Scalar fields in a non-commutative space

We discuss the λφ4 model in 2- and 3-dimensional non-commutative spaces. The mapping onto a Hermitian matrix model enables its non-perturbative investigation by Monte Carlo simulations. The numerical results reveal a phase where stripe patterns dominate. In d = 3 we show that in this phase the dispersion relation is deformed in the IR regime, in agreement with the property of UV/IR mixing. This "striped phase" also occurs in d = 2. For both dimensions we provide evidence that it persists in the simultaneous limit to the continuum and to infinite volume ("Double Scaling Limit"). This implies the spontaneous breaking of translation symmetryThis work was supported by the Mexican Consejo Nacional de Ciencia y Tecnología (CONACyT) through project 155905/10 “Física de Partículas por medio de Simulaciones Numéricas”, and by the Spanish MINECO (grant SEV-2012-0249