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

Pruisken, A. M. M.; Shankar, Ravi; Surendran, Naveen

doi:10.1209/0295-5075/82/47005

Cited by 8 publications

(13 citation statements)

References 18 publications

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“…The same behavior has recently been found in strong coupling studies of closely related two dimensional models of the instanton vacuum. 61 Analogous to Eq. (327) one therefore expects that βθ = a ln σ ′ xx (328) in the limit where σ ′ xx goes to zero.…”

Section: A Robust Quantization Of Hall Conductancementioning

confidence: 99%

θ renormalization, electron–electron interactions and super universality in the quantum Hall regime

Pruisken

Burmistrov

2007

Annals of Physics

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The renormalization theory of the quantum Hall effect relies primarily on the non-perturbative concept of θ renormalization by instantons. Within the generalized non-linear σ 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 β and γ 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.

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Section: A Robust Quantization Of Hall Conductancementioning

confidence: 99%

θ renormalization, electron–electron interactions and super universality in the quantum Hall regime

Pruisken

Burmistrov

2007

Annals of Physics

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“…non-linear sigma model in two dimensions it was shown that the ϑ vacuum generally displays massless excitations that propagate along the confining "edge" of the system [1][2][3][4][5]. This new aspect of the sigma model has come in many ways as a welcome surprise.…”

Section: Super Universality In a Series Of Investigations On The Grasmentioning

confidence: 99%

“…In applications to quantum A.M.M. Pruisken ( ) Institute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands e-mail: pruisken@science.uva.nl spin liquids [6,7], for example, the edge excitations describe the dynamics of the "dangling" quantum spins located at the "edges" of the spin chain [4,5]. In the context of quantum Hall liquids [8][9][10][11][12], on the other hand, these excitations are identically the same as those described by the theory of "chiral edge bosons" [2,3].…”

Section: Super Universality In a Series Of Investigations On The Grasmentioning

confidence: 99%

“…Within the grassmannian sigma model one finds, for example, that the SU(M + N) symmetry is spontaneously broken at the "edge" of the system whereas the critical correlations along the "edge" are identically the same for all values of M and N [2]. These unexpected features have motivated several studies where the idea of super universality has emerged [4,5,[20][21][22][23]. The essence of this idea is that the ϑ vacuum displays all the basic aspects of the quantum Hall effect for all non-negative values of M and N .…”

Section: Super Universality In a Series Of Investigations On The Grasmentioning

confidence: 99%

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Super Universality of the Quantum Hall Effect and the “Large N Picture” of the ϑ Angle

Pruisken

2009

Int J Theor Phys

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It is shown that the "massless chiral edge excitations" are an integral and universal aspect of the low energy dynamics of the ϑ vacuum that has historically gone unnoticed.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 super universal strong coupling feature of the ϑ 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 ϑ angle but fundamentally invalidate all the ideas, expectations and conjectures that are based on the historical "large N picture."

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“…For a single spin, the first term in the Berry phase in Eq. 7 can be written in the following way 59 ,…”

Section: Appendix A: Berry Phasementioning

confidence: 99%

Quantum spin quadrumer

Khatua¹,

Shankar²,

Ganesh³

2018

Phys. Rev. B

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A fundamental motif in frustrated magnetism is the fully mutually coupled cluster of N spins, with each spin coupled to every other spin. Clusters with N = 2 and 3 have been extensively studied as building blocks of square and triangular lattice antiferromagnets. In both cases, large-S semiclassical descriptions have been fruitfully constructed, providing insights into the physics of macroscopic magnetic systems. Here, we develop a semiclassical theory for the N = 4 cluster. This problem has rich mathematical structure with a ground state space that has non-trivial topology. We show that ground states are appropriately parametrized by a unit vector order parameter and a rotation matrix. Remarkably, in the low energy description, the physics of the cluster reduces to that of an emergent free spin-S spin and a rigid rotor. This successfully explains the spectrum of the quadrumer and its associated degeneracies. However, this mapping does not hold in the vicinity of collinear ground states due to a subtle effect that arises from the non-manifold nature of the ground state space. We demonstrate this by an analysis of soft fluctuations, showing that collinear states have a larger number of soft modes. Nevertheless, as these singularities only occur on a subset of measure zero, the mapping to a spin and a rotor provides a good description of the quadrumer. We interpret thermodynamic properties of the quadrumer that are accessible in molecular magnets, in terms of the rotor and spin degrees of freedom. Our study paves the way for field theoretic descriptions of systems such as pyrochlore magnets.

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

Cited by 8 publications

References 18 publications

θ renormalization, electron–electron interactions and super universality in the quantum Hall regime

θ renormalization, electron–electron interactions and super universality in the quantum Hall regime

Super Universality of the Quantum Hall Effect and the “Large N Picture” of the ϑ Angle

Quantum spin quadrumer

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