In this paper, we work on the pure and mixed cluster models with periodic boundary condition. The first purpose is to establish the concept of quantum cluster kink. We clarify that there are two types of cluster kinks since there are two types of ground states depending on the choice of cluster length, of which the first type exhibits symmetry breaking order and the second one string order. Simple pictures are constructed according to the pure cluster model, which facilitates us to introduce the quantity, cluster kink number. As we demonstrate, cluster kinks deriving from different sources can coexist, compete with each other, and lead to quantum phase transition in a mixed cluster model. The second purpose is to elucidate that the effect of ring frustration can be realized in the cluster model with symmetry breaking order, instead of the one with string order. The reason lies in that ring frustration can induce a huge ground-state degeneracy or a special extended-kink phase with gapless excitations in the former case. And, although ring frustration does not change the phase transition point, it can produce unusual ground state in the extended-kink phase, whose special properties are uncovered by the nonlocal scaling factor in the correlation function and the doubled degeneracies of the eigenvalues of the entanglement spectrum.
In this paper, we work on the pure and mixed cluster models with periodic boundary condition. The first purpose is to establish the concept of quantum cluster kink. We clarify that there are two types of cluster kinks since there are two types of ground states depending on the choice of cluster length, of which the first type exhibits symmetry breaking order and the second one string order. Simple pictures are constructed according to the pure cluster model, which facilitates us to introduce the quantity, cluster kink number. As we demonstrate, cluster kinks deriving from different sources can coexist, compete with each other, and lead to quantum phase transition in a mixed cluster model. The second purpose is to elucidate that the effect of ring frustration can be realized in the cluster model with symmetry breaking order, instead of the one with string order. The reason lies in that ring frustration can induce a huge ground-state degeneracy or a special extended-kink phase with gapless excitations in the former case. And, although ring frustration does not change the phase transition point, it can produce unusual ground state in the extended-kink phase, whose special properties are uncovered by the nonlocal scaling factor in the correlation function and the doubled degeneracies of the eigenvalues of the entanglement spectrum.
We study the quantum phase transition driven by a point impurity in a chain seamed with ring frustration. With strong coupling and light impurity, the system is in a topological extended-kink (TEK) phase, which exhibits gapless excitations in the bulk. With strong coupling and heavy impurity, the system is in a gapped kink bound state (KBS) phase. Two-point bulk and impurity correlations are defined to characterize the two phases. In the TEK phase, both the bulk and impurity correlations are long range and factorizable so that scaling functions can be parsed. The scaling functions relies on the distance scaled by the system's size. An impurity correlation length can be extracted from the impurity correlation. In the transition from TEK to KBS, the scaling function of the bulk correlation undergoes an abrupt steplike change. Meanwhile, the impurity correlation length decreases from a divergent value to a finite one. The ground state of the TEK phase retains a relatively high value of entanglement entropy due to the absence of symmetry breaking. However, spontaneous symmetry breaking occurs in the KBS phase, which induces antiferromagnetic order in the bulk and entangled spin configuration near the impurity.
We numerically study quantum phase transitions and dynamical properties in the one-dimensional cluster model with several interactions by using the time-evolving block decimation method for infinite systems and the exact diagonalization. First, boundaries among several quantum phases of the model are determined from energy gap and each phase is characterized by order parameters and the entanglement spectrum (ES). We confirm that in the model with open boundary condition the degeneracy of the lowest levels in the ES corresponds to that of the ground states. Then, using the time-dependent Bogoliubov transformation with open boundary condition, we investigate dynamical properties during an interaction sweep through the critical point which separates two topological phases involving four-fold degeneracy in the ground state. After a slow sweep across the critical point, we observe spatially periodic structures in the string correlation functions and the entanglement entropy. It is shown that the periodicities stem from the Bogoliubov quasiparticles generated near the critical point.
Abstract In quantum many-body systems with local interactions, the effects of boundary conditions are considered to be negligible, at least for sufficiently large systems. Here we show an example of the opposite. We consider a spin chain with two competing interactions, set on a ring with an odd number of sites. When only the dominant interaction is antiferromagnetic, and thus induces topological frustration, the standard antiferromagnetic order (expressed by the magnetization) is destroyed. When also the second interaction turns from ferro to antiferro, an antiferromagnetic order characterized by a site-dependent magnetization which varies in space with an incommensurate pattern, emerges. This modulation results from a ground state degeneracy, which allows to break the translational invariance. The transition between the two cases is signaled by a discontinuity in the first derivative of the ground state energy and represents a quantum phase transition induced by a special choice of boundary conditions.
Frustration in many-body spin systems is a situation where the spins cannot orient themselves to fully minimize the interactions with their neighbors. It is caused, in general, by competing interactions or by the lattice structure, as in the antiferromagnets on non-bipartite lattices. Over the years frustrated systems have been studied substantially, since frustration can destabilize the antiferromagnetic order, lead to different phases of matter and induce various exotic properties. Frustration can be caused also by a choice of boundary conditions, i.e. the lattice topology. For instance, a ring with an odd number of sites is non-bipartite, which makes antiferromagnets in this setting frustrated. The subject of this thesis are zero-temperature one-dimensional quantum systems exactly in this setting. The major content is discussing how such a simple setting can influence the antiferromagnetic order, by studying spin-1/2 chains with discrete symmetries, whose breaking is related to the onset of the order. After reviewing the known results on topological frustration, dealing with non-thermodynamic quantities, we develop an approach for studying symmetry breaking, that is both suitable for exact analytical computations and meaningful for discussing the order also in a finite system. It consists of the realization that many spin-1/2 chains without external magnetic fields, such as the quantum XY chain, possess anticommuting global symmetries when the number of lattice sites is an odd number, implying the ground-state degeneracy already in a finite system. In this framework, we discover that topological frustration, despite being induced only by the choice of boundary conditions and even/odd system size choice, can affect the magnetization in the thermodynamic limit and system's quantum phase transitions. We find that topological frustration can destroy local order, create a site-dependent magnetization that varies in space with an incommensurate pattern, induce a first-order quantum phase transition that is not present without frustration and modify the nature of a second-order transition, by destroying local order at both sides of the transition and preserving only non-local string order parameters. All these results indicate the incompleteness of the approach to quantum many-body systems, based on the Ginzburg-Landau theory, that tries to capture the properties of the system by taking the system size to infinity before computing the observables and neglects the influence of the chain length as a relevant scale. We find that topological frustration can affect also non-equilibrium properties, by considering a local quantum quench setup. Namely, we study the Loschmidt echo and find it displays qualitatively and quantitatively different behavior for antiferromagnetic rings with an even and odd number of sites. Most of all, the differences become clearer for large system sizes, thus allowing to distinguish in a simple out-of-equilibrium experiment a system made by a certain, large, number of spins from the one with a single additional spin. The thesis also contains a mathematical part. In studying order parameters of models mappable to free fermions, Toeplitz determinants with symbols that possess a part proportional to the delta function arise. We derive asymptotic formulas for this type of determinants.
How to impose boundary conditions is crucial to finite-cluster calculations of quantum spin systems, but only open-boundary clusters have been used in cluster-effective-field approximations up to now. In the present paper, periodic-boundary clusters are also considered for the formulation of cluster-effective-field approximations in frustrated quantum spin systems. Namely, the open-boundary-condition double-cluster approximation (OBC-DCA) and the periodic-boundary-condition double-cluster approximation (PBC-DCA) are applied to the one-dimensional S= 1/2 frustrated XXZ model at the ground state. These two approximations are compared using the coherent-anomaly method (CAM). Within the limitation of cluster sizes in exact-diagonalization calculations, the PBC-DCA can reproduce the true phase boundary of the Neel-dimer transition. On the other hand, the OBC-DCA severely underestimates the existence of magnetic orders as quantum fluctuation is increased. These findings suggest that previous cluster-effective-field studies based on open-boundary clusters should be reconsidered.
A bstract Quantum one-dimensional systems in their ordered phase admit kinks as elementary excitations above their symmetry-broken vacua. While the scattering properties of the kinks resemble those of quasiparticles, they have distinct locality features that are manifest in their entanglement content. In this work, we study the entanglement entropy of kink excitations. We first present detailed calculations for specific states of a spin-1/2 chain to highlight the salient features of these excitations. Second, we provide a field-theoretic framework based on the algebraic relations between the twist fields and the semilocal fields associated with the excitations, and we compute the Rényi entropies in this framework. We obtain universal predictions for the entropy difference between the excited states with a finite number of kinks and the symmetry-broken ground states, which do not depend on the microscopic details of the model in the limit of large regions. Finally, we discuss some consequences of the Kramers-Wannier duality, which relates the ordered and disordered phases of the Ising model, and we explain why, counterintuitively, no explicit relations between those phases are found at the level of entanglement.