Causality and quantum criticality in long-range lattice models

Maghrebi, Mohammad F.; Gong, Zhe-Xuan; Foss‐Feig, Michael; Gorshkov, Alexey V.

doi:10.1103/physrevb.93.125128

Cited by 71 publications

(95 citation statements)

References 78 publications

(141 reference statements)

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“…A similar possibility has been considered in a very recent work [66], dealing with causality in long-range critical systems. There two dispersion contributions ∝ p 2 and ∝ p β (typically used in RG treatments of long-range systems [48,65,84,85]) are present for every α in an effective action with a single mass term, allowing for a discussion of the correlation functions [66]. To make a comparison with the present paper, we notice that in our approach, at variance, the two contributions S D and S AN (the latter one containing the term ∝ p β ) derive respectively from the modes at the minimum of the energy spectrum and at the edges of the Brillouin zone, and that both the action terms have their own mass, scaling differently if α < 2.…”

Section: Discussionmentioning

confidence: 85%

“…To make a comparison with the present paper, we notice that in our approach, at variance, the two contributions S D and S AN (the latter one containing the term ∝ p β ) derive respectively from the modes at the minimum of the energy spectrum and at the edges of the Brillouin zone, and that both the action terms have their own mass, scaling differently if α < 2. Assuming our point of view for the implementation of the RG implies that the choice in [66] leads to mix the dispersions of the two sets of quasiparticles. This may be significant for α < 1 (a case not treated by the authors in [66]), where the energy of the ground state is still extensive in the thermodynamic limit.…”

Section: Discussionmentioning

confidence: 99%

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Effective theory and breakdown of conformal symmetry in a long-range quantum chain

et al. 2016

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We deal with the problem of studying the symmetries and the effective theories of long-range models around their critical points. A prominent issue is to determine whether they possess (or not) conformal symmetry (CS) at criticality and how the presence of CS depends on the range of the interactions. To have a model, both simple to treat and interesting, where to investigate these questions, we focus on the Kitaev chain with long-range pairings decaying with distance as power-law with exponent α. This is a quadratic solvable model, yet displaying non-trivial quantum phase transitions. Two critical lines are found, occurring respectively at a positive and a negative chemical potential. Focusing first on the critical line at positive chemical potential, by means of a renormalization group approach we derive its effective theory close to criticality. Our main result is that the effective action is the sum of two terms: a Dirac action SD, found in the short-range Ising universality class, and an "anomalous" CS breaking term SAN. While SD originates from low-energy excitations in the spectrum, SAN originates from the higher energy modes where singularities develop, due to the long-range nature of the model. At criticality SAN flows to zero for α > 2, while for α < 2 it dominates and determines the breakdown of the CS. Out of criticality SAN breaks, in the considered approximation, the effective Lorentz invariance (ELI) for every finite α. As α increases such ELI breakdown becomes less and less pronounced and in the short-range limit α → ∞ the ELI is restored. In order to test the validity of the determined effective theory, we compared the two-fermion static correlation functions and the von Neumann entropy obtained from them with the ones calculated on the lattice, finding agreement. These results explain two observed features characteristic of long-range models, the hybrid decay of static correlation functions within gapped phases and the area-law violation for the von Neumann entropy. The proposed scenario is expected to hold in other longrange models displaying quasiparticle excitations in ballistic regime. From the effective theory one can also see that new phases emerge for α < 1. Finally we show that at every finite α the critical exponents, defined as for the short-range (α → ∞) model, are not altered. This also shows that the long-range paired Kitaev chain provides an example of a long-range model in which the value of α where the CS is broken does not coincide with the value at which the critical exponents start to differ from the ones of the corresponding shortrange model. At variance, for the second critical line, having negative chemical potential, only SAN (SD) is present for 1 < α < 2 (for α > 2). Close to this line, where the minimum of the spectrum coincides with the momentum where singularities develop, the critical exponents change where CS is broken.

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Section: Discussionmentioning

confidence: 85%

Section: Discussionmentioning

confidence: 99%

Effective theory and breakdown of conformal symmetry in a long-range quantum chain

et al. 2016

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“…While a series of general results have been derived, allowing gradually tighter bounds [20][21][22][23][24][25][26][27], it is particularly informative to identify exactly solvable models that reproduce and explain the qualitative behaviour of the physical systems being studied. Up to now there are relatively few models with long-range interactions (limited mainly to longitudinal Ising models [28] and tunnelling bosons [29,30]) for which the transitions in behaviour are known.…”

Section: Introductionmentioning

confidence: 99%

Entanglement growth and correlation spreading with variable-range interactions in spin and fermionic tunneling models

et al. 2016

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We investigate the dynamics following a global parameter quench for two 1D models with variablerange power-law interactions: a long-range transverse Ising model, which has recently been realised in chains of trapped ions, and a long-range lattice model for spinless fermions with long-range tunnelling. For the transverse Ising model, the spreading of correlations and growth of entanglement are computed using numerical matrix product state techniques, and are compared with exact solutions for the fermionic tunnelling model. We identify transitions between regimes with and without an apparent linear light cone for correlations, which correspond closely between the two models. For long-range interactions (in terms of separation distance r, decaying slower than 1/r), we find that despite the lack of a light-cone, correlations grow slowly as a power law at short times, and that -depending on the structure of the initial state -the growth of entanglement can also be sublinear. These results are understood through analytical calculations, and should be measurable in experiments with trapped ions.

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“…For example, it is a well-known challenge to generalize Hastings' proof of the area law [35] to higher dimensions [38], and longrange interacting systems are in some sense similar to higherdimensional short-range interacting systems [23,24]. In addition, since ground states of gapped long-range interacting systems can have power-law decaying correlations [39][40][41], one would need to relax the condition of exponentially decaying correlations in the proof of Refs. [36,37] to algebraically decaying correlations.…”

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confidence: 99%

Entanglement Area Laws for Long-Range Interacting Systems

et al. 2017

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We prove that the entanglement entropy of any state evolved under an arbitrary 1=r α long-rangeinteracting D-dimensional lattice spin Hamiltonian cannot change faster than a rate proportional to the boundary area for any α > D þ 1. We also prove that for any α > 2D þ 2, the ground state of such a Hamiltonian satisfies the entanglement area law if it can be transformed along a gapped adiabatic path into a ground state known to satisfy the area law. These results significantly generalize their existing counterparts for short-range interacting systems, and are useful for identifying dynamical phase transitions and quantum phase transitions in the presence of long-range interactions. DOI: 10.1103/PhysRevLett.119.050501 Quantum many-body systems often have approximately local interactions, and this locality has profound effects on the entanglement properties of both ground states and the states created dynamically after a quantum quench. For example, the entanglement entropy, defined as the entropy of the reduced state of a subregion, often scales as the boundary area of the subregion for ground states of shortrange interacting Hamiltonians [1]. This "area law" of entanglement entropy is in sharp contrast to the behavior of thermodynamic entropy, which typically scales as the volume of the system. 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. As might be expected, α controls the extent to which the system respects notions of locality developed for short-range interacting systems: For α sufficiently small, it is well established [19] that locality may be completely lost, and for α sufficiently large there is ample numerical and analytical evidence [31][32][33][34] that area laws may persist. However, there is currently no general and rigorous understanding of when area laws do or do not survive the presence of long-range interactions.The modern understanding of area laws draws heavily from several rigor...

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Causality and quantum criticality in long-range lattice models

Cited by 71 publications

References 78 publications

Effective theory and breakdown of conformal symmetry in a long-range quantum chain

Effective theory and breakdown of conformal symmetry in a long-range quantum chain

Entanglement growth and correlation spreading with variable-range interactions in spin and fermionic tunneling models

Entanglement Area Laws for Long-Range Interacting Systems

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