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QUANTUM SPIN LIQUIDS: QUEST
PARTICLE
BY
FOR THE
ODD
YONG-BAEK KIM AND ARUN PARAMEKANTI
T
he world around us is full of materials which
exhibit interesting properties, from “familiar
stuff” such as fridge magnets, to more “complex
stuff” such as superconductors which display zero
resistance to current flow when cooled to low temperature.
Even a simple crystalline solid has the astonishing
property that it emerges as a perfectly periodic pattern of
atoms starting from a random soup of atoms in a liquid,
simply by cooling to the right temperature. Such states of
matter are called “Broken Symmetry” states. While the
fluctuating high temperature state looks the same when
viewed from any point within it, the low temperature
“Broken Symmetry” state (magnet or crystal) picks a
particular direction for magnetization, or a specific atomic
crystal pattern. Peering inside such a solid we see that not
all points in space or not all directions are equivalent.
This breaking of symmetries is accompanied by new
excitations. A periodic crystal has sharply defined modes
of collective atomic oscillations which can extend over the
entire crystal, which simply do not exist in the liquid.
Similarly a magnet has sharp modes associated with
collective periodic motions of the tiny atomic magnetic
moments, which again cease to exist in the random high
temperature state. A general lesson to take away is that
many materials exhibit broken symmetries, and states with
such broken symmetries possess new types of excitations.
In quantum mechanics, such “wave like” collective
excitations can acquire a discrete particulate character, so
that physicists talk of “phonons” in a vibrating crystal, and
“magnons” in an excited magnet.
It may appear that such ideas are applicable only to
materials, the study of which is part of Condensed Matter
Physics. However, physicists, informed by these
SUMMARY
Strongly correlated quantum condensed
matter phases often have unusual
excitations which bear no resemblance to
the constituent electrons which form such
states of matter. We review recent
developments which show that Quantum
Spin Liquids, made of insulating electrons in
solids, may support emergent particle-like
excitations with quantum numbers very
different from the electron.
examples, use the same language to understand such
complex questions as the origin of the Universe and the
origin of the slew of subatomic particles found in nature.
The premise is that our Universe has a “Broken
Symmetry” of some sort, and the properties of various
subatomic particles derive from such a Symmetry
Breaking principle. The search for the Higgs Boson C the
“God Particle” C at the Large Hadron Collider is a quest
for evidence of such symmetry breaking in the Universe.
In recent years, condensed matter physicists have started
to study other states of matter where quantum particles
with “strange properties” arise as sharp excitations at low
temperature. One example is the Fractional Quantum Hall
liquid of strongly interacting electrons confined to two
dimensions and moving in a large magnetic field C such a
quantum Hall state of ordinary charge-e electrons can
exhibit particle-like excitations which carry charge, even
though the constituent electron is, by itself, perfectly
robust against such splintering. More recent examples
include electrically insulating solids which support
excitations that slightly resemble electrons, i.e. they
behave as mobile spin-1/2 fermions which can transport
heat but they carry no charge, even while the constituent
electrons are themselves strongly interacting and truly
immobile in the insulator. Such insulators, driven by
strong electron-electron repulsion are termed Mott
Insulators (as opposed to Band Insulators like silicon), and
such a weird magnetic state of a Mott Insulator is an
example of a Quantum Spin Liquid. Understanding such
novel states of matter, the search for such strange quantum
excitations C our quest for an “Odd Particle” C and
uncovering the underlying order responsible for them, are
all topics at the forefront of research in physics. Here, we
briefly survey some recent developments in the field of
Quantum Spin liquids, highlighting recent Canadian
contributions to this field. For technical details and a
broader perspective, we refer the reader to reviews by by
Wen [1] and by Balents [2].
WHAT IS A QUANTUM SPIN LIQUID
Quantum spin liquid (QSL) phases are, broadly speaking,
paramagnetic (i.e., non-magnetic) insulating ground states
of strongly interacting electrons in a solid, where lattice
symmetries are not broken. While magnetically ordered
ground states of solids may be easily described by a
‘classical cartoon’ where we stick an arrow on each atom
Yong-Baek Kim
<ybkim@physics.
utoronto.ca> and
Arun Paramekanti
<arunp@physics.
utoronto.ca>,
Department of
Physics, University of
Toronto, Toronto,
Ontario, Canada
M5S 1A7 and
Canadian Institute for
Advanced Research,
Toronto, Ontario,
Canada M5G 1Z8
LA PHYSIQUE AU CANADA / Vol. 68, No. 2 ( avr. à juin 2012 ) C 71
QUANTUM SPIN LIQUIDS ... (KIM/PARAMEKANTI)
which shows its magnetic moment magnitude and direction,
there is no such simple cartoon picture for a spin liquid. Just as
for an ordinary liquid, where drawing a random pattern of
atoms yields a mere snapshot which does not capture the
thermal motions in the liquid or its ability to flow, cartoon
pictures with random orientations of the magnetic moments fail
to capture the quantum mechanical fluctuations and
superpositions of various magnetic moments.
We should therefore view Quantum Spin
Liquids as highly quantum entangled states of
magnetic moment configurations, with no
classical counterpart.
FRUSTRATING MAGNETIC ORDER
One key notion which seems to be crucial to realizing QSL
physics in higher dimensions is “frustration” C we expect
QSLs to win over magnetically ordered ground states only if
something frustrates magnetic ordering. The simplest example
of such frustration arises from lattice geometry. Various lattices
shown in Fig.1 (together with examples of materials), such as
A particularly well understood example of
such a QSL is a one dimensional (1D) Mott
insulator, where each site on a periodic chain
of atoms seats one electron (with spin-1/2 and
charge-e). While the strong electron-electron
interaction forbids electrons from moving
around freely, leading to something like a
‘traffic jam’, the kinetic motion of the
electrons allows them to jiggle slightly, and
this quantum jiggling can be shown to lead to
spins on neighbouring atoms preferring to
anti-align. Such a local tendency to “Néel
antiferromagnetism” does not, however,
extend
to
global
antiferromagnetic
order C such a global staggered pattern of Fig. 1 Materials based on triangular, kagomé, and hyperkagomé lattices which support
quantum spin liquid states.
moments across the entire lattice in unstable to
the tiniest quantum fluctuations. Thus, this 1D
Mott insulator is a spin liquid with “quantum
the 2D kagomé lattice, the 3D pyrochlore lattice, the 3D
mechanically melted” Néel order. What kind of excitations
hyperkagomé lattice C lattices made of corner sharing
does such a 1D spin liquid support Careful calculations show
motifs C turn out to not to favor magnetic orders. Indeed,
that such a 1D spin liquid has charge-zero spin-1/2 particle-like
many such geometrically frustrated materials tend to order, if
excitations which can propagate across the lattice, even while
they do, at temperatures much below the Curie-Weiss
[3]
the electrons themselves stay stuck locally . Such excitations,
temperature where one might expect magnetic ordering based
dubbed ‘spinons’, may be crudely viewed as mobile domain
on mean field theory, suggesting that the frustration induced by
walls between states with different orientations of
lattice geometry leads to strong fluctuations which suppress the
antiferromagnetic order. An alternative rough viewpoint is to
transition temperature . Aside from geometry, the exchange
view them as arising from splintering a magnon (which carries
paths for moments to interact in certain materials may involve
spin-1 and charge-zero) into two pieces, each of which carries
pairs of spins further apart than nearest neighbors even on a
spin-1/2.
bipartite lattice C in such cases, even simple bipartite lattices,
such as 2D honeycomb [4,5], may frustrate magnetic ordering.
Extending this idea to higher dimensions is complicated by
In other cases, proximity to an insulator-to-metal transition,
several issues. The fact that long range magnetic order is more
and the resulting enhancement of local charge fluctuations, is
stable in higher dimensions, means that such spin liquid states
found to lead to “ring-exchange”, where four or more spins
have to be able to compete against and beat out ordinary
cooperatively flip as several electrons cooperatively exchange
magnetically ordered states by having a lower energy. This
positions. Such ring-exchange frustrates magnetic order and
depends on the details of the Hamiltonian, and many simple
may promote QSL states [6].
model Hamiltonians in 2D or 3D simply end up having
magnetically ordered ground states. The second issue is that,
GAPPED SPIN LIQUIDS
unlike in 1D, there are typically no exact or controlled
analytical calculations in higher dimensions, while many
The simplest form of a QSL is one with a gap to all excitations,
numerical methods available to address such spin Hamiltonians
spinful excitations as well as spinless excitations. If the spin
also fail for models of direct relevance to materials. However,
gap is large, then one may effectively think of neighboring
recent detailed studies of numerically tractable toy models, as
spins on the lattice as having formed a singlet. However, there
well as experiments on a host of frustrated spin-1/2 magnets,
may be no unique pattern to which neighbors pair up into
appear to be uncovering such novel QSL states in 2D and 3D.
singlets. Instead, many many singlet patterns exist at this level,
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QUANTUM SPIN LIQUIDS ... (KIM/PARAMEKANTI)AA
so that such ‘dimer configurations’, while they significantly
lower the energy and entropy of the system compared to having
just unpaired spins, still possess a large degeneracy, with a
nonzero entropy per site. The approach to the eventual ground
states of such dimers is then via “dimer resonances” within the
restricted subspace of dimer configurations. This is the socalled “resonating valence bonds” picture of a QSL. Such
fluctuations, where the dimer patterns rearrange, correspond to
spinless fluctuations occurring below the spin gap scale.
Certain such “quantum dimer models”, for instance on the
triangular lattice [7], have been shown to have an eventual QSL
ground state where the dimers do not freeze into a regular
crystal even down to zero temperature. Such ground states may
be viewed as magnetic analogues of Helium-4, where singlet
bonds undergo significant zero point motion thus preventing
crystallization.
Going beyond such a simple model of dimers, one can study
microscopic yet numerically tractable spin-1/2 models using
quantum Monte Carlo calculations. Such studies show that the
triangular lattice antiferromagnetic XXZ model undergoes a
transition from a conventionally magnetically ordered state
into magnetic states analogous to supersolids [8] instead of
forming QSLs, while an extended kagome lattice spin-1/2
XXZ model undergoes a quantum ferromagnet to gapped QSL
transition [9]. Recent density-matrix renormalization group
calculations [10] on the nearest neighbor Heisenberg model on
the kagomé lattice indicate that its ground state may also be
such a QSL.
Ordinary correlations, such as spin-spin or dimer-dimer
correlations, all decay exponentially in such gapped QSLs and
they do not possess any broken symmetries. Characterizing
such QSLs properly thus requires going beyond simple ideas of
“Broken Symmetry” discussed in the Introduction. Such
gapped QSLs have been shown to possess, in addition to the
odd particles described above, ground state degeneracies which
depend on the topology of the manifold on which they reside.
Such degeneracies reflect a new form of “topological order” [1]
in the ground state. Such gapped QSLs also have a sharp notion
of a “topological entanglement entropy” which reflects the
topological order [11], and numerical simulations have
“measured” this entropy in the extended kagome lattice XXZ
model [12].
Such gapped 2D QSLs support two novel particle-like
excitations: a gapped bosonic spin-1/2 excitation called a
‘spinon’, and a vortex-like excitation called a ‘vison’. The
spinons carry, in addition to their spin quantum number, an
emergent Ising ‘electric charge’ (gauge charge), and interact
with each other via short range interactions, while the ‘vison’
carries an emergent Ising ‘magnetic flux’ [1,13]. This Ising
nature of its excitations leads to the name “ spin liquids” for
such QSLs. These excitations, the spinon and the vison, are
quite different from the original electrons which make up such
a Mott insulator; observing such odd particles in numerics or
experiments is a sure signature of a strange quantum state.
GAPLESS SPIN LIQUIDS
Aside from the Gapped Spin Liquids, discussed above, a less
well-understood and perhaps more interesting class is that of
Gapless Spin Liquids. Some of these spin liquids are
characterized by a gap to spinful excitations but have gapless
“dimer resonances” which survive down to the lowest energy.
Such gapless dimer fluctuations turn out to be simply
describable as “electromagnetic fluctuations” of an emergent
gauge field: an emergent “Photon”. The simple pyrochlore
spin-1/2 XXZ model is found to have such a novel photon
excitation [14,15]. The other, perhaps more experimentally
relevant form of a Gapless QSL is characterized by having both
gapless spinon excitations (which leads to algebraicallydecaying spin-spin correlations in space and time) as well as
gapless gauge fluctuations. Because of the algebraic behavior
of the spin-spin correlations, these spin liquid phases are often
called “critical spin liquids”, and the spinons and gauge fields
in such states are strongly coupled. One may think of such a
spin liquid phase as a superposition of valence bonds (or
“dimers") between two spins separated in varying distances. As
a result, there is no characteristic length scale in the spin
correlations, resulting in the algebraic form. The spinons in
such spin liquids can obey different types of statistics
depending on the microscopic details. When the spinons are
fermions, the gapless spinon excitations, for example, may
have a spinon Fermi surface [6] or possess a massless Dirac
fermion spectrum [16] similar to graphene. On the other hand, if
the spinons are bosonic, the corresponding spin liquid can have
gapless spinons only when the spin liquid phase is sitting at the
critical point. This is because if the bosonic spinons have
gapless spectrum, then they would generically be condensed
and form a magnetically ordered state. These two possibilities
are indeed explored in theoretical literatures to explain powerlaw-in-temperature specific heat observed in many candidate
spin liquid materials based on the triangular [17,18,6],
kagome [19,20,16], and 3D hyper-kagome lattices [21,22].
Gapless spin liquid phases with fermionic spinons may arise
most naturally when the Mott insulator is close a metalinsulator transition [23]. In this case, the fermionic spinons in
spin liquid Mott insulator inherit the Fermi surface of electrons
in the metallic side. This is thought to occur, for example, in the
case of organic salts C (BEDT-TTF)Cu(CN) [17,6] and
EtMeSb[Pd(dmit)] [18] based on triangular lattices, and
NaIrO [21,22] based on three-dimensional hyper-kagome lattice
(a 3D network of corner sharing triangles). Indeed these
materials show rather clear power-law specific heat and
constant spin susceptibility at low temperatures, consistent
with the presence of the Fermi surface. In addition, these Mott
insulators become metallic under moderately high external
pressure, indicating the proximity to a metal-insulator
transition.
TOPOLOGICAL MOTT INSULATOR
Recently the possibility of novel spin liquid phases with
fermionic spinons that have the so-called topological band
structure has been proposed [24,25]. The spinons in these phases
LA PHYSIQUE AU CANADA / Vol. 68, No. 2 ( avr. à juin 2012 ) C 73
QUANTUM SPIN LIQUIDS ... (KIM/PARAMEKANTI)
are gapped in the bulk as they possess the band-insulator-like
spectra. However, in analogy to the topological insulators of
electrons, the spectrum of such spinon band insulators can be
“trivial” or “topological". If the spectrum is “topological”, it
means that there is a protected gapless surface state despite a
bulk gap, and the surface spectrum in momentum space has an
odd number of linearly dispersing Dirac spinons. If the
spectrum is “trivial”, gapless surface states are not guaranteed.
Such topologically nontrivial spin liquid phases are called
topological Mott insulators or fractionalized topological
insulators.
Currently two different proposals exist to obtain topological
Mott insulators [24,25]. (i) One could start from an electronic
topological insulator with a strong spin-orbit coupling which is
essential for the existence of a “topological” band structure.
Increasing the on-site Hubbard interaction may lead to a
transition to a spin liquid Mott insulator. Within a slave-particle
formulation, where one represents the electron as a composite
of the charge-carrying boson and the spin-carrying fermionic
spinon, the transition from the spin liquid insulator to the band
insulator can be described as Bose condensation of the chargecarrying bosons while the fermionic spinons simply inherit
their topological band structure from the electrons [24].
(ii) Alternatively, one may start from spin models with
frustrated antiferromagnetic and ferromagnetic exchange
couplings. It has been recently shown that the slave-fermion
theory of such spin models can support a spin liquid with an
emergent spin-orbit coupling, hence the topological band
structure of the spinons [25]. In this case, the non-trivial
topology of the spinon spectra is emergent, and not inherited
from the electrons.
In either case, one obtains an insulator with a bulk gap but with
strange gapless spin-1/2 and charge-neutral surface states:
“surface spinons”. In such topological Mott insulators, there is
no electrical conduction. However, gapless surface spinons can
conduct heat while the bulk is a thermal insulator.
CONCLUSION
Condensed matter systems are made of a large number of
strongly interacting constituent particles, e.g. electrons, having
well-understood properties. Nevertheless, the quantum ground
state and excitations of such a system can be totally unlike its
constituents C differing in spin, charge, and statistics C and
quantum spin liquids are one particularly interesting example
of this emergent behavior. Synthesizing such novel materials,
and studying their properties experimentally and theoretically
is an exciting frontier in condensed matter physics.
Understanding the mechanisms for the appearance of exotic
new particles in such laboratory systems may even shed light
on the mechanisms for the emergence and properties of
elementary particles in our Universe.
ACKNOWLEDGEMENTS
We acknowledge research funding from NSERC of Canada
(YBK, AP) and CRC-Tier II programme (YBK).
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