A bstract In this work, we explore an unconventional class of problems in the study of (quantum) critical phenomena, termed “deep boundary criticality”. Traditionally, critical systems are analyzed with two types of perturbations: those uniformly distributed throughout the bulk, which can significantly alter the bulk criticality by triggering a nontrivial bulk renormalization group flow, and those confined to a boundary or subdimensional defect, which affect only the boundary or defect condition. Here, we go beyond this paradigm by studying quantum critical systems with boundary perturbations that decay algebraically (following a power law) into the bulk. By continuously varying the decay exponent, such perturbations can transition between having no effect on the bulk and strongly influencing bulk behavior. We investigate this regime using two prototypical models based on (1+1)D massless Dirac fermions. Through a combination of analytical and numerical approaches, we uncover exotic scaling laws in simple observables and observe qualitative changes in model behavior as the decay exponent varies.
While imaginary-time evolution has long served as a standard paradigm for ground-state preparation in numerical simulations and quantum devices, its intrinsic dynamical properties has been largely overlooked. Here, we investigate the short-imaginary-time critical dynamics in quantum systems with boundaries. A universal scaling theory is developed and verified in the two-dimensional quantum Ising model, uncovering rich dynamic critical behaviors dictated by boundary universality classes. For ordered initial states, the boundary order parameter
Ms decays with imaginary time
τ as
Ms∝τ−β1/νz, where
β1 denotes the boundary order parameter exponent, and
ν and
z correspond to the correlation length exponent and the dynamic exponent, respectively. For disordered initial states, the autocorrelation of the boundary order parameter is governed by a novel critical exponent
θ1, which is closely related to the critical initial slip behavior of
Ms characterized by the corresponding exponent
θ1′. In contrast to its positive bulk counterpart, the boundary initial-slip exponent
θ1′ is negative for the ordinary transition while remaining positive for the special transition. Although the static universality classes of
d-dimensional quantum phase transitions generally coincide with those of
(d+1)-dimensional classical phase transitions, we show that
θ1 does not follow this conventional quantum-classical mapping. We further discuss the implications of our results for more exotic forms of boundary criticality. Our findings provide new physical insights into boundary critical dynamics and offer a novel route for probing exotic boundary critical behaviors in quantum many-body systems.
The boundary entropy log(g) of a critical one-dimensional quantum system (or two-dimensional conformal field theory) is known to decrease under renormalization group (RG) flow of the boundary theory. We study instead the behavior of the boundary entropy as the bulk theory flows between two nearby critical points. We use conformal perturbation theory to calculate the change in g due to a slightly relevant bulk perturbation and find that it has no preferred sign. The boundary entropy log(g) can therefore increase during appropriate bulk flows. This is demonstrated explicitly in flows between minimal models. We discuss the applications of this result to D-branes in string theory and to impurity problems in condensed matter.
An emergent gauge field can control boundary critical behavior without changing the bulk quantum spin liquid. We investigate the boundary criticality in
(2+1)-dimensional
U(1) Dirac spin liquid, whose low-energy physics is described by massless quantum electrodynamics. Using a perturbative renormalization-group analysis in a half-space, we show that Neumann and Dirichlet boundary conditions for the emergent gauge field lead to distinct boundary universality classes. We determine the boundary scaling dimensions of the fundamental fields and gauge-invariant operators that provide observable signatures. We further propose a fermion-gauge lattice model with tunable boundary interactions as a microscopic setting for realizing and probing these boundary universality classes.
We study symmetry breaking quantum phase transitions in topological insulators and superconductors where the single electron gap remains open in the bulk. Specifically, we consider spontaneous breaking of the symmetry that protects the gapless boundary modes, so that in the ordered phase these modes are gapped. Here we determine the fate of the topological boundary modes right at the transition where they are coupled to the strongly fluctuating order parameter field. Using a combination of exact solutions and renormalization group techniques, we find that the surface fermionic modes either decouple from the bulk fluctuations, or flow to a strongly coupled fixed point which remains gapless. In addition, we study transitions where the critical fluctuations are confined only to the surface and find that in several cases the critical point is naturally supersymmetric. This allows a determination of critical exponents and points to an underlying connection between band topology and supersymmetry. Finally, we study the fate of gapless Majorana modes localized on point and line defects in topological superconductors at bulk criticality, which is analogous to a quantum impurity problem. Again, an interplay of topology and strong correlations causes these modes to remain gapless but in a strongly coupled state. Experimental candidates for realizing these phenomena are discussed.
The edge of a quantum critical system can exhibit multiple distinct types of boundary criticality. We use a numerical real-space renormalization group (RSRG) to study the boundary criticality of a 2d quantum Ising model with random exchange couplings and transverse fields, whose bulk exhibits an infinite randomness critical point. This approach enables an asymptotically numerically exact extraction of universal scaling data from very large systems with many thousands of spins that cannot be efficiently simulated directly. We identify three distinct classes of boundary criticality, and extract key scaling exponents governing boundary-boundary and boundary-bulk correlations and dynamics. We anticipate that this approach can be generalized to studying a broad class of (disordered) boundary criticality, including symmetry-enriched criticality and edge modes of gapless symmetry-protected topological states, in contexts were other numerical methods are restricted to one-dimensional chains.
Experimental progress in quantum simulators highlights the role of long-range (LR) interactions in reshaping quantum criticality and stabilizing exotic phases beyond the short-range (SR) paradigm. We study ferromagnetic long-range quantum
O(n) models with interactions decaying as
1/rd+σ and develop a perturbative renormalization-group expansion around the LR--SR boundary by setting
d=3−ε and
σ=2−δ. In this parametrization, the full interacting LR window
2d/3<σ<2 becomes
0<δ<2ε/3, and is therefore perturbatively controlled. A two-loop calculation yields explicit expressions, in terms of
ε,
δ, and
n, for the correlation-length exponent
ν and for the frequency and momentum anomalous dimensions
ηω and
ηk. The resulting exponents reduce to long-range Gaussian scaling at
σ=2d/3 and to SR quantum Wilson-Fisher scaling in the
σ→2 limit, thereby identifying
σ∗=2 as the LR--SR boundary within the controlled
3−ε expansion. Combining the RG results with scaling boundaries and classical LR analogies, we propose a
(d,σ) universality diagram for ferromagnetic long-range quantum
O(n) criticality and use it as an organizing framework for the phase diagram of long-range quantum spin chains.