Lessons From Random Matrix Theory for QCD at Finite Density

Splittorff, K.; Verbaarschot, J. J. M.

doi:10.1142/9789812838667_0014

Cited by 4 publications

(3 citation statements)

References 72 publications

(83 reference statements)

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“…More importantly, it suggests that the sign problem will be milder when the quark mass is larger than a certain critical value, set by the width of the quenched support. An essentially identical conclusion was reached for the three-colour RMT (see, e.g., [44] for a review), again despite the difference from our two-colour model. Next, we turn to the plot of ρ (N f =2,C) w (ξ;m 1 ,m 2 ) from eq.…”

Section: Re[ξ]supporting

confidence: 78%

Random matrix theory of unquenched two-colour QCD with nonzero chemical potential

Akemann

Kanazawa

Phillips

et al. 2011

J. High Energ. Phys.

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We solve a random two-matrix model with two real asymmetric matrices whose primary purpose is to describe certain aspects of quantum chromodynamics with two colours and dynamical fermions at nonzero quark chemical potential µ. In this symmetry class the determinant of the Dirac operator is real but not necessarily positive. Despite this sign problem the unquenched matrix model remains completely solvable and provides detailed predictions for the Dirac operator spectrum in two different physical scenarios/limits: (i) the ε-regime of chiral perturbation theory at small µ, where µ 2 multiplied by the volume remains fixed in the infinite-volume limit and (ii) the high-density regime where a BCS gap is formed and µ is unscaled. We give explicit examples for the complex, real, and imaginary eigenvalue densities including N f = 2 non-degenerate flavours. Whilst the limit of two degenerate masses has no sign problem and can be tested with standard lattice techniques, we analyse the severity of the sign problem for non-degenerate masses as a function of the mass split and of µ.On the mathematical side our new results include an analytical formula for the spectral density of real Wishart eigenvalues in the limit (i) of weak non-Hermiticity, thus completing the previous solution of the corresponding quenched model of two real asymmetric Wishart matrices.

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Section: Re[ξ]supporting

confidence: 78%

Random matrix theory of unquenched two-colour QCD with nonzero chemical potential

Akemann

Kanazawa

Phillips

et al. 2011

J. High Energ. Phys.

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“…In particular, we expect this for dense QCD-like theories with positivity-breaking external sources. 61 Indeed, in N f = 2 two-color QCD, j R = j L at θ = 0 is equivalent to j R = −j L at θ = π, and the similarity to the case considered above is evident. It would be interesting to study the impact of the sign problem on the singular value spectrum in more detail in analogy to the analysis of [108,109].…”

Section: C22 the Microscopic Limitmentioning

confidence: 64%

Singular values of the Dirac operator in dense QCD-like theories

Kanazawa

Wettig

Yamamoto

2011

J. High Energ. Phys.

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We study the singular values of the Dirac operator in dense QCD-like theories at zero temperature. The Dirac singular values are real and nonnegative at any nonzero quark density. The scale of their spectrum is set by the diquark condensate, in contrast to the complex Dirac eigenvalues whose scale is set by the chiral condensate at low density and by the BCS gap at high density. We identify three different low-energy effective theories with diquark sources applicable at low, intermediate, and high density, together with their overlapping domains of validity. We derive a number of exact formulas for the Dirac singular values, including Banks-Casher-type relations for the diquark condensate, Smilga-Stern-type relations for the slope of the singular value density, and Leutwyler-Smilga-type sum rules for the inverse singular values. We construct random matrix theories and determine the form of the microscopic spectral correlation functions of the singular values for all nonzero quark densities. We also derive a rigorous index theorem for non-Hermitian Dirac operators. Our results can in principle be tested in lattice simulations.Comment: 3 references added, version published in JHE

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“…This feature enables us to determine the magnitude of the chiral condensate in the QCD vacuum with considerable accuracy by matching the numerical results from lattice QCD simulations against the exact analytical results obtained in ChRMT [7]. In the past few years considerable progress has also been made in ChRMT at µ = 0 (see [8] for a review), but the high-density region µ Λ QCD is largely unexplored so far.…”

Section: Introductionmentioning

confidence: 96%

Chiral Lagrangian and spectral sum rules for two-color QCD at high density

Kanazawa¹,

Wettig²,

Yamamoto³

2010

Proceedings of the XXVII International Symposium on Lattice Field Theory — PoS(LAT2009)

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We report on our analytical study of two-color QCD with an even number of flavors at high baryon density. Based on the pattern of chiral symmetry breaking induced by BCS-type diquark pairing we construct the low-energy effective Lagrangian for the Nambu-Goldstone bosons. We also identify a new epsilon-regime at high baryon density and derive Leutwyler-Smilga-type spectral sum rules for the complex eigenvalues of the Dirac operator in terms of the fermion gap. Our results can in principle be tested in lattice QCD simulations.

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Lessons From Random Matrix Theory for QCD at Finite Density

Cited by 4 publications

References 72 publications

Random matrix theory of unquenched two-colour QCD with nonzero chemical potential

Random matrix theory of unquenched two-colour QCD with nonzero chemical potential

Singular values of the Dirac operator in dense QCD-like theories

Chiral Lagrangian and spectral sum rules for two-color QCD at high density

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