Time-reversal symmetry breaking without magnetism via directional scalar spin chirality Pavan Hosur, Stanford University a phase of matter that breaks time-reversal symmetry (TRS) but has no density of moments. Invariably, in TRS-breaking phases, one can find a set of total angular momentum operators such that their total expectation value is extensive. Thus, TRS-violation is always identified with static magnetism. The theoretical goal is to describe a phase that disobeys this basic synonymy. Directional scalar spin chiral order (DSSCO) Key idea: Discrete symmetry breaking is more robust against fluctuations than continuous symmetry breaking is. Thus, partially melt magnet order so that spin-rotation symmetry is restored but TRS remains broken. Classical magnetic order Conditions for DSSCO S = 0 due to 1D T = 0, clean Mermin-Wagner 2D T ≠ 0, clean Mermin-Wagner 3D Any T, field disorder Imry-Ma TRS q Kerr KS DSSCO only × × × × = 0 = 0 DSSCO + j x × × × × = 0 = 0 DSSCO + j y × × × × × × ≠ 0 ≠ 0 DSSCO + CDW × × × × × × ≠ 0 ≠ 0 For S > 1/2, H = H bi +H f|| +H f +H dis , where For S = 1/2, H = H ½ +H f +H dis , where 1. polar Kerr effect, a TRS-breaking-indicator, in the pseudogap phase of the underdoped cuprates, which is (i) untrainable by a magnetic field (ii) unchanged in sign on flipping the sample (iii) hysteretic above the onset temperature T K 2. NMR, which has not seen magnetic moments below T K 3. X-rays, which see charge ordering tendencies below T K 4. Nernst effect, which sees nematicity above T K 5. Transmission, which sees broken vertical reflections below T K (Refs: Xia 08; Karapetyan 12, 14; He 11; Blackburn 13; Comin 15; Tacon 14; Daou 10; Wu 15; Lubashevsky 14) Plausible phase diagram: The DSSCO forms at T D and coexists with charge order below T K . Together, they give a small Kerr effect with the observed symmetries and an immeasurably small NMR Knight shift (KS). Above T K , Kerr hysteresis upto T D and Nernst anisotropy upto T * suggest T D ~ T * . Detecting the DSSCO: If the chiral ordering direction is x,a j y current should produce a polar Kerr effect trainable by the current between T K and T D , but a j x current should not produce any Kerr effect. Any D 3D