Abstract Landau theory’s implicit assumption that microscopic details cannot affect the global phases has been challenged only recently in systems such as antiferromagnetic quantum spin chains with periodic boundary conditions. The new paradigm of topological frustration has been accordingly put forward. In this context, the quantum phases exhibit a dependence on the boundary conditions. In this work, we show that topological frustration modifies the zero temperature phase diagram of the XY chain in a transverse magnetic field by inducing new boundary quantum phase transitions. Here, by boundary quantum phase transitions we mean non-analyticities that are non-extensive in the number of particles. Such transitions are both of first and second order. Moreover, we come across the first case of second order boundary quantum phase transition characterized by a quartic dispersion relation. Our analytical results are supported by both analytical calculations and numerical investigations, and lay the foundation for understanding the phase diagram of frustrated model.
Abstract Landau theory’s implicit assumption that microscopic details cannot affect the global phases has been challenged only recently in systems such as antiferromagnetic quantum spin chains with periodic boundary conditions. The new paradigm of topological frustration has been accordingly put forward. In this context, the quantum phases exhibit a dependence on the boundary conditions. In this work, we show that topological frustration modifies the zero temperature phase diagram of the XY chain in a transverse magnetic field by inducing new boundary quantum phase transitions. Here, by boundary quantum phase transitions we mean non-analyticities that are non-extensive in the number of particles. Such transitions are both of first and second order. Moreover, we come across the first case of second order boundary quantum phase transition characterized by a quartic dispersion relation. Our analytical results are supported by both analytical calculations and numerical investigations, and lay the foundation for understanding the phase diagram of frustrated model.
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.
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.
Recently it was highlighted that one-dimensional antiferromagnetic spin models with frustrated boundary conditions, i.e. periodic boundary conditions in a ring with an odd number of elements, may show very peculiar behavior. Indeed the presence of frustrated boundary conditions can destroy the local magnetic orders presented by the models when different boundary conditions are taken into account and induce novel phase transitions. Motivated by these results, we analyze the effects of the introduction of frustrated boundary conditions on several models supporting (symmetry protected) topological orders, and compare our results with the ones obtained with different boundary conditions. None of the topological order phases analyzed are altered by this change. This observation leads naturally to the conjecture that topological phases of one-dimensional systems are in general not affected by topological frustration.
Abstract Ginzburg-Landau theory of continuous phase transitions implicitly assumes that microscopic changes are negligible in determining the thermodynamic properties of the system. In this work we provide an example that clearly contrasts with this assumption. In particular, we consider the 2-cluster-Ising model, a one-dimensional spin-1/2 system that is known to exhibit a quantum phase transition between a magnetic and a nematic phase. By imposing boundary conditions that induce topological frustration we show that local order is completely destroyed on both sides of the transition and that the two thermodynamic phases can only be characterized by string order parameters. Having proved that topological frustration is capable of altering the nature of a system's phase transition, this result is a clear challenge to current theories of phase transitions in complex quantum systems.
Abstract We studied quantum phase transitions in the antiferromagnetic dimerized spin- 1 2 XY chain and two-leg ladders. From analysis of several spin models we present our main result: the framework to deal with topological orders and hidden symmetries within the Landau paradigm. After mapping of the spin Hamiltonians onto the tight-binding models with Dirac or Majorana fermions and, when necessary, the mean-field approximation, the analysis can be done analytically. By utilizing duality transformations the calculation of nonlocal string order parameters is mapped onto the local order problem in some dual representation and done without further approximations. Calculated phase diagrams, phase boundaries, order parameters and their symmetries for each of the phases provide a comprehensive quantitative Landau description of the quantum critical properties of the models considered. Complementarily, the phases with hidden orders can also be distinguished by the Pontryagin (winding) numbers which we have calculated as well. This unified framework can be straightforwardly applied for various spin chains and ladders, topological insulators and superconductors. Applications to other systems are under way.
Abstract In this study, a spin-1/2 extended anisotropic XY chain has been introduced in which both time reversal and SU(2) symmetries are broken but Z 2 symmetry is preserved. Magnetic and topological phase diagrams in the parameter space have been drawn in the presence of transverse magnetic field. Entanglement measures like mutual information and quantum discord are also evaluated and it indicates that these transitions are second order in nature. Quantum phase transition is noted at zero magnetic field, as well as magnetic long range order is found to withstand magnetic field of any strength. Exact analytic results for spin-spin correlation functions have been obtained in terms of Jordan-Wigner fermionization. Existence of long range magnetic order has been investigated numerically by finding correlation functions as well as the Binder cumulant in the ground state. Dispersion relation, ground state energy, and energy gap are obtained analytically. In order to find the topologically nontrivial phase, sign of Pfaffian invariant and value of winding number have been evaluated. Both magnetic and topological phases are robust against the magnetic field and found to move coercively in the parameter space with the variation of its strength. Long range orders along two orthogonal directions and two different topological phases are found and their one-to-one correspondence has been found. Finally casting the spinless fermions onto Majorana fermions, properties of zero energy edge states are studied. Three different kinds of Majorana pairings are noted. In the trivial phase, next-nearest-neighbor Majorana pairing is found, whereas two different types of nearest-neighbor Majorana pairings are identified in the topological superconducting phase.