PROPOSAL SECTION C – PROPOSED RESEARCH October 17, 2005 1 Introduction Interacting electron systems have stimulated much of the work in condensed matter physics for several decades. Usually the system organizes itself into a ground state with nearly independent low-energy excitations. Examples are the Landau Fermi liquid[1] and incompressible fractional quantum Hall states[2, 3]. At the same time, it has been realized that quenched disorder plays a central role in electron systems. Early investigations[4] (with the exception of refs. [5]) concentrated on the effects of disorder in noninteracting electron systems. Recently it has become clear that disorder-induced spatial organization may lie at the heart of many strongly correlated systems[6]. This proposal focuses on systems in which the interplay of disorder and interactions conspires to create novel states which exist solely by virtue of this interplay. The theme of much of the research proposed below is to answer the following broad questions: (i) How does one create, understand, and characterize strongly correlated/quantum fluctuating states in mesoscopic systems? (ii) How does one understand the crossover from a mesoscopic to a bulk system near a bulk quantum phase transition? In particular, what are the signatures of quantum criticality in transport in a mesoscopic system? (iii) What are the possible ground states of a bosonic system in a strongly disordered environment? Is a metallic state[7] possible?(iv) Does quenched disorder qualitatively modify confinement/deconfinement in gauge theories as applied to condensed matter systems? The following subsections motivate and sharpen these questions, and explain how they pertain to the three major thrusts of this proposal: Mesoscopic systems, ν = 1 bilayer systems, and deconfinement in antiferromagnets and quantum dimer models. 1.1 Open questions for interacting electrons in mesoscopics The role of interelectron interactions in quantum dots[8, 9] has been clarified with the Universal Hamiltonian[10, 11] emerging as the low-energy effective theory for diffusive and ballistic/chaotic quantum dots in the three classical random matrix symmetry classes (the gaussian orthogonal– GOE, unitary–GUE, and symplectic–GSE[12]). The Universal Hamiltonian[10, 11] (henceforth called H U ) describes the physics of weak-coupling systems within the Thouless band (the energy window within the Thouless energy E T of E F ). States separated by ² ¿ E T are correlated with each other[4, 8, 9] by Random Matrix Theory (RMT)[12]. E T is the inverse ergodicization time of an electron in the dot. For a diffusive dot E T ≈ ¯ hD/L 2 , where D is the bulk diffusion constant and L is the linear size of the dot, while for a ballistic/chaotic dot E T ≈ ¯ hv F /L, where v F is the Fermi velocity. In 2D, δ ≈ ¯ h 2 /mL 2 is the single particle mean level spacing on the dot. The dimensionless Thouless number g = E T /δ is an important parameter of the dot. For large g and low energies 1
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PROPOSAL SECTION C – PROPOSED RESEARCH
October 17, 2005
1 Introduction
Interacting electron systems have stimulated much of the work in condensed matter physics for
several decades. Usually the system organizes itself into a ground state with nearly independent
low-energy excitations. Examples are the Landau Fermi liquid[1] and incompressible fractional
quantum Hall states[2, 3].
At the same time, it has been realized that quenched disorder plays a central role in electron
systems. Early investigations[4] (with the exception of refs. [5]) concentrated on the effects of
disorder in noninteracting electron systems. Recently it has become clear that disorder-induced
spatial organization may lie at the heart of many strongly correlated systems[6].
This proposal focuses on systems in which the interplay of disorder and interactions conspires to
create novel states which exist solely by virtue of this interplay. The theme of much of the research
proposed below is to answer the following broad questions: (i) How does one create, understand,
and characterize strongly correlated/quantum fluctuating states in mesoscopic systems? (ii) How
does one understand the crossover from a mesoscopic to a bulk system near a bulk quantum
phase transition? In particular, what are the signatures of quantum criticality in transport in a
mesoscopic system? (iii) What are the possible ground states of a bosonic system in a strongly
disordered environment? Is a metallic state[7] possible?(iv) Does quenched disorder qualitatively
modify confinement/deconfinement in gauge theories as applied to condensed matter systems?
The following subsections motivate and sharpen these questions, and explain how they pertain
to the three major thrusts of this proposal: Mesoscopic systems, ν = 1 bilayer systems, and
deconfinement in antiferromagnets and quantum dimer models.
1.1 Open questions for interacting electrons in mesoscopics
The role of interelectron interactions in quantum dots[8, 9] has been clarified with the Universal
Hamiltonian[10, 11] emerging as the low-energy effective theory for diffusive and ballistic/chaotic
quantum dots in the three classical random matrix symmetry classes (the gaussian orthogonal–
GOE, unitary–GUE, and symplectic–GSE[12]). The Universal Hamiltonian[10, 11] (henceforth
called HU ) describes the physics of weak-coupling systems within the Thouless band (the energy
window within the Thouless energy ET of EF ). States separated by ε ¿ ET are correlated with
each other[4, 8, 9] by Random Matrix Theory (RMT)[12]. ET is the inverse ergodicization time of
an electron in the dot. For a diffusive dot ET ≈ hD/L2, where D is the bulk diffusion constant and
L is the linear size of the dot, while for a ballistic/chaotic dot ET ≈ hvF /L, where vF is the Fermi
velocity. In 2D, δ ≈ h2/mL2 is the single particle mean level spacing on the dot. The dimensionless
Thouless number g = ET /δ is an important parameter of the dot. For large g and low energies
1
|ε − εF | ¿ ET , it can be shown that the following HU [10] (for the GOE case) contains all the
relevant interactions in the renormalization group (RG) sense[13] at weak-coupling
HU =∑
α,s
εαc†αscαs +U
2N2 − J ~S2 + λ
∑
α
c†α↑c†α↓
∑
β
cβ↓cβ↑ (1)
Here N is the total number of electrons on the dot and ~S is the total spin. The first term is the
kinetic energy, the second is the charging energy, and the third is the Stoner energy. The final,
superconducting, coupling is missing in the GUE (a dot with an orbital magnetic field but no spin-
orbit couplings), while both the last two terms are missing in the GSE (a dot with both an orbital
magnetic field and spin-orbit couplings). An important prediction of HU is the mesoscopic Stoner
effect[10, 11], which pertains to the submacroscopic magnetization induced by the Stoner term long
before the bulk Stoner transition (occuring at J = δ). This has been verified experimentally[14].
For λ = 0 HU has no quantum fluctuations and is tractable.
A single-particle perturbation (such as a small orbital magnetic field, or a small spin-orbit cou-
pling) which moves the system from one symmetry class to another[12, 15] enhances mesoscopic
(sample-to-sample) fluctuations of the interaction matrix elements and renders the Universal Hamil-
tonian inapplicable during the crossover[16]. Furthermore, quantum fluctuations in a single sample
are enhanced in the RMT crossover[17], effectively moving the system into the quantum critical
regime[18]. This regime can be easily accessed experimentally by tuning external control parame-
ters. Since this regime is dominated by many-body quantum fluctuations, it is qualitatively different
from the Universal Hamiltonian regime. Very little is known about such mesoscopic regimes.
The simplest illustration of such regimes occurs in a superconducting grain (with order param-
eter ∆ and mean level spacing δ), which has been extensively studied [19, 20, 21, 22, 23, 24, 25].
However, many open questions remain in the regime ∆ ' δ, which arises naturally for ultrasmall
grains[26, 27, 28, 29], and also when an orbital magnetic field suppressed ∆ (in the GOE→GUE
crossover). Comparison with Richardson’s exact solution[30, 31] shows that the mean-field solu-
tion becomes poor for ∆ ' δ[29], implying that quantum fluctuations, both of amplitude[32] and
phase[33, 34] become important.
In addition to characterizing fluctuating ground states, one needs to find experimental signatures
for such states. Coulomb Blockade (CB) is experimentally precise[35] and theoretically rich[36],
with many dimensionless ratios (U/δ, U/T , Γ/δ where U is the charging energy, and Γ is the
level width of a single dot level coupled to the leads) allowing for detailed tests of theory. The
simplest case of only a charging interaction[36] (the “orthodox” model) has been investigated by
RG[37] instanton methods[38, 39], phase functional[40] methods (for large number of channels),
bosonization (for a single channel)[41, 42, 43], numerics[44], and most recently a self-consistent
“slave rotor” method[47], which can smoothly interpolate between high-T and the low-T Kondo[48]
regime in quantum dots[49, 50, 51, 52, 53, 54, 55, 56]. Recently mesoscopic fluctuations[45, 46] of
CB have been explored.
In the orthodox model there are no quantum fluctuations in an isolated dot. Coupling to the
leads results in quantum fluctuations, and entangled Kondo-like states emerge[41, 49, 50, 51, 52,
53, 54, 55, 56]. Even richer states might emerge when dots with strong quantum fluctuations (even
when isolated) are coupled to the leads in CB.
Finally, the approach to the bulk limits of various mesoscopic phenomena remains unclear in
many cases. For example: (i) Preliminary calculations by the PI and Y. Gefen of the average
susceptibility for the Pomeranchuk transition[57, 58, 59] in the diffusive bulk indicate that the
critical coupling is identical to the clean limit, u∗ = −2 (spinless). However, in the standard replica
2
treatment of disordered interacting electrons[5], these non-s-wave Landau channels are neglected in
favor of s-wave diffuson/cooperon modes. The interplay between s-wave and non-s-wave slow modes
(near the Pomeranchuk transition) needs to be understood. (ii) The PI’s treatment[60] of Stoner +
Kondo interactions in a quantum dot, with no spatially dependent spin waves, suggests that as the
Stoner coupling J increases the Kondo scale ∆K is initially enhanced before being suppressed. An
enhanced ∆K is also found in the quantum-critical regime near the bulk Stoner transition[61, 62, 63].
A complementary treatment on the bulk Stoner side[64, 65] shows a suppression of the Kondo scale
due to spin waves. Both these behaviors are seen in the PI’s zero-dimensional treatment[60]. It
would be very useful to understand the nontrivial crossover between the finite system/bulk on the
one hand, and paramagnetic/ferromagnetic states on the other.
Summarizing, the following broad questions in mesoscopics appear to be presently open: (i)
How does one create, control, and characterize the ground states of strongly correlated/fluctuating
mesoscopic systems? (ii) Can one develop a generalized theory of Coulomb Blockade for such states
and find their signatures? (iii) Can one characterize the crossovers between mesoscopic systems
and the bulk near a bulk phase transition?
1.2 Open questions for the ν = 1 bilayer quantum Hall system
Experiments on bilayer quantum Hall systems have established a number of properties akin to
superfluidity[66]. Early theoretical investigations of these systems concluded that[67, 68, 121] when
the interlayer separation is small: (i) The system should be a superfluid at low temperatures. (ii)
There should be a finite-T Kosterlitz-Thouless transition when superfluidity is lost. (iii) The charge
carriers are “merons” which are half-skyrmions and have charges of ±e/2.
In the tunnelling geometry, current flows into the top layer and flows out the bottom layer, and
the interlayer voltage Vint shows a narrow (but not infinitely narrow[69, 70]) peak at zero bias. As
T decreases, the height of the peak increases and its width decreases. For a true superfluid one
expects an infinitely narrow Josephson-like peak at zero bias for T below the Kosterlitz-Thouless
transition temperature TKT . In the counterflow geometry currents flow in opposite directions in
the two layers (a neutral excitonic current). Both the longitudinal and Hall voltages on one layer
are activated (' e−∆/T )[71] and current flows throughout the sample. In a true superfluid the
counterflow current would decay within a Josephson length, and the bulk currents and voltages
would be strictly zero.
The presence of dissipation at the lowest measurable T in a bosonic system makes this state
extremely unusual. Despite much theoretical work[125, 126, 127, 128, 129, 130, 131, 132], a complete
explanation remains elusive. Disorder is believed to be central to a full understanding of this state.
Some of the many questions remain unanswered are: (i) What is the nature of the true ground
state at T = 0? Is it a superfluid state or a vortex liquid[131, 132], or equivalently, a vortex
metal[7]? (ii) What is the role, if any, of real spin[122, 123] as opposed to pseudospin? (iii) What
is the best description of the quantum phase transition[126, 127, 128, 129] between two widely
separated ν = 12 systems and the interlayer coherent state? What is the effect of disorder on this
transition?
1.3 Open questions for deconfined criticality with randomness
Confinement and deconfinement have proven to be fruitful concepts in the physics of correlated
electron systems. Many strongly interacting systems can be recast as gauge theories[72], and
phenomena such as fractionalization[73] and topological order[74] can be simply understood.
3
The study of two-dimensional quantum spin- 12 models[75, 76], and their descendants, the quan-
tum dimer models[77, 73, 79, 80, 81], has been very fertile: The nearest-neighbor hard-core quantum
dimer model on a triangular lattice supports an entire spin liquid deconfined phase[78], with nearly
free monomer (spin- 12) excitations. Other such models have been constructed since[82], and related
to the quantum Lifshitz theory[83]. Coming back to spin models, it has been conjectured[84, 85]
that the Neel to Valance Bond Solid (VBS) transition[75, 76] for the square lattice Heisenberg
antiferromagnet could be generically second order, contrary to the usual Landau rules. Further,
in this picture the critical region is best described in terms of spinons which are confined in either
phase. The root cause[84, 85] of these phenomena are the Haldane-Berry phase factors[86] associ-
ated with the tunnelling of a skyrmion (a “hedgehog” event) in an antiferromagnet. The spatial
dependence of these phase factors, together with the translation invariance of the clean lattice
leads to a paramagnetic phase breaking translation invariance, and the hedgehogs having to appear
in quadruplets[86]. This quadrupling makes hedgehogs irrelevant in the RG sense exactly at the
transition[84, 85], leading to spinon deconfinement.
Since real samples always have quenched randomness, one can ask whether randomness qual-
itatively changes the physics. There are strong reasons to believe that it does. Looking first at
critical points, by a quantum extention[87] of the Harris criterion[88], it is seen that disorder is
relevant in the absence of hedgehogs both at the dimer transitions described by the quantum Lif-
shitz theory[82], and at the conjectured deconfined critical point of the antiferromagnet[84, 85].
The important questions are: (i) What is the nature of the transition with quenched disorder?
In classical models, by mapping the random-bond disorder to an effective random-field disorder,
it is known that disorder renders first-order transitions second-order[89, 90, 91]. The opposite
might well occur here. (ii) If the critical point remains second-order, are hedgehogs irrelevant at
the disordered critical point? Recall that the lattice symmetries[86] that forced the hedgehogs to
occur quadrupled (on the square lattice) are broken by disorder. (iii) Away from the critical point,
quantum Griffiths singularities[92, 93, 94, 95] are expected. What form do these singularities take
near the disordered quantum Lifshitz critical point (assuming it exists)?
Coming now to the triangular lattice dimer spin liquid[78], it is known that the topological
degeneracy of the ground state is stable against weak disorder[97]. How about excitations? Disorder
will induce local VBS order, and spinons are confined in the presence of VBS order. So, one expects
disorder to induce a long-range potential between the excitations. The important questions here
are: (i) What is the nature of this disorder-induced interaction between excitations? (ii) Is there
a critical strength of disorder beyond which they “reconfine”? If so, does topological order get
destroyed at this critical disorder?
2 Results of previous grant-supported research
During the grant period July 2003 – present the PI and co-workers have made progress on meso-
scopic systems with disorder and interactions[98, 99, 100, 17, 60, 101, 102], aspects of fractional
quantum Hall edge states[103], and developed a coherent network model[104] of the bilayer quan-
tum Hall “superfluid” at ν = 1. Several other projects are in progress and will be described briefly
below. Six papers have been published, four are in press, and several are in preparation.
Results on mesoscopic systems: Before the current grant period the PI and co-workers
have shown that[13, 105] (i) The most natural starting point to describe ballistic/chaotic dots is
a Landau Fermi liquid with the disorder represented by scattering off the walls. (ii) HU is the
weak-coupling low-energy effective theory of the finite-size fermionic RG in the g → ∞ limit.(iii) In
4
ballistic/chaotic quantum dots a phase transition to a bulk Pomeranchuk phase[57, 58, 59] (with a
distorted Fermi surface) is possible as g → ∞.
During the current grant period the research carried out by the PI and co-workers on meso-
scopics can be broadly divided into; (i) Characterizing the mesoscopic Pomeranchuk transition
in ballistic/chaotic dots[98, 99, 100], (ii) Establishing[17] a generic connection between single-
particle crossovers between different symmetry classes by an external perturbation[12] and the
weak-coupling to quantum critical many-body crossover[18] in interacting quantum dots. (iii)
Treating[60] the competition between the Stoner interaction between electrons on a large dot and
the Kondo interaction[48] of these electrons with the “impurity” spin on a small dot[49, 50, 51].
This leads to a counterintuitive enhancement of the Kondo scale under Zeeman fields which may
have been seen in experiments[108]. (iv) Constructing a supersymmetric nonlinear sigma model to
investigate ballistic systems with random boundary transmission[101, 102].
(i). In ballistic/chaotic dots with Landau Fermi liquid interactions, by using large-N methods
the PI and co-workers showed[99] that there is a crossover between the weak-coupling regime
dominated by the Universal Hamiltonian and a many-body quantum critical regime[18]. We
computed[99] the behavior of the quasiparticle decay rate in the weak-coupling to quantum critical
crossover. For symmetry-breaking in an odd angular momentum Landau channel, each isloated
sample has a two-fold ground-state degeneracy. This results in a linear dependence of the Coulomb
Blockade peak position as a function of an external orbital magnetic field. If the dot is coupled
weakly to leads, via a Kondo-like effect[109, 110] it can spontaneously break time-reversal symme-
try and develop a large persistent current[99]. Separately, we carried out a numerical analysis of
the persistent current in the RMT limit with Fermi-liquid interactions and found a diamagnetic
persistent current[98] in the presence of even channel Landau interactions. Our numerical analysis
on the Robnik-Berry billiard[111] found[100] that most of the assumptions of our previous work
are qualitatively and semi-quantitatively correct. However, we do not find any window between
the mesoscopic strong-coupling phase and the bulk Pomeranchuk phase[57, 58, 59]. The absence of
this window was pointed out in ref. [112] in a related model. Also[100], the mesoscopic fluctuations
of the effective potential are larger than expected from RMT for symmetry-breaking in an even
Landau channel. Finally[100], significant symmetry-breaking can occur even for weak-coupling in
the even Landau channel case, enhancing experimental visibility.
(ii) Crossovers between different symmetry classes of RMT[12, 15] occur ubiquitously in quan-
tum dots, and can be tuned, for example, by an external orbital magnetic field (GOE→GUE), or
by changing the size of a 2D GaAs dot (GOE→GSE). Single-particle RMT crossovers are charac-
terized by an energy scale EX : States separated by ε ¿ EX exhibit correlations which are fully
crossed over, while for ε À EX they exhibit the correlations of the original symmetry class. Dur-
ing the crossover, extra correlations develop[15], mesoscopic fluctuations of the interaction matrix
elements become large, and HU is inapplicable[16]. Consider an order parameter Q consistent with
symmetry class I, but not with class II (e.g. a spin polarization is consistent with GOE but not
GSE). Label the relevant dimensionless coupling J with the critical point J ∗ at which a transition
to a macroscopic value of 〈Q〉 occurs. Using correlations in the RMT crossover[113], the PI has
shown[17] a generic connection between the single-particle scale EX and the energy scale EQCX
to cross over to the quantum critical regime of Q: EQCX = |J − J∗|EX . All physical correlators
in the quantum critical regime are given in terms of explicit scaling functions[17] by a nonpertur-
bative calculation. The PI considered two examples: (a) The Stoner transition in the crossover
between the GOE and a new symmetry class discovered by Aleiner and Fal’ko[114], and (b) A
superconducting grain in the GOE→GUE crossover. The experimental implication is that one can
5
tune access to the quantum critical regime externally. These results hold for diffusive as well as
ballistic/chaotic dots.
(iii). In a recent Kondo-like experiment conduction electrons live in a large dot[108], while the
Kondo “impurity” spin is a small dot with an odd number of electrons. The electrons on the large
dot are described by HU . The PI has investigated[60] the competition between Kondo screening
and the Stoner interaction which tries to polarize the dot electrons. There is a regime[60] in
which the polarization of the dot electrons is suppressed, while the Kondo energy scale is enhanced.
Most strikingly, a Zeeman coupling leads to a huge enhancement of the Kondo scale[60], which is
the opposite of what occurs when the conduction electrons are noninteracting[106]. Under a large
Zeeman field, the Kondo state collapses into a mesoscopically polarized dot with the impurity slaved
to the dot polarization. These effects have been seen in experiments[108], but the dot there is open
while the theoretical model assumes a closed dot. Other explanations[107] of the enhancement of
the Kondo scale rely on the presence of two impurity spins, whereas the enhancement occurs[108]
for a single impurity as well.
(iv). In many ballistic mesoscopic systems (such as quantum corrals[115], quantum dots with
many leads, and optical/microwave cavities with leaky walls[116, 117]) the coupling to the outer
world is confined to the boundary. When the average coupling is weak and is gaussian distributed
around its mean, the PI’s postdoc, I. Rozhkov, and the PI have constructed a supersymmetric non-
linear sigma model (for integrable[101], and chaotic billiards[102]) which can be used to study any
physical correlator. The random coupling to the boundary acts as a natural regulator, and allows
us to avoid the technical difficulties which have dogged previous attempts at constructing ballistic
nonlinear sigma models[118]. The result is a set of modes confined to the boundary and diffusing in
angle, representing whispering gallery modes[119] interfering and diffracting out of the dot through
its leaky walls. This describes a situation complementary to “dynamic localization”[120] in which
the modes diffuse in angular momentum in a closed billiard.
Results in the ν = 1 bilayer quantum Hall system: As described in the introduction, ex-
periments have shown that an unusual dissipative state exists at the lowest measurable T in inter-
layer coherent systems at ν = 1. Based on the known effects of disorder on incompressible states,
Herb Fertig and the PI have constructed[104] a model which explains some of these observations.
In the Efros picture[134] (supported by imaging experiments[135]), the ν = 1 system cannot screen
disorder due to the dopant layer linearly, since it is incompressible. The system breaks up into
compressible regions (with ν < 1 or ν > 1) and incompressible strips (with ν = 1) of typical width
a few magnetic lengths, and screens the disorder nonlinearly. This is an example of the generic
phenomenon of pattern formation by disorder[6].
Our model treats the spatial structure induced by disorder explicitly. Along an Efros strip
disorder induces many solitons[104] each of which ends in weakly localized merons or antimerons.
There are two main sources of dissipation. At a node, where two or more incompressible strips
meet, the effects of disorder are the least, and therefore the tunnelling term with characteristic
energy h is the most effective. Thermal/quantum fluctuations induce the solitons to pass through
the node, disrupting the tunnelling at the node and produce a viscosity for the tunnelling current,
which translates into an interlayer voltage. For T À h a perturbative classical treatment leads to
an interlayer voltage Vint ' h2T−3/2, and a tunnelling conductance proportional to the area of the
sample[104].
In the counterflow geometry dissipation is produced by merons/antimerons thermally hopping
across the Efros strips. In order for a nonzero Hall resistance to exist, the merons must be in a
liquid state[131, 132]. We show by a replica-RG analysis of the classical model that the liquid state
6
can occur even as T → 0 for sufficiently strong disorder[104]. In this liquid state, the barrier to
these meron-antimeron tunnelling events is of the same order as the charge gap. Since merons and
antimerons are vortices and antivortices, their tunnelling across the sample creates a longitudinal
gradient in the interlayer voltage Vint, which can be measured as a longitudinal voltage on a single
layer. The merons carry a dipole moment which ultimately[104] leads to the activated Hall resis-
tance. In our model RH has the same activation gap as the longitudinal resistance (experimentally
true in electron but not in hole samples[71]).
Miscellaneous and ongoing work: The projects below either do not fit neatly into a par-
ticular category or are in progress. (i) In collaboration with his former postdocs, Y. Joglekar and
H. Nguyen, the PI has carried out a calculation of collective edge modes[103] in ν = 13 and 2
5 ,
using the Hamiltonian formulation of the FQHE developed by R. Shankar and the PI[136]. (ii)
R. Shankar and the PI have revisited deconfinement[137] in 1 + 1 dimensions[138, 139, 141], and
we find that often the particles usually considered to be “free” are only half-asymptotic, which
means that they must appear with particles and antiparticles alternating[139]. The confinement-
deconfinement transition in the massive Schwinger model[139, 140] is in the 2D Ising universality
class[137], and there are truly free Majorana fermions which have a nonlocal connection to the
original Dirac fermions. Similar results obtain in the spin chain with frustrated antiferromagnetic
interactions[141, 142]. (iii) A graduate student, O. Zelyak, and the PI are analyzing the problem of
a ballistic/chaotic system penetrated by a point flux, and the persistent current resulting from this
flux[143]. We find that this problem seems to be quite different from that of a uniform magnetic
field pervading the dot (which has been noticed before[144]). We also find that there is a large
diamagnetic persistent current which is proportional to the number of electrons in the dot (which
has not been noticed before). We are also investigating chaotic annular geometries, using an annu-
lar generalization of the Robnik-Berry billiard[111]. (iv) Y. Gefen and the PI are considering the
Pomeranchuk transition in a diffusive system. Our goal is to compute mesoscopic fluctuations of
the order parameter, and the behavior of the quasiparticle decay rate as the system crosses over
into the quantum critical regime on the weak-coupling side. This work is in progress.
3 Proposed projects
3.1 Proposed Projects in Mesoscopics
Let us recall the broad issues to be addressed in mesoscopics: (i) How does one create, control, and
characterize the ground states of strongly correlated/fluctuating mesoscopic systems? (ii) Can one
develop a generalized theory of Coulomb Blockade for such states? (iii) Can one characterize the
crossovers between mesoscopic systems and the bulk near a bulk phase transition?
3.1.1 Competition between Kondo and Stoner interactions
The PI’s work on Stoner + Kondo[60] has several limitations: (i) It is carried out for a single impu-
rity, whereas the experiments have been carried out for the conduction electrons interacting with
both a single impurity and with two impurities. The enhancement of ∆K with a Zeeman coupling
EZ is also seen in the two-impurity case[108]. (ii) The PI’s calculation assumes equally spaced
energy levels and an equal coupling of each level to the impurity spin, thus ignoring mesoscopic
fluctuations. (iii) The calculation is carried out at T = 0. (iv) The most important limitation is
that the PI’s work is carried out in a closed dot. The real experiments are carried out with dots
with a reasonably strong coupling to the leads (∆K is inferred by conductance measurements[108]).
7
Research is proposed to remove these limitations. The PI’s calculation will be repeated for
T > 0. The result will be interesting for large total spin, since[10, 11] for small r = 1 − J/δ (J
is the Stoner coupling of Eq. (1)) the ground state spin is of order 1/r, while there are many
energetically close values of total spin which are populated at nonzero T . This is expected to
enhance the Kondo scale even more than at T = 0, until T ' δ at which point particle-hole
excitations will start frustrating the Kondo state. The T, EZ dependence of ∆K is expected to be
a good experimental signature of the nature of the state. Note that the large-N approach is not
accurate for T comparable to ∆K [106], but we will restrict ourselves to T of order δ ¿ ∆K
The bulk limit (L → ∞) of the Stoner + Kondo model in an isolated dot is extremely interesting
near the Stoner transition: In the seminal work of Larkin and Melnikov[61], as the bulk system
approaches the Stoner transition, the spin susceptibility diverges, and the impurity polarizes a
huge droplet of electrons[62]. Since the effective size of the impurity is now large, the conduction
electrons can couple to it via many channels with nontrivial spatial dependence. The number of
channels tends to ∞ as the Stoner transition is approached[62]. This physics is a special case of
the bose-Kondo problem[63, 64].
It is proposed to extend the PI’s calculation to the crossover to the bulk in a diffusive dot
beyond ET . This can be accomplished with exactly the same methods as used by the PI, except
that now the wavefunction correlators of the dot states are no longer controlled by RMT, but
become wavevector and energy dependent[150]. The emergence of Landau-damped spin waves and
their interaction with the impurity[61, 62], as well as the emergence of many channels coupling to
the impurity will be investigated. Since the calculation will be carried out for a finite system with a
spin magnetization (albeit submacroscopic), it may shed light on the physics of the Kondo impurity
in the bulk Stoner phase[64, 65], where ∆K is suppressed. In this context, recall that even in the
PI’s zero-dimensional calculation[60], ∆K is suppressed for large enough J . On the other hand, as
J → 0 one expects to recover the usual disordered Kondo treatments in the bulk limit[151]. This
calculation will serve to clarify diverse crossovers and how they tend in the bulk limit to phase
transitions.
The quantum dot interacting with two magnetic impurities[108, 107] is interesting for a variety
of reasons. In the two-impurity case, the experiment has been explained by assuming that an
antiferromagnetic RKKY interaction I exists between the two impurities[107], leading to a singlet.
The Zeeman field closes the singlet-triplet gap leading to an enhancement of the (singlet-triplet)
Kondo scale[107]. This latter explanation does not include the Stoner interaction, and is inoperative
in the single-impurity case, in which an enhancement of the Kondo scale is also observed[108]. The
two-impurity case is theoretically interesting even in the bulk[152, 153, 154, 155], where a phase
transition (as I decreases) is found between a phase in which the two impurities form a singlet with
no Kondo effect, and one with a Kondo effect of both impurities[152, 155]. Finally, the two-impurity
Kondo problem in the dot, when Stoner interactions are included, offers a rich, yet experimentally
realized[108] example of competing interactions.
It is proposed to address the two-impurity Kondo problem with Stoner interacting conduction
electrons by using the large-N [156, 157, 158, 159] (or slave boson[160]) approach. At mean-field
level in the bulk, this approach fails to recover the unstable fixed point, giving instead a first-order
transition[154]. A potential way to overcome this limitation is to look at the effective theory of
phase fluctuations of the Kondo order parameter[158, 159] around the mean field solution. The
usual mean-field will be used to identify the phases and describe the low-energy physics deep in
a particular phase, while a careful analysis of the phase fluctuations will be performed near the
transition. In order to distinguish the different ground states, the most convenient signatures occur
8
in transport[107], which will be addressed in subsection 3. 1. 4 on Coulomb Blockade.
3.1.2 GOE to GUE crossover in superconducting grains
The PI’s analysis[17] of the relation between the single-particle RMT GOE→GUE crossover and
the weak-coupling to quantum critical interacting crossover in superconducting grains is limited to
weak-coupling (when the magnetic flux has killed the order parameter) and depends solely on the
noninteracting 4-point ensemble averages in the single-particle crossover computed recently[113].
When a superconducting order parameter ∆ is present, even considering only thermodynamic
quantities (leaving the transport properties for the Coulomb Blockade subsection), a number of
issues need to be faced: (i) For an isolated grain, charge quantization means that the phase of
∆ must be fluctuating[33, 34]. (ii) Comparing the BCS mean-field solution with Richardson’s
exact solution[30, 31] shows that different values of ∆BCS are inferred depending on what one
measures[29]. These values can be very different as ∆BCS approaches the mean level spacing δ.
This means that as the orbital field increases and ∆ becomes small, both amplitude[32] and phase
fluctuations[33, 34] of ∆ become important. (iii) Finally, mesoscopic fluctuations[34] need to be
considered in the crossover regime in order to compare to experiments.
Some of the ingredients for a nonperturbative calculation of the thermodynamic properties in
the crossover regime exist in the literature cited above, but other important ones are proposed
below.
The first step is to compute the spectrum and wavefunction correlators of the mean-field Hamil-
tonian, which includes the weak magnetic flux, and ∆.
HX(φ, ∆) = HO + ∆∑
i
(c†i↑c†i↓ + ci↓ci↑) + αHA (2)
where HO is the noninteracting Hamiltonian with Orthogonal symmetry with eigenstates labelled
by i, j and HA is a normalized antisymmetric Hamiltonian[12] induced by the external magnetic flux
Φ and α ' Φ/Φ0. This is a double-crossover and the ensemble averages of the 4-point correlators of
wavefunctions will determine the effective action[17] of the fluctuations of ∆. It is proposed to use
numerical[113] and supersymmetry methods[15] to determine the wavefunction correlations. The
known results in the limits[113] will provide a test of our results.
The next step is to include phase[33, 34] and amplitude[32] fluctuations of ∆ in the effective
theory. The best way, presaging the CB proposed research below, is to introduce a very large
charging energy to fix the number of particles. Since the total charge consists of electrons which
are part of a Cooper pair as well as electrons which are not, there are two phases involved, the
phase φ conjugate to the total charge, and the phase φ∆ of the order parameter ∆ = |∆| exp iφ∆.
When ∆ À δ, one expects the two phases to be locked at low energies, but we wish to investigate
the quantum critical regime when ∆ has no expectation value, but only slow fluctuations of both
amplitude and phase. As in the treatment of Coulomb Blockade, instantons[38] in both φ and φ∆
are expected to play an important role in determining the physics.
The resulting effective theory will be analysed using RG[37] instanton[38, 39], and slave rotor[47]
methods. The PI’s postdoc, I. Rozhkov, is an expert in supersymmetry techniques and will assist
the PI in this project. The graduate student, O. Zelyak, will also participate in this project.
3.1.3 The mesoscopic Stoner effect with weak spin-orbit interactions in GaAs
In GaAs there are “intermediate” spin-orbit RMT universality classes[114]. In the simplest one, to-
tal spin is not conserved but Sz is. Deep into this crossover, one recovers the mesoscopic Ising-Stoner
9
effect[161]. In the PI’s previous work[17], the analysis was restricted to Sz = 0 and mesoscopic
fluctuations were neglected.
It is proposed to examine the distribution of the ground state Sz as a function of the Stoner J
in the above crossover. However, Sz is not directly measurable. To make contact with measurable
quantities, one needs to include an in-plane field[162]. Due to the spin-orbit coupling, this in-plane
field breaks time-reversal symmetry. One is again led to a double crossover from the GOE with
perturbations taking the system to the Aleiner-Falko intermediate symplectic class[114] and the
GUE. The dependence of CB peak position on in-plane field will be calculated, and the connection
with ground state Sz found. Since experiments can tune through this crossover by changing the size
of the dot[114], the results will be directly relevant for dots constructed from GaAs heterostructures.
Both the postdoc and the student will participate in this project.
3.1.4 Coulomb Blockade in strongly correlated and fluctuating states
The treatment of CB in the simplest case of the orthodox model reduces in the δ/T → 0 limit
(after integrating out fermions) to the analysis of the effective action of the phase φ conjugate to
the total dot charge[40]. For nonzero δ/T the recently developed slave rotor[47] formulation which
treats the fermionic degrees of freedom on the same level as the bosonic ones, appears promising.
The new feature in the research proposed below is the existence of other collective variables, which
possess their own bosonic degrees of freedom, which can often be reduced to a phase (or an SU(2)
nonabelian phase for spin-rotation invariant systems). The theme of this subsection is the interplay
of the charge degree of freedom with the other collective variables, and with the fermions.
Coulomb Blockade in the Mesoscopic Pomeranchuk Regime: As the PI and co-
workers have shown[99, 100], mesoscopic systems have ground state expectation values for the Fermi
surface distortion, characterized by the vector σ (which determines the single-particle states), even
in the weak-coupling regime (|u| < |u∗|). This is a result of mesoscopic fluctuations and explicit
rotational symmetry breaking[99, 100]. Typically, the ground states for N and N + 1 particles will
have different values of σ, and the quantum fluctuations of σ will be small for large g. When an ad-
ditional electron enters the dot all the single-particle states change in a ground-state to ground-state
transition, implying that the transition amplitude will suffer orthogonality catastrophe effects[163]
and be very small. The typical barrier EB between the two minima is expected to be of order ET .
The distribution of effective tunnelling strengths between the two minima will be computed using
the large-g mean field theory[99] developed by the PI and co-workers and RMT. This is related to
the Caldeira-Leggett problem of tunnelling between two nearly degenerate states in the presence
of a dissipative bath[109, 110]. The new feature is that the charge degrees of freedom (phase φ)
are coupled to the two minima of σ for T ¿ EB, and to large fluctuations of σ in the quantum
critical regime. Therefore one expects a generalization of the charge Kondo effect[41] at low T , and
nontrivial T dependence of the CB.
Coulomb Blockade for the Universal Hamiltonian with large S: While there has been
progress[146, 147, 148, 149] over the last few years in treating the CB problem for the Universal
Hamiltonian[10, 11], our knowledge is by no means complete. For an isolated dot, as the Stoner
coupling approaches δ (or r = (1−J/δ) ¿ 1), the ground state spin becomes large[10, 11] (' 1/r),
and there are many (of order 1/√
r) spin states within an energy δ. When the dot is coupled to
leads the spin has to change when particle number (charge) changes. However, there are strong
constraints on transitions between spin states because they have to be connected by a single-particle
operator (for weak tunnelling). This will couple the charge and spin degrees of freedom.
Since the spin of a dot coupled to leads is not conserved, one can decouple the Stoner interaction
10
by a Hubbard-Stratanovich exchange field h, and the charging interaction[38, 39] by a scalar V .
The imaginary-time action (suppressing the dot-lead coupling) is
S =
β∫
0
dt
(
h2
4J+
V 2
2U+ iV N0 +
∑
kss′
cks((∂t + εk − iV )δss′ −h
2· ~τss′)cks′
)
(3)
where N0 is related to the gate voltage, the single-particle dot states are (k, s), and ~τ is a Pauli spin
matrix. Now, one represents[38, 39] the non-constant part of V in terms of a phase V (t) = V0 + φ,
and constructs “neutral” d-fermions by the unitary transformation cks = eiφ(t)(R(t))ss′dks′ , where
the unitary SU(2) rotation operator R(t) is defined by the requirement |h(t)|τz = h(t) ·R(t)†~τR(t).
All the spin dynamics is now contained in the SU(2) nonabelian phase. Since the ground state
spin of the decoupled dot is large, h will have a nonzero expectation value, with the d fermions
subject to a Zeeman field in the z-direction. Integrating out the lead fermions, one obtains an
effective action which couples φ, R, and the dot fermions. Assuming equal coupling of all lead and
dot modes the key term is of the form
Γ∑
k,k′
β∫
0
dtdt′
(t − t′)ei(φ(t)−φ(t′))d(t′)R†(t′)R(t)d(t) (4)
In phase-only models[40] (valid in the limit δ/T → 0[39]) the dot fermions are also integrated out
with their free action to leading order. Since we want to access all the temperature regimes as well
as Stoner physics, we need to work at nonzero δ/T . The dynamics of this model will be investigated
by RG[37], instanton[38, 39], and slave-rotor[47] methods, suitably generalized. For example, in
the slave-rotor approach, one would decouple the integrand of Eq. (5) by