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arXiv:1909.01425 · Physical Review Research

Topological transition on the conformal manifold

Wenjie Ji, Shu-Heng Shao, Xiao-Gang Wen

Published Aug 26, 2020• 120 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevresearch.2.033317

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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 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.
We show that topology can protect exponentially localized, zero energy edge modes at critical points between one-dimensional symmetry-protected topological phases. This is possible even without gapped degrees of freedom in the bulk-in contrast to recent work on edge modes in gapless chains. We present an intuitive picture for the existence of these edge modes in the case of noninteracting spinless fermions with time-reversal symmetry (BDI class of the tenfold way). The stability of this phenomenon relies on a topological invariant defined in terms of a complex function, counting its zeros and poles inside the unit circle. This invariant can prevent two models described by the same conformal field theory (CFT) from being smoothly connected. A full classification of critical phases in the noninteracting BDI class is obtained: Each phase is labeled by the central charge of the CFT, c∈1/2N, and the topological invariant, ω∈Z. Moreover, c is determined by the difference in the number of edge modes between the phases neighboring the transition. Numerical simulations show that the topological edge modes of critical chains can be stable in the presence of interactions and disorder.
Topological holography is a conjectured correspondence between the symmetry charges and defects of a DDD-dimensional system with the anyons in a (D+1)(D+1)(D+1)-dimensional topological order: the symmetry topological field theory (SymTFT). Topological holography is conjectured to capture the topological aspects of symmetry in gapped and gapless systems, with different phases corresponding to different gapped boundaries (anyon condensations) of the SymTFT. This correspondence was previously considered primarily for bosonic systems, excluding many phases of condensed matter systems involving fermionic electrons. In this work, we extend the SymTFT framework to establish a topological holography correspondence for fermionic systems. We demonstrate that this fermionic SymTFT framework captures the known properties of 1+1D1+1D1+1D fermion gapped phases and critical points, including the classification, edge-modes, and stacking rules of fermionic symmetry-protected topological phases (SPTs), and computation of partition functions of fermionic conformal field theories (CFTs). Beyond merely reproducing known properties, we show that the SymTFT approach can additionally serve as a practical tool for discovering new physics, and use this framework to construct a new example of a fermionic intrinsically gapless SPT phase characterized by an emergent fermionic anomaly.
We propose a series of simple two-dimensional (2D) lattice interacting fermion models that we demonstrate at low energy describe bosonic symmetry-protected topological (SPT) states and quantum phase transitions between them. This is because due to interaction, the fermions are gapped both at the boundary of the SPT states and at the bulk quantum phase transition, thus these models at low energy can be described completely by bosonic degrees of freedom. We show that the bulk of these models is described by a Sp(N\mathrm{Sp}(NSp(N) principal chiral model with a topological $\mathrm{\ensuremath{\Theta}}$ term, whose boundary is described by a Sp(N\mathrm{Sp}(NSp(N) principal chiral model with a Wess-Zumino-Witten term at level 1. The quantum phase transition between SPT states in the bulk is tuned by a particular interaction term, which corresponds to tuning $\mathrm{\ensuremath{\Theta}}$ in the field theory, and the phase transition occurs at $\mathrm{\ensuremath{\Theta}}=\ensuremath{\pi}$. The simplest version of these models with N=1N=1N=1 is equivalent to the familiar O(4) nonlinear sigma model (NLSM) with a topological term, whose boundary is a (1+1)D(1+1)\text{D}(1+1)D conformal field theory with central charge c=1c=1c=1. After breaking the O(4) symmetry to its subgroups, this model can be viewed as bosonic SPT states with U(1), or Z2{Z}_{2}Z2​ symmetries, etc. All of these fermion models, including the bulk quantum phase transitions, can be simulated with the determinant quantum Monte Carlo method without the sign problem. Recent numerical results strongly suggest that the quantum disordered phase of the O(4) NLSM with precisely $\mathrm{\ensuremath{\Theta}}=\ensuremath{\pi}$ is a stable (2+1)D(2+1)\text{D}(2+1)D conformal field theory with gapless bosonic modes.
An attempt is made to show the topological transition on conformal manifold for quantum Ising model with longer range interaction. This model Hamiltonian system has different gapped phases with different topological indices and also different quantum critical lines depending on the presence and absence of the transverse field. We also present the central charge for the different regimes of the parameter space. We show explicitly on the interplay of criticality, topology and central charge (C) in the presence and absence of transverse field and also non-universal character of C. We show how the Lifshitz transition occurs in the presence of transverse field and changes the universality class of the central charge in presence of transverse field. We show explicitly the existence of conformal field theory (CFT) criticality and non-CFT criticality. We have presented an explicit calculations to find the relation between the polynomial function and the Anderson-pseudo spin model Hamiltonian. Our results are far more enrich than the existence results of noninteracting fermionic many body system. This work not only provides a new perspective in topological state of conformal field theory but also for low dimensional quantum many body system.
A bstract We use the ideas behind the duality web to construct numerous conformal field theories mediating the phase transitions between various symmetry broken and topological phases. In particular we obtain the full field theory version of the Kennedy Tasaki transformation, mapping a gapless theory mediating a topological phase transition of symmetry protected topological orders to a standard symmetry breaking one in a 1+1 dimensional ℤ 2 × ℤ 2 gauge theory. When we consider all possible discrete gauging operations, we obtain bosonic and fermionic webs with 9 critical theories per web, each connecting 4 separate gapped phases, some of them topological. Bosonization maps the two webs into each other. In addition to discussing the multi-critical theory connecting the four gapped phases in each phase diagram, we discuss the partially gapped theories connecting two of those four. Some of these are gapless symmetry protected topological phases.