Nearly Linear Light Cones in Long-Range Interacting Quantum Systems

Abstract

In non-relativistic quantum theories with short-range Hamiltonians, a velocity v can be chosen such that the influence of any local perturbation is approximately confined to within a distance r until a time t ∼ r/v, thereby defining a linear light cone and giving rise to an emergent notion of locality. In systems with power-law (1/r α ) interactions, when α exceeds the dimension D, an analogous bound confines influences to within a distance r only until a time t ∼ (α/v) log r, suggesting that the velocity, as calculated from the slope of the light cone, may grow exponentially in time. We rule out this possibility; light cones of power-law interacting systems are algebraic for α > 2D, becoming linear as α → ∞. Our results impose strong new constraints on the growth of correlations and the production of entangled states in a variety of rapidly emerging, long-range interacting atomic, molecular, and optical systems. Though non-relativistic quantum theories are not explicitly causal, Lieb and Robinson [1] proved that an effective speed limit emerges dynamically in systems with short-ranged interactions, thereby extending the notion of causality into the fields of condensed matter physics, quantum chemistry, and quantum information science. Specifically, they proved that when interactions have a finite range or decay exponentially in space, the influence of a local perturbation decays exponentially outside of a space-time region bounded by the line t = r/v, which therefore plays the role of a light cone [ Fig. 1(a)]. However, many of the systems to which non-relativistic quantum theory is routinely applied-ranging from frustrated magnets and spin glasses [2,3] The results of Lieb and Robinson were first generalized to power-law (1/r α ) interacting systems by Hastings and Koma [17], with the following picture emerging. For α > D, the influence of a local perturbation is bounded by a function ∝ e vt /r α , and while a light cone can still be defined as the boundary outside of which this function falls below some threshold value, yielding t ∼ log r, that boundary is logarithmic rather than linear [ Fig. 1(b)]. Improvements upon these results exist, revealing, e.g., that the light-cone remains linear at intermediate distance scales [12], but all existing bounds consistently predict an asymptotically logarithmic light cone. An immediate and striking consequence is that the maximum group velocity, defined by the slope of the light cone, grows exponentially with time, thus suggesting that the aforementioned processes -thermalization, entanglement growth after a quench, etc. -may in principle be sped up exponentially by the presence of long-range interactions. In this manuscript, we show that this scenario is not possible. While light cones can potentially be sub-linear for any finite α, thus allowing a velocity that grows with time, for α > 2D they remain bounded by a polynomial t ∼ r ζ , and ζ ≤ 1 approaches unity for increasing α [ Fig. 3(c)].Model and formalism.-We assume a generic spin model with time-independent Ham...

Entanglement Area Laws for Long-Range Interacting Systems

“…[43]. However, recent improvements to the long-range LiebRobinson bound [46] significantly improve the situation. The improved bound enables the following lemma to be derived (see [58]), which together with additional techniques described below leads to a proof of theorem 2.…”

“…We then use the powerful formalism of quasiadiabatic continuation [44] to relate such a state to the ground state of a spectrally gapped long-range interacting Hamiltonian. This strategy is made possible by the recent proof of Kitaev's small incremental entangling (SIE) conjecture [43,45], and by significant recent improvements in Lieb-Robinson bounds [4] for long-range interacting systems [46,47].…”

“…While the study of area laws originates from black hole physics [2,3], area laws have received considerable attention recently in the fields of quantum information and condensed matter physics. In particular, area laws are known to be closely related to the velocity of information propagation in quantum lattices [4], quantum critical phenomena [5], bulk-boundary correspondence [6], efficient classical simulation of quantum systems [7], topological order [8], and many-body localization [9].However, the description of many-body systems in terms of local interactions is often only an approximation, and not always a good one; in numerous systems of current interest, ranging from frustrated magnets and spin glasses [10,11] to atomic, molecular, and optical systems [12][13][14][15][16][17], longrangeinteractions are ubiquitous and lead to qualitatively new physics, e.g., giving rise to novel quantum phases and dynamical behaviors [18][19][20][21][22][23][24][25], and enabling speedups in quantum information processing [26][27][28][29][30]. Particles in these systems generally experience interactions that decay algebraically (∼1=r α ) in the distance (r) between them.…”

“…However, the description of many-body systems in terms of local interactions is often only an approximation, and not always a good one; in numerous systems of current interest, ranging from frustrated magnets and spin glasses [10,11] to atomic, molecular, and optical systems [12][13][14][15][16][17], longrangeinteractions are ubiquitous and lead to qualitatively new physics, e.g., giving rise to novel quantum phases and dynamical behaviors [18][19][20][21][22][23][24][25], and enabling speedups in quantum information processing [26][27][28][29][30]. Particles in these systems generally experience interactions that decay algebraically (∼1=r α ) in the distance (r) between them.…”

Kaleidoscope of quantum phases in a long-range interacting spin-1 chain

“…While long-range interacting classical models have been studied in considerable detail for some time [24][25][26][27][28] , there is a relative lack of in-depth studies of quantum phase transitions in long-range interacting systems, despite the emerging experimental prospects for studying both their equilibrium and nonequilibrium properties [15][16][17][18][29][30][31][32][33][34][35] . One reason is that many analytically solvable lattice models become intractable when interactions are no longer short-ranged, a well-known example being the spin-1/2 XXZ model.…”