off-critical regions and to a SU(3) × SU(3) free fermion theory with quadratic boundary terms at the critical point.

Griffiths-McCoy singularities in random quantum spin chains: Exact results

We consider random quantum (tight-binding, XX and Ising) spin chains in the off-critical region and study their Griffiths-McCoy singularities. These are obtained from the density of states of the low-energy excitations, which is calculated exactly by the Dyson-Schmidt method. In large finite systems the low-energy excitations are shown to follow the statistics of extremes and their distribution is given by the Fr\'echet form. Relation between the Dyson-Schmidt technique and the strong disorder renormalization group method is also discussed.

Griffiths-McCoy singularities in random quantum spin chains: exact results through renormalization.

The Ma-Dasgupta-Hu renormalization group (RG) scheme is used to study singular quantities in the Griffiths phase of random quantum spin chains. For the random transverse-field Ising spin chain we have extended Fisher's analytical solution to the off-critical region and calculated the dynamical exponent exactly. Concerning other random chains we argue by scaling considerations that the RG method generally becomes asymptotically exact for large times, both at the critical point and in the whole Griffiths phase. This statement is checked via numerical calculations on the random Heisenberg and quantum Potts models by the density matrix renormalization group method.

Random antiferromagnetic quantum spin chains: Exact results from scaling of rare regions

We study XY and dimerized XX spin-1/2 chains with random exchange couplings by analytical and numerical methods and scaling considerations. We extend previous investigations to dynamical properties, to surface quantities and operator profiles, and give a detailed analysis of the Griffiths phase. We present a phenomenological scaling theory of average quantities based on the scaling properties of rare regions, in which the distribution of the couplings follows a surviving random walk character. Using this theory we have obtained the complete set of critical decay exponents of the random XY and XX models, both in the volume and at the surface. The scaling results are confronted with numerical calculations based on a mapping to free fermions, which then lead to an exact correspondence with directed walks. The numerically calculated critical operator profiles on large finite systems (L<=512) are found to follow conformal predictions with the decay exponents of the phenomenological scaling theory. Dynamical correlations in the critical state are in average logarithmically slow and their distribution show multi-scaling character. In the Griffiths phase, which is an extended part of the off-critical region average autocorrelations have a power-law form with a non-universal decay exponent, which is analytically calculated. We note on extensions of our work to the random antiferromagnetic XXZ chain and to higher dimensions.

Characterizations of critical processes in liquid-liquid phase separation of the elastomeric protein-water system: microscopic observations and light scattering measurements.

Biological self-assembly process of tropoelastin in an extracellular space, viewed as a key step of the elastogenesis, can be mimicked by the temperature-dependent coacervation of the elastin-related polypeptide-water system. Early and late stages of the phase separation behavior of the bovine neck ligamental alpha-elastin-water system were examined respectively by the laser light scattering photometry and phase contrast microscopy. Changes in the hydrodynamic size of molecular assemblies and visible microcoacervate droplet size were traced as a function of the concentration of alpha-elastin and temperature. Near the critical point, alpha-elastin concentration of 0.11 mg/mL and temperature of 21.5 degrees C, the phase separation was initiated after fast increase of the hydrodynamic size of primary aggregates as scattering particles and followed by the appearance of larger microcoacervate droplets with a broad size distribution. Whereas in the off-critical region, slow decrease of the hydrodynamic size of primary particles induced phase separation with smaller droplets of a narrow size distribution. Observation of the phase separation processes in the alpha-elastin-water system with metal chlorides and hydrophobic synthetic model polypeptide-water system indicated that the fast and slow molecular assembly processes were based on the fundamental hydrophobic interactions and involvements of electrostatic interactions between charged amino acid residues, respectively.

Comparative study of the critical behavior in one-dimensional random and aperiodic environments

We consider cooperative processes (quantum spin chains and random walks) in one-dimensional fluctuating random and aperiodic environments characterized by fluctuating exponents omega>0. At the critical point the random and aperiodic systems scale essentially anisotropically in a similar fashion: length (L) and time (t) scales are related as t ~ log^{1/omega}. Also some critical exponents, characterizing the singularities of average quantities, are found to be universal functions of omega, whereas some others do depend on details of the distribution of the disorder. In the off-critical region there is an important difference between the two types of environments: in aperiodic systems there are no extra (Griffiths)-singularities.

Random transverse Ising spin chain and random walks

We study the critical and off-critical (Griffiths-McCoy) regions of the random transverse-field Ising spin chain by analytical and numerical methods and by phenomenological scaling considerations. Here we extend previous investigations to surface quantities and to the ferromagnetic phase. The surface magnetization of the model is shown to be related to the surviving probability of an adsorbing walk and several critical exponents are exactly calculated. Analyzing the structure of low energy excitations we present a phenomenological theory which explains both the scaling behavior at the critical point and the nature of Griffiths-McCoy singularities in the off-critical regions. In the numerical part of the work we used the free-fermion representation of the model and calculated the critical magnetization profiles, which are found to follow very accurately the conformal predictions for different boundary conditions. In the off-critical regions we demonstrated that the Griffiths-McCoy singularities are characterized by a single, varying exponent, the value of which is related through duality in the paramagnetic and ferromagnetic phases.

Random and aperiodic quantum spin chains: A comparative study

According to the Harris-Luck criterion the relevance of a fluctuating interaction at the critical point is connected to the value of the fluctuation exponent omega. Here we consider different types of relevant fluctuations in the quantum Ising chain and investigate the universality class of the models. At the critical point the random and aperiodic systems behave similarly, due to the same type of extreme broad distribution of the energy scales at low energies. The critical exponents of some averaged quantities are found to be a universal function of omega, but some others do depend on other parameters of the distribution of the couplings. In the off-critical region there is an important difference between the two systems: there are no Griffiths singularities in aperiodic models.