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The experimentally measured phase diagram of cuprate
superconductors in the temperature-applied magnetic field plane
illuminates key issues in understanding the physics of these materials.
At low temperature, the superconducting state gives way to a long-range
charge order with increasing magnetic field; both the orders coexist in
a small intermediate region. The charge order transition is strikingly
insensitive to temperature and quickly reaches a transition temperature
close to the zero-field superconducting $T_c$ . We argue that such a
transition along with the presence of the coexisting phase is difficult
to obtain in a weak coupling competing orders formalism. We demonstrate
that for some range of parameters there is an enlarged symmetry of the
strongly coupled charge and superconducting orders in the system
depending on their relative masses and the coupling strength of the two
orders. We establish that this sharp switch from the superconducting
phase to the charge order phase can be understood in the framework of a
composite SU(2) order parameter comprising the charge and
superconducting orders. Finally, we illustrate that there is a
possibility of the coexisting phase of the competing charge and
superconducting orders only when the SU(2) symmetry between them is
weakly broken due to biquadratic terms in the free energy. The relation
of this sharp transition to the proximity to the pseudogap quantum
critical doping is also discussed.
The main hindrance in the full understanding of the collective behavior
of interacting fermions is the limited availability of exact analytical
results. On top of it, the famous "fermion sign-problem" restricts their
investigation even numerically in Monte Carlo methods. But there are
some special cases where the quantum Monte Carlo simulations can be
sign-problem free. One such example is a spin fermion model with two
bands mimicking some features of the fermiology of the cuprates. If time
permits, I will explain how this model can be studied using a
determinant quantum Monte Carlo method in order to encapsulate various
instabilities arising out of spin fluctuations.
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