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arXiv:1808.07465 · Physical Review X

Adventure in Topological Phase Transitions in 3 + 1 -D: Non-Abelian Deconfined Quantum Criticalities and a Possible Duality

Zhen Bi, T. Senthil

Published May 20, 2019• 93 citations
arXivHTMLPDF

Abstract

Continuous quantum phase transitions beyond the conventional paradigm of fluctuations of a symmetry-breaking order parameter are challenging for theory.These phase transitions often involve emergent deconfined gauge fields at the critical points as demonstrated in phase transitions between different broken-symmetry states of 2 þ 1-dimensional quantum magnets, as well as those between symmetry-protected topological (SPT) phases.In this paper, we present several examples of deconfined quantum critical points between SPT phases in 3 þ 1-D for both bosonic and fermionic systems.These critical theories can be formulated as non-Abelian gauge theories either in the infrared-free regime or in the conformal window when they flow to the Banks-Zaks fixed points.We explicitly demonstrate several interesting quantum critical phenomena.We describe situations in which the same phase transition allows for multiple universality classes controlled by distinct fixed points.We exhibit the possibility-which we dub "unnecessary quantum critical points"-of stable generic continuous phase transitions within the same phase.We present examples of interaction-driven, band-theory-forbidden, continuous phase transitions between two distinct band insulators.The understanding we develop leads us to suggest an interesting possible 3 þ 1-D field theory duality between SUð2Þ gauge theory coupled to one massless adjoint Dirac fermion and the theory of a single massless Dirac fermion augmented by a decoupled topological field theory.

DOI
10.1103/physrevx.9.021034

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Continuous quantum phase transitions beyond the conventional paradigm of fluctuations of a symmetry-breaking order parameter are challenging for theory.These phase transitions often involve emergent deconfined gauge fields at the critical points as demonstrated in phase transitions between different broken-symmetry states of 2 þ 1-dimensional quantum magnets, as well as those between symmetry-protected topological (SPT) phases.In this paper, we present several examples of deconfined quantum critical points between SPT phases in 3 þ 1-D for both bosonic and fermionic systems.These critical theories can be formulated as non-Abelian gauge theories either in the infrared-free regime or in the conformal window when they flow to the Banks-Zaks fixed points.We explicitly demonstrate several interesting quantum critical phenomena.We describe situations in which the same phase transition allows for multiple universality classes controlled by distinct fixed points.We exhibit the possibility-which we dub "unnecessary quantum critical points"-of stable generic continuous phase transitions within the same phase.We present examples of interaction-driven, band-theory-forbidden, continuous phase transitions between two distinct band insulators.The understanding we develop leads us to suggest an interesting possible 3 þ 1-D field theory duality between SUð2Þ gauge theory coupled to one massless adjoint Dirac fermion and the theory of a single massless Dirac fermion augmented by a decoupled topological field theory.
The deconfined quantum critical point, a prototype Landau-forbidden transition, could exist, in principle, in the phase transitions involving a symmetry-protected topological phase; however, examples of such kinds of transitions in physical systems are rare beyond one-dimensional systems. Here, using a density-matrix renormalization-group calculation, we unveil a bosonic integer quantum Hall phase in a two-dimensional correlated honeycomb lattice, by full identification of its internal structure from the topological K\mathbf{K}K matrix. Moreover, we demonstrate that imbalanced periodic chemical potentials can destroy the bosonic integer quantum Hall state and drive it into a featureless trivial (Mott) insulator, where all physical observables evolve smoothly across the critical point. At the critical point, the entanglement entropy reveals a characteristic scaling behavior, which is consistent with the critical field theory as an emergent QED3{\mathrm{QED}}_{3}QED3​ with two flavors of Dirac fermions.
Decades of research have revealed a deep understanding of topological quantum matter with protected edge modes. We report that even richer physics emerges when tuning between two topological phases of matter whose respective edge modes are incompatible. The frustration at the edge leads to novel boundary physics, such as symmetry-breaking phases with exotic non-Landau transitions---even when the edge is zero-dimensional. As a minimal case study, we consider spin chains with ${\mathbb{Z}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{3}$ symmetry, exhibiting two nontrivial symmetry-protected topological (SPT) phases. At the bulk 1+1D critical transition between these SPT phases, we find two stable 0+1D boundary phases, each spontaneously breaking one of the Z3{\mathbb{Z}}_{3}Z3​ symmetries. Furthermore, we find that a single boundary parameter tunes a non-Landau boundary critical transition between these two phases. This constitutes a 0+1D version of an exotic phenomenon driven by charged vortex condensation known as deconfined quantum criticality. This work highlights the rich unexplored physics of criticality between nontrivial topological phases and provides insights into the burgeoning field of gapless topological phases.
We grow the web of dualities for conformal field theories (CFTs) in 2+1 spacetime dimensions to include quantum critical transitions of Dirac fermions. Our construction uses the seed duality, equating a free Dirac fermion with a complex boson coupled to a Chern-Simons gauge field, to express various Gross-Neveu-Yukawa (GNY) critical points of N fermions in terms of the same number of complex bosons and partner gauge fields. In some instances, monopole operators appear explicitly in the dual action, such as for the superconducting transition of a single Dirac fermion, which is a CFT with emergent supersymmetry. We match phase diagrams, symmetries and their anomalies. By further gauging certain symmetries, we derive offspring dualities. For example, starting with the duality for the regular GNY transition of 2 Dirac fermions, we arrive at the duality between a Neel-to-VBS deconfined quantum critical point and a gauged-Gross-Neveu transition. Finally, we use a large-N expansion for monopole operators, and compare with known scaling dimensions in the simplest single-flavour GNY transition.
Despite great successes in the study of gapped phases, a comprehensive understanding of the gapless phases and their transitions is still under development.In this paper, we study a general phenomenon in the space of (1 + 1)-dimensional critical phases with fermionic degrees of freedom described by a continuous family of conformal field theories (CFTs), also known as the conformal manifold.Along a one-dimensional locus on the conformal manifold, there can be a transition point, across which the fermionic CFTs on the two sides differ by stacking an invertible fermionic topological order (IFTO), point by point along the locus.At every point on the conformal manifold, the order and disorder operators have power-law two-point functions, but their critical exponents cross over with each other at the transition point, where stacking the IFTO leaves the fermionic CFT unchanged.We call this continuous transition on the fermionic conformal manifold a topological transition.By gauging the fermion parity, the IFTO stacking becomes a Kramers-Wannier duality between the corresponding bosonic CFTs.Both the IFTO stacking and the Kramers-Wannier duality are induced by the electromagnetic duality of the (2 + 1)-dimensional Z 2 topological order.We provide several examples of topological transitions, including the familiar Luttinger model of spinless fermions (i.e., the c = 1 massless Dirac fermion with the Thirring interaction) and a class of c = 2 examples describing U(1) × SU(2)-protected gapless phases.
Continuous phase transitions in equilibrium statistical mechanics were successfully described 50 years ago with the development of the renormalization group framework. This framework was initially developed in the context of phase transitions whose universal properties are captured by the long wavelength (and long time) fluctuations of a Landau order parameter field. Subsequent developments include a straightforward generalization to a class of T=0T = 0T=0 phase transitions driven by quantum fluctuations. In the last 2 decades it has become clear that there is a vast landscape of quantum phase transitions where the physics is not always usefully (or sometimes cannot be) formulated in terms of fluctuations of a Landau order parameter field. A wide class of such phase transitions - dubbed deconfined quantum critical points - involve the emergence of fractionalized degrees of freedom coupled to emergent gauge fields. Here I review some salient aspects of these deconfined critical points.
The spontaneous breaking of noninvertible symmetries can lead to exotic phenomena such as the coexistence of order and disorder. Here, we explore second-order phase transitions in one-dimensional spin chains between two phases that correspond to distinct patterns of noninvertible symmetry breaking. The critical point shares several features with well-understood examples of deconfined quantum critical points, such as enlarged symmetry and identical exponents for the two order parameters participating in the transition. Interestingly, such deconfined transitions involving noninvertible symmetries allow one to construct a whole family of similar critical points by gauging spin-flip symmetries. By employing gauging and bosonization, we characterize the phase diagram of our model in the vicinity of the critical point. We also explore proximate phases and phase transitions in related models, including a deconfined quantum critical point between invertible order parameters that is enforced by a noninvertible symmetry.