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arXiv:1610.01540 · SciPost Physics

Universal front propagation in the quantum Ising chain with domain-wall initial states

Viktor Eisler, Florian Maislinger, Hans Gerd Evertz

Published Dec 30, 2016• 75 citations
arXivHTMLPDF

Abstract

We study the melting of domain walls in the ferromagnetic phase of the transverse Ising chain, created by flipping the order-parameter spins along one-half of the chain. If the initial state is excited by a local operator in terms of Jordan-Wigner fermions, the resulting longitudinal magnetization profiles have a universal character. Namely, after proper rescalings, the profiles in the noncritical Ising chain become identical to those obtained for a critical free-fermion chain starting from a step-like initial state. The relation holds exactly in the entire ferromagnetic phase of the Ising chain and can even be extended to the zero-field XY model by a duality argument. In contrast, for domain-wall excitations that are highly non-local in the fermionic variables, the universality of the magnetization profiles is lost. Nevertheless, for both cases we observe that the entanglement entropy asymptotically saturates at the ground-state value, suggesting a simple form of the steady state.

DOI
10.21468/scipostphys.1.2.014

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We study the melting of domain walls in the ferromagnetic phase of the transverse Ising chain, created by flipping the order-parameter spins along one-half of the chain. If the initial state is excited by a local operator in terms of Jordan-Wigner fermions, the resulting longitudinal magnetization profiles have a universal character. Namely, after proper rescalings, the profiles in the noncritical Ising chain become identical to those obtained for a critical free-fermion chain starting from a step-like initial state. The relation holds exactly in the entire ferromagnetic phase of the Ising chain and can even be extended to the zero-field XY model by a duality argument. In contrast, for domain-wall excitations that are highly non-local in the fermionic variables, the universality of the magnetization profiles is lost. Nevertheless, for both cases we observe that the entanglement entropy asymptotically saturates at the ground-state value, suggesting a simple form of the steady state.
We study the time evolution of magnetization and entanglement for initial states with local excitations, created upon the ferromagnetic ground state of the XY chain. For excitations corresponding to a single or two well separated domain walls, the magnetization profile has a simple hydrodynamic limit, which has a standard interpretation in terms of quasiparticles. In contrast, for a spin-flip we obtain an interference term, which has to do with the nonlocality of the excitation in the fermionic basis. Surprisingly, for the single domain wall the hydrodynamic limit of the entropy and magnetization profiles are found to be directly related. Furthermore, the entropy profile is additive for the double domain wall, whereas in case of the spin-flip excitation one has a nontrivial behaviour.
We study effects of a transverse magnetic field on the properties of the one-dimensional spin-1/2 Ising-like antiferromagnet. In order to emphasize the relevant role of domain walls between opposite antiferromagnetic domains, the spin Hamiltonian is exactly mapped to the interacting fermion Hamiltonian, via the dual transformation as well as the Jordan-Wigner transformation. Within the generalized Hartree-Fock approximation, we find that the excitation spectrum exhibits characteristic features, which are explained as a result of the different roles of the transverse field and of the transverse components of the exchange interactions in the propagation of the domain wall. At a critical field where the excitation gap vanishes, a ground-state phase transition is found to take place in the same manner as that in the transverse Ising model. In addition, we discuss the relevance of our magnetization curve with regard to experimental data in CsCoCl 3 .
Abstract We revisit the out-of-equilibrium physics arising during the unitary evolution of a one-dimensional XXZ spin chain initially prepared in a domain wall state ∣ψ0⟩=∣⋯↑↑↓↓… ⟩\vert \psi _0\rangle =\vert \dots \uparrow \uparrow \downarrow \downarrow \dots \rangle∣ψ0​⟩=∣⋯↑↑↓↓…⟩ | ψ 0 ⟩ = | ⋯ ↑ ↑ ↓ ↓ ⋯ ⟩ . In absence of interactions, we review the exact lattice calculation of several conserved quantities, including e.g. the magnetization and the spin current profiles. At large distances x and times t, we show how these quantities allow for a ballistic scaling behavior in terms of the scaling variable ζ=x/t\zeta = x/tζ=x/t ζ = x / t , with exactly computable scaling functions. In such a limit of large space-time scales, we show that the asymptotic behavior of the system is suitably captured by the local occupation function of spinless fermionic modes, whose semi-classical evolution in phase space is given by a Euler hydrodynamic equation. Similarly, analytical results for the asymptotic fronts dynamics are obtained for the interacting chain via Generalized Hydrodynamics. In the last part of the work, we include large-scale quantum fluctuations on top of the semi-classical hydrodynamic background in the form of a conformal field theory that lives along the evolving Fermi contour. With this procedure, dubbed quantum generalized hydrodynamics, it is possible to obtain exact asymptotic results for the entanglement spreading during the melting dynamics.
We investigate the real-time dynamics of a critical spin-1/2 chain (XXZ model) prepared in an inhomogeneous initial state with different magnetizations on the left and right halves. Using the time-evolving block decimation method, we follow the front propagation by measuring the magnetization and entanglement entropy profiles. At long times, as in the free fermion case [Antal et al., Phys. Rev. E 59, 4912 (1999)], a large central region develops where correlations become time independent and translation invariant. The shape and speed of the fronts is studied numerically and we evaluate the stationary current as a function of initial magnetic field and as a function of the anisotropy $\ensuremath{\Delta}$. We compare the results with the conductance of a Tomonaga-Luttinger liquid, and with the exact free-fermion solution at $\ensuremath{\Delta}=0$. We also investigate the two-point correlations in the stationary region and find a good agreement with the ``twisted'' form obtained by Lancaster and Mitra [Phys. Rev. E 81, 061134 (2010)] using bosonization. Some deviations are nevertheless observed for strong currents.
We study the spreading of entanglement produced by the time evolution of a local fermionic excitation created above the ground state of the XXZ chain. The resulting entropy profiles are investigated via density-matrix renormalization group calculations, and compared to a quasiparticle ansatz. In particular, we assume that the entanglement is dominantly carried by spinon excitations traveling at different velocities, and the entropy profile is reproduced by a probabilistic expression involving the density fraction of the spinons reaching the subsystem. The ansatz works well in the gapless phase for moderate values of the XXZ anisotropy, eventually deteriorating as other types of quasiparticle excitations gain spectral weight. Furthermore, if the initial state is excited by a local Majorana fermion, we observe a nontrivial rescaling of the entropy profiles. This effect is further investigated in a conformal field theory framework, carrying out calculations for the Luttinger liquid theory. Finally, we also consider excitations creating an antiferromagnetic domain wall in the gapped phase of the chain, and find again a modified quasiparticle ansatz with a multiplicative factor.
We study the melting of a domain wall, prepared as a certain low-energy excitation above the ferromagnetic ground state of the XY chain. In a well-defined parameter regime the time-evolved magnetization profile develops sharp kinklike structures in the bulk, showing features of a phase transition in the hydrodynamic scaling limit. The transition is of a purely dynamical nature and can be attributed to the appearance of a negative effective mass term in the dispersion. The signatures are also clearly visible in the entanglement profile measured along the front region, which can be obtained by covariance-matrix methods despite the state being non-Gaussian.