We present a numerical study of a quantum phase transition from a spin-polarized to a topologically ordered phase in a system of spin-$1∕2$ particles on a torus. We demonstrate that this non-symmetry-breaking topological quantum phase transition (TOQPT) is of second order. The transition is analyzed via the ground state energy and fidelity, block entanglement, Wilson loops, and the recently proposed topological entropy. Only the topological entropy distinguishes the TOQPT from a standard QPT, and remarkably, does so already for small system sizes. Thus the topological entropy serves as a proper order parameter. We demonstrate that our conclusions are robust under the addition of random perturbations, not only in the topological phase, but also in the spin-polarized phase and even at the critical point.
We present a numerical study of a quantum phase transition from a spin-polarized to a topologically ordered phase in a system of spin-1∕2 particles on a torus. We demonstrate that this non-symmetry-breaking topological quantum phase transition (TOQPT) is of second order. The transition is analyzed via the ground state energy and fidelity, block entanglement, Wilson loops, and the recently proposed topological entropy. Only the topological entropy distinguishes the TOQPT from a standard QPT, and remarkably, does so already for small system sizes. Thus the topological entropy serves as a proper order parameter. We demonstrate that our conclusions are robust under the addition of random perturbations, not only in the topological phase, but also in the spin-polarized phase and even at the critical point.
We present an analytical study of the quantum phase transition between the topologically ordered toric-code-model ground state and the disordered spin-polarized state. The phase transition is induced by applying an external magnetic field, and the variation in topological order is detected via two nonlocal quantities: the Wilson loop and the topological R\'enyi entropy of order 2. By exploiting an equivalence with the transverse-field Ising model and considering two different variants of the problem, we investigate the field dependence of these quantities by means of an exact treatment in the exactly solvable variant and complementary perturbation theories around the limits of zero and infinite fields in both variants. We find strong evidence that the phase transition point between topological order and disorder is marked by a discontinuity in the topological R\'enyi entropy and that the two phases around the phase transition point are characterized by its different constant values. Our results therefore indicate that the topological R\'enyi entropy is a proper topological invariant: its allowed values are discrete and can be used to distinguish between different phases of matter.
This dissertation puts forward the concept of fidelity as a way to understand and characterize quantum phase transitions involving topological phases. It lies at the interphase between Condensed Matter and Quantum Phase Transitions. ❧ We find that the fidelity metric, which characterizes the degree of distinghuishability between two quantum states, can serve as a useful tool in the understanding of transitions involving topological phases. ❧ The methodology does not require a complete knowledge of the physical mechanism involved in the occurrence of the critical point, such as the identification of a local order parameter and a symmetry breaking mechanism, the fundamental pillars of critical phenomena. Topological phases are unique in that they cannot be classified under these methodologies, due to their lack of local order parameters. It is one of the purposes of this thesis to understand how well we can characterize the phase boundaries between these phases using fidelity concepts. ❧ In the first part, we introduce two quantum models that exhibit a phase transition between an ordered and a topological phase, as quantified by the topological entanglement entropy. In the thermodynamic limit, the different components of the fidelity metric exhibit a divergence, signalling the occurrence of the critical point. ❧ In the second part, we introduce models that have phase boundaries between two distinct topological phases, and we show that the fidelity is successful at identifying the occurrence of a critical region. Moreover, a finite size scaling analysis is carried out, from which we determine the value of critical exponents.
We study a quantum phase transition between a phase which is topologically ordered and one which is not. We focus on a spin model, an extension of the toric code, for which we obtain the exact ground state for all values of the coupling constant that takes the system across the phase transition. We compute the entanglement and the topological entropy of the system as a function of this coupling constant and show that the topological entropy remains constant all the way up to the critical point and jumps to zero beyond it. Despite the jump in the topological entropy, the transition is second order as detected via local observables.
A quantum phase transition (QPT), including both topological and symmetry breaking types, is usually induced by the change of global parameters, such as external fields or global coupling constants. In this work, we demonstrate the existence of QPT induced by the real-space topology of the system. We investigate the groundstate properties of the tight-binding model on a honeycomb lattice with the torus geometry based on exact results. It is shown that the ground state experiences a second-order QPT, exhibiting the scaling behavior, when the torus switches to a tube, which reveals the connection between quantum phase and the real-space topology of the system.
Abstract We use the topological entanglement entropy (TEE) as an efficient tool to fully characterize the Abelian phase of a Z 2 × Z 2 spin liquid emerging as the ground state of topological color code (TCC), which is a class of stabilizer states on the honeycomb lattice. We provide the fusion rules of the quasiparticle (QP) excitations of the model by introducing single- or two-body operators on physical spins for each fusion process which justify the corresponding fusion outcome. Besides this, we extract the TEE from Renyi entanglement entropy (EE) of the TCC, analytically and numerically by finite size exact diagonalization on the disk shape regions with contractible boundaries. We obtain that the EE has a local contribution, which scales linearly with the boundary length in addition to a topological term, i.e. the TEE, arising from the condensation of closed strings in the ground state. We further investigate the ground state dependence of the TEE on regions with non-contractible boundaries, i.e. by cutting the torus to half cylinders, from which we further identify multiple independent minimum entropy states (MES) of the TCC and then extract the U and S modular matrices of the system, which contain the self and mutual statistics of the anyonic QPs and fully characterize the topological phase of the TCC. Eventually, we show that, in spite of the lack of a local order parameter, TEE and other physical quantities obtained from the ground state wave function such as the entanglement spectrum (ES) and ground state fidelity are sensitive probes to study the robustness of a topological phase. We find that the topological order in the presence of a magnetic field persists until the vicinity of the transition point, where the TEE and fidelity drops to zero and the ES splits severely, signaling breakdown of the topological phase of the TCC.
The fundamental understanding of phases and their transitions has been a central theme in condensed matter physics. Until recently, it was largely believed that the Landau symmetry breaking principle was effective in distinguishing different phases of matter, with broken symmetries signaling the phase transition. But with the discovery of topological phases which are beyond the Landau symmetry breaking principle, the identification and classification of quantum phases at absolute zero has opened up new unexplored avenues thus leading to exciting theoretical discoveries further propelling technological advancement. Topological phases of matter are characterized by the notion of topological order and in this work we aim to explore and understand topological phases by introducing novel signatures which characterize topological order. The robustness of these phases to external perturbation makes them an ideal candidate to store and manipulate quantum information thus making them an unique and interesting prospect for realizing quantum computers. There have been several signatures to characterize intrinsic topological order, for instance the invariance of the topologically ordered state under local operators, the dependence of ground state degeneracy on the underlying manifold and its robustness to external perturbation, topological entanglement entropy related to the quantum dimension of the supers-selection sectors, the inability to construct the topologically ordered state from a product state via constant depth unitary transformations. With toric code as the toy model, we analyze the robustness of topological order on a manifold supporting open boundaries by computing some of the above signatures which effectively detect a topological to trivial phase transition. We then probe the existence of a quantum criticality between distinct topological phases obtained by varying the underlying manifold. In these scenarios, most of the above signatures turn out to be ineffective in detecting the distinct phases leading to the introduction of an non- local order parameter whose construction is facilitated by the phenomenon of anyon condensation. The signatures for quantitatively and qualitatively characterizing intrinsic topological order being highly scenario dependent and also with its definition for mixed states being elusive we introduce an operational definition based on concepts of topological error correction. We define a state to be topologically ordered if the errors in the state can be corrected by an error correction circuit of finite depth. To concretize the notion of topological to trivial phase transition in an open setting we turn to nonequilibrium phenomenon, for example: Directed Percolation, with the change in percolation rate driving a dynamical phase transition between absorbing and active states with the former being topologically ordered while the latter being topologically trivial. Additionally, we explore the notion of topological phase transitions between distinct topological phases obtained by varying underlying topology in an open setting, analogous to the closed setting discussed earlier. To summarize, we have introduced various mixed states which exhibit topological order and also an operational definition to quantify topological order applicable across multitude of scenarios. We extend the above operational definition to quantify and detect quantum phase transitions in the case of Symmetry Protected Topological (SPT) phases. To further validate the above notion, we consider the perturbed variants of the Su-Schrieffer–Heeger (SSH) models and detect quantum phase transitions to a high accuracy by employing the techniques from the framework of tensor networks. It is significant to note the distinction of the error correction algorithms applied earlier in the case of intrinsic topological order were independent of symmetry constraints while in the current scenario we impose additional symmetry constraints to accurately detect the phase transition. In addition, we also devise error correction strategies with respect to topologically trivial states to detect quantum phase transitions which do not involve topological phases. This gives rise to a very fundamental question on whether error correction statistics with a well defined error correction algorithm, not necessarily optimal, are capable of detecting a equivalence classes of phases and thereby acting as a reliable probe to effectively detect topological/quantum phase transitions? From theoretical and numerical end of the spectrum we shift gears to explore possible experimental platforms with an aim to realize some of the quantum many-body phenomenon discussed earlier. While there have been several innovative experimental avenues to realize the above, one such promising candidate has been ultracold polar molecules setups that offer additional degrees of freedom due to the ro-vibrational degrees of freedom. Based on the chemical reaction between atoms and molecules which results in a quantum Zeno-based blockade, we devise several optimal strategies to efficiently detect molecules using atom as probe, we further extend the above technique to entangle the internal states of molecules and atoms. In addition, we also present optimal strategies for dissipative state engineering using the atom-molecule interactions.