Quantum spin liquid: a tale of emergence from frustration

I give a review of the current status of the quantum spin liquid, emphasizing the connection to lattice gauge theory, de-confinement and the emergence of novel particles and gauge fields in the low energy effective theory The study of quantum spin liquid and frustrated magnetism is a field in many-body theory which has attracted a great deal of attention in recent years, due both to theoretical advances and the fact that promising candidate systems have been found experimentally. I would like to review some of the recent progress, but first begin with a bit of historical perspective. Louis Néel proposed the possibility of antiferromagnetic order back in 1936,[1] and by now Néel order is ubiquitous and seems second nature to condensed matter physicists. It is difficult for us to comprehend that Néel’s ideas was initially met with skepticism and Landau was among the doubters. This was briefly mentioned in the obituary for Néel written by Jacques Friedel.[2] ... Werner Heisenberg’s work on exchange had just shown that short-range effects should be expected. Between 1931 and 1933, Néel observed those effects in the susceptibility of iron and alloys and in the specific heat of nickel. Then, assuming that short-range interactions could be antiparallel, Néel developed the concept of anti ferromagnetism, in which two interpenetrating atomic lattices are treated in a molecular field approximation. ... To describe anti ferromagnetism Lev Landau and Cornelius Gorter suggested quantum fluctuations to mix Néel’s solution with that obtained by reversal of moments. ... Néel’s difficulties with antiferromagnetism and inconclusive discussions in the Strassbourg international meeting of 1939 fostered his skepticism about the usefulness of quantum mechanics; this was one of the few limitations of this superior mind. To paraphrase Friedel, we can say that Landau’s objection is that unlike a ferromagnet, Néel’s up-down arrangement of spins is not an eigenstate of the Heisenberg exchange Hamiltonian, because the mutual spin flip term will reduce the sublattice order and may eventually disorder the Néel state. After all, Néel’s solution is an uncontrolled mean-field theory, and the only exact solution known at the time is Bethe’s solution of the Heisenberg spin chain,[3] which does not show long range order. Therefore in the absence of direct experimental evidence, one has reasons to be skeptical. Well, Nature has sided with Néel, and except for one dimensional spin systems, almost all experimental spin systems find a way to order at low temperatures. (Disorder and random exchange may lead to spin freezing, but let us leave spin glass out of this discussion.) The possibility of quantum fluctuations destroying Néel order was raised again in 1973 by P.W. Anderson.[4] He focused on the singlet formation between two spins with energy gain of 17th International Conference on Recent Progress in Many-Body Theories (MBT17) IOP Publishing Journal of Physics: Conference Series 529 (2014) 012001 doi:10.1088/1742-6596/529/1/012001 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 −S(S + 1)J . The origin of the factor unity in (S + 1) is quantum mechanics and S = 12 enjoys the greatest energy gain. In a spin chain, a trial wave function of singlet dimers yields an energy of −8J per bond, which is lower than the Néel value of −4J . Indeed, we now understand Bethe’s solution as a linear superposition of singlets with varying range. Anderson suggested that a similar superposition may hold for higher dimensions, which he called the resonating valence bond (RVB) state. He further proposed that frustrated lattices such as the triangular lattice may help stabilize the RVB state vs Néel order. Unfortunately it was found that the Heisenberg model can partially relieve frustration by forming 120 degrees order.[5] The field laid dormant until 1987, when soon after the discovery of high temperature superconductors, Anderson argued that upon doping, the RVB state naturally forms a superconductor because the singlet pairs are now mobile and can be viewed as Cooper pairs.[6] This suggestion launched a great deal of theoretical activities. While there remains no consensus on the theory of high temperature superconductors, there has been a great deal of development in the “easier” problem of quantum disordered spin states, which we call spin liquids. The requirement for a spin liquid is that it is a charge insulator with an odd number of electrons per unit cell (i.e. a Mott insulator) which does not exhibit Néel order down to zero temperature, despite the presence of antiferromagnetic exchange. We now know that the spin liquid is more than just the absence of order. Theory predicts the emergence of new excitations such as particle carrying S = 12 and no charge, called spinons. Spinons may be fermions or bosons, and the excitation may gapped or gapless. Furthermore, the spinons do not live by themselves, but are coupled to an emergent gauge field, which may be U(1) or Z2. Current analytic theory is not powerful enough to tell us which of these possibilities is realized for a given Hamiltonian, but the different possibilities can be classified and their nature fully described. As we shall see, the spin liquid problem is closely related to lattice gauge theory coupled to matter field, and the spin liquid states correspond to the deconfined phases. The field of spin liquids received a powerful boost several years ago with the discovery of promising experimental candidates. They include the organic molecular compounds and a mineral called Herbertsmithite, which is a S = 12 Kagome lattice that can be synthesized in the laboratory.[7] The S = 12 Kagome lattice is a highly frustrated two dimensional lattice which has long been suspected to harbor a spin liquid ground state based on numerical work.[8] What is interesting is that in addition to using frustration to stabilize the spin liquid as proposed by Anderson, the organic materials have taught us a second route, i.e., the proximity to the Mott transition. The two families of organic material consist of molecular dimers (called ET and dmit as abbreviation) which carry a single electron each, and form an approximate triangular lattice. It can be modeled as a Hubbard model on a triangular lattice with one electron per unit cell.[9, 10] Deep in the Mott insulator regime, the low energy physics is described by a nearest-neighbor Heisenberg model and the system is expected to order into the 120 degree phase. The spin liquid materials turn out to be insulators which turn into a metal (dmit) or superconductor (ET) under modest pressure. The proximity to the Mott transition means that while the low energy excitations may still be described by the spin degrees of freedom, the Hamiltonian is much more complicated than the nearest-neighbor Heisenberg. For example, ring exchange terms may be important.[9] We therefore focus on the nature of the ground state, and leave aside the question of what spin Hamiltonian can stabilize it. These two quite distinct spin liquid candidates share the common feature that they both appear to be gapless. In the organic system, the spin susceptibility goes to a constant and the specific heat has a linear T term,[11] both characteristic of a metal with a Fermi surface, and most uncommon for an insulator. Thermo-conductivity κ was measured and it is found at least for the dmit salts that κ/T goes to a constant at low temperature.[12] All these strange properties are consistent with the emergence of fermonic spinons which form a Fermi surface. The situation with Herbertsmithite is less clear because there exists a significant amount of local 17th International Conference on Recent Progress in Many-Body Theories (MBT17) IOP Publishing Journal of Physics: Conference Series 529 (2014) 012001 doi:10.1088/1742-6596/529/1/012001