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The transport of heat and charge in cuprates was measured in single crystals of La_{2-x}Sr_xCuO_{4+\delta} (LSCO) across the doping phase diagram at low temperatures. In underdoped LSCO, the thermal conductivity is found to decrease with increasing magnetic field in the T goes to 0 limit, in striking contrast to the increase observed in all superconductors, including cuprates at higher doping. In heavily underdoped LSCO, where superconductivity can be entirely suppressed with an applied field, we show that a novel thermal metal-to-insulator transition takes place upon going from the superconducting state to the field-induced normal state.

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