We study the fate of quantum solids with a vanishing velocity of transverse phonons. Such floppy'' solids exhibit a cubic elastic nonlinearity that is relevant in spatial dimensions $d<4$ and drives the system to a strongly-coupled Lifshitz quantum critical point in $d=3$, that we analyze in an $ε= 4-d$ expansion. The associated phase transition is generically first-order, preempting the system's full access to the strongly-coupled fixed point. Nevertheless, the critical data of this quantum critical point is accessible within a Ginzburg region in the vicinity of the transition. This cubic nonlinearity vanishes exactly in $d=2$, where the leading interactions are instead quartic and are marginal, leading to weak logarithmic singularities. We discuss possible realizations of this crystal Lifshitz criticality'' in quantum materials.
The stability of a quantum critical point in the
O(N) universality class with respect to an elastic coupling, that preserves
O(N) symmetry, is investigated for isotropic elasticity in the framework of the renormalization group (RG) close to the upper critical dimension
d=3−ε. With respect to the Wilson-Fisher fixed point, we find that the elastic coupling is relevant in the RG sense for
1≤N≤4, and the crystal becomes microscopically unstable, i.e., a sound velocity vanishes at a finite value of the correlation length
ξ. For
N>4, an additional fixed point emerges that is located at a finite value of the dimensionless elastic coupling. This fixed point is repulsive and separates the flow to weak and strong elastic coupling. As the fixed point is approached the sound velocity is found to vanish only asymptotically as
ξ→∞ such that the crystal remains microscopically stable for any finite value of
ξ. The fixed point structure we find for the quantum problem is distinct from the classical counterpart in
d=4−ε, where the crystal always remains microscopically stable for finite
ξ.
The stability of a quantum critical point in the
O(N) universality class with respect to an elastic coupling, which preserves
O(N) symmetry, is investigated for isotropic elasticity in the framework of the renormalization group (RG) close to the upper critical dimension $d=3\ensuremath{-}\ensuremath{\epsilon}$. With respect to the Wilson-Fisher fixed point, we find that the elastic coupling is relevant in the RG sense for $1\ensuremath{\le}N\ensuremath{\le}4$, and the crystal becomes microscopically unstable, i.e., a sound velocity vanishes at a finite value of the correlation length $\ensuremath{\xi}$. For
N>4, an additional fixed point emerges that is located at a finite value of the dimensionless elastic coupling. This fixed point is repulsive and separates the flow to weak and strong elastic coupling. As the fixed point is approached, the sound velocity is found to vanish only asymptotically as $\ensuremath{\xi}\ensuremath{\rightarrow}\ensuremath{\infty}$, such that the crystal remains microscopically stable for any finite value of $\ensuremath{\xi}$. The fixed point structure we find for the quantum problem is distinct from the classical counterpart in $d=4\ensuremath{-}\ensuremath{\epsilon}$, where the crystal always remains microscopically stable for finite $\ensuremath{\xi}$.
We study a novel type of quantum criticality of the Lifshitz ${\ensuremath{\varphi}}^{4}$ theory below the upper critical dimension
du=z+dc=8, where the dynamic critical exponent
z=4 and the spatial upper critical dimension
dc=4. Two fixed points, one Gaussian and the other non-Gaussian, are identified with zero and finite interaction strengths, respectively. At zero temperature the particle density exhibits different power-law dependences on the chemical potential in the weak- and strong-interaction regions. At finite temperatures, critical behaviors in the quantum disordered region are mainly controlled by the chemical potential. In contrast, in the quantum critical region critical scalings are determined by temperature. The scaling ansatz remains valid in the strong-interaction limit for the chemical potential, correlation length, and particle density, while it breaks down in the weak-interaction one. Approaching the upper critical dimension, physical quantities develop logarithmic dependence on dimensionality in the strong-interaction region. These results are applied to spin-orbit coupled bosonic systems, leading to predictions testable by future experiments.
The critical dynamics at elastic phase transitions of second order is studied by the renormalization-group theory. The dynamical theory is based on a stochastic equation of motion for damped phonons. For systems with one-dimensional soft sectors usual dynamical scaling holds with the critical dynamical exponent
z=2. For two-dimensional sectors logarithmic corrections appear. Novel features are found for an isotropic-phonon model. The dynamical susceptibility and the characteristic frequency depend singularly on an irrelevant parameter, which, however, diverges at the fixed point. Consequently dynamical scaling breaks down; e.g., the characteristic frequency is no longer a homogeneous function of the wave number
k and the inverse correlation length ${\ensuremath{\xi}}^{\ensuremath{-}1}$. Instead it is a homogeneous function of these variables and an irrelevant parameter. Consequently, the critical dynamics can be characterized by the relaxational exponent $z=2+c\ensuremath{\eta}$ at
Tc, while in the hydrodynamic region the sound frequency is characterized by $z=2\ensuremath{-}\frac{1}{2}\ensuremath{\eta}$ and the damping by yet another exponent. This breakdown of scaling is also reflected by the fact that different fixing conditions, i.e., different choices of the frequency, lead to different values of
z. All of these apparently different transformations lead to the same modified dynamical scaling relations.
Deconfined quantum critical points (DQCPs) are putative phase transitions beyond the Landau paradigm with emergent fractionalized degrees of freedom. The original example of a DQCP is the spin-
21 quantum antiferromagnet on the square lattice which features a second-order transition between valence bond solid (VBS) and N\'eel order. The VBS order breaks a lattice symmetry, and the corresponding VBS order parameter may couple to lattice distortion modes (phonons) at appropriate momenta. We investigate a field-theoretic description of the DQCP in the presence of such a spin-lattice coupling. We show that treating phonons as classical lattice distortions leads to a relevant monopole-phonon interaction inducing an instability towards a distorted lattice by an analogous mechanism to the spin-Peierls instability in one dimension. Consequently, there is a breakdown of the DQCP which generally becomes a strong first-order transition. Taking into account the full quantum nature of the phonons, we argue that the continuous DQCP persists above a critical phonon frequency. Lastly, we comment on the connection to general gapless, deconfined gauge theories.
We consider spin-half quantum antiferromagnets in two spatial dimensions in the quantum limit, where the spins are in a valence bond solid (VBS) phase. The transition between two such VBS phases is studied. In some cases, an interesting second-order transition controlled by a fixed line with varying critical exponents is found. A specific example is provided by an antiferromagnetically coupled bilayer system on the honeycomb lattice where a continuous quantum phase transition can generically exist between two VBS phases. Furthermore, these critical points are deconfined, in the sense that gapped spin-
1∕2 spinon excitations emerge right at the transition. The low-energy physics of this critical point (up to marginally irrelevant interactions) contains just a free quadratically dispersing ``photon.'' The phase structure on one side of this continuous transition is very intricate, consisting of a series of infinitely closely spaced further transitions in a ``devil's staircase'' form. Analogies with previous examples of deconfined quantum criticality are emphasized. Closely related transitions in single layer systems are explored. These are second order only at some multicritical points. The solvable Rokshar-Kivelson point of quantum dimer models of single layer systems is found to correspond to a nongeneric multicritical point.