The Quantum Ising model is an exactly solvable model of quantum phase transition. This Letter gives an exact solution when the system is driven through the critical point at a finite rate. The evolution goes through a series of Landau-Zener level anticrossings when pairs of quasiparticles with opposite pseudomomenta get excited with a probability depending on the transition rate. The average density of defects excited in this way scales like a square root of the transition rate. This scaling is the same as the scaling obtained when the standard Kibble-Zurek mechanism of thermodynamic second order phase transitions is applied to the quantum phase transition in the Ising model.

We review recent theoretical work on two closely related issues: excitation of an isolated quantum condensed matter system driven adiabatically across a continuous quantum phase transition or a gapless phase, and apparent relaxation of an excited system after a sudden quench of a parameter in its Hamiltonian. Accordingly the review is divided into two parts. The first part revolves around a quantum version of the Kibble-Zurek mechanism including also phenomena that go beyond this simple paradigm. What they have in common is that excitation of a gapless many-body system scales with a power of the driving rate. The second part attempts a systematic presentation of recent results and conjectures on apparent relaxation of a pure state of an isolated quantum many-body system after its excitation by a sudden quench. This research is motivated in part by recent experimental developments in the physics of ultracold atoms with potential applications in the adiabatic quantum state preparation and quantum computation.

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 τQ 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 lengthξ proportional to the square root of the quench time τQ: ξ = √ τQ. 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 sameξ controls the entropy of entanglement, e.g. entropy of a block of L spins that are entangled with the rest of the system after the transition from the paramagnetic ground state induced by the quench. For large L, this entropy saturates at 1 6 log 2ξ , as might have been expected from the Kibble-Zurek mechanism. Close to the critical point, the entropy saturates when the block size L ≈ξ, but -in the subsequent evolution in the ferromagnetic phase -a somewhat larger length scale l = √ τQ ln τQ develops as a result of a dephasing process that can be regarded as a quantum analogue of phase ordering, and the entropy saturates when L ≈ 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 lengtĥ ξ, but (as for entropy of entanglement) in the following evolution length scale l 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 byξ. We also derive probability distribution for the number of kinks in a finite spin chain after the transition.

We introduce a one-dimensional model which interpolates between the Ising model and the quantum compass model with frustrated pseudospin interactions σ z i σ z i+1 and σ x i σ x i+1 , alternating between even/odd bonds, and present its exact solution by mapping to quantum Ising models. We show that the nearest neighbor pseudospin correlations change discontinuosly and indicate divergent correlation length at the first order quantum phase transition. At this transition one finds the disordered ground state of the compass model with high degeneracy 2 × 2 N/2 in the limit of N → ∞.

An infinite projected entangled pair state (iPEPS) is a tensor network ansatz to represent a quantum state on an infinite 2D lattice whose accuracy is controlled by the bond dimension D. Its real, Lindbladian or imaginary time evolution can be split into small time steps. Every time step generates a new iPEPS with an enlarged bond dimension D > D, which is approximated by an iPEPS with the original D. In Phys. Rev. B 98, 045110 (2018) an algorithm was introduced to optimize the approximate iPEPS by maximizing directly its fidelity to the one with the enlarged bond dimension D . In this work we implement a more efficient optimization employing a local estimator of the fidelity. For imaginary time evolution of a thermal state's purification, we also consider using unitary disentangling gates acting on ancillas to reduce the required D. We test the algorithm simulating Lindbladian evolution and unitary evolution after a sudden quench of transverse field hx in the 2D quantum Ising model. Furthermore, we simulate thermal states of this model and estimate the critical temperature with good accuracy: 0.1% for hx = 2.5 and 0.5% for the more challenging case of hx = 2.9 close to the quantum critical point at hx = 3.04438(2). arXiv:1811.05497v2 [cond-mat.str-el]

The dark soliton solution of the Gross-Pitaevskii equation in one dimension has two parameters that do not change the energy of the solution: the global phase of the condensate wave function and the position of the soliton. These degeneracies appear in the Bogoliubov theory as Bogoliubov modes with zero frequencies and zero norms. These "zero modes" cannot be quantized as the usual Bogoliubov quasiparticle harmonic oscillators. They must be treated in a nonperturbative way. In this paper I develop non-perturbative theory of zero modes. This theory provides non-perturbative description of quantum phase diffusion and quantum diffusion of soliton position. An initially well localized wave packet for soliton position is predicted to disperse beyond the width of the soliton.

When a system is driven across a quantum critical point at a constant rate its evolution must become non-adiabatic as the relaxation time τ diverges at the critical point. According to the Kibble-Zurek mechanism (KZM), the emerging post-transition excited state is characterized by a finite correlation lengthξ set at the timet =τ when the critical slowing down makes it impossible for the system to relax to the equilibrium defined by changing parameters. This observation naturally suggests a dynamical scaling similar to renormalization familiar from the equilibrium critical phenomena. We provide evidence for such KZM-inspired spatiotemporal scaling by investigating an exact solution of the transverse field quantum Ising chain in the thermodynamic limit.

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