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| By many measures, the three-dimensional pyrochlore lattice (shown here) is one of the most frustrated geometries for antiferromagnetism. It is actually quite common in a wide variety of materials, including both the crystal structure of the same name, A2B2X7 (in which both A and B sites form pyrochlore lattices), and the spinel structure, AB2X4, in which the B sites form the pyrochlore lattice. Such materials display a broad variety of physical phenomena, include ferromagnetism, antiferromagnetism, ferroelectricity, spin-driven lattice distortions, and spin liquid physics. We study this geometry in a range of theoretical and material contexts. |  |
Click below for specific descriptions:
The classical nearest-neighbor Heisenberg antiferromagnet on the pyrochlore lattice is very unusual in that it does not order magnetically at any non-zero temperature. This is a result of the very large classical degeneracy of ground states. One mechanism that can be important for small spin s in splitting this degeneracy is quantum fluctuations. The same large degeneracy which prevents order classically also makes theory of quantum fluctuations very tricky. Pioneering and tour-de-force work on the subject has been done by Chris Henley and collaborators (see this paper ). Unfortunately it turns out that it is necessary to go beyond the leading order (spin qave zero point energy) to fully split the ground state degeneracy, a task which at the time of writing (September 2006) is still not complete.
We have taken an alternative approach, in which rather than assume s is large (as in Henley's calculations), we instead assume spin fluctuations transverse to some preferred Ising axis are small. The approach has the advantage that it lets one treat arbitrary s, and includes the effects of quantum tunneling, which are non-perturbative in the other method. The two techniques can be compared in an overlapping limit of validity and are found (fortunately!) to agree. The result of our calculations is an effective "quantum Ising" Hamiltonian that describes the low energy states, and can usually be much more readily studied than the original Heisenberg model. Our approach is technically pretty tough (as is Henley's), and for the pyrochlore lattice requires carrying out 6th order degenerate perturbation theory!
| You can read a detailed account and analysis for the pyrochlore lattice in cond-mat/0607210 . A gentler version is in cond-mat/0608131, which also contains applications of the same technique to other lattices. Some of the findings are shown here for the half-magnetized states (corresponding to the plateaus seen in the spinel chromites). The left-hand figure shows the magnetic unit of the trigonal7 state, which is found to be the ground state for s>3/2. The colored dots are the minority spins aligned anti-parallel to the field (parallel spins are not shown). The right-hand figure shows the magnetic unit cell of the R state, in the same convention. |  |  |
It is now clear that such states (at least in all proposed cases in d>1) are described by emergent gauge-field structures, which is often exposed in non-standard quasiparticles (e.g. spin-1/2 neutral "spinons") and and global topological properties. Such a structure is accompanied by a Projective Symmetry Group (PSG - a concept due to Xiao-Gang Wen ) that gives a gauge-theoretic implementation of the unbroken physical symmetries of the state. These two ingredients may be thought of as characterizing some "fixed point" -- in the renormalization group (RG) sense -- that can potentially describe a phase or QCP of correlated quantum matter.
Our research in this area is aimed at bringing the recent theoretical advances into better contact with experiments. Of course, much of the early work on exotic states - notably slave particle mean field theories - was aimed at understanding the cuprate high-temperature superconductors. Views on the success of that particular program vary widely, and it seems appropriate, while not neglecting this important problem, to look more broadly at applications of exotic ideas to correlated materials. Aside from the cuprates, we are pursuing applications to various heavy fermion materials with QCPs, so-called "bad metals", and a variety of frustrated magnetic materials.
Click to read more about...
Such deconfined states of matter are so bizarre and counter-intuitive that many respectable researchers believed until recently that they were impossible in principle outside of the quantum Hall effect. Thus it came as a surprise to many when were able to construct two explicit examples of simple spin-1/2 models that could be proven to have such ground states! This paper postulated that an easy-axis Kagome spin-1/2 antiferromagnet provides an explicit example of a so-called Z2 spin liquid. In this model, we were able to find an exact ground state wavefunction, and more generally to define non-local "string" (dis-)order parameters that characterize the liquid state. By exact diagonalization calculations (carried out by Dong-Ning Sheng at CSUN), in this paper we were able to confirm this exotic state is stable over a wide range of phase space including a two-spin Heisenberg-type model. In this paper , we showed that a three-dimensional easy-axis spin-1/2 antiferromagnet on the pyrochlore lattice provides a simple and explicit example of a "U(1)" spin liquid in 3d. Like the Z2 spin liquid, this state supports deconfined spinons, but has in addition an emergent gapless transverse propagating "photon"-like excitation.
Specifically, several such QCPs were described. The most interesting is the transition from a Neél antiferromagnet to a valence bond solid state on a square lattice. This provides an answer to the old question of how antiferromagnetism is destroyed by frustration in a spin-1/2 magnet (though we described the situation for general spin, and with magnetic anisotropy). The mechanism for this transition is novel: the topological defects of either phase (skyrmions/vortices in the antiferromagnet and domain walls and their intersections in the valence bond solid) can be shown to carry unusual quantum numbers, such that when they proliferate as their associated order is destroyed, they induce the order of the neighboring phase. Another such deconfined QCP is the transition from a valence bond solid state to a Z2 spin liquid. We showed that at the QCP the system can be regarded as a deconfined U(1) (rather than Z2) state, with specific associated properties. Here is a long detailed paper on this problem. A less detailed but more pedagogical presentation is here .
These ideas have rather broad scope, and the group is actively pursuing other applications with an eye to experimental implications. Some immediate follow-ups are cond-mat/0408329 and cond-mat/0409470 . These papers, and related work, are described elsewhere on this site. A puzzling commentary by Robert Laughlin can be found in Science (Science, Vol 303, Issue 5663, 1475-1477) .
A single spin-1/2 chain is a proto-typical example of a gapless spin liquid state. It has power-law antiferromagnetic correlations, which in the ground state decay like 1/r, the distance along the chain. However, it should not be thought of just as a fluctuating antiferromagnet. Indeed, it also has power-law correlations in the staggered exchange energy per bound (i.e. oscillating with a period of two lattice spacings) - called staggered dimerization - with the same 1/r decay. In fact, this spin liquid state is completely understood, and the full critical theory (called the WZW SU(2) level 1 conformal field theory) is known as well as all its correlations. These two strong correlations make coupled spin chains very susceptible to staggered magnetization or dimerization ordering along the chains.
In an unfrustrated situation, magnetic exchange between chains is often the strongest coupling mechanism, leading to magnetic order (staggered along the chain) - since such interactions do not couple directly to the dimerization. Dimerization can result if spin-phonon coupling is stronger than inter-chain exchange. When the exchange is frustrated, however, the possibility is opened up for exchange-mediated dimerization to compete with staggered magnetization. We have studied this competition using renormalization group techniques in a variety of models. The actual calculations are a bit subtle - the basic mechanism is the generation of relevant (in the renormalization group sense) interactions between different chains by the naively irrelevant ones that are left from the bare inter-chain exchange interactions after cancellations due to frustration. Some examples:
| The Heisenberg antiferromaget on the checkerboard lattice, also called the planar pyrochlore and crossed-chains model, is the simplest model of corner-sharing tetrahedra. It has an anisotropic limit, when the dimensionless ratio of two exchange constants, JX/J is small, in which it consists of one-dimensional spin chains coupled weakly together in a frustrated fashion. We ( Oleg Starykh, Akira Furusaki , and LB) showed (in this paper (cond-mat/0503296) ) that in this limit the model enters a crossed dimer state with two-fold spontaneous symmetry breaking but no magnetic order. We proposed two possible phase diagrams describing the evolution of the ground state as a function of JX/J. | ||||
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Quantum wires are interesting theoretically because in one dimension, the Coulomb interaction between electrons has qualitatively significant effects, that require a description completely different from the "Fermi liquid theory" used for three-dimensional (and two-dimensional) metals. This is called Luttinger liquid theory, and leads to often dramatic effects (see e.g. earlier publications Kane et al , Bockrath et al , Yao et al and others). More recent explorations of the ramifications of strong Coulomb effects in quantum wires are described below (click!):
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| 36 pages of Cheianov+Zvonarev's paper. One page containing calculations was left off for symmetry of the image. | Our paper. There are 13 equations, several of which are definitions or final results. Our results are more general |
It is interesting that, though the Green's function decays exponentially in space in this regime, there is actually an enhanced power-law local (tunneling) density of states. We are currently pursuing other properties of the electron gas in the spin incoherent limit. In the same regime, an interesting result for an idealized 2-terminal transport measurement was obtained by Matveev, see cond-mat/0405542 . More recently (5/2/05), we discussed various aspects of electrical transport in the spin incoherent regime. Our ( Greg Fiete , Karyn Le Hur, and LB) paper studies electrical transport in such quantum wires in a variety of situations. Read the cond-mat .
| For solid state applications, one desires electron spins rather than photons. The trick is not only to create an entangled electron pair, but to spatially separate the two electrons of the pair (e.g. then exchanged between communicators). In this letter , we demonstrated that Luttinger liquid effects actually enable one to construct (in principle at least!) a device out of two quantum wires-e.g. carbon nanotubes- to extract and separate entangled electron pairs from a superconductor, and that the Luttinger liquid physics makes such a device more and more efficient as it is cooled. This work was featured in Physics Focus online . Less publicized work, here , calculated some detailed properties of the conductance of superconducting-nanotube junctions. |
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| One of the dramatic predictions of Luttinger liquid theory is "spin-charge separation": the elementary excitations of a one-dimensional metal are not at all like free electrons, but rather consist of collective modes carrying spin and charge separately and at different velocities! Our work showed that these different excitation branches could be observed experimentally by measuring tunneling conductance between two parallel quantum wires, as a function of voltage and a parallel magnetic field. We calculated this conductance explicitly, and its dependence upon the pertinent physical parameters. Beautiful experiments by Auslaender et. al. provide a rather dramatic confirmation of this theory. Relatively recent results are nicely summarized in Amir Yacoby's KITP talk from spring, 2004. |
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As an unfortunately commentary on the sometime inadequacies of the referee system, our theory paper was originally submitted to PRL (at the time of this cond-mat ), ultimately rejected from there and finally with difficulty published in Physical Review B. In the meantime, the experiments were published, as were a number of other theoretical works which received more recognition than our own, despite our precedent. Fortunately, no one was harmed in the process, and all persons involved seem to have continued with some success in physics :-)
In a couple ( [1], [2]) of papers with Reinhold Egger , we established a basic framework for these studies. This work provided a description for spin transport within a quantum wire, as well as exchange and electron transfer of spin through low-transparency contacts to the wire. We used it to study a simple model of a "spin-valve" device, finding that its efficiency is drastically reduced relative to its bulk counterpart by electron interaction effects. A number of other predictions were made for transport through ferromagnet-quantum wire-ferromagnet structures. At the time of writing, we still await the materials genius needed to pull off these experiments! But we have faith :-).
For specific topics, click below:
It has become well-known in recent years that the geometric structure (in Hilbert space) of Bloch wavefunctions influences the dynamics of quasiparticles. For non-interacting electrons, this is described by the Berry curvature, which plays the role of a dual (momentum and position interchanged) "artificial magnetic field" in momentum space. We have found that for interacting electrons, a counterpart, which we call the "artificial electric field" appears. Based on the Keldysh formalism, we derived an effective Boltzmann equation for a quasi-particle associated with a particular Fermi surface in an interacting Fermi liquid. This provides a many-body derivation of the Berry curvature effects in electron dynamics with spin-orbit coupling. Our Fermi liquid formulation completes the reinvention of modified band dynamics by introducing in addition the artificial electric field, which is related to Berry curvature in frequency and momentum space. With Ryuichi Shindou, we showed explicitly how the artificial electric field affects the renormalization factor and transverse conductivity of interacting U(1) Fermi liquids with non-degenerate bands. Accordingly, we also proposed a method of momentum resolved Berry's curvature detection in terms of angle resolved photoemission spectroscopy (ARPES). Analogous extensions to SU(2) Fermi liquids with doubly degenerate bands were also briefly mentioned.
Read cond-mat/0603089 .
In this paper , we calculated the oscillations of the spin relaxation rate with a perpendicular magnetic field in a two-dimensional electron gas with Rashba spin-orbit interaction and disorder. This is an extension of the well-known Dyakanov-Perel mechanism of spin relaxation, and we described behavior consistent with existing experiments in the Awschalom group at UCSB. In general, we feel that such microscopic understanding of the mechanisms of spin relaxation and decoherence is an important theoretical problem.
Prompted by intriguing experiments by Prof. Elizabeth Gwinn and collaborators, we studied the anomalous Hall effect in diluted magnetic semiconductors in the hopping transport regime. Since the Hall effect is ubiquitously used to experimentally determine hole concentrations in these materials, an understanding of the anomalous Hall effect is very important in interpreting these measurements. The strong spin-orbit coupling of the heavy and light hole bands in Ga1-xMnxAs is believed theoretically to lead to a strong anomalous contribution in clean samples. The majority of samples are, however, rather insulating, and display hopping-type conductivity over a broad range of temperatures. In this paper , Anton Burkov and I showed that in the hopping regime, the anomalous Hall contribution is large and is not directly proportional to the carrier density. In fact, it is proportional to the derivative of the density of states. Thus we predicted a mechanism for a sign change in the anomalous Hall conductivity if the Fermi level is moved through the center of the impurity band.
Although such quenched disorder provides some obvious mechanism for a complex ground state, it still leaves the fantastically challenging problem of understanding the structure of this and the low-lying excited states, and how the system moves between them. This problem remains basically unsolved for any of this sort of glass (and we understand even less of structural glasses!) outside of mean-field-theory, despite decades of work since the 1960s. The difficulty is that such systems have an extensive number of metastable states, connected in a non-trivial way with a broad distribution of energy barriers between them. Phenomenological theories have been developed based on this sort of picture. The problem is that "state" live in phase space, which is infinite dimensional, and it is problematic to combine this picture with the real-space structure of local degrees of freedom (spins etc.) that is of course crucial.
Our work (in collaboration with Pierre Le Doussal ) has been to try to derive the distribution of low energy metastable states, barriers, etc., from a microscopic calculation of a physically interesting model. In particular, we have studied the model of an elastic medium (like a charge density wave in a solid) pinned by quenched impurities. After several years of work with Pierre Le Doussal of ENS Paris, in this paper (cond-mat/0408048 - it is in press in Annals of Physics) on the statics, and this PRE and this EPL on the dynamics, we were able for the first time to derive the form (i.e. how barriers scale with length, how relaxation times are distributed statistically) of such distributions microscopically. We hope the structure we uncovered - a very different sort of statistical field theory - will be further useful in a more detailed understanding - putting quantitative numbers to the above forms - of these and other glassy systems.
We proposed (in 2005) that "Competing order" in the vortex core could be used to measure the mass of a vortex.
| We (with Subir Sachdev's group ) have formulated a theory of the quantum dynamics of vortices in two-dimensional superfluids (and superconductors) proximate to Mott insulators (see cond-mat/0408329 and cond-mat/0409470 ). The theory predicts modulations in the local density of states in the regions over which the vortices execute their quantum zero point motion. We use the spatial extent of such modulations in scanning tunnelling microscopy measurements (Hoffman et al., cond-mat/0201348) on the vortex lattice of BSCCO to estimate the inertial mass of a point vortex. We discuss other, more direct, experimental signatures of the vortex dynamics. |
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I hope to add some summary here. For now, read our conference review paper describing a theory competing orders and non-Landau-Ginzburg-Wilson criticality near Superfluid-Mott quantum phase transitions in boson systems. The paper also includes a particularly simple and explicit presentation of boson-vortex duality in two dimensions.
The idea of a "supersolid" - a seemingly-paradoxical phase of matter
that is both a solid and a superfluid - dates back to the 1960's, but
has been elusive in both models and experiments (but see these recent experiments ).
We showed, in this
cond-mat , that when strongly-interacting bosons are confined to a
two-dimensional triangular lattice, which frustrates their natural
crystalline ordering, they form such a novel supersolid phase.
Similar results were obtained simultaneously by two other groups (see
this
cond-mat and this one ).
We continued to think about some puzzling features of the "strong
coupling" supersolid. Its superfluidity is extremely weak:
we calculated that the superfluid density is about 25 times smaller
than the weakly-interacting superfluid on the same lattice! We also
found that there are actually not one but two supersolid
states with different symmetries, that are amazingly close in energy -
so close that we could not conclusively distinguish which of the two
is the ground state. We call them "ferrimagnetic" and
"antiferromagnetic" (the names come from a mapping of the problem to
an XXZ spin-1/2 model). Symmetry tells us that the ferrimagnetic
state should actually phase separate . However, from our
numerics, if this is the case, the difference in density of the two
components must be minute. We think now that we have some explanation
of these features.
The amazing
weakness of the superfluidity we found is an indication the system is close to a
quantum phase transition to a Mott insulator, where the superfluidity
would vanish. In fact, this assumption also seems to explain the
near-degeneracy of the two supersolid states.
We (Anton Burkov and LB) developed a general theory of the
quantum critical points in half-filled boson systems on the triangular
lattice. There are a number of interesting Mott transitions that can
occur, from the superfluid directly to a Mott state, or from different
supersolids to Mott states. All the transitions we found in this way
are "exotic", i.e defy the Landau-Ginzburg-Wilson (LGW) paradigm, and
can be thought of in one way or another as "deconfined" quantum
critical points.
One result of this work is a simple picture of the supersolid we
found earlier as a partially "melted" solid (Mott) state. For
instance, the ferrimagnetic supersolid below (left) can be thought of
by forming a solid of a density of 1/3 bosons on the blue sites, which
form a triangular sublattice of the original lattice. To get to
half-filling, the remaining bosons form a superfluid on the yellow
honeycomb lattice of remaining sites, which is then at 1/4-filling.
In the corresponding Mott state, the bosons on this honeycomb lattice form
a columnar Valence Bond Solid (VBS) state.
"Ferrimagnetic" supersolid state.
"Antiferromagnetic" supersolid state.
Density modulations in the "Ferrimagnetic" solid state, as
calculated by the dual vortex theory.
A cartoon of the ferrimagnetic solid. Blue sites form the triangular
sublattice solid. The 1/4-filled honeycomb lattice
is in green, and red ovals indicate valence bonds.
For much more detail,
read
the cond-mat.
The melting of this VBS state leads to the supersolid. This
quantum phase transition is quite interesting, because it requires
both the loss of VBS order and the collapse of the Mott gap
and onset of superfluidity. This is not possible in the LGW
framework. The way it happens, roughly, is that the VBS order
parameter is actually a vector -- which points along the + (-) x, y,
or z direction if the valence bonds occupy even (odd) columns along
the first, second, or third principal axis of the triangular lattice.
Near the quantum critical point, there are skyrmion defects in the
order parameter, and it turns out that these are "charged". Exactly
one extra/missing boson is bound to the center of a
skyrmion/antiskyrmion. So at the Mott transition, skyrmions condense,
destroying the VBS order and at the same creating the superfluid.
This is an example of "deconfined criticality".
