Dirk Schuricht

Dynamics in the Ising field theory after a quantum quench

We study the real-time dynamics of the order parameter σ(t) in the Ising field theory after a quench in the fermion mass, which corresponds to a quench in the transverse field of the corresponding transverse field Ising chain. We focus on quenches within the ordered phase. The long-time behaviour is obtained analytically by a resummation of the leading divergent terms in a form-factor expansion for σ(t). Our main result is the development of a method for treating divergences associated with working directly in the field theory limit. We recover the scaling limit of the corresponding result by Calabrese et al (2011 Phys. Rev. Lett. 106 227203), which was obtained for the lattice model. Our formalism generalizes to integrable quantum quenches in other integrable models.

Dynamical phase transitions after quenches in nonintegrable models

We investigate the dynamics following sudden quenches across quantum critical points belonging to different universality classes. Specifically, we use matrix product state methods to study the quantum Ising chain in the presence of two additional terms which break integrability. We find that in all models the rate function for the return probability to the initial state becomes a nonanalytic function of time in the thermodynamic limit. This so-called “dynamical phase transition” was first observed in a recent work by Heyl, Polkovnikov, and Kehrein [Phys. Rev. Lett. 110, 135704 (2013)] for the exactly-solvable quantum Ising chain, which can be mapped to free fermions. Our results for “interacting theories” indicate that nonanalytic dynamics is a generic feature of sudden quenches across quantum critical points. We discuss potential connections to the dynamics of the order parameter.

Luttinger-liquid universality in the time evolution after an interaction quench.

We provide evidence that the relaxation dynamics of one-dimensional, metallic Fermi systems resulting out of an abrupt amplitude change of the two-particle interaction has aspects which are universal in the Luttinger liquid sense: the leading long-time behavior of certain observables is described by universal functions of the equilibrium Luttinger liquid parameter and the renormalized velocity. We analytically derive those functions for the Tomonaga-Luttinger model and verify our hypothesis of universality by considering spinless lattice fermions within the framework of the density-matrix renormalization group.

Non-equilibrium current and relaxation dynamics of a charge-fluctuating quantum dot

We study the steady-state current in a minimal model for a quantum dot dominated by charge fluctuations and analytically describe the time evolution into this state. The current is driven by a finite-bias voltage V across the dot, and two different renormalization group methods are used to treat small-to-intermediate local Coulomb interactions. The corresponding flow equations can be solved analytically, which allows to identify all microscopic cutoff scales. Exploring the entire parameter space we find rich non-equilibrium physics which cannot be understood by simply considering the bias voltage as an infrared cutoff. For the experimentally relevant case of left-right asymmetric couplings, the current generically shows a power law suppression for large V. The relaxation dynamics towards the steady state features characteristic oscillations as well as an interplay of exponential and power law decay.

Quench dynamics of the Tomonaga–Luttinger model with momentum-dependent interaction

We study the relaxation dynamics of the one-dimensional Tomonaga–Luttinger model after an interaction quench, paying particular attention to the momentum dependence of the two-particle interaction. Several potentials of different analytical forms are investigated that all lead to universal Luttinger liquid (LL) physics in equilibrium. The steady-state fermionic momentum distribution shows universal behavior in the sense of the LL phenomenology. For generic regular potentials, the large time decay of the momentum distribution function toward the steady-state value is characterized by a power law with a universal exponent that depends only on the potential at zero momentum transfer. The commonly employed ad hoc procedure fails to give this exponent. In addition to quenches from zero to positive interactions, we also consider the abrupt changes in the interaction between two arbitrary values. Additionally, we discuss the appearance of a factor of two between the steady-state momentum distribution function and that obtained in equilibrium at equal two-particle interaction.

Strongly interacting Majorana modes in an array of Josephson junctions

An array of superconducting islands with semiconducting nanowires in the right regime provides a macroscopic implementation of Kitaevs toy model for Majorana wires. We show that a capacitive coupling between adjacent islands leads to an effective interaction between the Majorana modes. We demonstrate that, even though strong repulsive interaction eventually drives the system into a Mott insulating state, the competition between the (trivial) band insulator and the (trivial) Mott insulator leads to an interjacent topological insulating state for arbitrary strong interactions.

No attraction between spinons in the Haldane-Shastry model

While the Bethe Ansatz solution of the Haldane--Shastry model appears to suggest that the spinons represent a free gas of half-fermions, Bernevig, Giuliano, and Laughlin (BGL) (cond-mat/0011069, cond-mat/0011270) have concluded recently that there is an attractive interaction between spinons. We argue that the dressed scattering matrix obtained with the asymptotic Bethe Ansatz is to be interpreted as the true and physical scattering matrix of the excitations, and hence, that the result by BGL is inconsistent with an earlier result by Essler (cond-mat/9406081). We critically re-examine the analysis of BGL, and conclude that there is no interaction between spinons or spinons and holons in the Haldane--Shastry model.

Renormalization group analysis of the interacting resonant-level model at finite bias: Generic analytic study of static properties and quench dynamics

Using a real-time renormalization group method we study the minimal model of a quantum dot dominated by charge fluctuations, the two-lead interacting resonant level model, at finite bias voltage. We develop a set of RG equations to treat the case of weak and strong charge fluctuations, together with the determination of power-law exponents up to second order in the Coulomb interaction. We derive analytic expressions for the charge susceptibility, the steady-state current and the conductance in the situation of arbitrary system parameters, in particular away from the particle-hole symmetric point and for asymmetric Coulomb interactions. In the generic asymmetric situation we find that power laws can be observed for the current only as function of the level position (gate voltage) but not as function of the voltage. Furthermore, we study the quench dynamics after a sudden switch-on of the level-lead couplings. The time evolution of the dot occupation and current is governed by exponential relaxation accompanied by voltage-dependent oscillations and characteristic algebraic decay.

Many-Spinon States and the Secret Significance of Young Tableaux

We establish a one-to-one correspondence between the Young tableaux classifying the total spin representations of N spins and the exact eigenstates of the Haldane-Shastry model for a chain with N sites classified by the total spins and the fractionally spaced single-particle momenta of the spinons.

Exact results for SU(3) spin chains: Trimer states, valence bond solids, and their parent Hamiltonians

Beginning with the invention of the Bethe ansatz in 1931 1 as a method to solve the S = 1 Heisenberg chain with nearest neighbor interactions, a significant share of the entire effort in condensed matter physics has been devoted to the study of quantum spin chains. Faddeev and Takhtajan 2 discovered in 1981 that the elementary excitations now called spinons of the spin-1 / 2 Heisenberg chain carry spin 1 / 2 while the Hilbert space is spanned by spin flips, which carry spin 1. The fractional quantization of spin in spin chains is comparable to the fractional quantization of charge in quantized Hall liquids. 3 In 1982, Haldane 4 identified the O3 nonlinear sigma model as the effective low-energy field theory of SU2 spin chains, and argued that chains with integer spin possess a gap in the excitation spectrum, while a topological term renders half-integer spin chains gapless. The general methods—the Bethe ansatz and the use of effective field theories including bosonization—are complemented by a number of exactly solvable models, most prominently among them the Majumdar-Ghosh MG Hamiltonian 5 for the S = 1 dimer chain, the AKLT model 6 as a paradigm of the gapped S = 1 chain, and the HaldaneShastry model HSM. 7–9 In the HSM the wave functions for the ground state and single-spinon excitations are of a simple Jastrow form, elevating the conceptual similarity to quantized Hall states to a formal equivalence. One of the unique features of the HSM is that the spinons are free in the sense that they only interact through their half-Fermi statistics. 10–12 The HSM has been generalized from SU2 to SUn. 13–16 For the MG and the AKLT model, only the ground states are known exactly. Nonetheless, these models have amply contributed to our understanding of many aspects of spin chains, each of them through the specific concepts captured in its ground state. 17–24