A catalyst is a substance that enables otherwise impossible transformations between states of a system, without being consumed in the process. In this work, we apply the notion of catalysts to many-body quantum physics. In particular, we construct catalysts that enable transformations between different symmetry-protected topological (SPT) phases of matter using symmetric finite-depth quantum circuits. We discover a wide variety of catalysts, including GHZ-like states that spontaneously break the symmetry, gapless states with critical correlations, topological orders with symmetry fractionalization, and spin-glass states. These catalysts are all united under a single framework that has close connections to the theory of quantum anomalies, and we use this connection to put strong constraints on possible pure- and mixed-state catalysts. We also show how the catalyst approach leads to new insights into the structure of certain phases of matter, and to new methods to efficiently prepare SPT phases with long-range interactions or projective measurements.
A catalyst is a substance that enables otherwise impossible transformations between states of a system, without being consumed in the process. In this work, we apply the notion of catalysts to many-body quantum physics. In particular, we construct catalysts that enable transformations between different symmetry-protected topological (SPT) phases of matter using symmetric finite-depth quantum circuits. We discover a wide variety of catalysts, including GHZ-like states that spontaneously break the symmetry, gapless states with critical correlations, topological orders with symmetry fractionalization, and spin-glass states. These catalysts are all united under a single framework that has close connections to the theory of quantum anomalies, and we use this connection to put strong constraints on possible pure- and mixed-state catalysts. We also show how the catalyst approach leads to new insights into the structure of certain phases of matter, and to new methods to efficiently prepare SPT phases with long-range interactions or projective measurements.
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.
Dualities are central to our understanding of physics, in particular in the study of quantum phases of matter. In this dissertation, we explore exact lattice dualities that relate various topological phases of matter, which elucidates the interrelations between them. Moreover, searching for new dualities can even lead to the theoretical realization of new phases of matter. In the first half, we study local dualities, and how they can be used as quantum circuits to prepare Symmetry-Protected Topological (SPT) phases. We construct explicit quantum circuits that prepare fermionic SPT phases, which gives insight into the strongly interacting nature of these phases in contrast to previous free-fermion realizations. Moreover, we discuss a family of dualities, which not only generates the entanglement necessary to prepare SPT phases, but also generates anomalous symmetries which arise at transitions between such topological phases, giving new insights into topological quantum criticality. In the second half, we explore non-local dualities, exemplified by the famous Kramers-Wannier or Jordan-Wigner dualities in one dimension. we give a systematic method to generalize such non-local dualities to systems with different symmetries and to higher dimensions, which leads to explicit lattice models realizing new phases of matter. In particular, we uncover new families of fracton phases such as (i) Phases where immobile fracton excitations are emergent fermions (ii) Fracton phases that are enriched by spatial and/or time-reversal symmetries (iii) phases that "hybridizes" between conventional topological orders and fracton orders, some of which hosts non-abelian fracton excitations. Finally, we show that the two types of dualities can be related by an extra ingredient: measurement. This gives a practical method to realize non-local dualities in near-term quantum devices, paving a way to efficiently prepare a large class of topological phases of matter in experiments.
We develop a theoretical framework for the classification and construction of symmetry protected topological (SPT) phases, which are a special class of zero-temperature phases of strongly interacting gapped quantum many-body systems that exhibit topological properties. The framework unifies various proposals for the classification of SPT phases, including the group (super-)cohomology proposal, the (spin-)cobordism proposal, the Freed-Hopkins proposal, and the Kitaev proposal. The power of the framework is demonstrated in a number of applications: (1) the classification and construction of 3D fermionic SPT phases in Wigner-Dyson classes A and AII with glide symmetry, (2) the classification and construction of 3D bosonic SPT phases with space-group symmetries for all 230 space groups, (3) the derivation of a Mayer-Vietoris sequence relating the classification of SPT phases with and without reflection symmetry, and (4) an interpretation of the structure of general crystalline SPT phases via the Atiyah-Hirzebruch spectral sequence. The framework is based on Kitaev's idea that short-range entangled states form what are known as Ω-spectra in the sense of algebraic topology, and models the classification of SPT phases by what are called generalized cohomology theories.
We develop a theoretical framework for the classification and construction of symmetry protected topological (SPT) phases, which are a special class of zero-temperature phases of strongly interacting gapped quantum many-body systems that exhibit topological properties. The framework unifies various proposals for the classification of SPT phases, including the group (super-)cohomology proposal, the (spin-)cobordism proposal, the Freed-Hopkins proposal, and the Kitaev proposal. The power of the framework is demonstrated in a number of applications: (1) the classification and construction of 3D fermionic SPT phases in Wigner-Dyson classes A and AII with glide symmetry, (2) the classification and construction of 3D bosonic SPT phases with space-group symmetries for all 230 space groups, (3) the derivation of a Mayer-Vietoris sequence relating the classification of SPT phases with and without reflection symmetry, and (4) an interpretation of the structure of general crystalline SPT phases via the Atiyah-Hirzebruch spectral sequence. The framework is based on the idea that short-range entangled states form what are known as Ω-spectra in the sense of algebraic topology, and models the classification of SPT phases by what are called generalized cohomology theories.
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.
Symmetries play a central role in organizing many-body quantum phases of matter and in revealing universal low-energy behavior. Systems with gapless excitations are especially rich, as symmetries can act in intricate ways on low-energy degrees of freedom, while enhanced correlations and entanglement open possibilities not available in gapped settings. This thesis studies two settings in which symmetry considerations give rise to qualitatively new phenomena in gapless quantum matter and correlated electronic systems. In the first part of the thesis, I study how the framework developed for gapped topological phases can be generalized to gapless systems, especially those arising at topological phase transitions. I show that symmetry can refine a fixed conformal field theory universality class into distinct symmetry-enriched critical phases that cannot be smoothly connected without breaking symmetry or leaving the critical theory. In Chapter 1, I show that transitions between distinct symmetry-protected topological phases can exhibit exotic boundary critical behavior not seen in gapped systems. In Chapter 2, I show that distinct gapless symmetry-enriched phases can meet at an anomalous multicritical point. In Chapter 3, I show that spatial interfaces provide an operational indicator of these phases, and that distinct symmetry-enriched criticalities are separated by non-invertible conformal defects. Together, these results show that subtle distinctions in symmetry-enriched criticality have robust and physically detectable consequences. In the second part of the thesis, I study how symmetry and strong interactions govern charge carriers in correlated electron systems. In Chapter 4, I develop a microscopic theory of the Josephson diode effect in Josephson junctions, showing that nonreciprocal supercurrents arise universally from finite Cooper-pair momentum when time-reversal and inversion symmetry are simultaneously broken. In Chapter 5, I study spin-polaron formation in a flat-band ferromagnetic Hubbard model and show that binding occurs in a regime where the conventional mechanisms based on kinetic frustration or nontrivial band topology do not apply, but instead exotic composite charge carriers arise from quantum geometry.