Hank Thacker University of Virginia

Hank Thacker University of Virginia

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).

Lattice2008, Williamsburg, VA

Melting Instantons, Domain Walls, and Large N

CP(N-1) models on the lattice

Here z = N-component scalar, and U = U(1) gauge field

S N z x U x x z x h cx

*

, ( ) ( , ) ( ) .

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.

CP1, beta=1.6, Q = 1

CPN-1 instantons from overlap topological charge:

CP1, beta=1.6, Q = -2

CP2, beta=1.8, Q = 1

CPN-1 instantons from overlap topological charge:

CP9, beta=0.9, Q = -1

0.00.20.40.60.81.01.2

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Z Data

X D

ata

Y Data

Coherent structure: (CP5 50 50 12 20 . )

-1.5-1.0-0.50.00.51.01.5

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Z Data

X D

ata

Y Data

3D Graph 1

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

“Backbone” of coherent 1D regions is only 1 to 2 sites thick (~range of nonultralocality). Positive and negative regions everywhere close.

Plot sign(q(x)) for CP5 config:

cp1 cp2 cp3

cp5 cp9

Plot integrated q(x) in highest structure (within 2 sites of highest peak) for all configs with Q = +-1

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) )

Horvath, et al (2003): vacuum of 4D SU(3) gauge theory dominated by layered codimension one sheets of topological charge.

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.

.

E( )

large Nc

instantons

Large Nc behavior obtained from chiral Lagrangian arguments (Witten, 1979)

Confirmed by AdS/CFT duality (Witten, 1998)

Theta dependence, discrete vacua, and domain walls:

= (free energy)

2

( cos )1

transition between discrete “k-vacua”

( ) from fractionally charged Wilson loops

CP1 CP5

CP9

/ 2

( )

t

/ 2 / 2

large N

Instanton gas

( )

t

( )

t

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 Z Vt exp( / )2 2

giving large N result

Doing a Poisson transformation on the sum over we get

--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.

( )

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

-vacuum =

Wilson loop (charge= )

Q F d x A dxV C

z z( / ) ( / )2 22

/ 2

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.)

E( )

So Wilson loop is a boundary between vacua. 0 2 and q

/ 2

TC density in 2D

Two mechanisms for confinement of fractional charge:

Same vacuum inside and outside. Instantons are invisible to integer charged loop because . Fractional charge confined by random phases.

ei 1

Discrete vacua. Inside vacuum has units of background electric flux and energy . Flux string is quasi-stable even for where true vacuum = broken string.

/ 2

Instantons Large N

2

Static Potential for Integer Charged Loop:

CP1

CP5

CP9

Precise analogy between CPN-1 models in 2D and QCD in 4D (Luscher, 1978):

Identify Chern-Simons currents for the two theories.A A Tr

j A j A

Q j Q j

CS CS

CS CS

Wilson line integral over 3 - surface ("Wilson bag")

charged particle charged membrane

(= domain wall) (= domain wall)

RSTUVW

B B B B B3

2 [ ]

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.

In both cases, CS current correlator has massless pole ~1/q2

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:

QCD:

d

CP d

N

N N

11

35

1 3

zz:

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

QCD:

N

CP N

crit

Ncrit

12

1121:

(Lattice results for CPN-1 give Ncrit between 3 and 4, but note Luscher and Petcher, NPB (1983))

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.

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