First Order Phase Transition in the Anisotropic Quantum Orbital Compass Model

Abstract

We investigate the anisotropic quantum orbital compass model on an infinite square lattice by means of the infinite projected entangled-pair state algorithm. For varying values of the Jx and Jz coupling constants of the model, we approximate the ground state and evaluate quantities such as its expected energy and local order parameters. We also compute adiabatic time evolutions of the ground state, and show that several ground states with different local properties coexist at Jx = Jz. All our calculations are fully consistent with a first order quantum phase transition at this point, thus corroborating previous numerical evidence. Our results also suggest that tensor network algorithms are particularly fitted to characterize first order quantum phase transitions. Introduction.-When quantum many-body systems are cooled down close to zero temperature, important collective phenomena may occur [1]. A good example is provided by transition-metal oxides, whose physical properties have become of increasing interest in the last few years [2]. In these compounds the orbital degrees of freedom of the atomic electrons play a key role in determining properties such as metal-insulator transitions, high-temperature superconductivity and colossal magnetoresistance.The paradigmatic approach to these systems is based on the so-called orbital compass models [3,4], which have been the subject of many studies in the past both in the classical and quantum regimes. For these systems, JahnTeller effects produce an anisotropy of the pseudospin couplings which is intertwined with the orientation of the interaction bonds. The properties of these systems have attracted considerable attention since they are endowed with symmetries that effectively reduce the dimensionality of the system (the so-called dimensional reduction) [5,6]. Despite of their apparent simplicity, orbital compass models are relevant in a variety of contexts, such as in determining the physics of Mott insulators with orbital degrees of freedom [3] and the implementation of protected qubits for quantum computation in Josephson junction arrays [7]. These systems are also candidates to exhibit topological quantum order [8]. Furthermore, it was recently shown how to simulate these models using polar molecules in optical lattices and systems of trapped ions with state-of-the-art technology [9,10].Generally speaking, the symmetries in these systems involve large degeneracies in their energy spectra, which make their numerical simulation difficult [11]. This fact, together with the lack of exact solutions, makes it hard to elucidate their phase diagrams. In this paper we use a tensor product state (TPS) [12,13] or projected entangled-pair state (PEPS) [14] to study the twodimensional anisotropic quantum orbital compass model (AQOCM) and, in particular, to investigate whether its phase transition is of first order [15,16] or second order [17]. More specifically, we use the infinite PEPS (iPEPS)

Compass models: Theory and physical motivations

“…In the thermodynamic limit, these sets of 2 L degenerate states further merge at the point J x = J y to form bands of 2 L+1 degenerate states. Numerical and other analysis illustrates that the level crossing at J x = J y is related to a first-order (or discontinuous) transition of the lowest energy state as a function of (J x − J y ) (Chen et al, 2007;Dorier et al, 2005;Orús et al, 2009). The two sets of states for positive and negative values (J x − J y ) are related to one another by the global Ising type reflection symmetry of the 90…”

“…(1), it is clear that when |J z | exceeds |J x | there is a preferential orientation of the spins along the z axis (and, vice versa, when |J x | exceeds |J z an ordering along the x axis is preferred). The point J x = J z (a "self-dual" point for reasons which will be elaborated on later) marks a transition which has been studied by various other beautiful means and found to be first order (Chen et al, 2007;Dorier et al, 2005;Orús et al, 2009), similar to the 1D case (Brzezicki et al, 2007).…”

“…Particular forms for this global symmetry were written down in (Nussinov and Fradkin, 2005;Nussinov and Ortiz, 2009a;Orús et al, 2009). In essence, these correspond, e.g., to rotations in the internal pseudo-spin space about the T z axis by an angle of 90…”

Quantum phase transitions in exactly solvable one-dimensional compass models

“…In the latter case the operators include just one of the orthogonal pseudospin components at each bond and are Ising-like. This form of interactions is found as well in the compass models [28][29][30][31][32][33][34][35][36][37][38][39], and in the Kitaev model on the honeycomb lattice [40][41][42]. The interactions that are considered here are defined by the pseudospin operators T γ i for two active orbitals (for T = 1/2), and we define them as linear combinations of the Pauli matrices {σ These operators define the generalized compass model (GCM) considered in this paper.…”

“…Quantum phase transitions (QPTs) between different types of order were established in the 1D QCM [38], in a quantum compass ladder [39], and in the 2D QCM [29][30][31][32][33][34][35][36][37], when anisotropic interactions are varied through the isotropic point and the ground state switches between two different types of Ising nematic order dictated by either interaction. At the transition point itself, i.e., when the competing interactions are balanced, the ground state is highly degenerate and contains states which correspond to both relevant kinds of nematic order.…”