Emergent entanglement structures and self-similarity in quantum spin chains

Abstract

We introduce an experimentally accessible network representation for many-body quantum states based on entanglement between all pairs of its constituents. We illustrate the power of this representation by applying it to a paradigmatic spin chain model, the XX model, and showing that it brings to light new phenomena. The analysis of these entanglement networks reveals that the gradual establishment of quasi-long range order is accompanied by a symmetry regarding single-spin concurrence distributions, as well as by instabilities in the network topology. Moreover, we identify the existence of emergent entanglement structures, spatially localized communities enforced by the global symmetry of the system that can be revealed by model-agnostic community detection algorithms. The network representation further unveils the existence of structural classes and a cyclic self-similarity in the state, which we conjecture to be intimately linked to the community structure. Our results demonstrate that the use of tools and concepts from complex network theory enables the discovery, understanding and description of new physical phenomena even in models studied for decades. This article is part of the theme issue ‘Emergent phenomena in complex physical and socio-technical systems: from cells to societies’.

A network approach to atomic spectra

“…More generally, our results highlight the holistic character of network science, offering a promising path towards gaining further insights into the structure of complex physical systems. This complements previous applications in spin and Hubbard models [24,26]. Furthermore, complex physical systems such as atomic spectra, for which the solutions of the microscopic equations serve as a ground truth, can be considered a complementary and novel paradigm for testing algorithmic methods in network science [28], in contrast to, e.g.…”

“…The state space of quantum many-body systems grows exponentially with the number of particles [22,23], making the treatment of atoms with large numbers of electrons highly challenging, and generally necessitating approximate solutions [14][15][16]. In this context, the perspective offered by network science becomes particularly valuable for the predicting features of complex physical systems [24][25][26]. Alternatively, physical systems with known ground truth can serve as a complementary testbeds for evaluating algorithms in network science, in contrast to commonly used social benchmarks with unknown ground truth [27,28].…”

Emergent complex quantum networks in continuous-variables non-Gaussian states

“…In particular, emergent complex networks based on quantum mutual information have determined critical points for quantum phase transitions [44,51,52]. Likewise, complex network theory has been successful in determining self-similarity in entanglement structure of spin-chains [53] as well as new kinds of structured entanglement emerging from quantum cellular automata [54] on qubit/gate/ciruit-based quantum computers. Networks are naturally evoked in the quantum regime in relation to the quantum internet [55], where it is not clear yet if the best arrangement of its components will take a complex shape like the classical internet.…”

Virtual linear map algorithm for classical boost in near-term quantum computing

“…The cases that we consider correspond to B = 0 and B ≈ 1 (right below the phase transition exhibited by the model in the thermodynamic limit [35]). At finite size, the ground state exhibits a series of level crossings as a function of B, as a result of which the entanglement structure of the state changes notoriously [35,36]. We also use two different system sizes, N = 6 and N = 7 spins.…”