1 / 20

Hank Thacker University of Virginia

Melting Instantons, Domain Walls, and Large N. Hank Thacker University of Virginia. Lattice2008, Williamsburg, VA. References: J. Lenaghan, S. Ahmad, and HT Phys.Rev. D72:114511 (2005) Y. Lian and HT, Phys. Rev. D75:065031 (2007),

azize
Télécharger la présentation

Hank Thacker University of Virginia

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Melting Instantons, Domain Walls, and Large N Hank Thacker University of Virginia Lattice2008, Williamsburg, VA References: J. Lenaghan, S. Ahmad, and HT Phys.Rev. D72:114511 (2005) Y. Lian and HT, Phys. Rev. D75:065031 (2007), P. Keith-Hynes and HT, arXiv:0804.1534 [hep-lat] Phys. Rev. D (2008).

  2. CP(N-1) models on the lattice Here z = N-component scalar, and U = U(1) gauge field Following the approach of the Kentucky-Virginia collaboration in QCD, we studied the topological charge distribution in CPN-1 using q(x) constructed from overlap Dirac operator. (Overlap construction of q is crucial here.) Results: -- for CP1 and CP2, TC distributions are dominated by small instantons with locally quantized topological charge -- For CP3 and above, distributions are dominated by extended, coherent codimension 1 membranes (very similar to the results of Horvath, et. al in 4D SU(3) gauge theory). For N>4, no evidence for instantons or locally quantized topological charge.

  3. CPN-1 instantons from overlap topological charge: CP1, beta=1.6, Q = 1

  4. CP1, beta=1.6, Q = -2

  5. CPN-1 instantons from overlap topological charge: CP2, beta=1.8, Q = 1

  6. CP9, beta=0.9, Q = -1

  7. Plot sign(q(x)) for CP5 config: “Backbone” of coherent 1D regions is only 1 to 2 sites thick (~range of nonultralocality). Positive and negative regions everywhere close.

  8. Plot integrated q(x) in highest structure (within 2 sites of highest peak) for all configs with Q = +-1 cp1 cp2 cp3 cp5 cp9

  9. Horvath, et al (2003): vacuum of 4D SU(3) gauge theory dominated by layered codimension one sheets of topological charge. 2D slice of Q(x) distribution for 4D QCD Note: Topological charge distributed more-or-less uniformly throughout membrane, not concentrated in localized lumps. (Horvath, et al, Phys.Lett. B(2005) )

  10. large Nc instantons Theta dependence, discrete vacua, and domain walls: Witten (1979): In both 2D CPN-1 and 4D QCD, large-Nc arguments require a phase transition (cusp) at Contradicts instanton expansion, which gives smooth - dependence. . = (free energy) transition between discrete “k-vacua” Large Nc behavior obtained from chiral Lagrangian arguments (Witten, 1979) Confirmed by AdS/CFT duality (Witten, 1998)

  11. CP1 CP5 CP9 large N Instanton gas

  12. Behavior at large theta: Instantons vs. large N Write partition function as a sum over integer valued TC sectors: For dilute instanton gas which gives But for large N, global TC fluctuations are gaussian Doing a Poisson transformation on the sum over we get giving large N result

  13. --This shows that the sum over global winding number is dual to the sum over discrete k-vacua in the sense of Poisson resummation (local quantization of winding number replaced by steps in theta localized to domain walls). --The deviation of the instanton gas from quadratic behavior at large theta reflects the deviation from gaussian winding number fluctuations imposed by local quantization. --The large N “topological sandwich” vacuum delocalizes the topological charge onto extended membranes. This relaxes the constraints of local quantization and leads (for sufficiently large N) to purely gaussian fluctuations.

  14. -vacuum Wilson Loops and Domain Walls in 2D U(1) theories: On an open 2D surface with boundary, a theta term is equivalent to a Wilson loop of charge around the boundary Wilson loop (charge= ) = TC density in 2D So Wilson loop is a boundary between vacua. For CPN-1 can be obtained from area law for fractionally charged Wilson loops (P. Keith-Hynes and HT, arXiv:0804.1534 [hep-lat], Phys. Rev. D, 2008 and PKH talk at this conference.)

  15. Two mechanisms for confinement of fractional charge: Instantons Large N Same vacuum inside and outside. Instantons are invisible to integer charged loop because . Fractional charge confined by random phases. Discrete vacua. Inside vacuum has units of background electric flux and energy . Flux string is quasi-stable even for where true vacuum = broken string.

  16. Static Potential for Integer Charged Loop: CP1 CP5 CP9

  17. Precise analogy between CPN-1 models in 2D and QCD in 4D (Luscher, 1978): • Identify Chern-Simons currents for the two theories. In both cases, CS current correlator has massless pole ~1/q2 This analogy suggests that the coherent 1D structures in CPN-1 are charged particle world lines, and the 3D coherent structures in QCD are Wilson bags=excitation of Chern-Simons tensor on a 3-surface.

  18. A semiclassical estimate of the instanton melting point (Luscher, 1982) In both CPN-1 and QCD, classical instantons come in all sizes. In a semiclassical calculation, the form of the integral over instanton radius is dictated by asymptotic freedom: Monte Carlo results (Lian and HT, PRD(2007)) suggest that instantons “melt” at a value of N above which large instantons are favored over small ones, leading to delocalization of TC. The tipping point of the semiclassical integral gives an estimate (actually a lower bound) for the instanton melting point N=Ncrit of (Lattice results for CPN-1 give Ncrit between 3 and 4, but note Luscher and Petcher, NPB (1983))

  19. Conclusion: A heuristic view of the transition from instantons to the large N topological sandwich: -- For sufficiently small N, small instantons with dominate and contribute a positive contact term to the 2-point correlator (respecting negativity for x>0). -- As N is increased, instantons reach the point where they are no longer happy to be small, and they start to grow. -- Large instantons are not allowed to dominate due to the negativity of the 2-point topological charge correlator -- The growing instanton solves this problem by becoming a thin hollow “Wilson bag”. This shell is screened by an anti-bag of negative topological charge (analog of string-breaking), maintaining the negativity of the correlator. -- There is then no force between opposite walls of the screened bag and it expands indefinitely. Continuation of this process leads to layered, alternating sign membranes of topological charge filling the vacuum, as seen in the Monte Carlo configurations for both CPN and QCD.

More Related