The Valence Bond Glass phase

We show that a new glassy phase can emerge in presence of strong magnetic frustration and quantum fluctuations. It is a Valence Bond Glass. We study its properties solving the Hubbard-Heisenberg model on a Bethe lattice within the large N limit introduced by Affleck and Marston. We work out the phase diagram that contains Fermi liquid, dimer and valence bond glass phases. This new glassy phase has no electronic or spin gap (although a pseudo-gap is observed), it is characterized by long-range critical valence bond correlations and is not related to any magnetic ordering. As a consequence it is quite different from both valence bond crystals and spin glasses. The interplay of strong quantum fluctuations and geometrically frustrated magnetic interactions can give rise to new low temperature phases. As noticed by Anderson [1] a way to minimize the effect of frustration and obtain a low energy state is coupling the electrons in valence bonds. A very good variational wave function that is generically in competition with the antiferromagnetic (or more general magnetic) state can be obtained by forming a superposition of short range valence bonds that are arranged as dimers on the lattice. If no lattice symmetry is broken this corresponds to the (so called) resonating valence bond liquid (RVBL). In the last decades, this state has received a lot of attention in connection with the unusual physical behavior of the normal phase of underdoped high Tc superconductors [2]. Indeed Anderson [3] proposed that the holes created by doping the antiferromagnetic insulator (of the high Tc’s phase diagram) can gain substantial kinetic energy in the RVBL state and not in an antiferromagnetic background. As a consequence, doping favors the RVBL state which could then become the thermodynamic stable phase and be responsible for the unusual behavior of underdoped samples. Concomitantly, resonating valence bond ground states have been the focus of an intense activity [4] in the context of frustrated magnets. RVBL or spin liquids have been found for several models [4]. These states can undergo quantum phase transitions where lattice symmetries are spontaneously broken. This gives rise to valence bond crystals (VBC). Different models are known to lead to this type of ground states [4] characterized by long range dimer-dimer correlations. The situation in experiments is complicated by unavoidable magneto-elastic couplings: making the difference between induced and spontaneous dimerization is a difficult task. A first experimental example of spontaneously broken states has been apparently found in [5]. The aim of this work is to study a new kind of valence bond state: the valence bond glass (VBG). Similarly to VBC the arrangement of the dimers (or valence bonds) breaks the lattice symmetry. However, contrary to VBC, this corresponds to an amorphous dimerization and not crystalline one. Although VBG are analogous to spin glasses [6] they are physically quite different. In particular the spins do not freeze in a disordered profile. We expect that the VBG phase can arise in presence of strong magnetic frustration as one of the competing ground states. The addition of (little) quenched disorder will favor this phase. Depending on the system, the low temperature phase could be either a VBG or a spin glass. Actually, the spin glass phase is conjectured to exist even in absence of disorder on some frustrated lattices [7] (see however [8, 9]). In the following we shall investigate the properties of the valence bond glass phase focusing on the HubbardHeisenberg model within the largeN approximation introduced by Affleck and Marston [10]. The underlying lattice we shall focus on is a random regular graph with connectivity z. The reason for this choice is twofold. First, this type of graphs are on any finite lengthscale as Bethe It is a graph taken at random within the set of graphs whose N sites are all connected to z randomly choosen neighbors.