Continuously varying critical exponents in long-range quantum spin ladders

Abstract

We investigate the quantum-critical behavior between the rung-singlet phase with hidden string order and the Néel phase with broken SU(2)-symmetry in quantum spin ladders with algebraically decaying unfrustrated long-range Heisenberg interactions. To this end, we determine high-order series expansions of energies and observables in the thermodynamic limit about the isolated rung-dimer limit. This is achieved by extending the method of perturbative continuous unitary transformations (pCUT) to long-range Heisenberg interactions and to the calculation of generic observables. The quantum-critical breakdown of the rung-singlet phase then allows us to determine the critical phase transition line and the entire set of critical exponents as a function of the decay exponent of the long-range interaction. We demonstrate long-range mean-field behavior as well as a non-trivial regime of continuously varying critical exponents implying the absence of deconfined criticality contrary to a recent suggestion in the literature.

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

“…In this section, we provide a self-contained and comprehensive overview of linked-cluster expansions for long-range interacting systems [25,[29][30][31]34,35] using white graphs [152] in combination with Monte Carlo integration for the graph embedding. First, we introduce linked-cluster expansions (LCEs) and discuss perturbative continuous unitary transformations (pCUTs) [153,154] as a suitable high-order series expansion method.…”

“…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.…”

“…In contrast to many deterministic integration techniques, for which the standard error scales exponentially with the dimension of the underlying integral, the standard error of an integral calculated with Monte Carlo integration does not scale with the dimension of the underlying integral. We further chose to review this topic due to our personal involvement with the application and development of these methods [25,[29][30][31][32]34,35]. On the one hand, we explain in detail how classical Monte Carlo integration can enhance the capabilities of linked-cluster expansions (LCEs) with the pCUT+MC approach (a combination of the perturbative unitary transform approach (pCUT) and MC embedding).…”