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A. M. M. Pruisken

Publications and source records attributed to A. M. M. Pruisken.

At least 19 recordsLinked to original sources
arXiv2021

Super universality of dimerised $SU(N+M)$ spin chains

We explore the physics of the quantum Hall effect using the Haldane mapping of dimerised $SU(N+M)$ spin chains, the large $N$ expansion and the density matrix renormalization group technique. We show that while the transition is first order for $N+M >2$, the system at zero temperature nevertheless displays a continuously diverging length scale $ξ$ (correlation length). The numerical results for $(M, N) = (1,3), ~ (2, 2),~(1, 5)$ and $(1, 7)$ indicate that $ξ$ is a directly observable physical quantity, namely the spatial width of the edge states. We relate the physical observables of the quantum spin chain to those of the quantum Hall system (and, hence, the $\vartheta$ vacuum concept in quantum field theory). Our numerical investigations provide strong evidence for the conjecture of super universality which says the dimerised spin chain quite generally displays all the basic features of the quantum Hall effect, independent of the specific values of $M$ and $N$. For the cases at hand we show that the singularity structure of the quantum Hall plateau transitions involves a universal function with two scale parameters that may in general depend on $M$ and $N$. This includes not only the Hall conductance but also the ground state energy as well as the correlation length $ξ$ with varying values of $\vartheta \sim π$.

cond-mat.str-el↗
arXiv2009

The problem of "macroscopic charge quantization" in single-electron devices

In a recent Letter by the authors [I.S. Burmistrov and A.M.M. Pruisken, Phys. Rev. Lett. 101, 056801 (2008)] it was shown that single-electron devices (single electron transistor or SET) display "macroscopic charge quantization" which is completely analogous to the quantum Hall effect observed on very different electron systems. In this investigation we present more detail on these new findings. Based on the Ambegaokar-Eckern-Schoen (AES) theory of the Coulomb blockade we introduce a general response theory that probes the sensitivity of SET to changes in the boundary conditions. This response theory defines a new set of physical observables and we establish the contact with the standard results obtained from ordinary linear response theory. The response parameters generally define the renormalization behavior of the SET in the entire regime from weak coupling with large values of the tunneling conductance all the way down to the strong coupling phase where the system displays the Coulomb blockade. We introduce a general criterion for charge quantization that is analogous to the Thouless criterion for Anderson localization. We present the results of detailed computations on the weak coupling side of the theory, i.e. both perturbative and non-perturbative (instantons). Based on an effective theory in terms of quantum spins we study the quantum critical behavior of the AES model on the strong coupling side. Consequently, a unifying scaling diagram of the SET is obtained. This diagram displays all the super universal topological features of the theta-angle concept that previously arose in the theory of the quantum Hall effect.

cond-mat.mes-hall↗
arXiv2008

Super universality of the quantum Hall effect and the "large $N$ picture" of the $\vartheta$ angle

It is shown that the "massless chiral edge excitations" are an integral and universal aspect of the low energy dynamics of the $\vartheta$ vacuum that has historically gone unnoticed. Within the $SU(M+N)/S(U(M) \times U(N))$ non-linear sigma model we introduce an effective theory of "edge excitations" that fundamentally explains the quantum Hall effect. In sharp contrast to the common beliefs in the field our results indicate that this macroscopic quantization phenomenon is, in fact, a {\em super universal} strong coupling feature of the $\vartheta$ angle with the replica limit $M=N=0$ only playing a role of secondary importance. To demonstrate super universality we revisit the large $N$ expansion of the $CP^{N-1}$ model. We obtain, for the first time, explicit scaling results for the quantum Hall effect including quantum criticality of the quantum Hall plateau transition. Consequently a scaling diagram is obtained describing the cross-over between the weak coupling "instanton phase" and the strong coupling "quantum Hall phase" of the large $N$ theory. Our results are in accordance with the "instanton picture" of the $\vartheta$ angle but fundamentally invalidate all the ideas, expectations and conjectures that are based on the historical "large $N$ picture."

cond-mat.mes-hall↗
arXiv2008

Exact Haldane mapping for all $S$ and super universality in spin chains

The low energy dynamics of the anti-ferromagnetic Heisenberg spin $S$ chain in the semiclassical limit $S\to\infty$ is known to map onto the O(3) nonlinear $σ$ model with a $θ$ term in 1+1 dimension. Guided by the underlying dual symmetry of the spin chain, as well as the recently established topological significance of "dangling edge spins," we report an {\em exact} mapping onto the O(3) model that avoids the conventional large $S$ approximation altogether. Our new methodology demonstrates all the super universal features of the $θ$ angle concept that previously arose in the theory of the quantum Hall effect. It explains why Haldane's original ideas remarkably yield the correct answer in spite of the fundamental complications that generally exist in the idea of semiclassical expansions.

cond-mat.mes-hall↗
arXiv2007

Coulomb Blockade and Super Universality of the Theta-Angle

Based on the Ambegaokar-Eckern-Schon approach to the Coulomb blockade we develop a complete quantum theory of the single electron transistor. We identify a previously unrecognized physical observable q^\prime in the problem that, unlike the usual average charge (Q) on the island, is robustly quantized for any finite value of the tunneling conductance as the temperature goes to absolute zero. This novel quantity is fundamentally related to the non-symmetrized noise of the system. We present a unifying scaling diagram in the q^\prime - g^\prime plane where g^\prime denotes the conductance of the system. The results display all the super universal topological features of the theta-angle concept that previously arose in the theory of the quantum Hall effect.

cond-mat.mes-hall↗
arXiv2006

Observation of the quantized Hall insulator in the quantum critical regime of the two-dimensional electron gas

We have investigated the Hall resistance $R_H$ near the plateau-insulator transition of a two-dimensional electron gas in the quantum critical regime. High-field magnetotransport data taken on a low-mobility InGaAs/InP heterostructure with the plateau-insulator transition at a critical field $B_c$ of 17.2 T show that the Hall resistance $R_H$ is quantized at $h/e^2$ near the critical filling fraction ($ν_c$ = 0.55) when $T \to 0$. By making use of universal scaling functions extracted from the magnetotransport data we show that $R_H$ in the insulating phase in the limit $T \to 0$ is quantized at $h/e^2$ for all values of the scaling parameter $Δν/(T/T_0)^κ$ with $Δν= ν- ν_c$. However, as a function of $Δν$ (or magnetic field) the Hall resistance diverges in the limit $T \to 0$ for all values $ν< ν_c$.

cond-mat.mes-hall↗
arXiv2006

Quantum critical behaviour of the plateau-insulator transition in the quantum Hall regime

High-field magnetotransport experiments provide an excellent tool to investigate the plateau-insulator phase transition in the integral quantum Hall effect. Here we review recent low-temperature high-field magnetotransport studies carried out on several InGaAs/InP heterostructures and an InGaAs/GaAs quantum well. We find that the longitudinal resistivity $ρ_{xx}$ near the critical filling factor $ν_{c}$ ~ 0.5 follows the universal scaling law $ρ_{xx}(ν, T) \propto exp[-Δν/(T/T_{0})^κ]$, where $Δν=ν-ν_{c}$. The critical exponent $κ$ equals $0.56 \pm 0.02$, which indicates that the plateau-insulator transition falls in a non-Fermi liquid universality class.

cond-mat.mes-hall↗
arXiv2006

Massless excitations at $θ=π$ in the $CP^{N-1}$ model with large values of $N$

We study the instanton vacuum of the $CP^{N-1}$ model with large values of $N$ in 1+1 space-time dimensions. Unlike the longstanding claims which state that the theory always has a mass gap, we for the first time establish a complete {\em critical} theory for the transition at $θ= π$ obtained from a mapping onto the low temperature phase of the 1D Ising model. We derive a simple effective field theory in terms of 1D massless chiral fermions. Our results include, besides a diverging correlation length with an exponent $ν=1/2$, exact expressions for the $β$ functions. These expressions unequivocally demonstrate that the large $N$ expansion with varying $θ$ displays all the fundamental features of the quantum Hall effect.

cond-mat.mes-hall↗
arXiv2005

General topological features and instanton vacuum in quantum Hall and spin liquids

We introduce the concept of super universality in quantum Hall and spin liquids which has emerged from previous studies. It states that all the fundamental features of the quantum Hall effect are generically displayed as general topological features of the $θ$ parameter in nonlinear sigma models in two dimensions. To establish super universality in spin liquids we revisit the mapping by Haldane who argued that the anti ferromagnetic Heisenberg spin $s$ chain is effectively described by the O(3) nonlinear sigma model with a $θ$ term. By combining the path integral representation for the dimerized spin $s=1/2$ chain with renormalization group decimation techniques we generalise the Haldane approach to include a more complicated theory, the fermionic rotor chain, involving four different renormalization group parameters. We show how the renormalization group calculation technique can be used to lay the bridge between the fermionic rotor chain and the sigma model. As an integral and fundamental aspect of the mapping we establish the topological significance of the dangling spin at the edge of the chain which is in all respects identical to the massless chiral edge excitations in quantum Hall liquids. We consider various different geometries of the spin chain and show that for each of the different geometries correspond to a topologically equivalent quantum Hall liquid.

cond-mat.str-el↗
arXiv2005

Non-Fermi liquid criticality and super universality in the quantum Hall regime

We report the results of a microscopic theory, based on the topological concept of a $θ$ vacuum, which show that the Coulomb potential, unlike any finite ranged interaction potential, renders the longstanding problem of the plateau transitions in the quantum Hall regime non-Fermi liquid like. Our present results, which are of outstanding significance for quantum phase transitions in general and composite fermion ideas in particular, provide a novel understanding of the critical exponent values that have recently been (re)taken from a series of state-of-the-art quantum Hall samples.

cond-mat.mes-hall↗
arXiv2005

Comment on ``Topological Oscillations of the Magnetoconductance in Disordered GaAs Layers''

In a recent Letter, Murzin et. al. [Phys. Rev. Lett., vol. 92, 016802 (2004)] investigated "instanton effects" in the magneto resistance data taken from samples with heavily Si-doped GaAs layers at low temperatures. This topological issue originally arose in the development of a microscopic theory of quantum Hall effect some 20 years ago. The investigations by Murzin et. al., however, do not convey the correct ideas on scaling that have emerged over the years in the general theory of quantum transport.

cond-mat.mes-hall↗
arXiv2005

Theta renormalization, electron-electron interactions and super universality in the quantum Hall regime

The renormalization theory of the quantum Hall effect relies primarily on the non-perturbative concept of theta renormalization by instantons. Within the generalized non-linear sigma model approach initiated by Finkelstein we obtain the physical observables of the interacting electron gas, formulate the general (topological) principles by which the Hall conductance is robustly quantized and derive - for the first time - explicit expressions for the non-perturbative (instanton) contributions to the renormalization group beta- and gamma- functions. Our results are in complete agreement with the recently proposed idea of super universality which says that the fundamental aspects of the quantum Hall effect are all generic features the instanton vacuum concept in asymptotically free field theory.

cond-mat.mes-hall↗
arXiv2004

The instanton vacuum of generalized $CP^{N-1}$ models

It has recently been pointed out that the existence of massless chiral edge excitations has important strong coupling consequences for the topological concept of an instanton vacuum. In the first part of this paper we elaborate on the effective action for ``edge excitations'' in the Grassmannian $U(m+n)/U(m) \times U(n)$ non-linear sigma model in the presence of the $θ$ term. This effective action contains complete information on the low energy dynamics of the system and defines the renormalization of the theory in an unambiguous manner. In the second part of this paper we revisit the instanton methodology and embark on the non-perturbative aspects of the renormalization group including the anomalous dimension of mass terms. The non-perturbative corrections to both the $β$ and $γ$ functions are obtained while avoiding the technical difficulties associated with the idea of {\em constrained} instantons. In the final part of this paper we present the detailed consequences of our computations for the quantum critical behavior at $θ= π$. In the range $0 \leq m,n \lesssim 1$ we find quantum critical behavior with exponents that vary continuously with varying values of $m$ and $n$. Our results display a smooth interpolation between the physically very different theories with $m=n=0$ (disordered electron gas, quantum Hall effect) and $m=n=1$ (O(3) non-linear sigma model, quantum spin chains) respectively, in which cases the critical indices are known from other sources. We conclude that instantons provide not only a {\em qualitative} assessment of the singularity structure of the theory as a whole, but also remarkably accurate {\em numerical} estimates of the quantum critical details (critical indices) at $θ= π$ for varying values of $m$ and $n$.

cond-mat.mes-hall↗
arXiv2003

The effects of macroscopic inhomogeneities on the magneto transport properties of the electron gas in two dimensions

In experiments on electron transport the macroscopic inhomogeneities in the sample play a fundamental role. In this paper and a subsequent one we introduce and develop a general formalism that captures the principal features of sample inhomogeneities (density gradients, contact misalignments) in the magneto resistance data taken from low mobility heterostructures. We present detailed assessments and experimental investigations of the different regimes of physical interest, notably the regime of semiclassical transport at weak magnetic fields, the plateau-plateau transitions as well as the plateau-insulator transition that generally occurs at much stronger values of the external field only. It is shown that the semiclassical regime at weak fields plays an integral role in the general understanding of the experiments on the quantum Hall regime. The results of this paper clearly indicate that the plateau-plateau transitions, unlike the the plateau-insulator transition, are fundamentally affected by the presence of sample inhomogeneities. We propose a universal scaling result for the magneto resistance parameters. This result facilitates, amongst many other things, a detailed understanding of the difficulties associated with the experimental methodology of H.P. Wei et.al in extracting the quantum critical behavior of the electron gas from the transport measurements conducted on the plateau-plateau transitions.

cond-mat.mes-hall↗
arXiv2003

New Insights into the Plateau-Insulator Transition in the Quantum Hall Regime

We have measured the quantum critical behavior of the plateau-insulator (PI) transition in a low-mobility InGaAs/GaAs quantum well. The longitudinal resistivity measured for two different values of the electron density follows an exponential law, from which we extract critical exponents kappa = 0.54 and 0.58, in good agreement with the value (kappa = 0.57) previously obtained for an InGaAs/InP heterostructure. This provides evidence for a non-Fermi liquid critical exponent. By reversing the direction of the magnetic field we find that the averaged Hall resistance remains quantized at the plateau value h/e^2 through the PI transition. From the deviations of the Hall resistance from the quantized value, we obtain the corrections to scaling.

cond-mat.mes-hall↗
arXiv2003

The effect of carrier density gradients on magnetotransport data measured in Hall bar geometry

We have measured magnetotransport of the two-dimensional electron gas in a Hall bar geometry in the presence of small carrier density gradients. We find that the longitudinal resistances measured at both sides of the Hall bar interchange by reversing the polarity of the magnetic field. We offer a simple explanation for this effect and discuss implications for extracting conductivity flow diagrams of the integer quantum Hall effect.

cond-mat.mes-hall↗
arXiv2002

The $θ$ vacuum reveals itself as the fundamental theory of the quantum Hall effect. I. Confronting controversies

The $θ$ dependence of the Grassmannian $U(m+n)/U(m)\times U(n)$ non-linear $σ$ model is reexamined. This general theory provides an important laboratory for studying the quantum Hall effect, in the special limit $m=n=0$ (replica limit). We discover however that the quantum Hall effect is in fact independent of this limit and exists as a generic topological feature of the theory for all non-negative values of $m$ and $n$. The results are in concflict with many of the historical ideas and expectations on the basis of the large $N$ expansion of the $CP^{N-1}$ or $SU(N)/U(N-1)$ model

cond-mat.mes-hall↗