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In a recent comment [1], Armitage and Hu have suggested that our experiment observing dichroism in angle resolved photoemission (ARPES) [2] could not be conclusively interpreted as arising from time reversal symmetry breaking, arguing that our observations are likely due to structural effects. The concerns expressed by Armitage and Hu that our results could be due to a change in the mirror plane are as important as they are obvious. In fact the first part of their comment merely restates the results of Simon and Varma [3] about the relationship and contrast of effects due to time reversal symmetry breaking and those caused by crystallographic changes. In any test of time reversal symmetry one must ensure that parity alone is not inducing the observed changes. We have indeed considered this issue very carefully in the course of our study [2] and it is precisely the lack of temperature dependent structural changes significant enough to explain the magnitude of the observed dichroism that forced us to conclude that time reversal symmetry breaking is the only plausible explanation. Furthermore, recent experiments by Borisenko, et al. [4] confirm that changes in the mirror plane are unmeasurably small.
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When matter undergoes a phase transition from one state to another, usually a change in symmetry is observed, as some of the symmetries exhibited are said to be spontaneously broken. The superconducting phase transition in the underdoped high-Tc superconductors is rather unusual, in that it is not a mean-field transition as other superconducting transitions are. Instead, it is observed that a pseudo-gap in the electronic excitation spectrum appears at temperatures T* higher than Tc, while phase coherence, and superconductivity, are established at Tc (Refs. 1, 2). One would then wish to understand if T* is just a crossover, controlled by fluctuations in order which will set in at the lower Tc (Refs. 3, 4), or whether some symmetry is spontaneously broken at T* (Refs. 5-10). Here, using angle-resolved photoemission with circularly polarized light, we find that, in the pseudogap state, left-circularly polarized photons give a different photocurrent than right-circularly polarized photons, and therefore the state below T* is rather unusual, in that it breaks time reversal symmetry11. This observation of a phase transition at T* provides the answer to a major mystery of the phase diagram of the cuprates. The appearance of the anomalies below T* must be related to the order parameter that sets in at this characteristic temperature .
We study the behavior of spinless fermions in superconducting state, in which the phases of the superconducting order parameter depend on the direction of the link. We find that the energy of the superconductor depends on the phase differences of the superconducting order parameter. The solutions for the phases corresponding to the energy minimuma, lead to a topological superconducting state with the nontrivial Chern numbers. We focus our quantitative analysis on the properties of topological states of superconductors with different crystalline symmetry and show that the phase transition in the topological superconducting state is result of spontaneous breaking of time-reversal symmetry in the superconducting state. The peculiarities in the chiral gapless edge modes behavior are studied, the Chern numbers are calculated.
States of matter that break time-reversal symmetry are invariably associated with magnetism or circulating currents. Recently, one of us proposed a phase, the directional scalar spin chiral order (DSSCO), as an exception: it breaks time-reversal symmetry via chiral ordering of spins along a particular direction, but is spin-rotation symmetric. In this work, we prove the existence of this state via state-of-the-art density matrix renormalization group (DMRG) analysis on a spin-1 chain with nearest-neighbor bilinear-biquadratic interactions and additional third-neighbor ferromagnetic Heisenberg exchange. Despite the large entanglement introduced by the third-neighbor coupling, we are able to access system sizes up to $L=918$ sites. We find first order phase transitions from the DSSCO into the famous Haldane phase as well as a spin-quadrupolar phase where spin nematic correlations dominate. In the Haldane phase, we propose and demonstrate a method for detecting the topological edge states using DMRG that could be useful for other topological phases too.
The kagome lattice, which is composed of a network of corner-sharing triangles, is a structural motif in quantum physics first recognized more than seventy years ago. It has been gradually realized that materials which host such special lattice structures can exhibit quantum diversity, ranging from spin-liquid phases, topological matter to intertwined orders. Recently, charge sensitive probes have suggested that the kagome superconductors AV_3Sb_5 (A = K, Rb, Cs) exhibit unconventional chiral charge order, which is analogous to the long-sought-after quantum order in the Haldane model or Varma model. However, direct evidence for the time-reversal symmetry-breaking of the charge order remains elusive. Here we utilize state-of-the-art muon spin relaxation to probe the kagome charge order and superconductivity in KV_3Sb_5. We observe a striking enhancement of the internal field width sensed by the muon ensemble, which takes place just below the charge ordering temperature and persists into the superconducting state. Remarkably, the muon spin relaxation rate below the charge ordering temperature is substantially enhanced by applying an external magnetic field. We further show the multigap nature of superconductivity in KV_3Sb_5 and that the T_c/lambda_{ab}^{-2} ratio is comparable to those of unconventional high-temperature superconductors. Our results point to time-reversal symmetry breaking charge order intertwining with unconventional superconductivity in the correlated kagome lattice.
The noncentrosymmetric superconductor Re$_{24}$Ti$_{5}$, a time-reversal symmetry (TRS) breaking candidate with $T_c = 6$,K, was studied by means of muon-spin rotation/relaxation ($mu$SR) and tunnel-diode oscillator (TDO) techniques. At a macroscopic level, its bulk superconductivity was investigated via electrical resistivity, magnetic susceptibility, and heat capacity measurements. The low-temperature penetration depth, superfluid density and electronic heat capacity all evidence an $s$-wave coupling with an enhanced superconducting gap. The spontaneous magnetic fields revealed by zero-field $mu$SR below $T_c$ indicate a time-reversal symmetry breaking and thus the unconventional nature of superconductivity in Re$_{24}$Ti$_{5}$. The concomitant occurrence of TRS breaking also in the isostructural Re$_6$(Zr,Hf) compounds, hints at its common origin in this superconducting family and that an enhanced spin-orbital coupling does not affect pairing symmetry.
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