Dynamics of a Quantum Phase Transition: Exact Solution of the Quantum Ising Model

Dziarmaga, Jacek

doi:10.1103/physrevlett.95.245701

Cited by 642 publications

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“…The quantum Ising model quenches a state according to Hamiltonian (1) so, when g is changed in time, superpositions of different locations of a kink (e.g., α| · · · ↑↑↓↓↓↓ · · · +β| · · · ↑↑↑↓↓↓ · · · +γ| · · · ↑↑↑↑↓↓ · · · ) are allowed, and, indeed, inevitable 10,11 . Spreading of a localized kink will come about as a consequence of the kinetic term, gσ x n , in equation (1).…”

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confidence: 99%

Non-local quantum superpositions of topological defects

2011

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Topological defects (such as monopoles, vortex lines, or domain walls) mark locations where disparate choices of a broken symmetry vacuum elsewhere in the system lead to irreconcilable differences 1,2 . They are energetically costly (the energy density in their core reaches that of the prior symmetric vacuum) but topologically stable (the whole manifold would have to be rearranged to get rid of the defect). We show how, in a paradigmatic model of a quantum phase transition, a topological defect can be put in a nonlocal superposition, so that -in a region large compared to the size of its core -the order parameter of the system is "undecided" by being in a quantum superposition of conflicting choices of the broken symmetry. We dub such a topological Schrödinger cat state a 'Schrödinger kink', and devise a version of a double-slit experiment suitable for topological defects to describe one possible manifestation of the phenomenon. Coherence detectable in such experiments will be suppressed as a consequence of interaction with the environment. We analyze the environment-induced decoherence and discuss its role in symmetry breaking.Topological defects are the epitome of locality. An example of a defect occurs in the quantum Ising model where a lattice of spins interact ferromagnetically with their nearest neighbors, i.e., the Hamiltonian contains an interaction of the form −σ z n σ z n+1 for the n th spin on the lattice. The entire collection of spins achieves its lowest energy when they are all aligned. However, there are two choices for this lowest energy state, | · · · ↑↑↑↑ · · · or | · · · ↓↓↓↓ · · · . Both are energetically identical but each of them breaks the symmetry of the Hamiltonian, which has no preference between "up" and "down".A configuration that includes topological defects could arise, for example, in a driven phase transition, i.e., "a quench". In the Ising model, a quench can be induced when the transverse field strength g is changed in the HamiltonianWhen the external field is strong enough (g 1), it "wins" and all spins align with σ x . A decrease in g, however, will lead to a phase transition at g = 1 with the symmetry breaking term −σ z n σ z n+1 trying to align all the spins in their z-direction. The choice of whether the spins will point up or down is made locally. As a result of causality and the finite speed at which signals propagate, the size of these domains will be determined by rate of change of g 3-8 . This will lead to configurations, such as | · · · ↑↑↓↓ · · · , where the topological defect marks the location of a "switch" from one broken symmetry ground state to the other. In this one-dimensional example, the defect is a kink (a domain wall), but, in a sense, all the topological defects (monopoles, vortex lines, and so on) that exist in higher dimensions "look the same". A discussion of the consequences of this process for the post-transition state is beyond the scope of this work (but see, e.g., Refs. 9-11). We only note that the density of the resulting kinks will be finite.The...

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Non-local quantum superpositions of topological defects

2011

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“…The meeting points of these two transition lines at h = ±(J x + J y ) and J x = J y are multicritical points. Let us initiate our discussions with two types of quenching schemes: (i) quenching the magnetic field h as t/τ which we call transverse quenching [11,13], and (ii) the quenching of the interaction J x as t/τ which is referred to as anisotropic quenching [16]. In the process of anisotropic quenching, the system can be made to cross the multicritical points A and B shown in the phase diagram.…”

Section: A Spin Model: Transverse and Anisotropic Quenchingmentioning

confidence: 99%

“…The Kibble-Zurek scaling has been verified in various exactly solvable spin models and systems of interacting bosons [4,13,16,17,11,5]; it has been generalized to quenching through a multicritical point [31], across a gapless phase [20,23], and along a gapless line [26,28], and to systems with quenched disorder [14], white noise [18], infinite-range interactions [27], and edge states [30]. Studies have also been made to estimate the defect density for quenching with a non-linear form [21], an oscillatory variation of an applied magnetic field [32] or under a reversal of the magnetic field [33].…”

Section: Introductionmentioning

confidence: 99%

“…The non-equilibrium dynamics of a quantum system when quenched very fast [3] or slowly across a quantum critical point [4,5] has attracted the attention of several groups recently. The possibility of experimental realizations of quantum dynamics in spin-1 Bose condensates [6] and atoms trapped in optical lattices [7,8] has led to an upsurge in studies of related theoretical models [3,4,5,9,10,11,12,13,14,15,16,17,18,19,20,21,23,24,25,26,27,28,29,30,31,32,33].…”

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Defect Production Due to Quenching Through a Multicritical Point and Along a Gapless Line

Divakaran

Mukherjee

Dutta

et al. 2010

Quantum Quenching, Annealing and Computation

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The exciting physics of quantum phase transitions has been explored extensively in the last few years [1,2]. The non-equilibrium dynamics of a quantum system when quenched very fast [3] or slowly across a quantum critical point [4,5] has attracted the attention of several groups recently. The possibility of experimental realizations of quantum dynamics in spin-1 Bose condensates [6] and atoms trapped in optical lattices [7,8] has led to an upsurge in studies of related theoretical models [3,4,5,9,10,11,12,13,14,15,16,17,18,19,20,21,23,24,25,26,27,28,29,30,31,32,33].In this review, we concentrate on the dynamics of quantum spin chains swept across a quantum critical or multicritical point or along a gapless line by a slow (adiabatic) variation of a parameter appearing in the Hamiltonian of the system. Our aim is to find the scaling form of the density of defects (which, in our case, is the density of wrongly oriented spins) in the final state which is reached after the system is prepared in an initial ground state and then slowly quenched through a quantum critical point. The dynamics in the vicinity of a quantum critical point is necessarily non-adiabatic due to the divergence of the relaxation time of the underlying quantum system which forces the system to be infinitely sluggish; thus the system fails to respond to a change in a parameter of the Hamiltonian no matter how slow that rate of change may be! We first recall the Kibble-Zurek argument [34,35] which predicts a scaling form for the defect density following a slow quench through a quantum critical point. We assume that a parameter g of the Hamiltonian is varied in a linear fashion such that g − g c ∼ t/τ, where g = g c denotes the value of g at the quantum critical point and τ is the quenching time. Our interest is in the adiabatic limit, τ → ∞. The energy gap of the quantum Hamiltonian vanishes at the critical point as (g − g c ) νz whereas the relaxation time ξ τ , which is inverse of the gap, diverges at the critical point. It is clear that non-adiabaticity becomes important at a timet when the characteristic time scale of the quantum system (i.e., the relaxation time) is of the order of the inverse of the rate of change of the Hamiltonian; this yields

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“…We apply the standard Jordan-Wigner transformation, and the following procedure outlined in [22][23][24][25],…”

Section: Description Of the Modelmentioning

confidence: 99%

Non-Hermitian quantum annealing in the ferromagnetic Ising model

Nesterov¹,

Zepeda²,

Berman³

2013

Phys. Rev. A

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We developed a non-Hermitian quantum optimization algorithm to find the ground state of the ferromagnetic Ising model with up to 1024 spins (qubits). Our approach leads to significant reduction of the annealing time. Analytical and numerical results demonstrate that the total annealing time is proportional to ln N , where N is the number of spins. This encouraging result is important in using classical computers in combination with quantum algorithms for the fast solutions of NPcomplete problems. Additional research is proposed for extending our dissipative algorithm to more complicated problems.

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Dynamics of a Quantum Phase Transition: Exact Solution of the Quantum Ising Model

Cited by 642 publications

References 34 publications

Non-local quantum superpositions of topological defects

Non-local quantum superpositions of topological defects

Defect Production Due to Quenching Through a Multicritical Point and Along a Gapless Line

Non-Hermitian quantum annealing in the ferromagnetic Ising model

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