KITP - 12

Quantum Phase Transitions

Week 12
28th March - 1st April , 2005

Bloggers: Andrey Chubukov & Piers Coleman

Back to work after the March meeting! Incredible weather. A week of diverse discussions and seminars.

On Monday, Andrew Green from St Andrews gave a "Russian style" blackboard seminar, with a great deal of participation from the audience. He presented the idea that quantum critical points are infinitely sensitive to driving forces, so that a weak electric field for instance, once larger (in appropriate units) than the temperature, provides the new cut-off, and can, in principle, he argued, drive a quantum critical system out-of-equilibrium. The audience was extremely rowdy, but we learnt a lot.

On Tuesday, and later on Friday, Colin Broholm introduced us to a wonderful new array of experimental results on frustrated magnets, discussing the ideas that spin liquids are highly sensitive to glass formation, and presenting some beautiful new results on a valence-bond to antiferromagnet quantum critical point.

On Thursday, the group assembled to discuss the transition from itinerant, to localized behavior in one and two-band systems. Qimiao Si presented a diagram that attemped to link quantum critical points in heavy electron systems to those in frustrated antiferromagnets. To see this, read on!

Participants
Blackboard Seminar
Experimental Seminar
Thursday Discussion
Colin Broholm's second talk

Participants present. Click on participant to read questions that they have posed.

Abrahams, Elihu
Broholm, Colin
Chubukov, Andrey
Coleman, Piers
Eshrig, Mathias
Efetov, Kostya
Feldman, Dima
Geshkenbein, Vadim
Hanke, Werner
Kroha, Johannes
Larkin, Anatoli
Monien, Hartmut

Morr, Dirk
Pepin, Catherine
Posazhennikova, Anna
Schofield, Andrew
Schmalian, Joerg
Si, Qimiao
Weng, Zheng-Yu
Yakovenko,Victor

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Monday Blackboard Discussion.

Non-Linear Quantum Critical Transport and the Schwinger Mechanism[Aud][Cam]
Dr. Andrew Green, St Andrews U, UK

This was a ``Russian style'' seminar -- the talk lasted for nearly two hours and was constantly interrupted by questions. Nearly everyone was involved in the discussion.

Andrew discussed his work, done in collaboration with S. Sondhi from Princeton, on the universal conductivity near a quantum phase transition between a superconductor and an insulator (cond-mat/0501758). Previous works by Cha Fisher et. al, and by Damle and Sachdev focused on the scaling form of the conductivity as a function of frequency and temperature. Fisher considered the limit T=0 at a finite $\omega$, while Damle and Sachdev considered the opposite limit of omega

=0 and T finite. The conductivity is different in the two limits. That the two limits do not commute implies that the conductivity is a scaling function of omega

/T.

The key idea here, is that a quantum critical point is scale invariant, so that when a field energy scale arises, it replaces the temperature, producing a non-equilibrium distribution function.
Andy presented the key scaling equation

scaling form

for the conductivity, and then gave an in depth discussion about how this equation was motivated by a large N expansion for the non-equilibrium response of a phi

⁴ field theory, coupled to an electric field (via the vector potential).

phi^4

There were two key pieces of physics here - the scattering of the
bosons, and the "Schwinger mechanism", whereby excitations tunnel out of the vacuum in the presence of an electric field, at a rate proportional to exp(- Constant/E). The large N expansion was carried out by introducing replicas, and coupling the electric field to just one of the N replicas.

Greene and Sondhi argued that in the presence of a finite electric field, the conductivity becomes the scaling function of two variables, $\omega/T$ and $E/T$. They computed the conductivity at $\omega =T=0$ and a finite $E$ in the large $N$ limit, and argued that the conductivity is different from the results obtained by Fisher et al and by Damle and Sachdev. This intriguing result generated a lot of discussion and questions.

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Experimental Seminar, 12.30pm Tuesday, March 29th.

Glassy Phases in 2D Frustrated Quantum Magnets[Slides][Aud][Cam]

Dr. Colin Broholm, John Hopkins

Colin prefaced his talk by remarking that tremendous advances in neutron scattering are taking place, thanks to the development of accelerator driven spallation sources. He estimated that with the new spallation source at Oak Ridge, it will be possible to map out the inelastic neutron scattering spectrum in samples as small as 80mg. In his introduction to frustrated systems, Colin remarked that low connectivity plays an important role - this has the effect that order in one part of the system does not affect its surroundings.	The weak connectivity of the Kagome lattice is reminiscent of a one-dimensional system. It is this weak connectivity that gives rise to the "folding" zero modes of the classical Heisenberg magnet on this lattice.

	Colin reminded us that in Frustrated systems, there are really two axes - the frustration axis, where frustration and quantum fuctuations can drive the formation of spin liquids, and the disorder axis, whcih can drive formation of percolative random field magnets, and - perhaps - in the gase of spin liquids- the formation of glasses
Spin gap in "Kagome Sandwich" Ba₂Sn₂ZnGa₃Cr₇O₂₂	Kagome systems. Colin introduced two classes of Kagome systems. In both cases, interlayer coupling plays an important role. - a spin 1/2 ferrite with stronger interlayer coupling, in which corner-sharing tetrahedra form singlet complexes, generating a spin gap
	- a spin 3/2 chromium magnetoplumbite system with a weaker interlayer coupling, in which the spins ultimately develop a glassy state, with a small frozen moment, with slow, gapless spin excitations, and spin-wave excitations.
	Colin then turned to triangular magnets. We were all fascinated by a new spin-1 system, NiGa₂S₄ , where the S=1 Ni lie on a triangular lattice. This system has a -50K or so Weiss temperature, but does not order. Instead around 7K, it has a glassy like cusp in the susceptibility, where the spins freeze into a 120 degree structure, but there is no anomaly in the specific heat.
One of the ideas here, is that a transition takes place, in which Z₂vortices associated with the biaxial antiferromagnet bind in a sharp cross-over very similar to a Kosterlitz Thouless transition(first proposed by Kawamura and Miyashita. (Uniaxial 2D Heisenberg systems do not have topological defects, but biaxial ones can have Z_2 vortices - where each vortex is its own antiparticle. In a 2D Heisenberg system - these can't completely bind, because the spin correlation length is always finite, but since the correlation length is exponentially large, there is a very rapid cross-over from a floppy state with a high density of defects , to a "stiff" state with a low density of defects and a long correlation length - which resembles a KT transition). Broholm speculated that with a little disorder, this cross-over becomes a glass transition. Colin ended by remarking that glassy spin phases are most probably endemic to real, and relativedly clean spin systems near a quantum critical point. He noted that the results in NiGa₂S are very remininscnt to a glassy phase recently observed in underdoped YBCO. This is clearly an area with tremendous potential for theory input.

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Discussion: 4:30 pm Thursday, March 31st Founders Room.

Itinerant versus Mott-Hubbard behavior in one band and two band models. Since the early work of Mott and Hubbard,
it has been known that when Coulomb interactions become large, charge fluctuations in electronic systems are severely depressed. This has led to the notion that part of the Hilbert space for electrons becomes "locked away" in an upper or lower Hubbard band, projecting out the Hilbert space, leading to new classes of behavior in which spin fluctuations dominate over charge fluctuations. . Thus in one band systems one has the notion of an infinite U Hubbard model - where no two electrons can occupy one site, or a Heisenberg model, where electrons are localized and charge fluctuations are entirely suppressed. In two band-systems, the analogous behavior gives rise to the Kondo lattice model, where the f- or d-electron component of the band has become Mott-localized.

Nevertheless, even in such models, we know that the electrons can still "reconstitute" their Hilbert space by forming composite excitations that mimic the states that have been lost - thus a large Fermi surface is still possible, despite the loss of Hilbert space. This raises the fascinating question - how does the electron system make the transition from fully localized or partially localized behavior - a Mott insulator, or a magnetic Kondo lattice - to intinerant behavior, such as is found in a heavily doped Mott insulator, or a paramagnetic heavy electron system.

System	"Localized"	"Itinerant"
One-band infinite U Hubbard Model	Mott Insulator	Doped state with large FS
Two band Kondo lattice model	Magnetic Kondo lattice	Heavy electron paramagnet

A One band systems. Zheng Yu Weng began the discussion, posing the quesions:

When a Mott Insulator is doped, at what point does the Mott gap disappear?
How does the Mott gap disappear?

This led to a variety of interesting responses. Zheng Yu pointed out that one could still see the Mott gap in optical measurements of lightly doped cuprates, and that in LSCO, the optical signature of the Mott gap vanishes around 14% doping.

Werner Hanke turned to the issue of when, and where one might have a small, or large Fermi surface. He sketched the dispersion found in the Mott insulator, with the high lying bands corresponding to electron doping, the low lying bands to hole doping. (See across).

If the excitation spectrum were to remain rigid upon doping, he pointed out, one would expect a small "hole" pockets in the hole doped materials. However, once the chemical potential moves down into the hole bands, the interaction of the hole with the surrounding spin fluid, produces a markedly changed dispersion, with a large Fermi surface.

Joerg Schmalian pointed out that in EELS spectroscopy, one can see the excitation energy for transfering a hole from an oxygen p-orbital to a copper d-orbital - this energy is about 2eV. One can see this excitation in the doped systems -
it is always there - there are small differences between the predictions of models, for instance the spectral weight transfered between peaks is twice as much as for an infinite U Hubbard model. Schmalians main point was that the "high energy" Mott
gap is always there - its the changes in the low energy excitations that determine whether one has made a transition from
localized to itinerant behavior.

Andrei Chubukov cited some interesting experimental work on electron doped cuprates, Nd_2-x Ce_x Cu O₄, where optical measurements have been made out to 14% doping - just before the magnetic order disappears. Chubukov noted that as the doping proceeded, a new peak developed in the optical spectrum, which split off from the Mott gap, and moved to lower energies. He claimed that the position of this peak scales roughly with the magnetization - so that in the optical spectra there
were simulaneously - features characteristic of a Mott insulator and features characteristic of a spin density wave picture. The insulating featues disappear by the time one reaches 12.5%.

Werner Hanke said that this is exactly what one sees in Monte Carlo. Andy Schofield mentioned that this was also seen in his cluster DMFT calculations.

Two band models.

Qimiao Si described the contrasting situation in two band models relevant to heavy electron systems. Qimiao pointed out that one could imagine destroying the antiferromagnetic behavior in two ways:

by addiing frustration, and driving it to a spin liquid

by coupling the local moments to conduction electrons, and using the Kondo effect to screen the spins and ultimately melt the antiferromagnetic order into a heavy electron Fermi liquid with a large Fermi surface.

Si proposed that one might be able to link these two scenarios in a single phase diagram. He considered an Ising antiferromagnet. When the Kondo coupling of this AFM to the conduction fluid is small, it is clear that the spin fluctuations develop a gap, and the Fermi surface of the electron is small. By contrast, if the coupling is large, electrons bind to the local moments to form singlets. This state is a "Kondo insulator".
Adding a "hole" to such a singlet, creates a spin-full hole with an infinite U constraint. (You can only add one hole to each singlet). If you start out with one hole per site, and dope with electrons, then there are 1-x holes per site, or 1+x electrons - i.e, one has large Fermi surface.

Si argued that there might be two types of
antiferromagnet - a local moment antiferromagnet, with a small Fermi surface, and an antiferromagnetic liquid (AFL), formed from a spin-density wave instability of the large moment system.

This would then lead to two classes of quantum phase transition: QC1 - a spin density wave transition that is well described by Hertz-Moriya theory and QC2, where the phase transition would take place between a small and a large FS Fermi liquid.

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Tuning From Dimer to Spin Order: Informal second talk from Colin Broholm, JHU

Colin Broholm agreed to give an informal discussion on Friday morning about recent experiments (Stone et al, cond-mat/0503450 in which the group observes a field-induced phase transition between a dimer-ordered state to an antiferromagnetically ordered state. In the first part of this workshop we heard from Prokofiev who, theoretically, was unable to find a second-order transition from valence bond to antiferromagnet, casting some doubt on recent speculations of Senthil et al. Could it be that the experimentalists have found just such a type of transition?

Experiments were carried out on a S=1/2 system PHCC, involving coupled layers of Copper spins. Neutron experiments reveal a spin gap in this state, with interlayer dimers having formed. The dimers have a singlet-triplet gap. In a field, the triplet state splits, and the M=-1 state ultimately drops to zero energy - this is the point at which antiferromagnetism is thought to develop. Differential susceptibility measurements reveal that an "antiferromagnetic" dome is thought to form over a a finite field range. phase forms over a finite field range.

What is striking, is that in the vicinity of the valence-bond to antiferromagnet transition, there is a "wedge" of spin-liquid
phase - as if quantum criticality produces a spin-liquid phase between the antiferromagnet and the valence bond-state.

Colin mentioned that measurements currently underway suggest that pressure can also drive this transition - of course in this case, the entire triplet gap may collapse, providing a much higher degeneracy at the QCP.

vbtoafm

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