PHYS 578 Sp14

HW1 Prob 4

Lattice version of k^2

Nearest neighbor

g1@k_D := 4 Sin@k � 2D^2

Next-to-nearest neighbor (needed for this problem)

g2@k_D := 5 � 2 + Cos@2 kD � 6 - 8 Cos@kD � 3

Comparison (blue=NN, red=NNN)

Plot@8g1@kD, g2@kD<, 8k, -Pi, Pi<, AxesLabel ® 8k, g<, PlotStyle ® ThickD

-3 -2 -1 1 2 3

k

1

2

3

4

5

g

Aside: checking Taylor expansions to confirm that g2 is improved

(no k^4 term):

Series@g1@kD, 8k, 0, 6<D

k2

-k4

12

+k6

360

+ O@kD7

Series@g2@kD, 8k, 0, 6<D

k2

-k6

90

+ O@kD7

Functions for imaginary k

Along imaginary axis, showing issue with NNN

Plot@8-g1@I eeD, -g2@I eeD<, 8ee, 0, 2.7<, AxesLabel ® 8"E", g<, PlotStyle ® ThickD

0.5 1.0 1.5 2.0 2.5

E

2

4

6

8

10

12

g

Finding pole positions

Can solve analytically, though get pesky “ConditionalExpression” due to periodicity in k

Solve@g2@kD + w2 � 0, kD

::k ® ConditionalExpressionB

ArcTanB4 - 3 3 - w2 , - -24 + 8 3 3 - w2 + 3 w2 F + 2 Π C@1D, C@1D Î IntegersF>,

:k ® ConditionalExpressionBArcTanB4 - 3 3 - w2 , -24 + 8 3 3 - w2 + 3 w2 F +

2 Π C@1D, C@1D Î IntegersF>, :k ® ConditionalExpressionB

ArcTanB4 + 3 3 - w2 , - -24 - 8 3 3 - w2 + 3 w2 F + 2 Π C@1D, C@1D Î IntegersF>,

:k ® ConditionalExpressionBArcTanB4 + 3 3 - w2 , -24 - 8 3 3 - w2 + 3 w2 F +

2 Π C@1D, C@1D Î IntegersF>>

Copy the solutions by hand:

root1@w2_D = ArcTan@4 - Sqrt@3 H3 - w2LD, Sqrt@-24 + 8 Sqrt@3 H3 - w2LD + 3 w2DD;

root2@w2_D = ArcTan@4 + Sqrt@3 H3 - w2LD, Sqrt@-24 - 8 Sqrt@3 H3 - w2LD + 3 w2DD;

root3@w2_D = ArcTan@4 + Sqrt@3 H3 - w2LD, -Sqrt@-24 - 8 Sqrt@3 H3 - w2LD + 3 w2DD;

root4@w2_D = ArcTan@4 - Sqrt@3 H3 - w2LD, -Sqrt@-24 + 8 Sqrt@3 H3 - w2LD + 3 w2DD;

Trajectory of one of the pole positions (with Im(k)=Re(E)>0 and Re(k)>0)

Im(k) is blue, Re(k) is red

Note that it is real until w^2=3, and then becomes complex

2 soln1.nb

pole1 = PlotA8Im@root1@w2DD, Re@root1@w2DD<,8w2, 0, 4<, PlotRange ® 880, 4<, 8-0.5, 2.7<<,AxesLabel ® 9"w2", "Re,Im k"=, PlotStyle ® 88Blue, Thick<, 8Red, Thick<<E

1 2 3 4

w2

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

Re,Im k

This is the second pole position.

Due to the vagaries of ArcTan it jumps from positive to negative Im(k) at the critical w^2

PlotA8Im@root2@w2DD, Re@root2@w2DD<, 8w2, 0, 4<,AxesLabel ® 9"w2", "Re,Im k"=, PlotStyle ® ThickE

1 2 3 4

w2

-2

-1

1

2

Re,Im k

Third pole is the opposite of the second

soln1.nb 3

PlotA8Im@root3@w2DD, Re@root3@w2DD<, 8w2, 0, 4<,AxesLabel ® 9"w2", "Re,Im k"=, PlotStyle ® ThickE

1 2 3 4

w2

-2

-1

1

2

Re,Im k

Fourth pole is the opposite of the third

PlotA8Im@root4@w2DD, Re@root4@w2DD<, 8w2, 0, 4<,AxesLabel ® 9"w2", "Re,Im k"=, PlotStyle ® ThickE

1 2 3 4

w2

-2.0

-1.5

-1.0

-0.5

Re,Im k

4 soln1.nb

Combined final plot showing pole positions

pole2a =

PlotA8Im@root2@w2DD, Re@root2@w2DD<, 8w2, 0, 3<, PlotRange ® 880, 4<, 8-0.5, 2.7<<,AxesLabel ® 9"w2", "Re,Im k"=, PlotStyle ® 88Purple, Thick<, 8Red, Thick<<E;

pole2b = PlotA8Im@root3@w2DD, Re@root3@w2DD<, 8w2, 3, 4<,PlotRange ® 880, 4<, 8-0.5, 2.7<<, AxesLabel ® 9"w2", "Re,Im k"=,PlotStyle ® 88Blue, Thick<, 8Red, Thick<<E;

Show@pole1, pole2a, pole2bD

1 2 3 4

w2

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

Re,Im k

This is the largest value of E=Im(k) for the “second” pole

root2@0D �� N

0. + 2.63392 ä

Residue at poles

This is the residue, which vanishes at the “critical point” where w^2=3, and

changes sign for the second pole

denomres@ee_D = 8 Sinh@eeD � 3 - Sinh@2 eeD � 3;

Plot@denomres@eeD, 8ee, 0, 2.64<D

0.5 1.0 1.5 2.0 2.5

-6

-4

-2

2

soln1.nb 5

This shows the denominator of the residue for the two poles for 0 < w^2 < 3

PlotA8denomres@Im@root1@w2DDD, denomres@Im@root2@w2DDD<, 8w2, 0, 3<,PlotStyle ® Thick, AxesLabel ® 9"w2"=, PlotLabel ® "Denom. of residue"E

0.5 1.0 1.5 2.0 2.5 3.0

w2

-10

-5

Denom. of residue

For w^2>3, denominator is complex

8root1@4.0D, root2@4.0D, root3@4.0D, root4@4.0D<80.418498 + 2.15639 ä, 0.418498 - 2.15639 ä, -0.418498 + 2.15639 ä, -0.418498 - 2.15639 ä<

denomres@-I 8root1@4.D, root2@4.0D, root3@4.0D, root4@4.0D<D82.05431 + 4.49669 ä, -2.05431 + 4.49669 ä, 2.05431 - 4.49669 ä, -2.05431 - 4.49669 ä<

Propagator for w^2<3

prop@n4_, w2_D := Module@8e1, e2, den1, den2<, e1 = Im@root1@w2DD; e2 = Im@root2@w2DD;den1 = denomres@e1D; den2 = denomres@e2D; Exp@-e1 n4D � den1 + Exp@-e2 n4D � den2D

Recall that this is only the lattice propagator for integer values of n4,

but it is easier to plot for all real values.

Here it is shown for w^2=0.1, 1.0 and 3.0

In all cases the propagator is positive, despite the negative residue for the higher-energy pole.

6 soln1.nb

Plot@8prop@n4, 0.1D, prop@n4, 1D, prop@n4, 2.9D<, 8n4, 0, 3<, PlotStyle ® Thick,

AxesLabel ® 8"n4"<, PlotLabel ® "Propagator for w^2=0.1, 1, 2.9"D

0.5 1.0 1.5 2.0 2.5 3.0

n4

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Propagator for w^2=0.1, 1, 2.9

Propagator for w^2>3

prop2@n4_, w2_D :=

Module@8k1, k2, den1, den2<, k1 = root1@w2D; k2 = root3@w2D; den1 = denomres@-I k1D;den2 = denomres@-I k2D; Exp@I k1 n4D � den1 + Exp@I k2 n4D � den2D

Here is an example of the propagator becoming negative

prop2@3, 20D �� N

-0.0000104015 + 0. ä

Plots of propagator for w^2>3. Hard to see that it oscillates, but it does.

Plot@8prop2@n4, 3.1D, prop2@n4, 10.D, prop2@n4, 20.0D<, 8n4, 0, 5<,PlotStyle ® Thick, AxesLabel ® 8n4<, PlotLabel ® "Propagator for w^2=3.1, 10, 20"D

1 2 3 4 5

n4

0.01

0.02

0.03

0.04

Propagator for w^2=3.1, 10, 20

This shows the oscillation for w^2=10, by multiplying by the exponential envelope

soln1.nb 7

In[1113]:= Plot@prop2@n4, 10D Exp@Im@root1@10DD n4D, 8n4, 0, 5<, PlotStyle ® Thick,

AxesLabel ® 8n4<, PlotLabel ® "Propagator for w^2=10 with exponential removed"D

Out[1113]=

1 2 3 4 5

n4

-0.10

-0.05

0.05

0.10

Propagator for w^2=10 with exponential removed

8 soln1.nb