Topological insulators (TIs) are newly discovered states of matter with robust metallic surface states protected by the topological properties of the bulk wavefunctions [1][2][3][4][5][6]. A quantum phase transition (QPT) from a TI to a conventional insulator and a change in topological class can only occur when the bulk band gap closes [3]. In this work, we have utilized time-domain terahertz spectroscopy (TDTS) to investigate the low frequency conductance in (Bi 1−x In x ) 2 Se 3 as we tune through this transition by indium substitution. Above certain substitution levels we observe a collapse in the transport lifetime that indicates the destruction of the topological phase. We associate this effect with the threshold where states from opposite surfaces hybridize. The substitution level of the threshold is thickness dependent and only asymptotically approaches the bulk limit x ≈ 0.06 where a maximum in the midinfrared absorption is exhibited. This absorption can be identified with the bulk band gap closing and a change in topological class. The correlation length associated with the QPT appears as the evanescent length of the surface states. The observation of the thickness-dependent collapse of the transport lifetime shows the unusual role that finite size effects play in this topological QPT.The topological character of TIs is determined by the nature of their valence-band wave functions, which can be quantified by 4 Z 2 invariants. Fu and Kane have shown that for inversion symmetric crystals it is possible to evaluate these invariants directly with knowledge of the parity of Bloch wave functions for the occupied electronic states at high symmetry points in the Brillouin zone [10]. Although their argument is formulated for inversion symmetric systems, a material's topological classification does not require inversion or translation symmetry. Therefore the expectation is that the alloying of known TIs with lighter elements by reducing spin-orbit coupling or the tuning of lattice constant can cause the bulk band gap ∆ to close and invert at a quantum critical point where the topological class changes (See cartoon * Electronic address: npa@pha.jhu.edu Fig. 1a). This has been investigated in the thalliumbased ternary chalcogenide alloy TlBi(S 1−x Se x ) 2 [7-9], but thus far only with photoemission (Supplementary Information (SI) section B). Although signatures of topological surface state (TSS) conduction have been found in Bi 2 Se 3 [11-14], a demonstration that the surface transport changes dramatically when the band gap closes and the bulk changes topological class [15] would be strong evidence for the topological nature of these materials and is still lacking. In this regard, it was pointed out recently that indium (In) substitutes for bismuth to form a solid solution in Bi 2 Se 3 and that the non-topological end member In 2 Se 3 of the (Bi 1−x In x ) 2 Se 3 series shares the common rhombohedral D 5 3d structure with Bi 2 Se 3 [6]. In Ref.[6] a topological to trivial transition was observed in a range x ∼ 0.0...

The nature of the underdoped pseudogap regime of the high-temperature superconductors has been a matter of long-term debate [1][2][3]. On quite general grounds, one expects that due to their low superfluid densities and short correlation lengths, superconducting fluctuations will be significant for transport and thermodynamic properties in this part of the phase diagram [4,5]. Although there is ample experimental evidence for such correlations, there has been disagreement about how high in temperature they may persist, their role in the phenomenology of the pseudogap, and their significance for understanding high-temperature superconductivity [6][7][8][9][10]. In this work we use THz time-domain spectroscopy (TTDS) to probe the temporal fluctuations of superconductivity above the critical temperature (T c ) in La 2−x Sr x CuO 4 thin films over a doping range that spans almost the entire superconducting dome (x = 0.09 to 0.25). Signatures of the fluctuations persist in the conductivity in a comparatively narrow temperature range, at most 16 K above T c . Our measurements show that superconducting correlations do not make an appreciable contribution to the charge transport anomalies of the pseudogap in LSCO at temperatures well above T c .In general, continuous phase transitions are typified by fluctuations with correlation length and time scales that diverge near T c . Dynamical measurements like TTDS are a sensitive probe of the onset of superconductivity [11] and measure its temporal correlations on the time scales of interest. In the presence of superconducting vortices such highfrequency measurements are not affected by effects like vortex pinning, creep, and edge barriers that often complicate interpretation of low frequency and DC results. In this study, we investigate the fluctuation superconductivity in thin films of LSCO grown by molecular beam epitaxy (MBE). This synthesis technique provides exquisite control of the thickness and chemical composition of the films; the intrinsic chemical tunability of LSCO allows us to investigate essentially the entire phase diagram. For details on the films, see the 'Supplementary Information' (SI).In Figs. 1a and b, we show the real (σ 1 ) and imaginary (σ 2 ) parts of the THz conductivity measured at a number of different temperatures for optimally doped LSCO (x = 0.16) with T c =41 K. We obtain similar data at other doping levels. The spectra are easily understood in the limiting cases of high and low temperatures. Well above the onset of superconductivity, the real part of the conductivity is almost frequency independent and the imaginary part is small, consistent with the expectation for the behavior of a metal at frequencies well below the normal state scattering rate. At the lowest temperature the conductivity is consistent with that expected for a long-range ordered superconductor; σ 1 is small as most of the low frequency spectral weight has condensed into the ω = 0 delta function, and the frequency dependence of σ 2 is very close to 1/ω. Our principal interest...

We report the THz response of thin films of the topological insulator Bi2Se3. At low frequencies, transport is essentially thickness independent showing the dominant contribution of the surface electrons. Despite their extended exposure to ambient conditions, these surfaces exhibit robust properties including narrow, almost thickness-independent Drude peaks, and an unprecedentedly large polarization rotation of linearly polarized light reflected in an applied magnetic field. This Kerr rotation can be as large as 65• and can be explained by a cyclotron resonance effect of the surface states.Ordered states of matter are typically categorized by their broken symmetries. With the ordering of spins in a ferromagnet or the freezing of a liquid into a solid, the loss of symmetry distinguishes the ordered state from the disordered one. In contrast, topological states are distinguished by specific topological properties that are encoded in their quantum mechanical wavefunctions [1]. Frequently, a consequence of these properties is that there are robust "topologically protected" states on the sample's boundaries. The edge states of the quantum Hall effect (QHE) are the classic example [2]. In the last few years, it was realized that another class of such topological matter may exist in 3D band insulators with large spin-orbit interaction [3][4][5][6]. These so-called topological insulators are predicted to host robust surface states, which exhibit a number of interesting properties including spin helicity, immunity to back-scattering, and weak anti -localization. There are predictions of a number of unusual phenomena associated with these surface states, including a proximity-effect-induced exotic superconducting state with Majorana fermions bound to a vortex [7,8] and an axion electromagnetic response [9,10], and proposals for applications, such as their use in terahertz (THz) devices [11].Most of the signatures of topological behavior in these materials thus far have come from surface probes such as angle resolved photoemission (ARPES) and scanning tunneling spectroscopy [12][13][14][15][16][17]. These experiments have revealed that the surface states indeed show signatures of the predicted topological properties, such as a Diraclike dispersion, chiral spin textures, and the absence of backscattering. Direct observation of the topological behavior in transport has been hampered by the lack of a true bulk insulating state. Only recently have transport experiments started to distinguish the surface contribution from the bulk [18][19][20][21].As opposed to the case of the quantum Hall effect, in topological insulators, the quantization of the offdiagonal conductivity is not a requirement for the existence of the topological state. This, along with the problem of bulk conduction, has made finding a unique signature of this state difficult. It has been proposed that topological insulators may be characterized by their electrodynamic properties [9] due to the existence of an axionic term in the action ∆L = αθ dxdtE · B, where...