In this paper, we investigate signatures of topological phase transitions in interacting systems. We show that the key signature is the existence of a topologically protected level crossing, which is robust and sharply defines the topological transition, even in finite-size systems. Spatial symmetries are argued to play a fundamental role in the selection of the boundary conditions to be used to locate topological transitions in finite systems. We discuss the theoretical implications of these results, and utilize exact diagonalization to demonstrate its manifestations in the Haldane-Fermi-Hubbard model. Our findings provide an efficient way to detect topological transitions in experiments and in numerical calculations that cannot access the ground-state wave function.
In this paper, we investigate signatures of topological phase transitions in interacting systems. We show that the key signature is the existence of a topologically protected level crossing, which is robust and sharply defines the topological transition, even in finite-size systems. Spatial symmetries are argued to play a fundamental role in the selection of the boundary conditions to be used to locate topological transitions in finite systems. We discuss the theoretical implications of these results, and utilize exact diagonalization to demonstrate its manifestations in the Haldane-Fermi-Hubbard model. Our findings provide an efficient way to detect topological transitions in experiments and in numerical calculations that cannot access the ground-state wave function.
We demonstrate the existence of topological phase transitions in interacting, symmetry-protected quantum matter at finite temperatures. Using a combined numerical and analytical approach, we study a one-dimensional Su-Schrieffer-Heeger model with added Hubbard interactions, where no thermodynamic phase transition occurs at finite temperatures. The transition is signalled by a quantized, nonlocal bulk topological order parameter. It is driven by defects, which are enabled by the combination of interaction and thermal activation, with no counterpart in the noninteracting limit. The defects localize topological zero modes, which, when sufficiently abundant, cause the order parameter to vanish. This phenomenon, interpreted via bulk-boundary correspondence, reflects the loss of a topological edge mode at a well-defined critical temperature in the thermodynamic limit. Unlike zero-temperature topological transitions, these finite-temperature transitions lack thermodynamic signatures but remain observable in controlled quantum systems, such as ultracold fermionic atoms in optical lattices.
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.
For decades, Landau’s theory of phase transition has provided a successful classification for quantum and classical states of matter based on their symmetry broken patterns, except for certain exotic quantum states such as the fractional quantum Hall (FQH) effect. However, such exotic phenomena are crucial for a complete understanding of the nature. This thesis explores new physical principles emerging from topology and topological states. First, we use an example to demonstrate Landau’s theory by studying a pair-density wave system, where the symmetry-breaking paradigm is applicable. It turns out that such a system exhibits the Kosterlitz-Thouless (KT) transition, a phase transition driven by the proliferation of topological defects, i.e. vortex-antivortex pairs. For topological systems, where symmetry broken pattern cannot be used as a classification tool, we prove a theorem that two gapped systems with non-vanishing ground-state overlap must be adiabatically connected, and thus are necessarily in the same topological phase. This theorem provides a simple and generic approach to classify topological band insulators/superconductors, without the needs to calculate any known or yet-to-known topological indices. Once the overlap is found nonzero, the two systems must be topologically identical. After presenting a generic proof, the theorem is also verified through calculating the overlap for several milestone topological band insulators and certain interacting systems. Such an overlap technique is then generalized to (2+1)-D strongly-interacting topological systems at fixed points, including both symmetry-protected topological (SPT) states and intrinsic topological states like FQH. For interacting topological states, the main challenge of utilizing wave-function overlaps to classify them lies in the famous Anderson orthogonality catastrophe (AOC), which states that two different many-body wave functions must have zero overlap in the thermodynamic limit. In this thesis, we found that wave-function overlaps indeed carry critical information about the topological nature of quantum states and this information can be extracted from the finite-size scaling of the overlaps. In the finite-size scaling analysis, we found a universal topological response term as a sub-leading contribution. This term depends on both the central charge of the corresponding conformal field theory (CFT) and the Euler characteristics of the underlying manifolds on which the system is defined. This term reveals a fundamental connection between ground-state overlaps and CFT’s. In addition, surprisingly, the overlap between an intrinsic topological state and a topologically trivial product state shows a decay faster than the exponential behavior expected via a typical AOC analysis. Such finite-size scaling behaviors could be utilized to theoretically detect the gapless edge modes, and to distinguish the topology of quantum states or serve as a signature of topological phase transitions. Possible generalization to higher dimensions and generic non-fixed-point topological systems is also discussed.
We study strong correlation effects in topological insulators via the Lanczos algorithm, which we utilize to calculate the exact many-particle ground-state wave function and its topological properties. We analyze the simple, noninteracting Haldane model on a honeycomb lattice with known topological properties and demonstrate that these properties are already evident in small clusters. Next, we consider interacting fermions by introducing repulsive nearest-neighbor interactions. A first-order quantum phase transition was discovered at finite interaction strength between the topological band insulator and a topologically trivial Mott insulating phase by use of the fidelity metric and the charge-density-wave structure factor. We construct the phase diagram at T=0 as a function of the interaction strength and the complex phase for the next-nearest-neighbor hoppings. Finally, we consider the Haldane model with interacting hard-core bosons, where no evidence for a topological phase is observed. An important general conclusion of our work is that despite the intrinsic nonlocality of topological phases their key topological properties manifest themselves already in small systems and therefore can be studied numerically via exact diagonalization and observed experimentally, e.g., with trapped ions and cold atoms in optical lattices.
Topological phases of matter, characterized by robust edge states and quantized bulk invariants, have revolutionized condensed matter physics. Identifying and classifying their associated phase transitions, however, remains a significant challenge, particularly when conventional order parameters are absent. This paper introduces and explores the utility of determinantal quantities as powerful signatures for detecting and characterizing topological phase transitions. We demonstrate how determinants derived from the Bloch Hamiltonian, Green's functions, or related matrices can exhibit distinct, often singular, behaviors at critical points separating different topological phases. Through theoretical analysis and illustrative examples from various topological models, including the Su-Schrieffer-Heeger model, Haldane model, and Weyl semimetals, we show that these determinantal signatures provide a direct and robust link to the underlying topological invariants and spectral properties, such as band gap closures and inversions. The methodology offers a complementary and potentially more direct approach to unraveling the intricate nature of topological phase transitions, opening new avenues for both theoretical understanding and experimental detection in a wide range of quantum materials.
Characterizing topological phases for strongly interacting fermions in the mixed-state regime remains a major challenge. Here we introduce a general and numerically efficient framework to diagnose mixed-state topological phases in strongly interacting systems via the disorder parameter (DP) of the U(1) charge operator. Specifically, from the finite-size scaling of the second derivative of the DP generating function, we introduce the topological scaling indicator, which exhibits a characteristic linear scaling with the system's linear dimension for topological phases, a signature that vanishes upon transition into a topologically trivial phase. Crucially, we develop an efficient determinant Quantum Monte Carlo algorithm that facilitates the evaluation of this indicator in interacting systems. We apply our approach to two paradigmatic models: for the Kane-Mele-Hubbard model, we successfully map the interaction-driven transition from a quantum spin Hall insulator to a trivial Mott insulator. Furthermore, our method circumvents the limitations imposed by the severe sign problem in the Haldane-Hubbard model, enabling robust identification of the quantum anomalous Hall phase at accessible temperatures. This work provides a powerful and accessible tool for the numerical exploration of topological phenomena in interacting mixed states, opening a pathway to study systems previously inaccessible due to computational obstacles.