Dynamical properties of quantum many-body systems with long-range interactions

Monte Carlo Based Techniques for Quantum Magnets with Long-Range Interactions

“…Interestingly, a third regime for strong long-range interactions σ ≤ 0 (σ ≤ −2) was found where the Goldstone modes become gapped via a generalised Higgs mechanism [321]. A recent large-scale QMC study [38] investigating the dynamic properties of the non-frustrating staggered antiferromagnetic square lattice Heisenberg model confirmed these three scenarios and found evidence that the Higgs regime already occurs in the regime σ ≤ 0.2 when the Hamiltonian is still extensive [38]. Let us mention at this point that previous results from linear spin-wave theory [22,322] already indicated the existence of a sublinear dispersion in the staggered antiferromagnetic long-range Heisenberg chain a decade earlier.…”

“…For a critical coupling ratio λ = J /J ⊥ , the system undergoes a QPT breaking the continuous SU(2) symmetry of the Hamiltonian from a ground state adiabatically connected to the product singlet state towards an antiferromagnetic Néel ground state with gapless magnon excitations (Nambu-Goldstone modes). For long-range interactions, we expect that these Goldstone modes are altered [38,321] and that the criticality of the system changes as a function of the decay exponent σ. We expect short-range criticality until σ ≥ σ * = 2 − η SR , a non-trivial regime of continuously varying criticality in σ uc ≤ σ < σ * with σ uc = 4/3, and a long-range mean-field regime for σ < σ uc [20,21,244,326].…”

“…Meanwhile, a similar historic development took place regarding the study of QPTs under the influence of long-range interactions. By virtue of pioneering RG studies [20,21], the numerical investigation of long-range interacting magnetic systems has been triggered [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. In particular, Monte Carlo-based techniques became a popular tool to gain quantitative insight into these long-range interacting quantum magnets [22,25,26,[29][30][31][32][33][34][35][36][37][38][39][40].…”

“…By virtue of pioneering RG studies [20,21], the numerical investigation of long-range interacting magnetic systems has been triggered [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. In particular, Monte Carlo-based techniques became a popular tool to gain quantitative insight into these long-range interacting quantum magnets [22,25,26,[29][30][31][32][33][34][35][36][37][38][39][40]. On the one hand, this includes high-order series expansion techniques, where classical Monte Carlo integration is applied for the graph embedding scheme, allowing extracting energies and observables in the thermodynamic limit [25,29].…”

“…In this review, we are mainly interested in the description and discussion of two Monte Carlo-based numerical techniques, which were successfully used to study the lowenergy physics of long-range interacting quantum magnets, in particular with respect to the quantitative investigation of QPTs [22,25,[29][30][31][32][34][35][36][37][38]40]. The success of Monte Carlo techniques in this field is due to the occurrence of high-dimensional sums and integrals that commonly arise in the formulation of many-particle statistics.…”