Charge-density-wave and pair-density-wave orders in underdoped cuprates

This thesis is motivated by the charge orders and associated symmetry breaking observed in the enigmatic pseudogap phase of underdoped cuprates. The theoretical approach we take is an itinerant electron one, in which we assume a two dimensional metal interacts with soft spin-density-wave collective modes at the onset of antiferro-magnetism. This so called "spin-fermion model" has been well studied for over twenty-five years, and the leading instability of the spin-fermion model has been shown to be d-wave superconductivity. In this thesis we show in two parts that the physics of the spin-fermion model is far more richer than originally thought, and provides an explanation for the incommensurate charge-density-wave order, the symmetry breaking including the polar Kerr effect, and anomalous angular resolved photoemission spectra (ARPES) observed in underdoped cuprates.;This thesis is divided into two parts. In Part I, we analyze with in the spin-fermion model the incommensurate charge-density-wave (CDW) order with momentum (0, Q) and (Q, 0) consistent with the experiments. In Chapter 2 we first show that the normal state is unstable toward forming a CDW condensate, provided that the magnetic correlation length xi exceeds a certain critical value. We find that the resulting CDW order parameter has a d-wave form factor, which has been directly confirmed by STM experiments. In Chapter 3 we derive a Ginzburg-Landau (GL) functional for CDW order parameters with momenta both (0, Q) and (Q, 0), and show that the ground state breaks both time-reversal symmetry T and lattice rotational symmetry C4. Moreover, by extending our analysis beyond mean-field level, we show that the system breaks T and C4 at higher temperatures than the onset temperature of the primary CDW order. Using this state with broken T as an input, in Chapter 4 we derive the polar Kerr angle theta K, which is defined as the rotation of polarization plane of the incident light off reflection. We show that thetaK is indeed nonzero as experiments found, and discuss some peculiar features of the Kerr effect data.;In Part II, we discuss the pair-density-wave (PDW) order in the spin-fermion model. The PDW order is defined as a superconducting order with nonzero total momentum, which is also known as an FFLO order. This order has been proposed to explain some features of the pseudogap state of cuprates. In Chapter 5 we find that PDW also emerges in the spin-fermion model, and moreover, it is degenerate with the CDW order. We argue that this degeneracy emerges from a particle-hole SU(2) pseudo-spin symmetry, first discovered by Metlitski and Sachdev. Furthermore, due to the SU(2) symmetry, we show that CDW and PDW orders, each being a U(1) order, forms a larger SO(4) symmetry manifold. Building on our results in Part 1, we derive a GL theory for the SO(4)-symmetric CDW/PDW order parameters and classify the ground state. Subsequently in Chapter 6 we study the properties of the ground state at the weak- coupling limit, the "orthogonal" CDW/PDW state, and argue that it explains several important features of the ARPES data taken in the pseudogap. Finally, we present a full theoretical phase diagram including all orders.