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arXiv:0808.0916 · Physical Review Letters

Finite-Size Scaling of Classical Long-Ranged Ising Chains and the Criticality of Dissipative Quantum Impurity Models

Stefan Kirchner, Qimiao Si A, Kevin Ingersent

Published Apr 23, 2009• 28 citations
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Abstract

Motivated in part by quantum criticality in dissipative Kondo systems, we revisit the finite-size scaling of a classical Ising chain with 1/r;{2-} interactions. For 1/2<<1, the scaling of the dynamical spin susceptibility is sensitive to the degree of "winding" of the interaction under periodic boundary conditions. Infinite winding yields the expected mean-field behavior, whereas without any winding the scaling is of an interacting omega/T form. The contrast with the behavior of the Bose-Fermi Kondo model suggests a breakdown of a mapping from the quantum model to a classical one due to the smearing of the Kondo spin flips by the continuum limit taken in this mapping.

DOI
10.1103/physrevlett.102.166405

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Motivated in part by quantum criticality in dissipative Kondo systems, we revisit the finite-size scaling of a classical Ising chain with 1/r;{2-} interactions. For 1/2<<1, the scaling of the dynamical spin susceptibility is sensitive to the degree of "winding" of the interaction under periodic boundary conditions. Infinite winding yields the expected mean-field behavior, whereas without any winding the scaling is of an interacting omega/T form. The contrast with the behavior of the Bose-Fermi Kondo model suggests a breakdown of a mapping from the quantum model to a classical one due to the smearing of the Kondo spin flips by the continuum limit taken in this mapping.
We study finite-size scaling for the magnetic observables of an impurity residing at the end point of an open quantum Ising chain with transverse magnetic field, realized by locally rescaling the field by a factor $\ensuremath{\mu}\ensuremath{\ne}1$. In the homogeneous chain limit at $\ensuremath{\mu}=1$, we find the expected finite-size scaling for the longitudinal impurity magnetization, with no specific scaling for the transverse magnetization. At variance, in the classical impurity limit $\ensuremath{\mu}=0$, we recover finite scaling for the longitudinal magnetization, while the transverse one basically does not scale. We provide both analytic approximate expressions for the magnetization and the susceptibility as well as numerical evidences for the scaling behavior. At intermediate values of $\ensuremath{\mu}$, finite-size scaling is violated, and we provide a possible explanation of this result in terms of the appearance of a second, impurity-related length scale. Finally, by going along the standard quantum-to-classical mapping between statistical models, we derive the classical counterpart of the quantum Ising chain with an end-point impurity as a classical Ising model on a square lattice wrapped on a half-infinite cylinder, with the links along the first circle modified as a function of $\ensuremath{\mu}$.
Studies of non-Fermi-liquid properties in heavy fermions have led to the current interest in the Bose-Fermi Kondo model. Here we use a dynamical large-NNN approach to analyze an $\mathrm{S}\mathrm{U}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{S}\mathrm{U}(\ensuremath{\kappa}N)$ generalization of the model. We establish the existence in this limit of an unstable fixed point when the bosonic bath has a sub-Ohmic spectrum ($|\ensuremath{\omega}{|}^{1\ensuremath{-}ϵ}\mathrm{s}\mathrm{g}\mathrm{n}\ensuremath{\omega}$, with 0<ϵ<10<ϵ<10<ϵ<1). At the quantum-critical point, the Kondo scale vanishes and the local spin susceptibility (which is finite on the Kondo side for $\ensuremath{\kappa}<1$) diverges. We also find an $\ensuremath{\omega}/T$ scaling for an extended range (15 decades) of $\ensuremath{\omega}/T$. This scaling violates (for $ϵ\ensuremath{\ge}1/2$) the expectation of a naive mapping to certain classical models in an extra dimension; it reflects the inherent quantum nature of the critical point.
Using free-fermionic techniques we study the entanglement entropy of a block of contiguous spins in a large finite quantum Ising chain in a transverse field, with couplings of different types: homogeneous, periodically modulated and random. We carry out a systematic study of finite-size effects at the quantum critical point, and evaluate subleading corrections both for open and for periodic boundary conditions. For a block corresponding to a half of a finite chain, the position of the maximum of the entropy as a function of the control parameter (e.g. the transverse field) can define the effective critical point in the finite sample. On the basis of homogeneous chains, we demonstrate that the scaling behavior of the entropy near the quantum phase transition is in agreement with the universality hypothesis, and calculate the shift of the effective critical point, which has different scaling behaviors for open and for periodic boundary conditions.
We study the out-of-equilibrium dynamics of one-dimensional quantum Ising-like systems, arising from sudden quenches of the Hamiltonian parameter ggg driving quantum transitions between disordered and ordered phases. In particular, we consider quenches to values of ggg around the critical value gc{g}_{c}gc​, and mainly address the question whether, and how, the quantum transition leaves traces in the evolution of the transverse and longitudinal magnetizations during such a deep out-of-equilibrium dynamics. We shed light on the emergence of singularities in the thermodynamic infinite-size limit, likely related to the integrability of the model. Finite systems in periodic and open boundary conditions develop peculiar power-law finite-size scaling laws related to revival phenomena, but apparently unrelated to the quantum transition, because their main features are generally observed in quenches to generic values of ggg. We also investigate the effects of dissipative interactions with an environment, modeled by a Lindblad equation with local decay and pumping dissipation operators within the quadratic fermionic model obtainable by a Jordan-Wigner mapping. Dissipation tends to suppress the main features of the unitary dynamics of closed systems. We finally address the effects of integrability breaking, due to further lattice interactions, such as in anisotropic next-to-nearest-neighbor Ising (ANNNI) models. We show that some qualitative features of the post-quench dynamics persist, in particular, the different behaviors when quenching to quantum ferromagnetic and paramagnetic phases, and the revival phenomena due to the finite size of the system.
Abstract For the inverse square long-range ferromagnetic Ising chain in a transverse field, the thermal phase boundary of the floating Kosterlitz–Thouless phase is obtained for several values of the transverse field down to the quantum critical point. The sharp domain walls in the classical model are increasingly smeared out by the transverse field, which is evidenced by a pronounced broadening of the non-universal bump in the specific heat. The discernability of KT critical scaling in finite-size simulations is discussed.
We study the criticality of a long-range quantum ferromagnetic Ising chain with algebraically decaying interactions $1/{r}^{\ensuremath{\alpha}}$ via the fidelity susceptibility based on the exact diagonalization and the density-matrix renormalization-group techniques. We find that critical exponents change monotonically from the mean-field universality class to the short-range Ising universality class for intermediate $\ensuremath{\alpha}$, which are consistent with recent results obtained from renormalization-group techniques. In addition, we determine the critical values for $1.8\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}3$ from the finite-size scaling of the fidelity susceptibility. Our work provides very nice numerical data from the fidelity susceptibility for the quantum long-range ferromagnetic Ising chain.