Exact Ground States for One- and Two-dimensional Quantum Spin Systems PDF Download

Author: Harald Niggemann
Publisher:
ISBN:
Category :
Languages : en
Pages : 135

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Author: John B. Parkinson
Publisher: Springer Science & Business Media
ISBN: 3642132898
Category : Science
Languages : en
Pages : 159

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The topic of lattice quantum spin systems is a fascinating and by now well established branch of theoretical physics. Based on a set of lectures, this book has a level of detail missing from others, and guides the reader through the fundamentals of the field.

Author: Elliott H. Lieb
Publisher: Springer Science & Business Media
ISBN: 3662063905
Category : Science
Languages : en
Pages : 645

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Book Description
This is the third Selecta of publications of Elliott Lieb, the first two being Stabil ity of Matter: From Atoms to Stars, edited by Walter Thirring, and Inequalities, edited by Michael Loss and Mary Beth Ruskai. A companion fourth Selecta on Statistical Mechanics is also edited by us. Elliott Lieb has been a pioneer of the discipline of mathematical physics as it is nowadays understood and continues to lead several of its most active directions today. For the first part of this selecta we have made a selection of Lieb's works on Condensed Matter Physics. The impact of Lieb's work in mathematical con densed matter physics is unrivaled. It is fair to say that if one were to name a founding father of the field, Elliott Lieb would be the only candidate to claim this singular position. While in related fields, such as Statistical Mechanics and Atomic Physics, many key problems are readily formulated in unambiguous mathematical form, this is less so in Condensed Matter Physics, where some say that rigor is "probably impossible and certainly unnecessary". By carefully select ing the most important questions and formulating them as well-defined mathemat ical problems, and then solving a good number of them, Lieb has demonstrated the quoted opinion to be erroneous on both counts. What is true, however, is that many of these problems turn out to be very hard. It is not unusual that they take a decade (even several decades) to solve.

Author: David Cooper
Publisher: Elsevier
ISBN: 0080543499
Category : Science
Languages : en
Pages : 837

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Valence bond (VB) theory, which builds the descriptions of molecules from those of its constituent parts, provided the first successful quantum mechanical treatments of chemical bonding. Its language and concepts permeate much of chemistry, at all levels. Various modern formulations of VB theory represent serious tools for quantum chemical studies of molecular electronic structure and reactivity. In physics, there is much VB-based work (particularly in semi-empirical form) on larger systems. Importance of TopicThe last decade has seen significant advances in methodology and a vast increase in the range of applications, with many new researchers entering the field.Why This TitleValence Bond Theory succeeds in presenting a comprehensive selection of contributions from leading valence bond (VB) theory researchers throughout the world. It focuses on the vast increase in the range of applications of methodology based on VB theory during the last decade and especially emphasizes recent advances.

Author: W J Caspers
Publisher: World Scientific
ISBN: 9814590924
Category : Science
Languages : en
Pages : 231

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Book Description
This book is about spin systems as models for magnetic materials, especially antiferromagnetic lattices. Spin-systems are well-defined models, for which, in special cases, exact properties may be derived. These special cases are for the greater part, one- dimensional and restricted in their applicability, but they may give insight into general properties that also exist in higher dimension. This work pays special attention to qualitative differences between spin lattices of different dimensions. It also replaces the traditional picture of an (ordered) antiferromagnetic state of a Heisenberg system by one which makes use of the space of quantum spin states.

Author: Amanda M Young
Publisher:
ISBN: 9781369311044
Category :
Languages : en
Pages :

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In this dissertation, we analyze spectral properties of frustration-free quantum spin systems, with an emphasis on multi-dimensional models. The focus of this analysis is two-fold. First, we determine the existence or non-existence of a spectral gap above the ground state for a class of multi-dimensional quantum spin systems. Second, we determine a set of criteria for which the spectral gap above the ground state energy of a frustration-free quantum spin system is stable in the presence of local perturbations. We begin by providing the mathematical formulation of quantum spin systems, including a discussion of the thermodynamic limit. We formalize the construction of infinite volume systems when we introduce Lieb-Robinson bounds. When these bounds hold, the dynamics of the finite volume quantum spin systems extend to the thermodynamic limit, and a well-defined infinite volume Hamiltonian, known as the GNS Hamiltonian, is guaranteed to exist. Determining if a quantum spin system is gapped or gapless is a question of the spectrum of the GNS Hamiltonian. It is gapped if there is an interval above the ground state energy that does not contain any points of the spectrum. Otherwise, it is gapless. We show that to prove a gap for the GNS Hamiltonian of a frustration-free system, it suffices to obtain a uniform lower bound for the spectral gap in a sequence of finite volumes. To obtain these results we apply the martingale method, which was first introduced for frustration-free quantum spin systems with open boundary conditions by Nachtergaele in 1996 [48]. We extend this method to hold for periodic boundary conditions, and apply it to the class of pure finitely correlated state models with periodic boundary conditions.We then study the the one-species, multi-dimensional PVBS models and determine the exact collection of model parameters for which the models are gapped in the thermodynamic limit. The PVBS models are a class of particle hopping models where the model parameters dictate the direction a particle prefers to move. We consider these models on two types of infinite volumes. First, we take the thermodynamic limit of the PVBS model to the d-dimensional integer lattice. Second, we consider the PVBS model on a half space of the integer lattice where the boundary is defined by a hyperplane. We find two types of closures of the spectral gap. First, there is a bulk closure due to a critical change in the set of model parameters. The second type of closure is only found in the case of the hyperplane boundary, and is due to the creation of edge states across the boundary. This is inherently different from the bulk closure, as away from the boundary the model still behaves like a gapped system. We then transition to considering the stability of gapped ground state phases. Stability results establish that if a system is gapped, it will remain gapped under the addition of small perturbations. The two main technical tools we use to prove such a result are Lieb-Robinson bounds and the spectral flow. Common to both tools is that they express locality properties for observables under a global evolution. We generalize this notion by introducing the theory of quasi-local maps. We apply our theory to prove the stability result from [47], which holds for both one-dimensional andmulti-dimensional quantum spin systems. The result from [47], which we extend to hold for a larger class of models and boundary conditions, states that a uniformly gapped family of frustration-free Hamiltonians with topologically ordered ground states will remain gapped in the presence of an exponentially decaying perturbation. We conclude with applying the stability result the finitely correlated state models with periodic boundary conditions.

Author: Hung-the Diep
Publisher: World Scientific
ISBN: 9814440752
Category : Science
Languages : en
Pages : 644

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Book Description
This book covers all principal aspects of currently investigated frustrated systems, from exactly solved frustrated models to real experimental frustrated systems, going through renormalization group treatment, Monte Carlo investigation of frustrated classical Ising and vector spin models, low-dimensional systems, spin ice and quantum spin glass. The reader can — within a single book — obtain a global view of the current research development in the field of frustrated systems.This new edition is updated with recent theoretical, numerical and experimental developments in the field of frustrated spin systems. The first edition of the book appeared in 2005. In this edition, more recent works until 2012 are reviewed. It contains nine chapters written by researchers who have actively contributed to the field. Many results are from recent works of the authors.The book is intended for postgraduate students as well as researchers in statistical physics, magnetism, materials science and various domains where real systems can be described with the spin language. Explicit demonstrations of formulas and full arguments leading to important results are given where it is possible to do so.

Author: Laurens Vanderstraeten
Publisher: Springer
ISBN: 3319641913
Category : Science
Languages : en
Pages : 229

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This thesis develops new techniques for simulating the low-energy behaviour of quantum spin systems in one and two dimensions. Combining these developments, it subsequently uses the formalism of tensor network states to derive an effective particle description for one- and two-dimensional spin systems that exhibit strong quantum correlations. These techniques arise from the combination of two themes in many-particle physics: (i) the concept of quasiparticles as the effective low-energy degrees of freedom in a condensed-matter system, and (ii) entanglement as the characteristic feature for describing quantum phases of matter. Whereas the former gave rise to the use of effective field theories for understanding many-particle systems, the latter led to the development of tensor network states as a description of the entanglement distribution in quantum low-energy states.

Author: Sei Suzuki
Publisher: Springer
ISBN: 3642330398
Category : Science
Languages : en
Pages : 407

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Book Description
Quantum phase transitions, driven by quantum fluctuations, exhibit intriguing features offering the possibility of potentially new applications, e.g. in quantum information sciences. Major advances have been made in both theoretical and experimental investigations of the nature and behavior of quantum phases and transitions in cooperatively interacting many-body quantum systems. For modeling purposes, most of the current innovative and successful research in this field has been obtained by either directly or indirectly using the insights provided by quantum (or transverse field) Ising models because of the separability of the cooperative interaction from the tunable transverse field or tunneling term in the relevant Hamiltonian. Also, a number of condensed matter systems can be modeled accurately in this approach, hence granting the possibility to compare advanced models with actual experimental results. This work introduces these quantum Ising models and analyses them both theoretically and numerically in great detail. With its tutorial approach the book addresses above all young researchers who wish to enter the field and are in search of a suitable and self-contained text, yet it will also serve as a valuable reference work for all active researchers in this area.

Author: Masuo Suzuki
Publisher: World Scientific
ISBN: 9789810220235
Category : Science
Languages : en
Pages : 540

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Book Description
This book presents a systematic and coherent approach to phase transitions and critical phenomena, namely the coherent-anomaly method (CAM theory) based on cluster mean-field approximations. The first part gives a brief review of the CAM theory and the second part a collection of reprints covering the CAM basic calculations, the Blume-Emery-Griffiths model, the extended Baxter model, the quantum Heisenberg model, zero-temperature phase transitions, the KT-transition, spin glasses, the self-avoiding walk, contact processes, branching processes, the gas-liquid transition and even non-equilibrium phase transitions.