Type equation here.

Time-reversal symmetry breaking withoutmagnetism via directional scalar spin chiralityPavan Hosur, Stanford University

a phase of matter that breaks time-reversal symmetry (TRS) but has nodensity of moments. Invariably, in TRS-breaking phases, one can find aset of total angular momentum operators such that their totalexpectation value is extensive. Thus, TRS-violation is always identifiedwith static magnetism. The theoretical goal is to describe a phase thatdisobeys this basic synonymy.

Directional scalar spin chiral order (DSSCO)

Key idea: Discrete symmetry breaking is more robust against fluctuationsthan continuous symmetry breaking is. Thus, partially melt magnet orderso 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 𝑴𝒙 𝑴𝒚 𝑴𝒛 𝑹𝒙𝟐 𝑹𝒚

𝟐 𝑹𝒛𝟐 qKerr KS

DSSCO only × × × × = 0 = 0

DSSCO + jx × × × × = 0 = 0

DSSCO + jy × × × × × × ≠ 0 ≠ 0

DSSCO + CDW × × × × × × ≠ 0 ≠ 0

For S > 1/2, H = Hbi+Hf||+Hf+Hdis, where

For S = 1/2, H = H½+Hf+Hdis, where

1. polar Kerr effect, a TRS-breaking-indicator, in the pseudogap phaseof the underdoped cuprates, which is (i) untrainable by a magneticfield (ii) unchanged in sign on flipping the sample (iii) hystereticabove the onset temperature TK

2. NMR, which has not seen magnetic moments below TK

3. X-rays, which see charge ordering tendencies below TK

4. Nernst effect, which sees nematicity above TK

5. Transmission, which sees broken vertical reflections below TK(Refs: Xia 08; Karapetyan 12, 14; He 11; Blackburn 13; Comin 15; Tacon 14; Daou 10; Wu 15; Lubashevsky 14)

Plausible phase diagram: The DSSCOforms at TD and coexists with chargeorder below TK. Together, they give asmall Kerr effect with the observedsymmetries and an immeasurably smallNMR Knight shift (KS). Above TK, Kerrhysteresis upto TD and Nernstanisotropy upto T* suggest TD ~ T*.

Detecting the DSSCO: If the chiral

ordering direction is x, a jy currentshould produce a polar Kerr effecttrainable by the current between TK

and TD, but a jx current should notproduce any Kerr effect.

Any D

3D