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.
In this paper, we discuss strong coupling limits of topological quantum critical points (TQCPs) where quantum phase transitions between two topological distinct superconducting states take place. We illustrate that while superconducting phases on both sides of TQCPs spontaneously break the same symmetries, universality classes of critical states can be identified only when global symmetries in topological states are further specified. In dimensions
d=2and3, we find that continuous
(d+1)th-order transitions at weakly interacting TQCPs that were pointed out previously in the presence of the emergent Lorentz symmetry can be terminated by strongly interacting fixed points of Majorana fields. For
2d time reversal symmetry breaking TQCPs, termination points are supersymmetric with
N=4Nf=1 (where
Nf is the number of four-component Dirac fermions and
N is the number of two-component real fermions) beyond which transitions are discontinuous first-order ones. For
2d time reversal symmetric TQCPs without other global symmetries, termination points of
(d+1)th-order continuous transition lines are generically conformal invariant without supersymmetry. Beyond these strong coupling fixed points, there are first-order discontinuous transitions as far as the protecting symmetry is not spontaneously broken but no direct transitions if the protecting symmetry is spontaneously broken in the presence of strong interactions. In this case, termination points can become supersymmetric with
N=4Nf=2 only when interacting Majorana fermions further display an emergent global
U(1) symmetry. In
3d, strong coupling termination points can be further effectively represented by new emergent gapless real bosons weakly coupled with free gapless Majorana fermions. However, in
1d, time reversal symmetric
(d+1)th continuous transition lines of TQCPs are terminated by simple free Majorana fermion fixed points.
The quantum phase transition between two clean, noninteracting topologically distinct gapped states in three dimensions is governed by a massless Dirac fermion fixed point, irrespective of the underlying symmetry class, and this constitutes a remarkably simple example of superuniversality. For a sufficiently weak disorder strength, we show that the massless Dirac fixed point is at the heart of the robustness of superuniversality. We establish this by considering both perturbative and nonperturbative effects of disorder. The superuniversality breaks down at a critical strength of disorder, beyond which the topologically distinct localized phases become separated by a delocalized diffusive phase. In the global phase diagram, the disorder controlled fixed point where superuniversality is lost, serves as a multicritical point, where the delocalized diffusive and two topologically distinct localized phases meet and the nature of the localization-delocalization transition depends on the underlying symmetry class. Based on these features, we construct the global phase diagrams of noninteracting, dirty topological systems in three dimensions. We also establish a similar structure of the phase diagram and the superuniversality for weak disorder in higher spatial dimensions. By noting that
1/r2 power-law correlated disorder acts as a marginal perturbation for massless Dirac fermions in any spatial dimension
d, we have established a general renormalization group framework for addressing disorder driven critical phenomena for fixed spatial dimension
d>2.
In this article, we study quantum critical phenomena in surfaces of symmetry-protected topological matter, i.e., surface topological quantum criticality. A generic phase boundary of gapless surfaces in a symmetry-protected state shall be a codimension one manifold in an interaction parameter space of dimension
Dp (where
p refers to the parameter space) where the value of
Dp further depends on bulk topologies. In the context of fermionic topological insulators that we focus on,
Dp depends on the number of half-Dirac cones
N. We construct such manifolds explicitly for a few interaction parameter spaces with various
Dp values. Most importantly, we further illustrate that in cases with
Dp=3 and 6, there are submanifolds of fixed points that dictate the universalities of surface topological quantum criticality. These infrared stable manifolds are associated with emergent symmetries in the renormalization-group-equation flow naturally appearing in the loop expansion. Unlike in the usual order-disorder quantum critical phenomena, typically governed by an isolated Wilson-Fisher fixed point, we find in the one-loop approximation surface topological quantum criticalities are naturally captured by conformal manifolds where the number of marginal operators uniquely determines their codimensions. Isolated strong-coupling fixed points also appear, usually as the endpoints in the phase boundary of surface topological quantum phases. However, their extreme infrared instabilities along multiple directions suggest that they shall be related to multicritical surface topological quantum critical phenomena rather than generic surface topological quantum criticality. We also discuss and classify higher-loop symmetry-breaking effects, which can either distort the conformal manifolds or further break the conformal manifolds down to a few distinct fixed points.
Generic non-magnetic disorder effects on topological quantum critical points (TQCP), which intervene between the three-dimensional (3D) topological insulator and an ordinary insulator, are investigated in this work. We first show that, in such 3D TQCP, any backward-scattering process mediated by chemical-potential-type impurity is always cancelled by its time-reversal ( -reversal) counter-process because of the nontrivial Berry phase supported by these two processes in the momentum space. However, this cancellation can be generalized into only those backward-scattering processes that conserve a certain internal degree of freedom, i.e. the parity density, while the 'absolute' stability of the TQCP against any non-magnetic disorders is required by the bulk-edge correspondence. Motivated by this, we further derive the self-consistent Born-phase diagram and the quantum conductivity correction in the presence of generic non-magnetic disorder potentials. The distinctions and similarities between the case with only the chemical-potential-type disorder and that with the generic non-magnetic disorders are summarized.
Algebraic/Dirac spin liquids (DSLs) are a class of critical quantum ground states that do not have a quasi-particle description. DSLs and related spin liquid phases often arise in strongly frustrated quantum spin systems, in which strong correlations and quantum fluctuations among constituent spins persist down to zero temperature. In this work, we analyze Mott insulating phases of SU(3) S U ( 3 ) fermions on a kagome lattice which may realize a DSL phase, described at low energies by (2 + 1)d ( 2 + 1 ) d quantum electrodynamics (QED _3 3 ) with N_f=6 N f = 6 Dirac fermions. By analyzing the action of physical symmetries on the operators of the QED _3 3 theory, we conclude that the low energy DSL is a quantum critical point that can be accessed by tuning a single microscopic parameter. Aided by the emergent symmetry and anomalies of the low energy effective theory, we conjecture and present supporting arguments that the SU(3) S U ( 3 ) Kagome magnet DSL is an unnecessary quantum critical point, lying completely within a single phase.