156 research outputs found
Time evolution of the Luttinger model with nonuniform temperature profile
No full textWe study the time evolution of a one-dimensional interacting fermion system described by the Luttinger model starting from a nonequilibrium state defined by a smooth temperature profile T (x). As a specific example we consider the case when T (x) is equal to T-L (T-R) far to the left (right). Using a series expansion in epsilon = 2(T-R -T-L)/(T-L + T-R), we compute the energy density, the heat current density, and the fermion two-point correlation function for all times t >= 0. For local (delta-function) interactions, the first two are computed to all orders, giving simple exact expressions involving the Schwarzian derivative of the integral of T (x). For nonlocal interactions, breaking scale invariance, we compute the nonequilibrium steady state (NESS) to all orders and the evolution to first order in epsilon. The heat current in the NESS is universal even when conformal invariance is broken by the interactions, and its dependence on T-L,T-R agrees with numerical results for the XXZ spin chain. Moreover, our analytical formulas predict peaks at short times in the transition region between different temperatures and show dispersion effects that, even if nonuniversal, are qualitatively similar to ones observed in numerical simulations for related models, such as spin chains and interacting lattice fermions
Steady States and Universal Conductance in a Quenched Luttinger Model
No full textWe obtain exact analytical results for the evolution of a 1+1-dimensional Luttinger model prepared in a domain wall initial state, i.e., a state with different densities on its left and right sides. Such an initial state is modeled as the ground state of a translation invariant Luttinger Hamiltonian Hλ with short range non-local interaction and different chemical potentials to the left and right of the origin. The system evolves for time t > 0 via a Hamiltonian Hλ′ which differs from Hλ by the strength of the interaction. Asymptotically in time, as t→ ∞, after taking the thermodynamic limit, the system approaches a translation invariant steady state. This final steady state carries a current I and has an effective chemical potential difference μ+- μ- between right- (+) and left- (−) moving fermions obtained from the two-point correlation function. Both I and μ+- μ- depend on λ and λ′. Only for the case λ= λ′= 0 does μ+- μ- equal the difference in the initial left and right chemical potentials. Nevertheless, the Landauer conductance for the final state, G= I/ (μ+- μ-) , has a universal value equal to the conductance quantum e2/ h for the spinless case
Inhomogeneous conformal field theory out of equilibrium
No full textWe study the non-equilibrium dynamics of conformal field theory (CFT) in 1+1
dimensions with a smooth position-dependent velocity explicitly breaking
translation invariance. Such inhomogeneous CFT is argued to effectively
describe 1+1-dimensional quantum many-body systems with certain inhomogeneities
varying on mesoscopic scales. Both heat and charge transport are studied,
where, for concreteness, we suppose that our CFT has a conserved U
current. Based on projective unitary representations of diffeomorphisms and
smooth maps in Minkowskian CFT, we obtain a recipe for computing the exact
non-equilibrium dynamics in inhomogeneous CFT when evolving from initial states
defined by smooth inverse-temperature and chemical-potential profiles
and . Using this recipe, the following exact analytical
results are obtained: (i) the full time evolution of densities and currents for
heat and charge transport, (ii) correlation functions for components of the
energy-momentum tensor and the U current as well as for any primary field,
and (iii) the thermal and electrical conductivities. The latter are computed by
direct dynamical considerations and alternatively using a Green-Kubo formula.
Both give the same explicit expressions for the conductivities, which reveal
how inhomogeneous dynamics opens up the possibility for diffusion as well as
implies a generalization of the Wiedemann-Franz law to finite times within CFT.Comment: 34 pages, LaTeX, 1 figure; updated and revised, final published
version; added dedication to Krzysztof Gaw\k{e}dzk
Stability of the classical catenoid and Darboux-P\"oschl-Teller potentials
No full textWe revisit the stability (instability) of the outer (inner) catenoid
connecting two concentric circular rings and give an explicit new construction
of the unstable mode of the inner catenoid by studying the spectrum of an
exactly solvable one-dimensional Schr\"odinger operator with an asymmetric
Darboux-P\"oschl-Teller potential.Comment: 9 pages, LaTeX, 2 figures; minor updates and corrections; final
published versio
Interacting fermions and non-equilibrium properties of one-dimensional many-body systems
No full textRecent experimental progress on ultracold atomic gases have opened up the possibility to simulate many-body systems out of equilibrium. We consider such a system described by the Luttinger model, which is a model of interacting fermions in one spatial dimension. It is well known that the Luttinger model is exactly solvable using bosonization. This also remains true for certain extensions of the model, e.g., where, in addition, the fermions are coupled to phonons. We give a self-contained account of bosonization, together with complete proofs, and show how this can be used to solve the Luttinger model and the above fermion-phonon model rigorously. The main focus is on non-equilibrium properties of the Luttinger model. We use the exact solution of the Luttinger model, with non-local interactions, to study the evolution starting from a non-uniform initial state with a position-dependent chemical potential. The system is shown to reach a current-carrying final steady state, in which the universal value of the electrical conductance, known from near-to-equilibrium settings, is recovered. We also study the effects of suddenly changing the interactions and show that the final state has memory of the initial state, which is, e.g., manifested by non- equilibrium exponents in its fermion two-point correlation functions.QC 20161003</p
Non-equilibrium dynamics of exactly solvable quantum many-body systems
No full textRecent experimental advances on ultracold atomic gases and trapped ions have made it possible to simulate exactly solvable quantum systems of interacting particles. In particular, the feasibility of making rapid changes, so-called quantum quenches, to such set-ups has allowed experimentalists to probe non-equilibrium phenomena in closed interacting quantum systems. This, in turn, has spurred a considerable theoretical interest in quantum many-body systems out of equilibrium. In this thesis, we study non-equilibrium properties of quantum many-body systems in the framework of exactly solvable quantum field theory in one spatial dimension. Specific systems include interacting fermions described by the Luttinger model and effective descriptions of spin chains using conformal field theory (CFT). Special emphasis is placed on heat and charge transport, studied from the point of view of quench dynamics, and, in particular, the effects of breaking conformal symmetries on transport properties. Examples include the Luttinger model with non-local interactions, breaking Lorentz and scale invariance, and inhomogeneous CFT, which generalizes standard CFT in that the usual propagation velocity v is replaced by a function v(x) that depends smoothly on the position x, breaking translation invariance. The quench dynamics studied here is for quantum quenches between, in general, different smooth inhomogeneous systems. An example of this is the so-called smooth-profile protocol, in which the initial state is defined by, e.g., smooth inhomogeneous profiles of inverse temperature and chemical potential, and the time evolution is governed by a homogeneous Hamiltonian. Using this protocol, we compute exact analytical results for the full time evolution of the systems mentioned above. In particular, we derive finite-time results that are universal in the sense that the same relations between the non-equilibrium dynamics and the initial profiles hold for any unitary CFT. These results also make clear that heat and charge transport in standard CFT are purely ballistic. Finally, we propose and study an inhomogeneous CFT model with v(x) given by a random function. We argue that this model naturally emerges as an effective description of one-dimensional quantum many-body systems with certain static random impurities. Using tools from wave propagation in random media, we show that such impurities lead to normal and anomalous diffusive contributions to heat transport on top of the ballistic one known from standard CFT.QC 20181119</p
Exact Dirac-Bogoliubov-de Gennes Dynamics for Inhomogeneous Quantum Liquids
No full textWe study inhomogeneous 1+1-dimensional quantum many-body systems described by
Tomonaga-Luttinger-liquid theory with general propagation velocity and
Luttinger parameter varying smoothly in space, equivalent to an inhomogeneous
compactification radius for free boson conformal field theory. This model
appears prominently in low-energy descriptions, including for trapped ultracold
atoms, while here we present an application to quantum Hall edges with
inhomogeneous interactions. The dynamics is shown to be governed by a pair of
coupled continuity equations identical to inhomogeneous Dirac-Bogoliubov-de
Gennes equations with a local gap and solved by analytical means. We obtain
their exact Green's functions and scattering matrix using a Magnus expansion,
which generalize previous results for conformal interfaces and quantum wires
coupled to leads. Our results explicitly describe the late-time evolution
following quantum quenches, including inhomogeneous interaction quenches, and
Andreev reflections between coupled quantum Hall edges, revealing a remarkably
universal dependence on details at stationarity or at late times out of
equilibrium.Comment: 7 pages + SM, RevTeX, 1 figure; reorganized version with updates and
typos corrected; final published versio
Breaking of Huygens–Fresnel principle in inhomogeneous Tomonaga–Luttinger liquids
No full textTomonaga–Luttinger liquids (TLLs) can be used to effectively describe one-dimensional quantum many-body systems such as ultracold atoms, charges in nanowires, superconducting circuits, and gapless spin chains. Their properties are given by two parameters, the propagation velocity and the Luttinger parameter. Here we study inhomogeneous TLLs where these are promoted to functions of position and demonstrate that they profoundly affect the dynamics: in general, besides curving the light cone, we show that propagation is no longer ballistically localized to the light-cone trajectories, different from standard homogeneous TLLs. Specifically, if the Luttinger parameter depends on position, the dynamics features pronounced spreading into the light cone, which cannot be understood via a simple superposition of waves as in the Huygens–Fresnel principle. This is the case for ultracold atoms in a parabolic trap, which serves as our main motivation, and we discuss possible experimental observations in such systems
Construction by bosonization of a fermion-phonon model
No full textWe discuss an extension of the (massless) Thirring model describing interacting fermions in one dimension which are coupled to phonons and where all interactions are local. This fermion-phonon model can be solved exactly by bosonization.We present a construction and solution of this model which is mathematically rigorous by treating it as a continuum limit of a Luttinger-phonon model. A self-contained account of the mathematical results underlying bosonization is included, together with complete proofs.</p
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