We prove the existence of a finite temperature Z 2 phase transition for the topological charge ordering within the fully frustrated XY model. Our method enables a proof of the topological charge confinement within the conventional XY models from a rather general vista. One of the complications that we face is the non-exact equivalence of the continuous (angular) XY model and its discrete topological charge dual. In reality, the energy spectra of the various topological sectors are highly nested, much unlike that suggested by the discrete dual models. We surmount these difficulties by exploiting the reflection positivity symmetry that this periodic flux phase model possesses. The techniques introduced here may prove binding of topological charges in numerous models and might be applied to examine transitions associated with various topological defects, e.g., the confinement of disclinations in the isotropic to nematic transition.
We prove the existence of a finite temperature Z 2 phase transition for the topological charge ordering within the fully frustrated XY model. Our method enables a proof of the topological charge confinement within the conventional XY models from a rather general vista. One of the complications that we face is the non-exact equivalence of the continuous (angular) XY model and its discrete topological charge dual. In reality, the energy spectra of the various topological sectors are highly nested, much unlike that suggested by the discrete dual models. We surmount these difficulties by exploiting the reflection positivity symmetry that this periodic flux phase model possesses. The techniques introduced here may prove binding of topological charges in numerous models and might be applied to examine transitions associated with various topological defects, e.g., the confinement of disclinations in the isotropic to nematic transition.
The topological defects of Spin(n+1) nematics in two spatial dimensions, known as disclinations, are characterized by the ${\ensuremath{\pi}}_{1}(\mathbb{R}{P}^{n})={\text{Z}}_{2}$ homotopy group for $n\ensuremath{\ge}2$. We argue that incompressible quantum liquids of disclinations can exist as stable low-temperature phases and host composite quasiparticles which combine a fractional amount of fundamental Z2 charge with a unit of topological charge. The fourfold topological ground state degeneracy on a torus admits a fermionic or semionic quasiparticle exchange statistics. The topological nontriviality of these states is visible in the existence of protected gapless edge modes. While the fermionic nematic and gapped Z2 spin liquids have equivalent topological orders, they are still thermodynamically distinct due to having different edge modes, in analogy to the topologically nontrivial and trivial states of quantum spin-Hall systems. The analysis proceeds by recasting the Z2 gauge theory of spin nematics as a continuum limit theory with a larger gauge structure. Nematic fractionalization parallels that of a quantum Hall liquid, but the large gauge symmetry restores the time-reversal symmetry and restricts the quasiparticle fusion rules and statistics. The conclusions from field theory analysis are complemented with the construction of plausible host microscopic models.
We study the phase diagram of the two-dimensional fully frustrated XY model (FFXY) and of two related models, a lattice discretization of the Landau-Ginzburg-Wilson Hamiltonian for the critical modes of the FFXY model, and a coupled Ising-XY model. We present Monte Carlo simulations on square lattices L × L , L ≲ 10 3 . We show that the low-temperature phase of these models is controlled by the same line of Gaussian fixed points as in the standard XY model. We find that, if a model undergoes a unique transition by varying temperature, then the transition is of first order. In the opposite case we observe two very close transitions: a transition associated with the spin degrees of freedom and, as temperature increases, a transition where chiral modes become critical. If they are continuous, they belong to the Kosterlitz-Thouless and to the Ising universality class, respectively. Ising and Kosterlitz-Thouless behavior is observed only after a preasymptotic regime, which is universal to some extent. In the chiral case, the approach is nonmonotonic for most observables, and there is a wide region in which finite-size scaling is controlled by an effective exponent ν eff ≈ 0.8. This explains the result ν ≈ 0.8 of many previous studies using smaller lattices.
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.
The two-component cold atom systems with anisotropic hopping amplitudes can be phenomenologically described by a two-dimensional Ising-XY coupled model with spatial anisotropy. At low temperatures, theoretical predictions [Phys. Rev. A 72 053604 (2005)] and [arXiv: 0706.1609] indicate the existence of a topological ordered phase characterized by Ising and XY disorder but with 2XY ordering. However, due to ergodic difficulties faced by Monte Carlo methods at low temperatures, this topological phase has not been numerically explored. We propose a linear cluster updating Monte Carlo method, which flips spins without rejection in the anisotropy limit but does not change the energy. Using this scheme and conventional Monte Carlo methods, we succeed in revealing the nature of topological phases with half-vortices and domain walls. In the constructed global phase diagram, Ising and XY-type transitions are very close to each other and differ significantly from the schematic phase diagram reported earlier. We also propose and explore a wide range of quantities, including magnetism, superfluidity, specific heat, susceptibility, and even percolation susceptibility, and obtain consistent and reliable results. Furthermore, we observed first-order transitions characterized by common intersection points in magnetizations for different system sizes, as opposed to the conventional phase transition where Binder cumulants of various sizes share common intersections. The critical exponents of different types of phase transitions are reasonably fitted. The results are useful to help cold atom experiments explore the half-vortex topological phase.
We present a numerical study of a blockade Hamiltonian with a local Z2 symmetry. For such models, the existence of a topologically ordered ground state has been proven rigorously for weak Rabi driving [T. F. Maier et al., PRX Quantum 6, 030340 (2025)]. However, the phase diagram beyond weak driving is still an open question. We perform large-scale infinite-size density matrix renormalization group (iDMRG) simulations on a cylinder and map out the full phase diagram of the model. We find a single continuous phase transition between the topologically ordered phase and a trivial paramagnetic phase, and confirm numerically the stability of the topologically ordered phase up to Rabi drives comparable to the characteristic energy scale of the system. Furthermore, we compute the excitation gaps and show that the charge gap remains open across the phase transition and charges become confined in the trivial phase. On the other hand, the gap in the symmetric sector closes at the transition, consistent with the condensation of flux excitations.
We use a topological approach to describe the frustration- and field-induced phase transitions exhibited by the infinite-range XY model on the AB2 chain, including noncollinear spin structures. For this purpose, we have computed the Morse number and the Euler characteristic, as well as other topological invariants, which are found to behave similarly as a function of the energy level in the context of Morse theory. In particular, we use a method based on an analogy with statistical mechanics to compute the Euler characteristic, which proves to be quite feasible. We also introduce topological energies which help us to clarify several properties of the transitions, both at zero and finite temperatures. In addition, we establish a nontrivial direct connection between the thermodynamics of the systems, which have been solved exactly under the saddle-point approach, and the topology of their configuration space. This connection allows us to identify the nondegeneracy condition under which the divergence of the density of Jacobian's critical points [jl(E)] at the critical energy of a topology-induced phase transition, proposed by Kastner and Schnetz [Phys. Rev. Lett. 100, 160601 (2008)] as a necessary criterion, is suppressed. Finally, our findings and those available in the literature suggest that the cusplike singularity exhibited both by the Euler characteristic and the topological contribution for the entropy at the critical energy, put together with the divergence of jl(E) , and emerge as necessary and sufficient conditions for the occurrence of the finite-temperature topology-induced phase transitions examined in this work. The general character of this proposal should be subject to a more rigorous scrutiny.