Phase boundary and finite temperature crossovers of the quantum Ising model in two dimensions

Abstract

We revisit the two-dimensional quantum Ising model by computing renormalization group flows close to its quantum critical point. The low but finite temperature regime in the vicinity of the quantum critical point is squashed between two distinct non-Gaussian fixed points: the classical fixed point dominated by thermal fluctuations and the quantum critical fixed point dominated by zero-point quantum fluctuations. Truncating an exact flow equation for the effective action we derive a set of renormalization group equations and analyze how the interplay of quantum and thermal fluctuations, both non-Gaussian in nature, influences the shape of the phase boundary and the region in the phase diagram where critical fluctuations occur. The solution of the flow equations makes this interplay transparent: we detect finite temperature crossovers by computing critical exponents and we confirm that the power law describing the finite temperature phase boundary as a function of control parameter is given by the correlation length exponent at zero temperature as predicted in an ǫ-expansion with ǫ = 1 by Sachdev, Phys. Rev. B 55, 142 (1997).PACS numbers: 05.10. Cc, 73.43.Nq, 71.27.+a The quantum Ising model serves as a prime textbook example to illustrate fundamental aspects of quantum phase transitions. 1,2,3,4,5 The quantum Ising Hamiltonian has the form,where J is a ferromagnetic exchange coupling, the sum i j runs over pairs of nearest neighbor sites, and the quantum degrees of freedom are represented by the operators σ z,x i which reside on a site i of a hypercubic lattice in d dimensions and reduce to the Pauli matrices in the basis where σ z is diagonal. 2 The parameter h is the external transverse magnetic field which induces quantum-mechanical tunneling events that flip the orientation of the Ising spins. The relevant parameter of Eq. (1) is the ratioδ ∼ J/h. For largeδ the ground state is ferromagnetically ordered and spontaneously breaks the discrete Z 2 Ising symmetry while for smallerδ the spins in the ground state remain disordered. The two phases are separated by a second order quantum phase transition at a criticalδ c . At finite temperature the formation of spin order is hindered and theδ at which the order sets in is increased leading to a line of second order phase transitions T c δ that terminates at the quantum critical point (QCP) T c δ c = 0. Since the phase diagram of the quantum Ising model exhibits many generic features of physical systems in vicinity of their QCPs, it is important to understand it in detail.Various finite temperature properties of compounds modelled by the quantum Ising model were measured experimentally in three dimensions. 6 Theoretically, the corresponding phase diagrams were investigated by Sachdev within analytical approaches. 2,7 These rely on the effective continuum field theory to which an expansion around the upper critical dimension is applied. In two dimensions, the quantum Ising model describes a strongly coupled lattice system. Its ground state was recently analyzed nume...

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