In this paper, we analytically compute the quench dynamics of simple product states that evolve under a spin-1/2 XY-model with power-law interactions in a staggered magnetic field. We use linear spin-wave theory as well as a low-energy theory that accounts for strong phase fluctuations. This allows us to provide an analytic expression for the dynamics of phase-type correlations with short-range interactions, where the linear spin-wave theory is known to fail. For density-type correlations the low-energy theory and the linear spin-wave theory give the same result. Our analysis is valid for arbitrary magnetic field strength and in particular provides an analytical understanding of the quench dynamics near criticality. Our results can thus provide useful tools for the experimental observation of excitation gaps and critical behavior.
In this paper, we analytically compute the quench dynamics of simple product states that evolve under a spin-1/2 XY-model with power-law interactions in a staggered magnetic field. We use linear spin-wave theory as well as a low-energy theory that accounts for strong phase fluctuations. This allows us to provide an analytic expression for the dynamics of phase-type correlations with short-range interactions, where the linear spin-wave theory is known to fail. For density-type correlations the low-energy theory and the linear spin-wave theory give the same result. Our analysis is valid for arbitrary magnetic field strength and in particular provides an analytical understanding of the quench dynamics near criticality. Our results can thus provide useful tools for the experimental observation of excitation gaps and critical behavior.
The nature and spectrum of elementary excitations are defining features of a many-body system. In this study, we used a Rydberg quantum simulator to demonstrate a form of spectroscopy, called quench spectroscopy, that probes these low-energy excitations. We illustrated the method on a two-dimensional simulation of the spin-1/2 dipolar XY model. Through microscopic measurements of the spatial spin correlation dynamics following a quench, we extracted the dispersion relation of the elementary excitations for both ferro- and antiferromagnetic couplings. The ferromagnet exhibits elementary excitations behaving as linear spin waves, whereas in the antiferromagnet, spin waves appear to decay, suggesting the presence of strong nonlinearities. Our demonstration highlights the importance of power-law interactions on the excitation spectrum of a many-body system.
Slow quenches of the magnetic field across the paramagnetic-ferromagnetic phase transition of spin systems produce heat. In systems with short-range interactions the heat exhibits universal power-law scaling as a function of the quench rate, known as Kibble-Zurek scaling. In this work we analyze slow quenches of the magnetic field in the Lipkin-Meshkov-Glick (LMG) model, which describes fully connected quantum spins. We analytically determine the quantum contribution to the residual heat as a function of the quench rate δ by means of a Holstein-Primakoff expansion about the mean-field value. Unlike in the case of short-range interactions, scaling laws in the LMG model are only found for a ramp starting or ending at the critical point. If instead the ramp is symmetric, as in the typical Kibble-Zurek scenario, then the number of excitations exhibits a crossover behavior as a function of δ and tends to a constant in the thermodynamic limit. Previous, and seemingly contradictory, theoretical studies are identified as specific limits of this dynamics. Our results can be tested on several experimental platforms, including quantum gases and trapped ions.
We study the quenching dynamics of a many-body system in one dimension described by a Hamiltonian that has spatial periodicity. Specifically, we consider a spin-21 chain with equal xx and yy couplings and subject to a periodically varying magnetic field in the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ direction or, equivalently, a tight-binding model of spinless fermions with a periodic local chemical potential, having period 2q, where q is a positive integer. For a linear quench of the strength of the magnetic field (or chemical potential) at a rate $1/\ensuremath{\tau}$ across a quantum critical point, we find that the density of defects thereby produced scales as $1/{\ensuremath{\tau}}^{q/(q+1)}$, deviating from the $1/\sqrt{\ensuremath{\tau}}$ scaling that is ubiquitous in a range of systems. We analyze this behavior by mapping the low-energy physics of the system to a set of fermionic two-level systems labeled by the lattice momentum k undergoing a nonlinear quench as well as by performing numerical simulations. We also show that if the magnetic field is a superposition of different periods, the power law depends only on the smallest period for very large values of $\ensuremath{\tau}$, although it may exhibit a crossover at intermediate values of $\ensuremath{\tau}$. Finally, for the case where a zz coupling is also present in the spin chain, or equivalently, where interactions are present in the fermionic system, we argue that the power associated with the scaling law depends on a combination of q and the interaction strength.
We present a systematic study of the anisotropic spin-1/2 Heisenberg model in staggered magnetic fields in two dimensions. To mimic real materials, we consider a system of coupled, antiferromagnetic chains, whose interchain interaction can be either ferro- or antiferromagnetic. When the staggered field is commensurate with the magnetic interactions, an energy gap opens immediately and follows a power law as a function of the applied field, similar to the situation in one dimension. When the field competes with the interactions, a quantum phase transition (QPT) occurs from a gapless, magnetically ordered phase at low fields to a gapped, disordered regime. We use a continuous-time Monte Carlo method to compute the staggered moment of the ordered phases and the spin gap of the disordered phases. We deduce the phase diagrams as functions of the anisotropy ratio and the applied field, and calculate the scaling behavior of the models in both quantities. We show that in the competitive case, the staggered field acts to maintain a regime of quasi-1D behavior around the QPT, and we discuss as a consequence the nature of the crossover from one-dimensional (1D) to 2D physics.
Long-range interacting spin systems are ubiquitous in physics and exhibit a variety of ground-state disorder-to-order phase transitions. We consider a prototype of infinite-range interacting models known as the Lipkin-Meshkov-Glick model describing the collective interaction of N spins and investigate the dynamical properties of fluctuations and correlations after a sudden quench of the Hamiltonian. Specifically, we focus on critical quenches, where the initial state and/or the postquench Hamiltonian are critical. Depending on the type of quench, we identify three distinct behaviors where both the short-time dynamics and the stationary state at long times are effectively thermal, quantum, and genuinely nonequilibrium, characterized by distinct universality classes and static and dynamical critical exponents. These behaviors can be identified by an infrared effective temperature that is finite, zero, and infinite (the latter scaling with the system size as N^{1/3}), respectively. The quench dynamics is studied through a combination of exact numerics and analytical calculations utilizing the nonequilibrium Keldysh field theory. Our results are amenable to realization in experiments with trapped-ion experiments where long-range interactions naturally arise.
We investigate the ground-state properties of the spin-1/2 XXZ model with power-law-decaying ($1/{r}^{\ensuremath{\alpha}}$) interactions, which describe spins interacting with long-range transverse (XX) ferromagnetic interactions and longitudinal (Z) antiferromagnetic interactions, or hard-core bosons with long-range repulsion and hopping. The long-range nature of the couplings allows us to quantitatively study the spectral, correlation, and entanglement properties of the system by making use of linear spin-wave theory, supplemented with density-matrix renormalization group in one-dimensional systems. Our most important prediction is the existence of three distinct coupling regimes, depending on the decay exponent $\ensuremath{\alpha}$ and number of dimensions d: (1) a short-range regime for $\ensuremath{\alpha}>d+{\ensuremath{\sigma}}_{c}$ (where ${\ensuremath{\sigma}}_{c}=1$ in the gapped N\'eel antiferromagnetic phase exhibited by the XXZ model, and ${\ensuremath{\sigma}}_{c}=2$ in the gapless XY ferromagnetic phase), sharing the same properties as those of finite-range interactions ($\ensuremath{\alpha}=\ensuremath{\infty}$); (2) a long-range regime $\ensuremath{\alpha}<d$, sharing the same properties as those of the infinite-range interactions ($\ensuremath{\alpha}=0$) in the thermodynamic limit; and (3) a most intriguing medium-range regime for $d<\ensuremath{\alpha}<d+{\ensuremath{\sigma}}_{c}$, continuously interpolating between the finite-range and the infinite-range behavior. The latter regime is characterized by elementary excitations with a long-wavelength dispersion relation $\ensuremath{\omega}\ensuremath{\approx}{\mathrm{\ensuremath{\Delta}}}_{g}+c{k}^{z}$ in the gapped phase, and $\ensuremath{\omega}\ensuremath{\sim}{k}^{z}$ in the gapless phase, exhibiting a continuously varying dynamical exponent $z=(\ensuremath{\alpha}\ensuremath{-}d)/{\ensuremath{\sigma}}_{c}$. In the gapless phase of the model the z exponent is found to control the scaling of fluctuations, the decay of correlations, and a universal subdominant term in the entanglement entropy, leading to a very rich palette of behaviors for ground-state quantum correlations beyond what is known for finite-range interactions.