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arXiv:2311.17799 · Physical Review Letters

Topological Quantum Criticality from Multiplicative Topological Phases

R. Flores-Calderón, Elio J. König, Ashley M. Cook

Published Mar 17, 2025• 8 citations
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Abstract

Symmetry-protected topological phases (SPTs) characterized by short-range entanglement include many states essential to the understanding of topological condensed matter physics, and the extension to gapless SPTs provides essential understanding of their consequences. In this work, we identify a fundamental connection between gapless SPTs and recently introduced multiplicative topological phases, demonstrating that multiplicative topological phases are an intuitive and general approach to realizing concrete models for gapless SPTs. In particular, they are naturally well suited to realizing higher-dimensional, stable, and intrinsically gapless SPTs through the combination of canonical topological insulator and semimetal models with critical gapless models in symmetry-protected tensor product constructions, opening avenues to far broader and deeper investigation of topology via short-range entanglement.

DOI
10.1103/physrevlett.134.116602

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Symmetry-protected topological phases (SPTs) characterized by short-range entanglement include many states essential to the understanding of topological condensed matter physics, and the extension to gapless SPTs provides essential understanding of their consequences. In this work, we identify a fundamental connection between gapless SPTs and recently introduced multiplicative topological phases, demonstrating that multiplicative topological phases are an intuitive and general approach to realizing concrete models for gapless SPTs. In particular, they are naturally well suited to realizing higher-dimensional, stable, and intrinsically gapless SPTs through the combination of canonical topological insulator and semimetal models with critical gapless models in symmetry-protected tensor product constructions, opening avenues to far broader and deeper investigation of topology via short-range entanglement.
Abstract Symmetry-protected topological phases of matter have challenged our understanding of condensed matter systems and harbour exotic phenomena promising to address major technological challenges. Considerable understanding of these phases of matter has been gained recently by considering additional protecting symmetries, different types of quasiparticles, and systems out of equilibrium. Here, we show that symmetries could be enforced not just on full Hamiltonians, but also on their components. We construct a large class of previously unidentified multiplicative topological phases of matter characterized by tensor product Hilbert spaces similar to the Fock space of multiple particles. To demonstrate our methods, we introduce multiplicative topological phases of matter based on the foundational Hopf and Chern insulator phases, the multiplicative Hopf and Chern insulators (MHI and MCI), respectively. The MHI shows the distinctive properties of the parent phases as well as non-trivial topology of a child phase. We also comment on a similar structure in topological superconductors as these multiplicative phases are protected in part by particle-hole symmetry. The MCI phase realizes topologically protected gapless states that do not extend from the valence bands to the conduction bands for open boundary conditions, which respects to the symmetries protecting topological phase. The band connectivity discovered in MCI could serve as a blueprint for potential multiplicative topology with exotic properties.
Symmetry, topology and entanglement are three indispensable concepts for our understanding of quantum many-body systems. In this dissertation, we demonstrate the use and interplay of these concepts with four distinct topics. The first part focuses on two-dimensional (2D) symmetry protected topological (SPT) phases with lattice rotation symmetries. We start by discussing a prototypical example that exhibits interesting topological responses, and then investigate general aspects of rotation protected topology. In the second part, we report on a Hartree-Fock calculation of possible symmetry-broken phases in magic angle bilayer graphene at charge neutrality, assuming that the valley U(1) symmetry is preserved. Our solutions include both correlated insulators and nematic topological semimetals, whose relative stability can be tuned by weak strains. In the third part, inspired by the recent studies on gapless SPT states, we examine the fate of magnetic impurities at 2+1D quantum critical points. We introduced a series of large-N solvable impurity models which, as a consequence of topology, demonstrate intermediate coupling fixed points. In the last part, we develop a diagrammatic method to compute the entanglement properties of random mixed states which are obtained by partial tracing random pure states and are relevant for a variety of problems ranging from chaotic quantum dynamics to black hole physics.
We study several aspects of the realization of global symmetries in highly entangled phases of quantum matter. Examples include gapped topological ordered phases, gapless quantum spin liquids, and non-Fermi liquid phases. An insightful window into such phases is provided by recent developments in the theory of short-ranged entangled symmetry protected topological (SPT) phases. First, they generate useful no-go constraints on how global symmetry may be implemented in a highly entangled phase. Possible symmetry implementation in gapped topological phases and some proposed gapless spin/Bose liquids are examined in this light. We show that some previously proposed spin liquid states for 2D quantum magnets do not in fact have consistent symmetry implementation unless they occur as the surface of a 3D SPT phase. A second SPT-based insight into highly entangled states is the development of a view point of such states as SPT phases of one of the emergent excitations. We describe this in the specific context of time-reversal symmetric 3D U(1) quantum spin liquids with an emergent photon. Different such spin liquids are shown to be equivalent to different SPT insulating phases of the emergent monopole excitation of such phases. The highly entangled states also in turn enrich our understanding of SPT phases. We use the insights obtained from our results to provide an explicit construction of bosonic SPT phases in 3D in a system of coupled layers. This includes construction of a time-reversal symmetric SPT state that is not currently part of the cohomology classification of such states.
One problem of paramount importance in physics over various fields such as elementary particle physics, condensed matter physics, and statistical mechanics has been to make classification of phases of matter, including nontrivial structures of the vacuum. The triumphant history of theory on the phase classification and the phase transition can be traced back to Landau, which is based on the spontaneous breaking of symmetry. It is known, however, that other kinds of phases exist that cannot be characterized by conventional arguments of symmetry breaking, namely, topological phases. Topological phases are gapped phases at zero temperature without symmetry breaking. Examples of the topological phases include a quantum phase in a fractional quantum Hall system, where the ground state exhibits degeneracy, and an anyon, a quasi-particle with nontrivial statistical nature, neither boson nor fermion, emerges as a low-energy excitation. In contemporary physics of the 21st century, it has been revealed that, among various topological states, there exists a nontrivial ground state that does not have ground state degeneracy and yet can be distinguished from a trivial state. This nontrivial ground state, i.e., a topological insulator, has been recognized first in free electron systems [1–3] and later extended to more general quantum many-body systems with interactions including bosonic (spin) systems. Now, such a generalized situation is called a symmetry protected topological (SPT) phase [4–7]. From the point of a modern view of the SPT phase, nontrivial states, such as the Haldane state proposed in antiferromagnetic quantum spin systems in the 1980’s [8] and the ground state in the Affleck-Kennedy-Lieb-Tasaki model [9], belong to the SPT phase. A remarkable feature of the SPT phase is that, if the system has a surface, gapless surface states appear and they are stable against disturbances which do not break the symmetry. One way to understand these surface states is the quantum anomaly on the surface. That is, as long as the surface is concerned, the quantum anomaly arises, but the whole system has no quantum anomaly, and the symmetry is preserved due to cancellation from the bulk contribution. This canceling mechanism of the quantum anomaly has been long known as the anomaly inflow in elementary particle physics [10]. In fact, the classification of the SPT phases can be identified as the classification of the quantum anomaly through the bulk-boundary correspondence. This type of the quantum anomaly closely related to the SPT phase can provide us with a useful theoretical device to constrain phases that can or cannot be realized under the ’t Hooft anomaly matching condition [11–13]. It is quite difficult to scrutinize nonperturbatively interacting systems, but the ’t Hooft anomaly matching condition can be applied to those systems, and possible phases can be constrained. Nowadays, our understanding of the SPT phases has been so developed that applications of the anomaly matching condition can cover systems with finite groups and/or extending objects [14–16]. In this way, the concept of the topological phases has been essential by now in a wide range of physics, particularly in elementary particle physics, condensed matter physics, and statistical mechanics, as mentioned in the beginning. This special section is a collection of papers related to the topological aspects of physics including the topological phases. We would like to express our sincere gratitude to all the authors who made excellent contributions.
Abstract Classifying and understanding phases of matter is an important task in condensed matter physics. The class of ‘conventional’ symmetry-broken phases is well understood in terms of Landau’s theory. A paradigmatic example is the Z2-symmetric Ising model with symmetric (paramagnetic) and symmetry-broken (ferromagnet) phases, which can be distinguished by measuring the magnetization as a local order parameter. Topological phases of matter, however, are less understood and no complete classification is known. These lecture notes discuss schemes that provide an understanding and characterization of certain topological phases of matter in models with local Hamiltonians. The main focus is on symmetry-protected topological (SPT) phases in one-dimensional bosonic systems. On the basis of the entanglement properties of such systems, a motivation is given for the matrix product state (MPS) representation of ground states. Using the MPS framework, it is demonstrated how SPT phases can be classified using projective representations of the symmetries.
Symmetry-protected topological (SPT) states are short-range entangled states with a symmetry GGG. They belong to a new class of quantum states of matter which are classified by the group cohomology Hd+1(G,R/Z){\mathcal{H}}^{d+1}(G,\mathbb{R}/\mathbb{Z})Hd+1(G,R/Z) in ddd-dimensional space. In this paper, we propose a class of symmetry-protected topological invariants that may allow us to fully characterize SPT states with a symmetry group GGG [i.e., allow us to measure the cocycles in Hd+1(G,R/Z){\mathcal{H}}^{d+1}(G,\mathbb{R}/\mathbb{Z})Hd+1(G,R/Z) that characterize the SPT states]. We give an explicit and detailed construction of symmetry-protected topological invariants for 2+12+12+1D SPT states. Such a construction can be directly generalized to other dimensions.