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arXiv:1812.01660 · Journal of Statistical Mechanics Theory and Experiment

Critical behaviour of the quasi-periodic quantum Ising chain

Philip J. D. Crowley, Chris R. Laumann, Anushya Chandran

Published Aug 1, 2022• 8 citations
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Abstract

Abstract The interplay of correlated spatial modulation and symmetry breaking leads to quantum critical phenomena intermediate between those of the clean and randomly disordered cases. By performing a detailed analytic and numerical case study of the quasi-periodically (QP) modulated transverse field Ising chain, we provide evidence for the conjectures of reference (Crowley et al 2018 Phys. Rev. Lett. 120 175702) regarding the QP-Ising universality class. In the generic case, we confirm that the logarithmic wandering coefficient w governs both the macroscopic critical exponents and the energy-dependent localisation length of the critical excitations. However, for special values of the phase difference Δ between the exchange and transverse field couplings, the QP-Ising transition has different properties. For Δ = 0, a generalised Aubry–André duality prevents the finite energy excitations from localising despite the presence of logarithmic wandering. For Δ such that the fields and couplings are related by a lattice shift, the wandering coefficient w vanishes. Nonetheless, the presence of small couplings leads to non-trivial exponents and localised excitations. Our results add to the rich menagerie of quantum Ising transitions in the presence of spatial modulation.

DOI
10.1088/1742-5468/ac815d

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Abstract The interplay of correlated spatial modulation and symmetry breaking leads to quantum critical phenomena intermediate between those of the clean and randomly disordered cases. By performing a detailed analytic and numerical case study of the quasi-periodically (QP) modulated transverse field Ising chain, we provide evidence for the conjectures of reference (Crowley et al 2018 Phys. Rev. Lett. 120 175702) regarding the QP-Ising universality class. In the generic case, we confirm that the logarithmic wandering coefficient w governs both the macroscopic critical exponents and the energy-dependent localisation length of the critical excitations. However, for special values of the phase difference Δ between the exchange and transverse field couplings, the QP-Ising transition has different properties. For Δ = 0, a generalised Aubry–André duality prevents the finite energy excitations from localising despite the presence of logarithmic wandering. For Δ such that the fields and couplings are related by a lattice shift, the wandering coefficient w vanishes. Nonetheless, the presence of small couplings leads to non-trivial exponents and localised excitations. Our results add to the rich menagerie of quantum Ising transitions in the presence of spatial modulation.
Unlike random potentials, quasiperiodic modulation can induce localization-delocalization transitions in one dimension. In this Letter, we analyze the implications of this for symmetry breaking in the quasiperiodically modulated quantum Ising chain. Although weak modulation is irrelevant, strong modulation induces new ferromagnetic and paramagnetic phases which are fully localized and gapless. The quasiperiodic potential and localized excitations lead to quantum criticality that is intermediate to that of the clean and randomly disordered models with exponents of ν=1^{+} (exact) and z≈1.9, Δ_{σ}≈0.16, and Δ_{γ}≈0.63 (up to logarithmic corrections). Technically, the clean Ising transition is destabilized by logarithmic wandering of the local reduced couplings. We conjecture that the wandering coefficient w controls the universality class of the quasiperiodic transition and show its stability to smooth perturbations that preserve the quasiperiodic structure of the model.
We study quantum phase transitions in transverse-field Ising spin chains in which the couplings are random but hyperuniform, in the sense that their large-scale fluctuations are suppressed. We construct a one-parameter family of disorder models in which long-wavelength fluctuations are increasingly suppressed as a parameter $\ensuremath{\alpha}$ is tuned. For $\ensuremath{\alpha}=0$, one recovers the familiar infinite-randomness critical point. For $0<\ensuremath{\alpha}<1$, we find a line of infinite-randomness critical points with continuously varying critical exponents; however, the Griffiths phases that flank the critical point at $\ensuremath{\alpha}=0$ are absent at any $\ensuremath{\alpha}>0$. When $\ensuremath{\alpha}>1$, randomness is a dangerously irrelevant perturbation at the clean Ising critical point, leading to a state we call the critical Ising insulator. In this state, thermodynamics and equilibrium correlation functions behave as in the clean system. However, all finite-energy excitations are localized, thermal transport vanishes, and autocorrelation functions remain finite in the long-time limit. We characterize this line of hyperuniform critical points using a combination of perturbation theory, renormalization-group methods, and exact diagonalization.
We employ an adaptation of a strong-disorder renormalization group technique in order to analyze the ferro–paramagnetic quantum phase transition of Ising chains with aperiodic but deterministic couplings under the action of a transverse field. In the presence of marginal or relevant geometric fluctuations induced by aperiodicity, for which the critical behavior is expected to depart from the Onsager universality class, we derive analytical and asymptotically exact expressions for various critical exponents (including the correlation length and the magnetization exponents, which are not easily obtainable by other methods), and shed light onto the nature of the ground-state structures in the neighborhood of the critical point. The main results obtained by this approach are confirmed by finite-size scaling analyses of numerical calculations based on the free-fermion method.
Abstract Motivated by the recent surge of transverse-field experiments on quasi-one-dimensional (1D) antiferromagnets Sr(Ba)Co 2 V 2 O 8 , we investigate the quantum phase transition (QPT) in a Heisenberg–Ising chain under a combination of two in-plane inter-perpendicular transverse fields and a four-period longitudinal field, where the in-plane transverse field is either uniform or staggered. We show that the model can be unitary mapped to the 1D transverse-field Ising model (1DTFIM) when the x and y components of the spin interaction and the four-period field are absent. When these two terms are present, following both analytical and numerical efforts, we demonstrate that the system undergoes a second-order QPT with increasing transverse fields, where the critical exponents as well as the central charge fall into the universality of 1DTFIM. Our results naturally identify the 1DTFIM universality of 1D quantum phase transitions observed in the existed experiments in Sr(Ba)Co 2 V 2 O 8 with transverse field applied along either [ ] or [ ] direction. Upon varying the tuning parameters, a critical surface with 1DTFIM universality is determined and silhouetted to exhibit the general presence of the universality in a much wider scope of models than conventional understanding. Our results provide a broad guiding framework to facilitate the experimental realization of 1DTFIM universality in real materials.
Abstract Motivated by experimental results on compounds like LiHo x Y 1 − x F 4 , we consider an Ising chain with random bonds in the simultaneous presence of random transverse and longitudinal fields. We study the low-energy properties of the model at zero temperature by the strong disorder renormalisation group method. In the absence of random longitudinal fields, the model showcases a trivial quantum-ordered and quantum-disordered fixed-point and a non-trivial infinite disorder critical point. In the absence of random transverse fields, the behaviour is dictated by the classical random-field Ising fixed-point. In the simultaneous presence of both a longitudinal and transverse random field, the RG trajectories are attracted to a set of disordered fixed-points, in which the disorder is either due to random quantum fluctuations, or due to classical random-field effects. Between the two regimes there is a smooth cross-over, which becomes sharp at the infinite disorder fixed-point. This local separatrix defines the relevant scaling direction, where the correlation-length is shown to diverge with an exponent ν h ≈ 1 .
We present here our study of the adiabatic quantum dynamics of a random Ising chain across its quantum critical point. The model investigated is an Ising chain in a transverse field with disorder present both in the exchange coupling and in the transverse field. The transverse field term is proportional to a function $\ensuremath{\Gamma}(t)$ which, as in the Kibble-Zurek mechanism, is linearly reduced to zero in time with a rate ${\ensuremath{\tau}}^{\ensuremath{-}1}$, $\ensuremath{\Gamma}(t)=\ensuremath{-}t∕\ensuremath{\tau}$, starting at $t=\ensuremath{-}\ensuremath{\infty}$ from the quantum disordered phase $(\ensuremath{\Gamma}=\ensuremath{\infty})$ and ending at t=0t=0t=0 in the classical ferromagnetic phase $(\ensuremath{\Gamma}=0)$. We first analyze the distribution of the gaps, occurring at the critical point ${\ensuremath{\Gamma}}_{c}=1$, which are relevant for breaking the adiabaticity of the dynamics. We then present extensive numerical simulations for the residual energy Eres{E}_{\mathrm{res}}Eres​ and density of defects ${\ensuremath{\rho}}_{k}$ at the end of the annealing, as a function of the annealing inverse rate $\ensuremath{\tau}$. Both the average ${E}_{\mathrm{res}}(\ensuremath{\tau})$ and ${\ensuremath{\rho}}_{k}(\ensuremath{\tau})$ are found to behave logarithmically for large $\ensuremath{\tau}$, but with different exponents, $[{E}_{\mathrm{res}}(\ensuremath{\tau})∕L{]}_{\mathrm{av}}\ensuremath{\sim}1∕{\mathrm{ln}}^{\ensuremath{\zeta}}(\ensuremath{\tau})$ with $\ensuremath{\zeta}\ensuremath{\approx}3.4$, and $[{\ensuremath{\rho}}_{k}(\ensuremath{\tau}){]}_{\mathrm{av}}\ensuremath{\sim}1∕{\mathrm{ln}}^{2}(\ensuremath{\tau})$. We propose a mechanism for $1∕{\mathrm{ln}}^{2}\ensuremath{\tau}$ behavior of $[{\ensuremath{\rho}}_{k}{]}_{\mathrm{av}}$ based on the Landau-Zener tunneling theory and on a Fisher's-type real-space renormalization group analysis of the relevant gaps. The model proposed shows therefore a paradigmatic example of how an adiabatic quantum computation can become very slow when disorder is at play, even in the absence of any source of frustration.