The expectation value of the twist operator has been widely used as a polarization-based index for gapped and gapless phases in interacting quantum many-body systems. Although numerous studies support this usage in specific settings and rigorous results have established the validity of the criterion in important settings, the precise assumptions required for it to sharply distinguish gapped and gapless phases under more general conditions have not been fully clarified. In this work, we clarify the logical status of polarization-based indices by formulating symmetry-based statements under explicitly stated assumptions. We identify the role of ground-state degeneracy in the statements for gapped systems and clarify the distinct assumptions required to exclude gapless scenarios that could otherwise mimic gapped behavior in the thermodynamic limit. Building on this controlled framework, we construct a meaningful extension beyond one dimension, emphasizing that such an extension is nontrivial and cannot be obtained by a straightforward generalization of the one-dimensional twist operator. Our results delineate the regime in which polarization-based quantities are justified as sharply defined many-body indices.
The expectation value of the twist operator has been widely used as a polarization-based index for gapped and gapless phases in interacting quantum many-body systems. Although numerous studies support this usage in specific settings and rigorous results have established the validity of the criterion in important settings, the precise assumptions required for it to sharply distinguish gapped and gapless phases under more general conditions have not been fully clarified. In this work, we clarify the logical status of polarization-based indices by formulating symmetry-based statements under explicitly stated assumptions. We identify the role of ground-state degeneracy in the statements for gapped systems and clarify the distinct assumptions required to exclude gapless scenarios that could otherwise mimic gapped behavior in the thermodynamic limit. Building on this controlled framework, we construct a meaningful extension beyond one dimension, emphasizing that such an extension is nontrivial and cannot be obtained by a straightforward generalization of the one-dimensional twist operator. Our results delineate the regime in which polarization-based quantities are justified as sharply defined many-body indices.
We propose several general necessary conditions for quantum many-body system in one dimension respecting U(1) symmetry to be gapped. We show that the ground-state expectation value of topologically trivial twisting operators must approach unity in the thermodynamic limit with a certain finite-size scaling. Equivalently, its violation can indicate gaplessness of U(1)-symmetric Hamiltonians. The topological triviality of such a twisting operator enables us to derive infinitely many other gaplessness indicators by static structure factor to any order in real experiments, which are impossible to obtain by earlier topologically nontrivial twisting operators. We also apply analytic and numerical calculations to test the efficiency and consistency of our results.
Twisting-operator approach to identifying gaplessness in SU(N) fermionic systems
We propose a general necessary condition for a spinful fermion chain with SU(2) spin-rotation symmetry to be gapped. Specifically, we prove that the expectation value of a properly defined fermionic twisting operator asymptotically approaches unity in any gapped phase with finite ground-state degeneracy, with finite-size corrections bounded by O(1/L). Consequently, a non-unity value in the thermodynamic limit provides a sufficient criterion for identifying gapless fermion chains. We confirm this criterion using the s-wave Bardeen-Cooper-Schrieffer (BCS) Hamiltonian and determinant quantum Monte Carlo (DQMC) simulations of interacting Hubbard models. We further extend the twisting-operator approach to SU(N)-symmetric fermionic systems, where the gapped ground-state sector must be SU(N)-singlet and ⟨U^M⟩=1+O(1/L) by a fermionic twisting operator U^ with a suitably chosen integer M.
Identifying whether a many-body ground state is gapped or gapless is a fundamental yet challenging problem, especially in quantum spin liquids. In this work, we develop a gap-diagnostic scheme based on the polarization amplitude defined via a twist operator, evaluated within the infinite density-matrix renormalization group (iDMRG) framework. As a benchmark, analysis of the spin-1/2 XXZ chain demonstrates that the polarization amplitude clearly distinguishes the gapless Tomonaga-Luttinger liquid from the gapped Néel phase. We then extend this framework to infinite cylinders of the spin-1/2 XY-Jχ model on the square lattice. We find that the polarization amplitude sharply detects the transition between the gapless XY phase and the gapped chiral spin liquid phase. These results show that polarization amplitudes provide a strong energy-gap diagnostic in two-dimensional frustrated quantum magnets, including quantum spin liquids.
The Schrödinger equation is nearly a century old, yet we are still in the midst of uncovering the remarkable phenomena emerging in many-body quantum systems. From superconductivity to anyonic quasiparticles, nature consistently surprises with its rich self-organization. To elucidate and grasp this variety, it is paramount to understand the phases of matter that can occur in many-body ground states, as well as their emergent collective excitations. Of particular interest are topological phases of matter, characterized by exotic excitations or edge phenomena. There exist by now several universal frameworks for gapped systems, i.e., those with an energy gap above the ground state. However, in the last decade, a multitude of gapless quantum wires---effectively one-dimensional systems---have been reported to be topologically non-trivial. A framework for their understanding and classification is missing. In addition to ground state order---topological or otherwise---a more complete picture involves the properties of excitations above the ground state. Alas, little is known about excitations beyond the universal low-energy regime. In part, this is due to a lack of analytical and numerical methods able to describe excitations at finite energies, especially in strongly-interacting systems beyond one dimension. In this thesis, we address these issues: firstly, we build a general understanding of topological phases in one dimension, including both gapped and gapless cases. In particular, we unify previously studied examples into a single framework. Secondly, we develop a novel numerical method for obtaining spectral functions in two dimensions---these give direct insight into the properties of excitations and are moreover experimentally measurable. Using this numerical method, we uncover a variety of robust properties of excitations at finite energies. Part I of this thesis concerns gapped and gapless topological phases in one dimension. In Chapter 2, we first treat the case of non-interacting fermions. Therein, we review the known classification of gapped phases before extending it to the gapless case, showing that exponentially-localized Majorana zero modes can still emerge at the edge when the bulk is gapless. Interacting gapped phases are discussed in Chapter 3, with a focus on symmetry-protected topological order. These have already been classified; our contribution is to provide a non-technical review of this classification as well as showing that many paradigmatic model Hamiltonians can be related to one another. Finally, Chapter 4 introduces the notion of symmetry-enriched quantum criticality, which we propose as a framework for classifying gapless phases. The key message is that in the presence of symmetries, a universality class can divide into distinct phases, characterized by the symmetry action on the low-energy scaling operators. This includes gapless topological phases, with examples hiding in plain sight; we clarify their stability and reinterpret previously studied examples. Part II studies the excitations above the ground states of two-dimensional quantum spin models. The main object of our study is the dynamic spin structure factor; this type of spectral function is reviewed in the first part of Chapter 5. The second part of this chapter introduces a novel matrix-product-state-based algorithm to efficiently compute it, opening a new window on the dynamics of two-dimensional quantum systems. We benchmark this numerical method in Chapter 6 on the exactly-solvable Kitaev model---a paradigmatic topological model realizing a quantum spin liquid. By adding non-integrable Heisenberg perturbations, we identify the first unequivocal theoretical realization of a proximate spin liquid: the ground state becomes conventionally ordered, yet the high-energy spectral properties are structurally similar to those of the nearby Kitaev spin liquid. The latter agrees with aspects of recent inelastic neutron scattering experiments on alpha-RuCl3. In Chapter 7, we turn to one of the oldest models in many-body quantum physics: the spin-1/2 Heisenberg antiferromagnet on the square lattice. Despite its venerable history, there is still disagreement about the physical origin of high-energy spectral features which low-order spin wave theory cannot account for. We provide a simple picture for this strongly-interacting-magnon feature by connecting it to a simple Ising limit. Lastly, Chapter 8 discusses the stability of quasiparticles---collective excitations behaving like a single emergent entity, of which magnons are a prime example. These are often known to be stable at the lowest energies and are presumed to decay whenever this is seemingly allowed by energy and momentum conservation. However, we show that strong interactions can prevent this from happening. We numerically confirm this principle of avoided decay in the (slightly-detuned) Heisenberg antiferromagnet on the triangular lattice. Moreover, we can even identify its fingerprints in existing experimental data on Ba3CoSb2O9 and superfluid helium. In this thesis, we thus enlarge our understanding of quantum phases and their excitations. The identification of the key principles of gapless topological phases in one dimension calls for direct analogues in higher dimensions, waiting to be uncovered. With regard to the robust properties of the excitations identified in this thesis, we are hopeful that these can be extended into a theory of quasiparticle properties away from the universal low-energy regime.
The symmetry and the locality are the two major sources of various general theorems in quantum many-body systems. We demonstrate that, in gapped phases of a U(1) symmetric Hamiltonian with finite-range interactions, the bulk properties such as the expectation value of local operators, the ground-state energy and the excitation gap, and the static and low-frequency dynamical responses in general do not depend on the U(1) phase of the twisted boundary condition in the limit of the large system size. Specifically, their dependence on the twisted angle is exponentially suppressed with the linear dimension of the system. Our argument is solely based on the exponential decay of various types of equal-time correlation functions and does not assume any details of the Hamiltonian, meaning that the statement applies quite generally regardless of the dimensionality or the interaction strength of the system.
We propose a general necessary condition for a spin chain with SO(3) spin-rotation symmetry to be gapped. Specifically, we prove that the ground state(s) of an SO(3)-symmetric gapped spin chain must be spin singlet(s), and the expectation value of a twisting operator asymptotically approaches unity in the thermodynamic limit, where finite-size corrections are inversely proportional to the system size. This theorem provides (i) supporting evidence for various conjectured gapped phases and (ii) a sufficient criterion for identifying gapless spin chains. We verify our theorem by numerical simulations for a variety of spin models and show that it offers a novel efficient way to identify gapless phases in spin chains with spin-rotation symmetry.