One of the important characteristics of topological phases of matter is the topology of the underlying manifold on which they are defined. In this paper, we present the sensitivity of such phases of matter to the underlying topology, by studying the phase transitions induced due to the change in the boundary conditions. We claim that these phase transitions are accompanied by broken symmetries in the excitation space and to gain further insight we analyze various signatures like the ground state degeneracy, topological entanglement entropy while introducing the open-loop operator whose expectation value effectively captures the phase transition. Further, we extend the analysis to an open quantum setup by defining effective collapse operators, the dynamics of which cool the system to distinct steady states both of which are topologically ordered. We show that the phase transition between such steady states is effectively captured by the expectation value of the open-loop operator.
One of the important characteristics of topological phases of matter is the topology of the underlying manifold on which they are defined. In this paper, we present the sensitivity of such phases of matter to the underlying topology, by studying the phase transitions induced due to the change in the boundary conditions. We claim that these phase transitions are accompanied by broken symmetries in the excitation space and to gain further insight we analyze various signatures like the ground state degeneracy, topological entanglement entropy while introducing the open-loop operator whose expectation value effectively captures the phase transition. Further, we extend the analysis to an open quantum setup by defining effective collapse operators, the dynamics of which cool the system to distinct steady states both of which are topologically ordered. We show that the phase transition between such steady states is effectively captured by the expectation value of the open-loop operator.
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.
Although the topological order is known as a quantum order in quantum many-body systems, it seems that there is not a one-to-one correspondence between topological phases and quantum phases. As a well-known example, it has been shown that all one-dimensional (1D) quantum phases are topologically trivial [X. Chen et al., Phys. Rev. B 83, 035107 (2011)]. By such facts, it seems a challenging task to understand when a quantum phase transition between different topological models necessarily reveals different topological classes of them. In this paper, we make an attempt to consider this problem by studying a phase transition between two different quantum phases which have a universal topological phase. We define a Hamiltonian as interpolation of the toric code model with Z2 topological order and the color code model with ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ topological order on a hexagonal lattice. We show such a model is exactly mapped to many copies of 1D quantum Ising model in transverse field by rewriting the Hamiltonian in a new complete basis. Consequently, we show that the universal topological phase of the color code model and the toric code model reflects in the 1D nature of the phase transition. We also consider the expectation value of Wilson loops by a perturbative calculation and show that behavior of the Wilson loop captures the nontopological nature of the quantum phase transition. The result on the point of phase transition also shows that the color code model is strongly robust against the toric code model.
We investigate the impact of an isotropic antiferromagnetic Heisenberg perturbation on the toric code, focusing on the resulting quantum phase transition and the nature of the phase that emerges beyond topological order. Using neural-network quantum states (NQS), we compute ground states over a wide range of Heisenberg couplings while fully respecting the exact symmetries of the model. In the weak-coupling regime, the numerical results are in excellent agreement with an effective low-energy description derived from a Schrieffer-Wolff (SW) transformation, providing analytic control over the perturbative breakdown of topological order. We show that the Heisenberg perturbation only renormalizes local operators at low orders, whereas mixing between topological sectors occurs only at a perturbative order proportional to the system size. At intermediate values of the Heisenberg interaction, the topological phase breaks down. We estimate the critical point through a combination of the fidelity susceptibility and the logarithmic susceptibility of non-contractible Wilson loops for various system sizes. Furthermore, we utilize the topological entanglement entropy to provide a comprehensive characterization of the phase transition. Beyond the transition, an antiferromagnetic ±X/±Z Néel phase emerges, characterized by a fourfold-degenerate symmetry-broken manifold, which is explicitly probed using staggered-magnetization-based diagnostics. Our results show how local two-spin interactions, which naturally arise in realistic implementations of the toric code, drive the breakdown of topological order. Moreover, we establish the SW approach as a systematic framework for analyzing such perturbations in combination with variational many-body methods.
The interplay of the constituents of interacting many-body systems may reveal emergent properties on the macroscopic scale which are not inherent to the individual constituents. These properties are expressed in macroscopic observables \ndescribing the state — denoted as the phase of the system. Continuous phase transition between phases are generically manifested in critical behavior, for example, a divergence of a macroscopic observable. The identification of the present phase of a system and the classification of critical phenomena into universality classes are exciting challenges of condensed-matter physics. \nIn this thesis, we use entanglement entropies as macroscopic quantities for the characterization of phases of quantum matter and critical theories. For groundstates of quantum many-body systems the entanglement entropy is a measure of \nthe amount of entanglement between two subsystems. The generic dependence of the entanglement entropy on the size and shape of the subsystems is contained in the well-known boundary law — stating a scaling of the entanglement entropy with the boundary between the two subsystems. We numerically investigate how the coefficient of this dependence reflects quantum phase transitions in simple spin-half bilayer models. \nSubleading terms to the boundary law such as a logarithmic contribution provide universal numbers for the criticality of field theories. We examine free and interacting theories from an entanglement entropy perspective in order to assess the role of the coefficient of the logarithmic correction induced by corners in the subsystems. Beyond its universality, this coefficient also quantifies degrees of freedom of low-lying excitations in the conformal field theory describing a critical point. \nA constant contribution to the boundary law indicates the presence of so-called topological order in the ground state of a many-body system. This extremely useful property can also be identified in classical counterparts of entanglement entropy which we study at the example of various toric code models. To this endeavor, we have designed Monte Carlo techniques which allow for an efficient numerical computation of the constant contribution. In particular, we analyze via entanglement entropies under which conditions remnants of topological order are present in the quantum system at finite temperature and at perturbations from a magnetic field. The major motivation behind this effort is to use topological order for the robust storage of a quantum information — a basic need for the construction of quantum \ncomputers.
Quantum phase transitions between different topologically ordered phases exhibit rich structures and are generically challenging to study in microscopic lattice models. In this paper, we propose a tensor-network solvable model that allows us to tune between different symmetry enriched topological (SET) phases. Concretely, we consider a decorated two-dimensional toric code model for which the ground state can be expressed as a two-dimensional tensor-network state with bond dimension D=3 and two tunable parameters. We find that the time-reversal (TR) symmetric system exhibits three distinct phases: (i) an SET toric code phase in which anyons transform nontrivially under TR, (ii) a toric code phase in which TR does not fractionalize, and (iii) a topologically trivial phase that is adiabatically connected to a product state. We characterize the different phases using the topological entanglement entropy and a membrane order parameter that distinguishes the two SET phases. Along the phase boundary between the SET toric code phase and the toric code phase, the model has an enhanced U(1) symmetry and the ground state is a quantum critical loop gas wavefunction whose squared norm is equivalent to the partition function of the classical O(2) model. By duality transformations, this tensor-network solvable model can also be used to describe transitions between SET double-semion phases and between ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T}$ symmetry protected topological phases in two dimensions.
Entanglement entropy has proven to be a powerful probe of phenomena such as quantum chaos and phase transitions. Pseudo entropy is a recently proposed time-like generalization of an entanglement measure, motivated by de Sitter holography. In this work, we find that pseudo entropy can also serve as a novel probe for distinguishing topological phases of matter. For this, we consider the Su--Schrieffer--Heeger model as a representative example and investigate the averaged excess entropy ΔS12, defined as the difference between pseudo entropy and the average entanglement entropy, across the topological-to-trivial and trivial-to-topological phase transitions. When the two states are in the same phase, we find that $ ΔS_{12}$ is non-positive under periodic boundary conditions, while for open boundary conditions, it is non-positive only when the system is sufficiently large. Moreover, we analyze ground-state quench protocols for topology-crossing quenches and find that the imaginary pseudo entropy tracks the critical times predicted by the Fisher zeros.