Topological degeneracy and pairing in a one-dimensional gas of spinless Fermions

We revisit the low energy physics of one dimensional spinless fermion liquids, showing that with sufficiently strong interactions the conventional Luttinger liquid can give way to a strong pairing phase. While the density fluctuations in both phases are described by a gapless Luttinger liquid, single fermion excitations are gapped only in the strong pairing phase. Smooth spatial Interfaces between the two phases lead to topological degeneracies in the ground state and low energy phonon spectrum. Using a concrete microscopic model, with both single particle and pair hopping, we show that the strong pairing state is established through emergence of a new low energy fermionic mode. We characterize the two phases with numerical calculations using the density matrix renormalization group. In particular we find enhancement of the central charge from $c=1$ in the two Luttinger liquid phases to $c=3/2$ at the critical point, which gives direct evidence for an emergent critical Majorana mode. Finally, we confirm the existence of topological degeneracies in the low energy phonon spectrum, associated with spatial interfaces between the two phases

Ferromagnetic transition in a one-dimensional spin-orbit-coupled metal and its mapping to a critical point in smectic liquid crystals

We study the quantum phase transition between a paramagnetic and ferromagnetic metal in the presence of Rashba spin-orbit coupling in one dimension. Using bosonization, we analyze the transition by means of renormalization group, controlled by an $\varepsilon$-expansion around the upper critical dimension of two. We show that the presence of Rashba spin-orbit coupling allows for a new nonlinear term in the bosonized action, which generically leads to a fluctuation driven first-order transition. We further demonstrate that the Euclidean action of this system maps onto a classical smectic-A -- C phase transition in a magnetic field in two dimensions. We show that the smectic transition is second-order and is controlled by a new critical point.Comment: 16 pages, 4 figures, 1 tabl

Measurement-Induced Phase Transitions in the Dynamics of Entanglement

We define dynamical universality classes for many-body systems whose unitary evolution is punctuated by projective measurements. In cases where such measurements occur randomly at a finite rate $p$ for each degree of freedom, we show that the system has two dynamical phases: `entangling' and `disentangling'. The former occurs for $p$ smaller than a critical rate $p_c$, and is characterized by volume-law entanglement in the steady-state and `ballistic' entanglement growth after a quench. By contrast, for $p > p_c$ the system can sustain only area-law entanglement. At $p = p_c$ the steady state is scale-invariant and, in 1+1D, the entanglement grows logarithmically after a quench. To obtain a simple heuristic picture for the entangling-disentangling transition, we first construct a toy model that describes the zeroth R\'{e}nyi entropy in discrete time. We solve this model exactly by mapping it to an optimization problem in classical percolation. The generic entangling-disentangling transition can be diagnosed using the von Neumann entropy and higher R\'{e}nyi entropies, and it shares many qualitative features with the toy problem. We study the generic transition numerically in quantum spin chains, and show that the phenomenology of the two phases is similar to that of the toy model, but with distinct `quantum' critical exponents, which we calculate numerically in $1+1$D. We examine two different cases for the unitary dynamics: Floquet dynamics for a nonintegrable Ising model, and random circuit dynamics. We obtain compatible universal properties in each case, indicating that the entangling-disentangling phase transition is generic for projectively measured many-body systems. We discuss the significance of this transition for numerical calculations of quantum observables in many-body systems.Comment: 17+4 pages, 16 figures; updated discussion and results for mutual information; graphics error fixe

Superconductivity near a ferroelectric quantum critical point in ultralow-density Dirac materials

The experimental observation of superconductivity in doped semimetals and semiconductors, where the Fermi energy is comparable to or smaller than the characteristic phonon frequencies, is not captured by the conventional theory. In this paper, we propose a mechanism for superconductivity in ultralow-density three-dimensional Dirac materials based on the proximity to a ferroelectric quantum critical point. We derive a low-energy theory that takes into account both the strong Coulomb interaction and the direct coupling between the electrons and the soft phonon modes. We show that the Coulomb repulsion is strongly screened by the lattice polarization near the critical point even in the case of vanishing carrier density. Using a renormalization group analysis, we demonstrate that the effective electron-electron interaction is dominantly mediated by the transverse phonon mode. We find that the system generically flows towards strong electron-phonon coupling. Hence, we propose a new mechanism to simultaneously produce an attractive interaction and suppress strong Coulomb repulsion, which does not require retardation. For comparison, we perform same analysis for covalent crystals, where lattice polarization is negligible. We obtain qualitatively similar results, though the screening of the Coulomb repulsion is much weaker. We then apply our results to study superconductivity in the low-density limit. We find strong enhancement of the transition temperature upon approaching the quantum critical point. Finally, we also discuss scenarios to realize a topological $p$-wave superconducting state in covalent crystals close to the critical point

Fate of the one-dimensional Ising quantum critical point coupled to a gapless boson

The problem of a quantum Ising degree of freedom coupled to a gapless bosonic mode appears naturally in many one dimensional systems, yet surprisingly little is known how such a coupling affects the Ising quantum critical point. We investigate the fate of the critical point in a regime, where the weak coupling renormalization group (RG) indicates a flow toward strong coupling. Using a renormalization group analysis and numerical density matrix renormalization group (DMRG) calculations we show that, depending on the ratio of velocities of the gapless bosonic mode and the Ising critical fluctuations, the transition may remain continuous or become fluctuation-driven first order. The two regimes are separated by a tri-critical point of a novel type.Comment: 8 pages, 8 figures; published versio