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arXiv:0706.1832 · Physical Review B

Adiabatic quantum dynamics of a random Ising chain across its quantum critical point

Tommaso Caneva, Rosario Fazio, Giuseppe E. Santoro

Published Oct 23, 2007• 145 citations
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Abstract

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 $t=0$ 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 ${E}_{\mathrm{res}}$ 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.

DOI
10.1103/physrevb.76.144427

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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 \ensuremathΓ(t)\ensuremath{\Gamma}(t)\ensuremathΓ(t) which, as in the Kibble-Zurek mechanism, is linearly reduced to zero in time with a rate \ensuremathτ\ensuremath−1{\ensuremath{\tau}}^{\ensuremath{-}1}\ensuremathτ\ensuremath−1, \ensuremathΓ(t)=\ensuremath−t∕\ensuremathτ\ensuremath{\Gamma}(t)=\ensuremath{-}t∕\ensuremath{\tau}\ensuremathΓ(t)=\ensuremath−t∕\ensuremathτ, starting at t=\ensuremath−\ensuremath∞t=\ensuremath{-}\ensuremath{\infty}t=\ensuremath−\ensuremath∞ from the quantum disordered phase (\ensuremathΓ=\ensuremath∞)(\ensuremath{\Gamma}=\ensuremath{\infty})(\ensuremathΓ=\ensuremath∞) and ending at t=0t=0t=0 in the classical ferromagnetic phase (\ensuremathΓ=0)(\ensuremath{\Gamma}=0)(\ensuremathΓ=0). We first analyze the distribution of the gaps, occurring at the critical point \ensuremathΓc=1{\ensuremath{\Gamma}}_{c}=1\ensuremathΓc​=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 \ensuremathρk{\ensuremath{\rho}}_{k}\ensuremathρk​ at the end of the annealing, as a function of the annealing inverse rate \ensuremathτ\ensuremath{\tau}\ensuremathτ. Both the average Eres(\ensuremathτ){E}_{\mathrm{res}}(\ensuremath{\tau})Eres​(\ensuremathτ) and \ensuremathρk(\ensuremathτ){\ensuremath{\rho}}_{k}(\ensuremath{\tau})\ensuremathρk​(\ensuremathτ) are found to behave logarithmically for large \ensuremathτ\ensuremath{\tau}\ensuremathτ, but with different exponents, [Eres(\ensuremathτ)∕L]av\ensuremath∼1∕ln\ensuremathζ(\ensuremathτ)[{E}_{\mathrm{res}}(\ensuremath{\tau})∕L{]}_{\mathrm{av}}\ensuremath{\sim}1∕{\mathrm{ln}}^{\ensuremath{\zeta}}(\ensuremath{\tau})[Eres​(\ensuremathτ)∕L]av​\ensuremath∼1∕ln\ensuremathζ(\ensuremathτ) with \ensuremathζ\ensuremath≈3.4\ensuremath{\zeta}\ensuremath{\approx}3.4\ensuremathζ\ensuremath≈3.4, and [\ensuremathρk(\ensuremathτ)]av\ensuremath∼1∕ln2(\ensuremathτ)[{\ensuremath{\rho}}_{k}(\ensuremath{\tau}){]}_{\mathrm{av}}\ensuremath{\sim}1∕{\mathrm{ln}}^{2}(\ensuremath{\tau})[\ensuremathρk​(\ensuremathτ)]av​\ensuremath∼1∕ln2(\ensuremathτ). We propose a mechanism for 1∕ln2\ensuremathτ1∕{\mathrm{ln}}^{2}\ensuremath{\tau}1∕ln2\ensuremathτ behavior of [\ensuremathρk]av[{\ensuremath{\rho}}_{k}{]}_{\mathrm{av}}[\ensuremathρk​]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.
Based on the strong-disorder renormalization group method, a microscopic mechanism of defect formation in the quantum annealing of the random transverse-field Ising chain is proposed, which represents the annealing process as a gradual aggregation of strongly coupled spin clusters. The ferromagnetic ground state of clusters is either preserved or get excited in pairwise fusions of clusters, depending on the effective annealing rate of the fusion, the latter events being responsible for the appearance of defects in the final state. A consequence of the theory is that, although the Griffiths-McCoy phases surrounding the critical point are gapless, they are still effectively gapped from the point of view of quantum annealing. Thereby we provide an explanation of the finiteness of gap outside of the critical point, which was implicit in an early approach to the problem by Kibble-Zurek scaling [Dziarmaga, Phys. Rev. B {\bf 74}, 064416 (2006)]. Furthermore, by identifying the accessible excitations, we refine the functional form of the vanishing of the gap at the critical point. The defect density in the final state is found to decrease with the annealing time τττ, as n(τ)∼ln⁡−2(τln⁡2τ)n(τ)\sim \ln^{-2}\left(\fracτ{\ln^2τ}\right)n(τ)∼ln−2(ln2ττ​) for large τττ. In addition to this, our approach gives access also to the density of defects at intermediate times of the annealing process.
Abstract We develop a time-dependent real-space renormalization-group approach which can be applied to Hamiltonians with time-dependent random terms. To illustrate the renormalization-group analysis, we focus on the quantum Ising Hamiltonian with random site- and time-dependent (adiabatic) transverse-field and nearest-neighbour exchange couplings. We demonstrate how the method works in detail, by calculating the off-critical flows and recovering the ground state properties of the Hamiltonian such as magnetization and correlation functions. The adiabatic time allows us to traverse the parameter space, remaining near-to the ground state which is broadened if the rate of change of the Hamiltonian is finite. The quantum critical point, or points, depend on time through the time-dependence of the parameters of the Hamiltonian. We, furthermore, make connections with Kibble–Zurek dynamics and provide a scaling argument for the density of defects as we adiabatically pass through the critical point of the system.
We analyze mechanisms for universal out-of-equilibrium dynamics near criticality by exploring the effect of randomized quantum resetting (QR) under a finite-time quench across a quantum phase transition. Using the transverse-field Ising chain as a generic model and exploiting its exact solution, QR is found to cause a crossover of the scaling of the topological defect density with the time scale τττ of the quench, from Kibble-Zurek to anti-Kibble-Zurek scaling as τττ increases. This reflects a competition between non-adiabatic quench-driven excitations and QR, giving rise to local minima of the defect densities at optimal annealing times. These times and the corresponding local minima are shown to scale as universal power laws with the rate of QR. Additional results for the scaling of the mean excess energy suggest that a system driven across a quantum critical point exhibits the same scaling behavior under a linear quench with QR as with uncorrelated noise.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.
Quantum Ising model in one dimension is an exactly solvable example of a quantum phase transition. We investigate its behavior during a quench caused by a gradual turning off of the transverse bias field. The system is then driven at a fixed rate characterized by the quench time ${\ensuremath{\tau}}_{Q}$ across the critical point from a paramagnetic to ferromagnetic phase. In agreement with Kibble-Zurek mechanism (which recognizes that evolution is approximately adiabatic far away, but becomes approximately impulse sufficiently near the critical point), quantum state of the system after the transition exhibits a characteristic correlation length $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$ proportional to the square root of the quench time ${\ensuremath{\tau}}_{Q}$: $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}=\sqrt{{\ensuremath{\tau}}_{Q}}$. The inverse of this correlation length is known to determine average density of defects (e.g., kinks) after the transition. In this paper, we show that this same $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$ controls the entropy of entanglement, e.g., entropy of a block of LLL spins that are entangled with the rest of the system after the transition from the paramagnetic ground state induced by the quench. For large LLL, this entropy saturates at $\frac{1}{6}\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, as might have been expected from the Kibble-Zurek mechanism. Close to the critical point, the entropy saturates when the block size $L\ensuremath{\approx}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, but---in the subsequent evolution in the ferromagnetic phase---a somewhat larger length scale $l=\sqrt{{\ensuremath{\tau}}_{Q}}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\tau}}_{Q}$ develops as a result of a dephasing process that can be regarded as a quantum analog of phase ordering, and the entropy saturates when $L\ensuremath{\approx}l$. We also study the spin-spin correlation using both analytic methods and real time simulations with the Vidal algorithm. We find that at an instant when quench is crossing the critical point, ferromagnetic correlations decay exponentially with the dynamical correlation length $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, but (as for entropy of entanglement) in the following evolution length scale lll gradually develops. The correlation function becomes oscillatory at distances less than this scale. However, both the wavelength and the correlation length of these oscillations are still determined by $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$. We also derive probability distribution for the number of kinks in a finite spin chain after the transition.
Dynamical correlations of the spin and the energy density are investigated in the critical region of the random transverse-field Ising chain by numerically exact calculations in large finite systems (L ≤ 128). The spin-spin autocorrelation function is found to decay proportional to (log t) −2xm and (log t) −2xs m in the bulk and on the surface, respectively, with xm and x s m the bulk and surface magnetization exponents, respectively. On the other hand the critical energy-energy autocorrelation functions have a power law decay, which are characterized by novel critical exponents ηe ≈ 2.2 in the bulk and η s e ≈ 2.5 at the surface, respectively. The numerical results are compared with the predictions of a scaling theory. 05.50.+q, 64.60.Ak, 68.35.Rh Typeset using REVTEX 1 The asymptotic behavior of the time-dependent correlation functions for interacting many-body systems turned out a very difficult subject of theoretical research. Exact results in this field are scarce, one can mention the one-dimensional spin 1/2 XY-model [1] and the Ising chain in a transverse-field [2]. Both models can be mapped onto a system of non-interacting fermions, where the equal-position correlation functions are calculated by the Pfaffian method utilizing the theory of Töplitz determinants. In this Letter we consider- at the first time- the critical dynamical correlations of an interacting quantum system in the presence of quenched (i.e. time-dependent) disorder. It has recently become clear that quenched disorder has rather different effects on phase transitions in quantum systems [3] than on those thermally driven phase transitions. For example, in the Griffiths phase, which is situated at the disordered side of the critical point, the susceptibility has an essential singularity in classical systems, whereas in a quantum system the corresponding singularity is stronger, it is in a power law form. Here we consider the prototype of random quantum systems the one-dimensional random transverse-field Ising model defined by the Hamiltonian: