Topological holography is a holographic principle that describes the generalized global symmetry of a local quantum system in terms of a topological order in one higher dimension. This framework separates the topological data from the local dynamics of a theory and provides a unified description of the symmetry and duality in gapped and gapless phases of matter. In this work, we develop the topological holographic picture for (1+1) d d quantum phases, including both gapped phases as well as a wide range of quantum critical points, including phase transitions between symmetry protected topological (SPT) phases, symmetry enriched quantum critical points, deconfined quantum critical points, and intrinsically gapless SPT phases. Topological holography puts a strong constraint on the emergent symmetry and the anomaly for these critical theories. We show how the partition functions of these critical points can be obtained from dualizing (orbifolding) more familiar critical theories. The topological responses of the defect operators are also discussed in this framework. We further develop a topological holographic picture for conformal boundary states of (1+1) d d rational conformal field theories. This framework provides a simple physical picture to understand conformal boundary states and also uncovers the nature of the gapped phases corresponding to the boundary 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.
We utilize the topological holographic framework to characterize and gain insights into the nature of quantum critical points and gapless phases in fermionic quantum systems. Topological holography is a general framework that describes the generalized global symmetry and the symmetry charges of a local quantum system in terms of a slab of a topological order, termed as the symmetry topological field theory (SymTFT), in one higher dimension. In this work, we consider a generalization of the topological holographic picture for (1+1) d fermionic quantum phases of matter. We discuss how spin structures are encoded in the SymTFT, and we establish the connection between the formal fermionization formula in quantum field theory and the choice of fermionic gapped boundary conditions of the SymTFT. We demonstrate the identification and the characterization of the fermionic gapped phases and phase transitions through detailed analysis of various examples, including the fermionic systems with Z 2 F , Z 2 × Z 2 F , Z 4 F , and the fermionic version of the noninvertible Rep ( S 3 ) symmetry. Our work uncovers many exotic fermionic gapped phases, quantum critical points, and gapless phases. These include gapped phases with fermionic noninvertible Rep ( S 3 ) symmetry, two kinds of fermionic symmetry-enriched quantum critical points, a fermionic gapless symmetry-protected topological phase, and a fermionic gapless spontaneous symmetry-breaking phase that breaks the fermionic noninvertible symmetry.
We outline a holographic framework that attempts to unify Landau and beyond-Landau paradigms of quantum phases and phase transitions. Leveraging a modern understanding of symmetries as topological defects/operators, the framework uses a topological order to organize the space of quantum systems with a global symmetry in one lower dimension. The global symmetry naturally serves as an input for the topological order. In particular, we holographically construct a String Operator Algebra (SOA) which is the building block of symmetric quantum systems with a given symmetry G in one lower dimension. This exposes a vast web of dualities which act on the space of G-symmetric quantum systems. The SOA facilitates the classification of gapped phases as well as their corresponding order parameters and fundamental excitations, while dualities help to navigate and predict various corners of phase diagrams and analytically compute universality classes of phase transitions. A novelty of the approach is that it treats conventional Landau and unconventional topological phase transitions on an equal footing, thereby providing a holographic unification of these seemingly-disparate domains of understanding. We uncover a new feature of gapped phases and their multi-critical points, which we dub fusion structure, that encodes information about which phases and transitions can be dual to each other. Furthermore, we discover that self-dual systems typically posses emergent non-invertible, i.e., beyond group-like symmetries. We apply these ideas to 1+1d 1 + 1 d quantum spin chains with finite Abelian group symmetry, using topologically-ordered systems in 2+1d 2 + 1 d . We predict the phase diagrams of various concrete spin models, and analytically compute the full conformal spectra of non-trivial quantum phase transitions, which we then verify numerically.
Two global symmetries are holoequivalent if their algebras of local symmetric operators are isomorphic. A holoequivalent class of global symmetries is described by a topological order (TO) in one higher dimension (called symmetry TO), which leads to a symmetry/topological-order (Symm/TO) correspondence. We establish that (1) for systems with a symmetry described by symmetry TO
M, their gapped and gapless states are classified by condensable algebras
A, formed by elementary excitations in
M with trivial self-/mutual statistics. These so-called
A states can describe symmetry breaking orders, symmetry protected topological orders, symmetry enriched topological orders, gapless critical points, etc. in a unified way. (2) The local low-energy properties of an
A state can be calculated from its reduced symmetry TO
M/A, using holographic modular bootstrap (holoMB) which takes
M/A as an input. Here
M/A is obtained from
M by condensing excitations in
A. Notably, an
A state must be gapless if
M/A is nontrivial. This provides a unified understanding of the emergence and symmetry protection of gaplessness that applies to symmetries that are anomalous, higher-form, and/or noninvertible. (3) The relations between condensable algebras constrain the structure of the global phase diagram. We find that, for $1+1\mathrm{D} {\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{\ensuremath{'}}$ symmetry with mixed anomaly, there is a stable continuous transition (deconfined quantum critical point) between the
Z2-breaking-${\mathbb{Z}}_{2}^{\ensuremath{'}}$-symmetric phase and the
Z2-symmetric-${\mathbb{Z}}_{2}^{\ensuremath{'}}$-breaking phase. The critical point is the same as a
Z4 symmetry breaking critical point. (4)
1+1D bosonic systems with
S3 symmetry have four gapped phases with unbroken symmetries
S3,Z3,Z2, and
Z1. We find a duality between two transitions ${S}_{3}\ensuremath{\leftrightarrow}{\mathbb{Z}}_{1}$ and ${\mathbb{Z}}_{3}\ensuremath{\leftrightarrow}{\mathbb{Z}}_{2}$: they are either both first order or both (stably) continuous, and in the latter case, they are described by the same conformal field theory (CFT). (5) The gapped and gapless states for
1+1D bosonic systems with anomalous
S3 symmetries are obtained as well. For example, anomalous
S3(1) and
S3(2) symmetries can have symmetry protected chiral gapless states with only symmetric irrelevant and marginal operators.
Classifying gapless phases of matter remains a fundamental challenge in condensed matter theory. The authors pioneer here a systematic classification of 1+1D bosonic gapless symmetry-protected topological phases via the Symmetry Topological Field Theory (SymTFT) framework. They derive, from physical considerations, the algebra of symmetry operators in a 1+1D bosonic gSPT and show that it precisely matches the structure of condensable algebras in the SymTFT. This study marks the first systematic classification of gapless quantum phases in the SymTFT paradigm.