Time-reversal symmetry breaking and spontaneous anomalous Hall effect in Fermi fluids

We study the spontaneous nonmagnetic time-reversal symmetry breaking in a two-dimensional Fermi liquid without breaking either the translation symmetry or the U(1) charge symmetry. Assuming the low-energy physics is described by fermionic quasiparticle excitations, we identified an “emergent” local $\text{U}{\left(1\right)}^N$ symmetry in momentum space for an $N$-band model. For a large class of models, including all one-band and two-band models, we found that the time-reversal and chiral symmetry breaking can be described by the $\text{U}{\left(1\right)}^N$ gauge theory associated with this emergent local $\text{U}{\left(1\right)}^N$ symmetry. This conclusion enables the classification of the time-reversal symmetry-breaking states as types I and II, depending on the type of accompanying spatial symmetry breaking. The properties of each class are studied. In particular, we show that the states breaking both time reversal and chiral symmetries are described by spontaneously generated Berry phases. We also show examples of the time-reversal symmetry-breaking phases in several different microscopically motivated models and calculate their associated Hall conductance within a mean-field approximation. The fermionic nematic phase with time-reversal symmetry breaking is also presented and the possible realizations in strongly correlated models such as the Emery model are discussed.

Figure 1

(Color online) Fermi surfaces of phases with Pomeranchuk instability with [(a)-(d)] $l=$2, 3, 4, and 5. Odd $\ell$ states break the $\mathbf{I}$ and $\mathbf{T}$ symmetries but preserve $\mathbf{I}\mathbf{T}$.

Figure 2

(Color online) The phase diagram with exact relative U(1) symmetry. Thick lines are first-order phase boundaries and others are second-order ones. The black dot is a tricritical point. The dashed circle is a critical point where the first-order phase boundary meets two second-order phase transitions.

Figure 3

(Color online) The Fermi surface in (a) ${\alpha }_1$, (b) ${\beta }_1$, (c) ${\alpha }_2$, (d) ${\beta }_2$, (e) ${\alpha }_3$, and (f) ${\beta }_3$ phases in the $\ell =2$ channel of a two-band model. The dashed (solid) lines are the Fermi surfaces of the normal (symmetry broken) phases. The small arrows in (b), (d), and (f) show the relative phases of the fermions in the two bands.

Figure 4

(Color online) The phase diagram with broken relative U(1) symmetry. The notation is the same as in Fig. 2. The circle represents a bicritical point where a second-order phase boundary and two first-order boundaries meet.

Figure 5

(Color online) (a) is the flux state in a simple triangular lattice and (b) shows the Fermi surfaces. The two dashed lines are the Fermi surface of the states with flux 0 and $\pm \pi$ (these two states can be transferred into each other by a gauge transformation), and the solid line is for flux $\pm \pi /10$. The rotational symmetry is threefold for flux $\pm \pi /10$ and sixfold for flux 0 and $\pi$.

Figure 6

(Color online) (a) The vector field $\left(n_x,n_y\right)$ defined in Eq. (B2) for the honeycomb lattice, which has $n_z=0$ and (b) the vector field $\left(n_x,n_z\right)$ of the crossed-chain lattice, which has $n_y=0$. The blue line marks a Brillouin zone. For the honeycomb lattice there are two degeneracy points with monopole flux $\pm \pi$ and for the crossed-chain lattice, there is one with monopole flux $\pm 2\pi$.

Figure 7

The $\mathbf{T}$ symmetry-breaking (flux) state in (a) the crossed-chain lattice and [(b)–(d)] the Emery lattice. (b) is the Varma ${\theta }_{\text{II}}$ loop state. (a) and (c) are type II $\mathbf{T}$-breaking states, (b) is type I, and (d) is a type I-type II mixed state.

Figure 8

(Color online) Different possible schematic phase diagrams for thermal phase transitions from the ${\beta }_2$ or ${\beta }_3$ phases to the normal phase. Here, “Inter” stands for the intermediate phase which may be the $\alpha$ phase or the isotropic type II time-reversal symmetry-breaking phase depending on microscopic details, although the mean-field approach can only predict (b) and (c) and requires the intermediate to be the $\alpha$ phase. The points indicated by $B_{cp}$, $T_{cp}$, and $M_{cp}$ are the bicritical, tricritical, and multicritical points. The thin lines are second-order transitions and the thick lines stand for first-order transitions. The convention is the same as in Fig. 2. In (a), the multicritical point is a KT phase-transition point. The phase boundary between the tricritical and the multicritical points has continuous varying exponent, and all other thin lines in these phase diagrams are Ising transitions. In (c), the phase boundaries (or part of the phase boundary) may be first order in general.