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arXiv:2112.13868 · Physical review. B./Physical review. B

Superuniversality of dimerized SU(N+M) spin chains

Adrianus M. M. Pruisken, Bimla Danu, Ravi Shankar

Published Apr 6, 2022• 1 citation
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Abstract

We explore the physics of the quantum Hall effect using the Haldane mapping of dimerized $\mathrm{SU}(N+M)$ 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 $\ensuremath{\xi}$ (correlation length). The numerical results for $(N,M)=(3,1),\phantom{\rule{4pt}{0ex}}(2,2),\phantom{\rule{4pt}{0ex}}(5,1)$, and $(7,1)$ indicate that $\ensuremath{\xi}$ 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 $\ensuremath{\vartheta}$ vacuum concept in quantum field theory). Our numerical investigations provide strong evidence for the conjecture of superuniversality which says that the dimerized spin chain quite generally displays all the basic features of the quantum Hall effect, independent of the specific values of $N$ and $M$. 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 $N$ and $M$. This includes not only the Hall conductance but also the ground state energy as well as the correlation length $\ensuremath{\xi}$ with varying values of $\ensuremath{\vartheta}\ensuremath{\sim}\ensuremath{\pi}$.

DOI
10.1103/physrevb.105.155111

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We explore the physics of the quantum Hall effect using the Haldane mapping of dimerized SU(N+M)\mathrm{SU}(N+M)SU(N+M) spin chains, the large NNN expansion, and the density matrix renormalization group technique. We show that while the transition is first order for N+M>2N+M>2N+M>2, the system at zero temperature nevertheless displays a continuously diverging length scale \ensuremathξ\ensuremath{\xi}\ensuremathξ (correlation length). The numerical results for (N,M)=(3,1),(2,2),(5,1)(N,M)=(3,1),\phantom{\rule{4pt}{0ex}}(2,2),\phantom{\rule{4pt}{0ex}}(5,1)(N,M)=(3,1),(2,2),(5,1), and (7,1)(7,1)(7,1) indicate that \ensuremathξ\ensuremath{\xi}\ensuremathξ 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 \ensuremathϑ\ensuremath{\vartheta}\ensuremathϑ vacuum concept in quantum field theory). Our numerical investigations provide strong evidence for the conjecture of superuniversality which says that the dimerized spin chain quite generally displays all the basic features of the quantum Hall effect, independent of the specific values of NNN and MMM. 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 NNN and MMM. This includes not only the Hall conductance but also the ground state energy as well as the correlation length \ensuremathξ\ensuremath{\xi}\ensuremathξ with varying values of \ensuremathϑ\ensuremath∼\ensuremathπ\ensuremath{\vartheta}\ensuremath{\sim}\ensuremath{\pi}\ensuremathϑ\ensuremath∼\ensuremathπ.
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We use the density-matrix renormalization-group (DMRG) algorithm and finite-size scaling to study a supersymmetric (SUSY) spin chain that models plateau transitions in the integer quantum Hall effect. To illustrate the method, we first present results for the S=1 antiferromagnetic spin chain. Different scaling behavior of the dimerization operator is obtained for the Haldane phase, the dimerized phase, and at the critical point separating the two phases. We then investigate the supersymmetric spin chain. As the on-site Hilbert space has infinite dimension, it is non-compact. We present DMRG calculations for supersymmetric truncations of the on-site Hilbert space and also for non-supersymmetric truncations. These two cases scale differently, in accord with a Lieb-Schultz-Mattis type theorem.
The density-matrix renormalization-group technique is employed to show explicitly that S=2S=2S=2 quantum spin chains belong to the same universality class as S=1S=1S=1 spin chains. The dispersion of the single-magnon band for periodic chains is shown to be $\sqrt{{\ensuremath{\Delta}}^{2}+{\ensuremath{\omega}}_{q}^{2}}$, where $\ensuremath{\Delta}$ is the Haldane gap and ${\ensuremath{\omega}}_{q}=v|\mathrm{sin}(\ensuremath{\pi}\ensuremath{-}q)|$ with $v=\ensuremath{\Delta}\ensuremath{\xi}$, $\ensuremath{\xi}$ as the correlation length, $\ensuremath{\pi}\ensuremath{-}q=2\ensuremath{\pi}n/L$, LLL as the chain length, and nnn as an integer. Moreover, the existence of spin-1/21/21/2 topological excitations due to the breaking of one singlet bond is demonstrated by finite-size scaling of correlation functions decaying as $\mathrm{exp}(\ensuremath{-}r/\ensuremath{\xi})/\sqrt{L}$. These stringent numerical tests, as well as the repulsive interactions between magnons, provide strong evidence in favor of the existence of a universality class for integer quantum spin chains conjectured by Haldane.
Quantum Hall inter-plateau transitions are physical exemplars of quantum phase transitions. Near each of these transitions, the measured electrical conductivity scales with the same correlation length and dynamical critical exponents, i.e., the critical points are superuniversal. In apparent contradiction to these experiments, prior theoretical studies of quantum Hall phase transitions within the framework of Abelian Chern-Simons theory coupled to matter found correlation length exponents that depend on the value of the quantum critical Hall conductivity. Here, we use non-Abelian bosonization and modular transformations to theoretically study the phenomenon of superuniversality. Specifically, we introduce a new effective theory that has an emergent U(N)U(N)U(N) gauge symmetry with any N>1N>1N>1 for a quantum phase transition between an integer quantum Hall state and an insulator. We then use modular transformations to generate from this theory effective descriptions for transitions between a large class of fractional quantum Hall states whose quasiparticle excitations have Abelian statistics. We find the correlation length and dynamical critical exponents are independent of the particular transition within a controlled 't Hooft large NNN expansion, i.e., superuniversal! We argue that this superuniversality could survive away from this controlled large NNN limit using recent duality conjectures.
A careful study of the supersymmetric version of Pruisken’s nonlinear σ model for the integer quantum Hall effect is presented. The lattice regularized model is cast in Hamiltonian form by taking the anisotropic limit and interpreting the topological density as an alternating sum of Wess-Zumino terms. It is argued that the relevant large-scale physics of the model is preserved by projection of the quantum Hamiltonian on its sector of degenerate strong-coupling ground states. For values of the Hall conductivity close to e 2 /2h (mod e 2 /h), where a delocalization transition occurs, this yields the Hamiltonian of a quantum superspin chain which is closely related to an anisotropic version of the Chalker-Coddington model. The relation implies that the ratio of magnetic length over potential correlation length is an irrelevant parameter at the transition. The superspin chain resembles a 1d isotropic antiferromagnet with spin 1/2. It has an alternating structure which however permits an invariance under translation by one site. The conductance coefficients of a quantum Hall system with N small contacts translate into N-superspin correlation functions which are governed by conformal invariance. The superspin formalism provides a framework for studying the crossover from classical to quantum percolation. It does not however encompass the frequency-dependent correlations of wave amplitudes at criticality.
In this thesis we study various one-dimensional quantum spin systems with SU(2) and SU(N) symmetry. We investigate the short-distance behavior of the SU(2) Heisenberg model in the limit of large spin and show that there exists an extended regime where perturbation theory, in the form of spin-wave theory, can be successfully applied. The reason for this perturbative regime stems from the asymptotic freedom of the nonlinear sigma model onto which the Heisenberg model can be mapped. When considering observables which respect the rotational invariance of the model, we observe a cancellation of infrared divergences in the perturbative expansions, leading to a meaningful description of correlation functions. We then turn to the study of SU(N) models. Building on the representation theory of the SU(N) group and on the matrix product state (MPS) formalism, we introduce a generic method to construct Affleck-Kennedy-Lieb-Tasaki (AKLT) models having edge states described by any self-conjugate irreducible representation (irrep) of SU(N). A simple example is given by a spin-1 AKLT model having spin-1 edge states. The phase transition between this model and the original AKLT model is shown to be continuous and to correspond to a topological phase transition described by the SU(2) level-1 Wess-Zumino-Witten conformal field theory universality class. In addition we study an SU(3) AKLT model for the 3-box symmetric irrep at each site. We demonstrate that the edge states are adjoint edge irreps, we extract its correlation length and provide a useful construction as an optimal MPS. Finally, we develop a density matrix renormalization group algorithm based on standard Young tableaus and subduction coefficients to make full use of the non-abelian symmetry and to investigate the SU(3) Heisenberg model with 3-box symmetric irrep at each site. We show that the model has a finite gap above the singlet ground state, in agreement with an extension of the Haldane conjecture to SU(3) chains in the fully symmetric irreps. We also argue that there are five branches of elementary excitations living in four different irreps, each of which is gapped.