condensed matter - Spontaneous Time Reversal Symmetry Breaking?

It is known that you can break P spontaneously--- look at any chiral molecule for an example. Spontaneous T breaking is harder for me to visualize. Is there a well known condensed matter system which is uncontroversial example where T is broken spontaneously?

I remember vaguely articles of Wen, Wilczek, and Zee from 1989 or so on standard High Tc hopping models, electrons which singly-occupy lattice sites, double-occupation repulsion, small amount of p-doping (holes running around), where they made the claim that T is spontaneously broken. Unfortunately I didn't understand how this happened or if it actually happened. If somebody understands the Zee example, that's good, but I would be happy with any example.

I am not looking for explicit T breaking, only spontaneous T breaking. I would also like an example where the breaking is thermodynamically significant in the large system limit, so mesoscopic rings with permanent currents caused by electron discreteness is not a good example.

Answer

The simplest example in condensed matter physics that spontaneously breaks time reversal symmetry is a ferromagnet. Because spins (angular momentum) change sign under time reversal, the spontaneous magnetization in the ferromagnet breaks the symmetry. This is a macroscopic example.

The chiral spin liquid (Wen-Wilczek-Zee) mentioned in the question is a non-trivial example that breaks time reversal but with out any spontaneous magnetization. Its order parameter is the spin chirality $E_{123}=\mathbf{S}_1\cdot(\mathbf{S}_2\times\mathbf{S}_3)$, which measures the Berry curvature (effective magnetic field) in the spin texture. Because $E_{123}$ also changes sign under time reversal, so the T symmetry is broken by spontaneous development of the spin chirality. Chiral spin liquid can be consider as a condensation of the skyrmion which carries the quantum of spin chirality but is spin neutral as a whole.

In fact, within the spin system, one can cook up any order parameter consisting of odd number of spin operators ($\mathbf{S}_1$ for ferromagnets and $E_{123}$ for chiral spin liquid are both examples of such constructions). Then by ordering such order parameter, the time reversal symmetry can be broken spontaneously.

Beyond the spin system, it is still possible to break time reversal symmetry by the development of orbital angular momentum (loop current) ordering. Just think of spins and loop currents are both angular momenta, what can be done with spins can also be done with loop currents. Indeed, the spinless fermion system can break the time reversal symmetry using the loop current (Note the word "spinless", so there is no spin SU(2) nor spin-orbit coupling involved in the following discussion). Simply consider the spinless fermion $c_i$ on a square lattice coupling to a U(1) gauge field $a_{ij}$, the Hamiltonian reads

$$H=-t\sum_{\langle ij\rangle}e^{ia_{ij}}c_i^\dagger c_j+g\sum_\square \prod_{\langle ij\rangle\in\partial\square}e^{ia_{ij}}+h.c.$$ With zero flux per plaquette and with the filling of 1/2 fermion per site, the system has a fermi surface and the fermi level rest on a Van Hove singularity, which is very unstable energetically. The fermions wish to develop any kind of order as long as a it helps to open a gap at the fermi level, such that the fermi energy can be reduced. It is found that the stagger flux is a solution, in which the U(1) flux $\pm\phi$ goes through the plaquette alternately following the checkboard pattern. The corresponding gauge connection is $a_{i,i+x}=0, a_{i,i+y}=(\phi/2)(-)^{i_x+i_y}$. One can show that the energy dispersion for the fermion is given by $$E=\pm\sqrt{\cos^2k_x+\cos^2k_y+2\cos\frac{\phi}{2}\cos k_x\cos k_y},$$ which removes the Van Hove singularity and opens up a pseudo gap (like Dirac cones) as long as $\phi\neq 0$. Therefore driven by the fermi energy, $\phi$ wishes to grow toward the maximum flux $\pi$. However due to the $g$ term in the Hamiltonian, the development of stagger flux consumes magnetic energy (the energy of orbital angular momentum), which grows as $\phi^2$ for small $\phi$. The competition between the fermi energy $t$ and the magnetic energy $g$ will eventually agree on a saddle point value for $\phi$ which is between 0 and $\pi$ and its specific value can be tuned by the $t/g$ ratio. In terms of fermions, the stagger flux $\phi$ is interpreted as loop currents alternating between clockwise and counterclockwise around each plaquette following the check board pattern. Such a state is also call the orbital antiferromagnet (an antiferromagnetic arrangement of orbital angular momentum) or d-wave density wave (DDW) in high-Tc context.

Here $\phi$ serves as the order parameter of the stagger flux state. Because $\phi$ changes sign under time reversal symmetry (like any other magnetic flux), the spontaneous development of the stagger flux pattern in the spinless fermion system will break the time reversal symmetry. In solid-state materials, such phenomenon has not been observed due to the too small $t/g$ ratio which is unable to drive $\phi$ away from 0. However considering the fast development of cold atom physics, the spontaneous time reversal symmetry broken in spinless fermion system may be realized in the future in the optical lattice.