Correlations of quantum curvature and variance of Chern numbers

We analyse the correlation function of the quantum curvature in complex quantum systems, using a random matrix model to provide an exemplar of a universal correlation function. We show that the correlation function diverges as the inverse of the distance at small separations. We also define and analyse a correlation function of mixed states, showing that it is finite but singular at small separations. A scaling hypothesis on a universal form for both types of correlations is supported by Monte-Carlo simulations. We relate the correlation function of the curvature to the variance of Chern integers which can describe quantised Hall conductance.

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A soluble random-matrix model for relaxation in quantum systems

Journal of Statistical Physics ◽

10.1007/bf01015321 ◽

1988 ◽

Vol 51 (1-2) ◽

pp. 77-94 ◽

Cited By ~ 16

Author(s):

Pier A. Mello ◽

Pedro Pereyra ◽

Narendra Kumar

Keyword(s):

Matrix Model ◽

Random Matrix ◽

Quantum Systems ◽

Random Matrix Model

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Worldvolume approach to the tempered Lefschetz thimble method

Progress of Theoretical and Experimental Physics ◽

10.1093/ptep/ptab010 ◽

2021 ◽

Author(s):

Masafumi Fukuma ◽

Nobuyuki Matsumoto

Keyword(s):

Molecular Dynamics ◽

Monte Carlo ◽

Matrix Model ◽

Random Matrix ◽

Gradient Flow ◽

Monte Carlo Algorithm ◽

Hybrid Monte Carlo ◽

Random Matrix Model ◽

Sign Problem ◽

Multimodal Problems

Abstract As a solution towards the numerical sign problem, we propose a novel Hybrid Monte Carlo algorithm, in which molecular dynamics is performed on a continuum set of integration surfaces foliated by the antiholomorphic gradient flow (“the worldvolume of an integration surface”). This is an extension of the tempered Lefschetz thimble method (TLTM), and solves the sign and multimodal problems simultaneously as the original TLTM does. Furthermore, in this new algorithm, one no longer needs to compute the Jacobian of the gradient flow in generating a configuration, and only needs to evaluate its phase upon measurement. To demonstrate that this algorithm works correctly, we apply the algorithm to a chiral random matrix model, for which the complex Langevin method is known not to work.

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Extremal correlators and random matrix theory

Journal of High Energy Physics ◽

10.1007/jhep04(2021)214 ◽

2021 ◽

Vol 2021 (4) ◽

Author(s):

Alba Grassi ◽

Zohar Komargodski ◽

Luigi Tizzano

Keyword(s):

Field Theory ◽

Gauge Theory ◽

Matrix Model ◽

Effective Field Theory ◽

Random Matrix ◽

Correlation Functions ◽

Matrix Theory ◽

Effective Field ◽

Random Matrix Model ◽

Large Charge

Abstract We study the correlation functions of Coulomb branch operators of four-dimensional $$ \mathcal{N} $$ N = 2 Superconformal Field Theories (SCFTs). We focus on rank-one theories, such as the SU(2) gauge theory with four fundamental hypermultiplets. “Extremal” correlation functions, involving exactly one anti-chiral operator, are perhaps the simplest nontrivial correlation functions in four-dimensional Quantum Field Theory. We show that the large charge limit of extremal correlators is captured by a “dual” description which is a chiral random matrix model of the Wishart-Laguerre type. This gives an analytic handle on the physics in some particular excited states. In the limit of large random matrices we find the physics of a non-relativistic axion-dilaton effective theory. The random matrix model also admits a ’t Hooft expansion in which the matrix is taken to be large and simultaneously the coupling is taken to zero. This explains why the extremal correlators of SU(2) gauge theory obey a nontrivial double scaling limit in states of large charge. We give an exact solution for the first two orders in the ’t Hooft expansion of the random matrix model and compare with expectations from effective field theory, previous weak coupling results, and we analyze the non-perturbative terms in the strong ’t Hooft coupling limit. Finally, we apply the random matrix theory techniques to study extremal correlators in rank-1 Argyres-Douglas theories. We compare our results with effective field theory and with some available numerical bootstrap bounds.

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Fixed-phase correlation-function quantum Monte Carlo calculations for ground and excited states of helium in neutron-star magnetic fields

Physical Review A ◽

10.1103/physreva.87.032515 ◽

2013 ◽

Vol 87 (3) ◽

Cited By ~ 9

Author(s):

Dirk Meyer ◽

Sebastian Boblest ◽

Günter Wunner

Keyword(s):

Monte Carlo ◽

Correlation Function ◽

Neutron Star ◽

Magnetic Fields ◽

Excited States ◽

Quantum Monte Carlo ◽

Phase Correlation ◽

Monte Carlo Calculations ◽

Fixed Phase

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Chern Numbers and Green's Functions in Solid State Physics

International Journal of Modern Physics B ◽

10.1142/s021797929700071x ◽

1997 ◽

Vol 11 (11) ◽

pp. 1389-1410

Author(s):

Xiao-Rong Wu-Morrow ◽

Cecile Dewitt-Morette ◽

Lev Rozansky

Keyword(s):

Periodic Potential ◽

Solid State Physics ◽

Hall Conductance ◽

Energy Eigenvalues ◽

Finite Dimensional ◽

Topological Quantum ◽

Chern Numbers ◽

Density Response ◽

Physical Quantities ◽

Noninteracting Electron

Using the energy Green's function formulation proposed by Niu 1 for particle densities, we construct and clarify the nature of the topological invariant assigned to the Hall conductance in the Hall system of 2-dimensional noninteracting electron gas; we identify this topological quantum number explicitly as the first Chern number of a complex vector bundle over a 2-torus parametrized by the magnetic potential (a1, a2); the fibres are finite dimensional spaces spanned by eigenfunctions of the system with energy eigenvalues below the Fermi energy. Other cases can be treated by a similar procedure, namely, by recognizing that some physical quantities are integrals of curvatures defined on a nontrivial finite dimensional complex bundle. Therefore, in suitable units, they take integer values. We treat, as an example, the electron density response to a dilation of a periodic potential. The integer in this case is the number of Bloch bands. The quantization of the Hall conductance and density response is also shown in the presence of disorder.

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