arXiv subjects

Alexander G. Abanov

Publications and source records attributed to Alexander G. Abanov.

At least 55 records · Page 3Linked to original sources

arXiv2013

Counting free fermions on a line: a Fisher-Hartwig asymptotic expansion for the Toeplitz determinant in the double-scaling limit

We derive an asymptotic expansion for a Wiener-Hopf determinant arising in the problem of counting one-dimensional free fermions on a line segment at zero temperature. This expansion is an extension of the result in the theory of Toeplitz and Wiener-Hopf determinants known as the generalized Fisher-Hartwig conjecture. The coefficients of this expansion are conjectured to obey certain periodicity relations, which renders the expansion explicitly periodic in the "counting parameter". We present two methods to calculate these coefficients and verify the periodicity relations order by order: the matrix Riemann-Hilbert problem and the Painleve V equation. We show that the expansion coefficients are polynomials in the counting parameter and list explicitly first several coefficients.

cond-mat.str-el↗

arXiv2013

Characterizing correlations with full counting statistics: classical Ising and quantum XY spin chains

We propose to describe correlations in classical and quantum systems in terms of full counting statistics of a suitably chosen discrete observable. The method is illustrated with two exactly solvable examples: the classical one-dimensional Ising model and the quantum spin-1/2 XY chain. For the one-dimensional Ising model, our method results in a phase diagram with two phases distinguishable by the long-distance behavior of the Jordan-Wigner strings. For the quantum XY chain, the method reproduces the previously known phase diagram.

cond-mat.str-el↗

arXiv2012

FQHE interferometers in strong tunneling regime. The role of compactness of edge fields

We consider multiple-point tunneling in the interferometers formed between edges of electron liquids with in general different filling factors in the regime of the Fractional Quantum Hall effect (FQHE). We derive an effective matrix Caldeira-Leggett models for the multiple tunneling contacts connected by the chiral single-mode FQHE edges. It is shown that the compactness of the Wen- Fröhlich chiral boson fields describing the FQHE edge modes plays a crucial role in eliminating the spurious non-locality of the electron transport properties of the FQHE interferometers arising in the regime of strong tunneling.

cond-mat.str-el↗

arXiv2012

On the effective hydrodynamics of FQHE

We present here a classical hydrodynamic model of a two-dimensional fluid which has many properties of Fractional Quantum Hall effect. This model incorporates the FQHE relation between the vorticity and density of the fluid and exhibits the Hall viscosity and Hall conductivity found in FQHE liquids. We describe the relation of the model to the Chern-Simons-Ginzburg-Landau theory of FQHE and show how Laughlin's wave function is annihilated by the quantum velocity operator.

cond-mat.str-el↗

arXiv2012

Hydrodynamics of cold atomic gases in the limit of weak nonlinearity, dispersion and dissipation

Dynamics of interacting cold atomic gases have recently become a focus of both experimental and theoretical studies. Often cold atom systems show hydrodynamic behavior and support the propagation of nonlinear dispersive waves. Although this propagation depends on many details of the system, a great insight can be obtained in the rather universal limit of weak nonlinearity, dispersion and dissipation (WNDD). In this limit, using a reductive perturbation method we map some of the hydrodynamic models relevant to cold atoms to well known chiral one-dimensional equations such as KdV, Burgers, KdV-Burgers, and Benjamin-Ono equations. These equations have been thoroughly studied in literature. The mapping gives us a simple way to make estimates for original hydrodynamic equations and to study the interplay between nonlinearity, dissipation and dispersion which are the hallmarks of nonlinear hydrodynamics.

cond-mat.quant-gas↗

arXiv2011

Quantum fluctuations of one-dimensional free fermions and Fisher-Hartwig formula for Toeplitz determinants

We revisit the problem of finding the probability distribution of a fermionic number of one-dimensional spinless free fermions on a segment of a given length. The generating function for this probability distribution can be expressed as a determinant of a Toeplitz matrix. We use the recently proven generalized Fisher--Hartwig conjecture on the asymptotic behavior of such determinants to find the generating function for the full counting statistics of fermions on a line segment. Unlike the method of bosonization, the Fisher--Hartwig formula correctly takes into account the discreteness of charge. Furthermore, we check numerically the precision of the generalized Fisher--Hartwig formula, find that it has a higher precision than rigorously proven so far, and conjecture the form of the next-order correction to the existing formula.

cond-mat.str-el↗

arXiv2011

Soliton solutions of Calogero model in harmonic potential

A classical Calogero model in an external harmonic potential is known to be integrable for any number of particles. We consider here reductions which play a role of "soliton" solutions of the model. We obtain these solutions both for the model with finite number of particles and in a hydrodynamic limit. In the latter limit the model is described by hydrodynamic equations on continuous density and velocity fields. Soliton solutions in this case are finite dimensional reductions of the hydrodynamic model and describe the propagation of lumps of density and velocity in the nontrivial background.

cond-mat.str-el↗

arXiv2010

Observation of shock waves in a strongly interacting Fermi gas

We study collisions between two strongly interacting atomic Fermi gas clouds. We observe exotic nonlinear hydrodynamic behavior, distinguished by the formation of a very sharp and stable density peak as the clouds collide and subsequent evolution into a box-like shape. We model the nonlinear dynamics of these collisions using quasi-1D hydrodynamic equations. Our simulations of the time-dependent density profiles agree very well with the data and provide clear evidence of shock wave formation in this universal quantum hydrodynamic system.

cond-mat.quant-gas↗

arXiv2010

Phase transitions in full counting statistics for periodic pumping

We discuss the problem of full counting statistics for periodic pumping. The probability generating function is usually defined on a circle of the "physical" values of the counting parameter, with its periodicity corresponding to charge quantization. The extensive part of the generating function can either be an analytic function on this circle or have singularities. These two cases may be interpreted as different thermodynamic phases in time domain. We discuss several examples of phase transitions between these phases for classical and quantum systems. Finally, we prove a criterion for the "analytic" phase in the problem of a quantum pump for noninteracting fermions.

cond-mat.mes-hall↗

arXiv2009

Factorization of quantum charge transport for non-interacting fermions

We show that the statistics of the charge transfer of non-interacting fermions through a two-lead contact is generalized binomial, at any temperature and for any form of the scattering matrix: an arbitrary charge-transfer process can be decomposed into independent single-particle events. This result generalizes previous studies of adiabatic pumping at zero temperature and of transport induced by bias voltage.

cond-mat.mes-hall↗

arXiv2006

Flat spin wave dispersion in a triangular antiferromagnet

The excitation spectrum of a S=1/2 2D triangular quantum antiferromagnet is studied using 1/S expansion. Due to the non-collinearity of the classical ground state significant and non-trivial corrections to the spin wave spectrum appear already in the first order in 1/S in contrast to the square lattice antiferromagnet. The resulting magnon dispersion is almost flat in a substantial portion of the Brillouin zone. Our results are in quantitative agreement with recent series expansion studies by Zheng, Fjaerestad, Singh, McKenzie, and Coldea [PRL 96, 057201 (2006) and cond-mat/0608008].

cond-mat.str-el↗

arXiv2006

Nonlinear Dynamics of Quantum Systems and Soliton Theory

We show that space-time evolution of one-dimensional fermionic systems is described by nonlinear equations of soliton theory. We identify a space-time dependence of a matrix element of fermionic systems related to the {\it Orthogonality Catastrophe} or {boundary states} with the $τ$-function of the modified KP-hierarchy. The established relation allows to apply the apparatus of soliton theory to the study of non-linear aspects of quantum dynamics. We also describe a {\it bosonization in momentum space} - a representation of a fermion operator by a Bose field in the presence of a boundary state.

nlin.SI↗

arXiv2006

Quantum Shock Waves - the case for non-linear effects in dynamics of electronic liquids

Using the Calogero model as an example, we show that the transport in interacting non-dissipative electronic systems is essentially non-linear. Non-linear effects are due to the curvature of the electronic spectrum near the Fermi energy. As is typical for non-linear systems, propagating wave packets are unstable. At finite time shock wave singularities develop, the wave packet collapses, and oscillatory features arise. They evolve into regularly structured localized pulses carrying a fractionally quantized charge - {\it soliton trains}. We briefly discuss perspectives of observation of Quantum Shock Waves in edge states of Fractional Quantum Hall Effect and a direct measurement of the fractional charge.

cond-mat.str-el↗

arXiv2005

Topological calculation of the phase of the determinant of a non self-adjoint elliptic operator

We study the zeta-regularized determinant of a non self-adjoint elliptic operator on a closed odd-dimensional manifold. We show that, if the spectrum of the operator is symmetric with respect to the imaginary axis, then the determinant is real and its sign is determined by the parity of the number of the eigenvalues of the operator, which lie on the positive part of the imaginary axis. It follows that, for many geometrically defined operators, the phase of the determinant is a topological invariant. In numerous examples, coming from geometry and physics, we calculate the phase of the determinants in purely topological terms. Some of those examples were known in physical literature, but no mathematically rigorous proofs and no general theory were available until now.

math-ph↗

arXiv2005

Hydrodynamics of correlated systems. Emptiness Formation Probability and Random Matrices

A hydrodynamic approach is used to calculate an asymptotics of the Emptiness Formation Probability - the probability of a formation of an empty space in the ground state of a quantum one-dimensional many body system. Quantum hydrodynamics of a system is represented as a Euclidian path integral over configurations of hydrodynamic variables. In the limit of a large size of the empty space, the probability is dominated by an instanton configuration, and the problem is reduced to the finding of an instanton solution of classical hydrodynamic equations. After establishing a general formalism, we carry out this calculation for several simple systems -- free fermions with an arbitrary dispersion and Calogero-Sutherland model. For these systems we confirm the obtained results by comparison with exact results known in Random Matrix theory. We argue that the nonlinear hydrodynamic approach might be useful even in cases where the linearized hydrodynamics fails.

cond-mat.str-el↗

arXiv2005

Quantum Hydrodynamics, Quantum Benjamin-Ono Equation, and Calogero Model

Collective field theory for Calogero model represents particles with fractional statistics in terms of hydrodynamic modes -- density and velocity fields. We show that the quantum hydrodynamics of this model can be written as a single evolution equation on a real holomorphic Bose field -- quantum integrable Benjamin-Ono equation. It renders tools of integrable systems to studies of nonlinear dynamics of 1D quantum liquids.

cond-mat.str-el↗

arXiv2005

Asymptotics of Toeplitz Determinants and the Emptiness Formation Probability for the XY Spin Chain

We study an asymptotic behavior of a special correlator known as the Emptiness Formation Probability (EFP) for the one-dimensional anisotropic XY spin-1/2 chain in a transverse magnetic field. This correlator is essentially the probability of formation of a ferromagnetic string of length $n$ in the antiferromagnetic ground state of the chain and plays an important role in the theory of integrable models. For the XY Spin Chain, the correlator can be expressed as the determinant of a Toeplitz matrix and its asymptotical behaviors for $n \to \infty$ throughout the phase diagram are obtained using known theorems and conjectures on Toeplitz determinants. We find that the decay is exponential everywhere in the phase diagram of the XY model except on the critical lines, i.e. where the spectrum is gapless. In these cases, a power-law prefactor with a universal exponent arises in addition to an exponential or Gaussian decay. The latter Gaussian behavior holds on the critical line corresponding to the isotropic XY model, while at the critical value of the magnetic field the EFP decays exponentially. At small anisotropy one has a crossover from the Gaussian to the exponential behavior. We study this crossover using the bosonization approach.

cond-mat.str-el↗

arXiv2005

Quantum Electrical Dipole in Triangular Systems: a Model for Spontaneous Polarity in Metal Clusters

Triangular symmetric molecules with mirror symmetry perpendicular to the 3-fold axis are forbidden to have a fixed electrical dipole moment. However, if the ground state is orbitally degenerate and lacks inversion symmetry, then a ``quantum'' dipole moment does exist. The system of 3 electrons in D_3h symmetry is our example. This system is realized in triatomic molecules like Na_3. Unlike the fixed dipole of a molecule like water, the quantum moment does not point in a fixed direction, but lies in the plane of the molecule and takes quantized values +/- mu_0 along any direction of measurement in the plane. An electric field F in the plane leads to a linear Stark splitting +/- mu_0 F}. We introduce a toy model to study the effect of Jahn-Teller distortions on the quantum dipole moment. We find that the quantum dipole property survives when the dynamic Jahn-Teller effect is included, if the distortion of the molecule is small. Linear Stark splittings are suppressed in low fields by molecular rotation, just as the linear Stark shift of water is suppressed, but will be revealed in moderately large applied fields and low temperatures. Coulomb correlations also give a partial suppression.

cond-mat.str-el↗