Complex-network description of thermal quantum states in the Ising spin chain

Abstract

We use network analysis to describe and characterize an archetypal quantum system -an Ising spin chain in a transverse magnetic field. We analyze weighted networks for this quantum system, with link weights given by various measures of spin-spin correlations such as the von Neumann and Rényi mutual information, concurrence, and negativity. We analytically calculate the spinspin correlations in the system at an arbitrary temperature by mapping the Ising spin chain to fermions, as well as numerically calculate the correlations in the ground state using matrix product state methods, and then analyze the resulting networks using a variety of network measures. We demonstrate that the network measures show some traits of complex networks already in this spin chain, arguably the simplest quantum many-body system. The network measures give insight into the phase diagram not easily captured by more typical quantities, such as the order parameter or correlation length. For example, the network structure varies with transverse field and temperature, and the structure in the quantum critical fan is different from the ordered and disordered phases. I. INTRODUCTIONNetwork analysis is a powerful technique to characterize the structure of connections between agents in a network [1,2]. Studies have shown that classical systems as diverse as the brain and the Internet have a complex network structure [3][4][5][6][7][8]. Quantum systems also show a wide variety of complexity emerging due to inter-particle interactions. Like classical systems, quantum systems have an interconnected web of correlations, and network analysis provides a powerful set of tools to study them. However, while complex networks are ubiquitous in classical systems with a sufficiently rich set of interacting components, it is an open question what the minimal interacting quantum many-body system is in which complex network structures can appear.In this paper, we address this question by studying the network of correlations that arises in the simplest of interacting quantum models, the one-dimensional transverse field Ising model (TIM). We introduce and calculate networks whose links are weighted by various measures of correlations and entanglement, and quantify their complexity. The emergence of network complexity illuminates the richness of the quantum system.Earlier works have studied complex networks in the context of quantum systems, but by enforcing complex network structure in the Hamiltonian, e.g, in interactions [9][10][11][12][13][14]. However, there is no need for this explicit enforcement, as one finds network structure already in quantum states even for simple models such as the nearest-neighbor TIM.

Holographic local quench and effective complexity

“…tem can be represented as a network of nodes (lattice sites) connected by links weighted with the corresponding intersite mutual information. The degree of randomness is then associated with the clustering of the network, which is minimal near the quantum critical point [54], [55]. N is a direct analogy of that, with the reserve that here we deal with a continuum limit, and the links are weighted with deviations of entanglement entropy from the vacuum state instead of mutual information.…”

Detecting quantum critical points in the t-$$t'$$ Fermi-Hubbard model via complex network theory

“…By associating the quantum state of the t-t ′ Hubbard model with a weighted network of inter-site mutual information, for different values of the next-neighbor hopping t ′ , we have found a set of transition lines in the U-t ′ plane of the model parametric space, where characteristics of the network have a clearly distinguishable feature. Such a behavior was previously shown to be an indication of quantum phase transitions in different one-dimensional models 45,46 . The modern experimental understanding of the putative QCP in cuprates tells that it must be associated with the emergence of the pseudogap phase 4 .…”

“…Recently, a novel approach to phase transitions in quantum lattice models based on complex network theory has been suggested 45,46 . It was noticed that a particular structure that can be computed with relative ease and appears to be very sensitive to reconfigurations of the quantum state is the network of quantum mutual information.…”

“…At the same time, such a network contains information of quantum correlations which could be very important to understand the dynamics of strongly correlated systems. In the cases of the transverse field Ising and the Bose-Hubbard models in 1d, it was demonstrated that certain characteristics of the mutual information network can be used to detect quantum phase transitions 45,46 . Namely, behavior of the following functions upon changing parameters of the models has been studied:…”

The topological Dirac equation of networks and simplicial complexes

“…This approach includes the study of quantum critical dynamics defined on complex networks instead of the traditionally investigated lattices [25,26], or quantum dynamics on non-commutative geometry of graphs [27], and approaches that associate quantum-oscillators to the nodes of a network and couple them to a single probe whose frequency can be modulated freely [28][29][30]. The second class of approaches uses networks as "abstract computational objects" that can represent quantum states or the entanglement present in many-body systems [31][32][33][34][35][36][37][38][39]. This has lead to the definition of Von Neumann entropy of networks [33], and its generalizations based on the spectral properties of networks [35].…”