Shannon-Rényi entropies and associated participation spectra quantify how much a many-body wave-function is localized in a given configuration basis. Using these tools, we present an analysis of the ground-state wave functions of various quantum spin systems in one and two dimensions. General ideas and a review of the current status of this field are first given, with a particular emphasis on universal subleading terms characterizing different quantum phases of matter, and associated transitions. We highlight the connection with the related entanglement entropies and spectra when this is possible. In a second part, new results are presented for the participation spectra of interacting spin models, mostly based on quantum Monte Carlo simulations, but also using perturbation theory in some cases. For full antiferromagnetic systems, participation spectra are analyzed in terms of ferromagnetic domain walls which experience a repulsive pairwise interaction. This confinement potential is either linear for long-range Néel order, or logarithmic for quasi-long-range order. The case of subsystems is also analyzed in great detail for a 2d dimerized Heisenberg model undergoing a quantum phase transition between a gapped paramagnet and a Néel phase. Participation spectra of line shaped (1d) sub-systems are quantitatively compared with finite temperature participation spectra of ansatz effective boundary (1d) entanglement Hamiltonians. While short-range models describe almost perfectly the gapped side, the Néel regime is best compared using long-range effective Hamiltonians. Spectral comparisons performed using Kullback-Leibler divergences, a tool potentially useful for entanglement spectra, provide a quantitative way to identify both the best boundary entanglement Hamiltonian and temperature.
Shannon-Rényi entropies and associated participation spectra quantify how much a many-body wave-function is localized in a given configuration basis. Using these tools, we present an analysis of the ground-state wave functions of various quantum spin systems in one and two dimensions. General ideas and a review of the current status of this field are first given, with a particular emphasis on universal subleading terms characterizing different quantum phases of matter, and associated transitions. We highlight the connection with the related entanglement entropies and spectra when this is possible. In a second part, new results are presented for the participation spectra of interacting spin models, mostly based on quantum Monte Carlo simulations, but also using perturbation theory in some cases. For full antiferromagnetic systems, participation spectra are analyzed in terms of ferromagnetic domain walls which experience a repulsive pairwise interaction. This confinement potential is either linear for long-range Néel order, or logarithmic for quasi-long-range order. The case of subsystems is also analyzed in great detail for a 2d dimerized Heisenberg model undergoing a quantum phase transition between a gapped paramagnet and a Néel phase. Participation spectra of line shaped (1d) sub-systems are quantitatively compared with finite temperature participation spectra of ansatz effective boundary (1d) entanglement Hamiltonians. While short-range models describe almost perfectly the gapped side, the Néel regime is best compared using long-range effective Hamiltonians. Spectral comparisons performed using Kullback-Leibler divergences, a tool potentially useful for entanglement spectra, provide a quantitative way to identify both the best boundary entanglement Hamiltonian and temperature.
Universal features in the scalings of Shannon-R\'enyi entropies of many-body ground states are studied for interacting spin-21 systems across (2+1)-dimensional O(3) critical points, using quantum Monte Carlo simulations on dimerized and plaquettized Heisenberg models on the square lattice. Considering both full systems and line-shaped subsystems, SU(2) symmetry breaking on the N\'eel ordered side of the transition is characterized by the presence of a logarithmic term in the scaling of Shannon-R\'enyi entropies, which is absent in the disordered gapped phase. Such a difference in the scalings allows to capture the quantum critical point using Shannon-R\'enyi entropies for line-shaped subsystems of length L embedded in $L\ifmmode\times\else\texttimes\fi{}L$ tori, as the smaller subsystem entropies are numerically accessible to much higher precision than for the full system. Most interestingly, at the quantum phase transition an additive subleading constant ${b}_{\ensuremath{\infty}}^{*\mathrm{line}}=0.41(1)$ emerges in the critical scaling of the line Shannon-R\'enyi entropy ${S}_{\ensuremath{\infty}}^{\mathrm{line}}$. This number appears to be universal for 3d O(3) criticality, as confirmed for the finite-temperature transition in the 3d antiferromagnetic spin-21 Heisenberg model. Additionally, the phases and phase transition can be detected in several features of the participation spectrum, consisting of the diagonal elements of the reduced density matrix of the line subsystem. In particular, the N\'eel ordering transition can be simply understood in the Sz basis by a confinement mechanism of ferromagnetic domain walls.
Using quantum Monte Carlo simulations, we compute the participation (Shannon-Rényi) entropies for groundstate wave functions of Heisenberg antiferromagnets for one-dimensional (line) subsystems of length L L embedded in two-dimensional ( L\times L L × L ) square lattices. We also study the line entropy at finite temperature, i.e. of the diagonal elements of the density matrix, for three-dimensional ( L\times L\times L L × L × L ) cubic lattices. The breaking of SU(2) symmetry is clearly captured by a universal logarithmic scaling term l_q\ln L l q ln L in the Rényi entropies, in good agreement with the recent field-theory results of Misguish, Pasquier and Oshikawa . We also study the dependence of the log prefactor l_q l q on the Rényi index q q for which a transition is detected at q_c\simeq 1 q c ≃ 1 .
At the interface of quantum information and condensed matter physics, the study of entanglement in quantum many-body systems requires a new toolset which combines concepts from each. This thesis introduces a set of computational methods to study phases and phase transitions in lattice models of quantum systems, using the Renyi entropies as a means of quantifying entanglement. The scaling of entanglement entropy can give valuable insight into the phase of a condensed matter system. It can be used to detect exotic types of phases, to pinpoint transitions between phases, and can give us universal information about a system. \nThe first approach in this thesis is a technique to measure entanglement in finite size \nlattice systems using zero-temperature quantum Monte Carlo simulations. The algorithm \nis developed, implemented, and used to explore anomalous entanglement scaling terms in \nthe spin-1/2 Heisenberg antiferromagnet. \nIn the second part of this thesis, a new and complementary numerical technique is introduced to study entanglement not just in finite size systems, but as we approach the thermodynamic limit. This “numerical linked-cluster expansion” is used to study two different systems at their quantum critical points — continuous phase transitions occurring at zero temperature, at which these systems exhibit universal properties. Remarkably, these universal properties can be reflected in the scaling of entanglement. \nEntanglement offers a new perspective on condensed matter systems, one which takes us closer to genuinely understanding what goes on in these materials at the quantum mechanical level. This thesis demonstrates the first steps in developing an extensive list of computational tools that can be used to study entanglement over a wide range of interacting quantum many-body systems. With the ever increasing computational power available, it may be only a matter of time before these tools are used to create a comprehensive framework for the characterization of condensed matter phases and phase transitions.
One of the most important predictions of quantum mechanics is the existence of quantum entanglement. Since the inception of quantum theory, the study of entanglement has become ubiquitous in several branches of physics, from high energy to condensed matter physics and quantum information. To this avail, different measures of entanglement in quantum systems have been introduced. The most successful ones are the von Neumann and the Rényi entanglement entropies, which have been heavily studied both in lattice models and in quantum field theories (QFTs). A different, more complete characterization of entanglement is provided by the entanglement (or modular) Hamiltonian, defined as the logarithm of the reduced density matrix. One of the key property of this operator is that, under generic conditions, in 1+1-dimensional conformal field theories (CFTs) presents a local and few-body structure. Beyond just being of theoretical interest, this property has been recently used to efficiently construct the ground state of spin chains in quantum simulators. A recently introduced generalisation of the entanglement Hamiltonian is the negativity Hamiltonian, a non-Hermitian operator which is more useful for mixed states and tripartite entanglement. In this thesis we study these quantifications in many-body quantum systems and in QFTs. In the first part of the thesis, we focus on fermionic systems at equilibrium, and we obtain analytical expressions for the entanglement and the negativity Hamiltonians in the massless Dirac fermion QFT, both in presence of boundaries and at finite temperature. We compare these predictions in the continuum QFT with numerical calculations on the lattice, showing that a proper comparison requires a non-trivial continuum limit of the lattice expression, which we detail. We also investigate a lattice realisation of a non-unitary CFT, the bc-ghost theory, showing that the entanglement Hamiltonian presents an additional term with respect to the unitary case. In the second part, we study the entanglement entropies in an integrable QFT, the massless renormalisation group flow from the tricritical to the critical Ising CFT, demonstrating that the entropy is monotonically decreasing under the renormalisation group flow, as expected from the entropic c-theorem. In the third part we consider out-of-equilibrium problems. We study the melting of a domain wall configuration, computing analytically the entanglement Hamiltonian and showing that the insertion of a defect dramatically modifies the entanglement properties, changing the growth of the entropies from logarithmic to linear in time. Finally, we study generic quantum quenches in free fermionic theories, using the celebrated quasiparticle picture to obtain a prediction for the evolution of the entanglement Hamiltonian during the quench.
The interplay of the constituents of interacting many-body systems may reveal emergent properties on the macroscopic scale which are not inherent to the individual constituents. These properties are expressed in macroscopic observables \ndescribing the state — denoted as the phase of the system. Continuous phase transition between phases are generically manifested in critical behavior, for example, a divergence of a macroscopic observable. The identification of the present phase of a system and the classification of critical phenomena into universality classes are exciting challenges of condensed-matter physics. \nIn this thesis, we use entanglement entropies as macroscopic quantities for the characterization of phases of quantum matter and critical theories. For groundstates of quantum many-body systems the entanglement entropy is a measure of \nthe amount of entanglement between two subsystems. The generic dependence of the entanglement entropy on the size and shape of the subsystems is contained in the well-known boundary law — stating a scaling of the entanglement entropy with the boundary between the two subsystems. We numerically investigate how the coefficient of this dependence reflects quantum phase transitions in simple spin-half bilayer models. \nSubleading terms to the boundary law such as a logarithmic contribution provide universal numbers for the criticality of field theories. We examine free and interacting theories from an entanglement entropy perspective in order to assess the role of the coefficient of the logarithmic correction induced by corners in the subsystems. Beyond its universality, this coefficient also quantifies degrees of freedom of low-lying excitations in the conformal field theory describing a critical point. \nA constant contribution to the boundary law indicates the presence of so-called topological order in the ground state of a many-body system. This extremely useful property can also be identified in classical counterparts of entanglement entropy which we study at the example of various toric code models. To this endeavor, we have designed Monte Carlo techniques which allow for an efficient numerical computation of the constant contribution. In particular, we analyze via entanglement entropies under which conditions remnants of topological order are present in the quantum system at finite temperature and at perturbations from a magnetic field. The major motivation behind this effort is to use topological order for the robust storage of a quantum information — a basic need for the construction of quantum \ncomputers.
In this work, building on state-of-the-art quantum Monte Carlo simulations, we perform systematic finite-size scaling of both entanglement and participation entropies for long-range Heisenberg chain with unfrustrated power-law decaying interactions. We find distinctive scaling behaviors for both quantum entropies in the various regimes explored by tuning the decay exponent α, thus capturing nontrivial features through logarithmic terms, beyond the case of linear Nambu-Goldstone modes. Our systematic analysis reveals that the quantum entanglement information, hidden in the scaling of the two studied entropies, can be obtained to the same level of order parameters and other usual finite-size observables of quantum many-body lattice models. The analysis and results obtained here can readily apply to more quantum criticalities in 1D and 2D systems.