renormalization group flow of point defects in one dimensioncmworksh/ldqs/talks/satoshi.pdf ·...

Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 1 / 29

Renormalization Group Flow of Point Defectsin One Dimension

Satoshi OhyaHarish-Chandra Research Institute

October 10, 2011

Based on:SO, M. Sakamoto, M. Tachibana, Prog. Theor. Phys. 125 (2011) 225, arXiv:1005.4676.SO, arXiv:1104.5481.

http://dx.doi.org/10.1143/PTP.125.225

http://arxiv.org/abs/1005.4676

http://arxiv.org/abs/1104.5481

Introduction

Introduction• Introduction andmotivation• Running boundaryconditions: trivial example• Set up

U(2) Family of BoundaryConditions

Exact RG Flow ofBoundary Conditions

Generalization to QuantumWire Junctions

Summary and Perspective


Introduction and motivation 1©







• Slowly moving particle cannot resolve the structure of short-rangescatterers (such as impurities or defects).

• (Much) below the physical cutoff scale a, any short-rangeinteraction could be approximated by a point interaction.

x

Ene

rgy

Scal

e

a

λ�a−−−→

x

short-range interactionlong-wavelength limit−−−−−−−−−−−→ point interaction

(boundary condition)

Introduction and motivation 2©







Question: Do there exist any universality classes of short-range in-teractions whose long-wavelength limits appear to be the same?

a a

same point interaction?

a a

λ�a λ�a

λ�a λ�a

Yes. The universality classes do exist.It is described by running boundary conditions; that is, RG flow ofboundary conditions.

Running boundary conditions: trivial example 1©







• Consider the Hamiltonian H =− d2

dx2 +2gδ (x). As is well known,δ -function potential is described by the boundary conditions:

ψ(0+) = ψ(0−),

ψ ′(0+)−ψ ′(0−) = g(

ψ(0+)+ψ(0−))

.

• Next consider the momentum flow of S-matrix elements:

V (x) = 2gδ (x)

xeikx

R(k)e−ikx

T (k)eikx

−1 k→0←

R(k;g) =g

ik−g

k→∞→ 0

0 k→0←

T (k;g) =ik

ik−g

k→∞→ 1

• IR limit: no transmission, reflection with phase shift π (eiπ =−1)⇒ Dirichlet boundary condition ψ(0+) = 0 = ψ(0−).

• UV limit: no reflection, perfect transmission⇒ Free boundary condition ψ(0+) = ψ(0−) & ψ ′(0+) = ψ ′(0−).








• Consider the Hamiltonian H =− d2

dx2 +2gδ (x). As is well known,δ -function potential is described by the boundary conditions:

ψ(0+) = ψ(0−),

ψ ′(0+)−ψ ′(0−) = g(

ψ(0+)+ψ(0−))

.

• Next consider the momentum flow of S-matrix elements:

V (x) = 2gδ (x)

xeikx

R(k)e−ikx

T (k)eikx −1 k→0← R(k;g) =

gik−g

k→∞→ 0

0 k→0← T (k;g) =

ikik−g

k→∞→ 1

• IR limit: no transmission, reflection with phase shift π (eiπ =−1)⇒ Dirichlet boundary condition ψ(0+) = 0 = ψ(0−).

• UV limit: no reflection, perfect transmission⇒ Free boundary condition ψ(0+) = ψ(0−) & ψ ′(0+) = ψ ′(0−).








• RG argument: Consider momentum rescaling k→ ket

R(ket ;g) =g

iket −g

=ge−t

ik−ge−t = R(k; g(t))

T (ket ;g) =iket

iket −g

=ik

ik−ge−t = T (k; g(t))

• Corresponding running boundary conditions are:

ψ(0+) = ψ(0−),

• UV limit t→ ∞ (g→ 0):

• IR limit t→−∞ (g→ ∞):









R(ket ;g) =g

iket −g=

ge−t


T (ket ;g) =iket

iket −g=

ikik−ge−t = T (k; g(t))

where g(t) is the running coupling constant given by

g(t) = ge−t , −∞ < t < ∞.


ψ(0+) = ψ(0−),

• UV limit t→ ∞ (g→ 0):

• IR limit t→−∞ (g→ ∞):









R(ket ;g) =g

iket −g=

ge−t


T (ket ;g) =iket

iket −g=



g(t) = ge−t , −∞ < t < ∞.


ψ(0+) = ψ(0−),

ψ ′(0+)−ψ ′(0−) = g(t)(

ψ(0+)+ψ(0−))

.

• UV limit t→ ∞ (g→ 0):

• IR limit t→−∞ (g→ ∞):









R(ket ;g) =g

iket −g=

ge−t


T (ket ;g) =iket

iket −g=



g(t) = ge−t , −∞ < t < ∞.


ψ(0+) = ψ(0−),

ψ ′(0+)−ψ ′(0−) = 0.

• UV limit t→ ∞ (g→ 0): Free boundary conditions.

• IR limit t→−∞ (g→ ∞):









R(ket ;g) =g

iket −g=

ge−t


T (ket ;g) =iket

iket −g=



g(t) = ge−t , −∞ < t < ∞.


ψ(0+) = ψ(0−),

1g(t)

(

ψ ′(0+)−ψ ′(0−))

= ψ(0+)+ψ(0−).


• IR limit t→−∞ (g→ ∞):









R(ket ;g) =g

iket −g=

ge−t


T (ket ;g) =iket

iket −g=



g(t) = ge−t , −∞ < t < ∞.


ψ(0+) = ψ(0−),

0 = ψ(0+)+ψ(0−).


• IR limit t→−∞ (g→ ∞): Dirichlet boundary conditions.








•

Whole

RG flow

Dirichlet BC (IR fixed point)

Free BC (UV fixed point)

• Main goal of this talk is to derive the above RG flow diagram.








• Whole RG flow

Dirichlet BC (IR stable fixed point)

Neumann BC (UV stable fixed point)

Fixed point with 1 relevant direction

• Main goal of this talk is to derive the above RG flow diagram.

Set up







• Spinless one-particle quantum mechanics on a line;

• Single localized potential centered at the origin;

• Long-wavelength limit.

⇒ A particle we consider would freely propagate in the bulk yetinteract only at the origin.

Time-independent Schrodinger equation describing this situation mustbe as follows:

Hψ(x) = Eψ(x), H =−d2

dx2 , x 6= 0.

Theory space (space of parameters which characterize all possiblepoint interactions)

=

parameter space of 2-dimensional unitary group U(2)

Plan of the Talk







Introduction

U(2) Family of Boundary Conditions

Exact RG Flow of Boundary Conditions

Generalization to Quantum Wire Junctions



Introduction

U(2) Family of BoundaryConditions• U(2) family of boundaryconditions• S-matrix• Boundary bound states• Phase diagram





U(2) family of boundary conditions 1©

Introduction






• What is the all possible point interactions in quantum mechanics?

• Standard argument: point interactions consistent with probabilityconservation [cf. Cheon et al. (2000)].

• Probability current must be continuous even at the position of pointinteraction:

j(0+) = j(0−),

where

j(x) =−i[

ψ ′∗(x)ψ(x)−ψ∗(x)ψ ′(x)]

.

• Note: j(0+) = j(0−) is equivalent to the requirement of

◦ self-adjointness (hermiticity) of the Hamiltonian H;

◦ unitary time evolution e−iHt of the system.

http://arxiv.org/abs/quant-ph/0008123


Introduction






• Derivation of U(2) family of boundary conditions:

j(0+) = j(0−)


Introduction







j(0+) = j(0−)

⇔ ψ ′∗(0+)ψ(0+)−ψ∗(0+)ψ ′(0+) = ψ ′∗(0−)ψ(0−)−ψ∗(0−)ψ ′(0−)


Introduction







j(0+) = j(0−)

⇔ ψ ′∗(0+)ψ(0+)−ψ∗(0+)ψ ′(0+) = ψ ′∗(0−)ψ(0−)−ψ∗(0−)ψ ′(0−)

⇔

(

ψ ′(0+)−ψ ′(0−)

)†·

(

ψ(0+)ψ(0−)

)

=

(

ψ(0+)ψ(0−)

)†·

(

ψ ′(0+)−ψ ′(0−)

)


Introduction







j(0+) = j(0−)

⇔ ψ ′∗(0+)ψ(0+)−ψ∗(0+)ψ ′(0+) = ψ ′∗(0−)ψ(0−)−ψ∗(0−)ψ ′(0−)

⇔

(

ψ ′(0+)−ψ ′(0−)

)†·

(

ψ(0+)ψ(0−)

)

=

(

ψ(0+)ψ(0−)

)†·

(

ψ ′(0+)−ψ ′(0−)

)

⇔

∣

∣

∣

∣

(

ψ(0+)ψ(0−)

)

− iL0

(

ψ ′(0+)−ψ ′(0−)

)∣

∣

∣

∣

2=

∣

∣

∣

∣

(

ψ(0+)ψ(0−)

)

+ iL0

(

ψ ′(0+)−ψ ′(0−)

)∣

∣

∣

∣

2

(L0: arbitrary length scale)


Introduction







j(0+) = j(0−)

⇔ ψ ′∗(0+)ψ(0+)−ψ∗(0+)ψ ′(0+) = ψ ′∗(0−)ψ(0−)−ψ∗(0−)ψ ′(0−)

⇔

(

ψ ′(0+)−ψ ′(0−)

)†·

(

ψ(0+)ψ(0−)

)

=

(

ψ(0+)ψ(0−)

)†·

(

ψ ′(0+)−ψ ′(0−)

)

⇔

∣

∣

∣

∣

(

ψ(0+)ψ(0−)

)

− iL0

(

ψ ′(0+)−ψ ′(0−)

)∣

∣

∣

∣

2=

∣

∣

∣

∣

(

ψ(0+)ψ(0−)

)

+ iL0

(

ψ ′(0+)−ψ ′(0−)

)∣

∣

∣

∣

2

(L0: arbitrary length scale)

⇔

(

ψ(0+)ψ(0−)

)

− iL0

(

ψ ′(0+)−ψ ′(0−)

)

= U[(

ψ(0+)ψ(0−)

)

+ iL0

(

ψ ′(0+)−ψ ′(0−)

)]

(U ∈U(2))


Introduction






• U(2) family of boundary conditions

(1−U)

(

ψ(0+)ψ(0−)

)

− iL0(1+U)

(

ψ ′(0+)ψ ′(0−)

)

=~0, U ∈U(2).

• Parameterization of U ∈U(2)

U = ∑j=±

eiα j Pj, P± =1±~e ·~σ

2

• 4 independent parameters

α+,α− ∈ [0,2π),

~e = (ex,ey,ez) with e2x + e2

y + e2z = 1

• We wish to know the RG flow of the parameters {α+,α−,ex,ey,ez}.

Physical Quantities: S-matrix

Introduction






• Reflection and transmission coefficients (k > 0)

xeikx

R−(k)e−ikx

T−(k)eikx

xe−ikx

R+(k)eikx

T+(k)e−ikx

• S-matrix

S(k) =

(

R+(k) T−(k)T+(k) R−(k)

)

= ∑j=±

ikL j−1ikL j +1

Pj

where

L± = L0 cotα±2

Physical Quantities: Boundary bound states

Introduction






• Bound (antibound) state =simple pole of S(k) lying onthe positive (negative)imaginary k-axis

• Bound/antibound state energyat a pole k = i/L±:

E± =

(

iL±

)2

=−1

L2±

0 Rek

Imk

×(bound state)

×(antibound state)

• Bound/antibound state wave function at a pole k = i/L±:

ψ±(x) ∝ exp(

−|x|L±

)

(

L± = L0 cotα±2

, L0 > 0)

⇒

{

normalizable bound state for 0 < α± < πnon-normalizable antibound state for π < α± < 2π

Phase diagram

Introduction






2 bound states0 antibound state

0 bound state2 antibound states

1 bound state1 antibound state

1 bound state1 antibound state

0 π 2π

π

2π

α+

α−

Exact RG Flow of BoundaryConditions

Introduction


Exact RG Flow ofBoundary Conditions• Exact RG flow ofboundary conditions• Symmetry classification




Exact RG flow of boundary conditions 1©

Introduction






• Renormalization group transformationSince L0 is an arbitrary parameter, any physical quantities must beindependent of the choice of L0. The lack of dependence of L0 canbe expressed as the invariance of the theory under the RGtransformation

Rt : L0 7→ L(t) := L0e−t , −∞ < t < ∞.

• Renormalization group equationAny change of L0 must be equivalent to changes in the U(2)parameters. This is expressed as the RG equation

S(k;g j,L0) = S(k; g j(t), L(t)),

where g j(t) = {α±(t), e j(t)} are the running U(2) parameters.


Introduction






• Since S(k;g j,L0) does not have t in any way, we must have

∂∂ t

S(k;gi,L0)

∣

∣

∣

∣

gi,L0

= 0 =∂∂ t

S(k; gi(t), L(t))∣

∣

∣

∣

gi,L0

.

• The first equality is trivial.

• But the second equality leads to the following homogeneous RGequation

(

−L∂

∂ L+ ∑

g j=α±,e j

βg j(g j(t))∂

∂ g j

)

S(k; g j(t), L(t)) = 0.

• β -functions are defined by

βg j(g j(t)) =∂ g j(t)

∂ t

∣

∣

∣

∣

g j ,L0

with g j(0) = g j.


Introduction






• Exact β -functions:

βα±(α±(t)) =−sin α±(t),

βe j(e j(t)) = 0, (⇐ exactly marginal).

βα±

α±0

π2πUV limit

(t→+∞)

IR limit(t→−∞)

IR limit(t→−∞)

UV limit(t→+∞)

• α∗± = 0: UV fixed point

• α∗± = π: IR fixed point

• 22 = 4 fixed points on T 2 = {(α+,α−) | 0≤ α± < 2π}


Introduction






α+

α−

0 π 2π

π

2π

Arrows indicate the directionstoward the infrared.

• Neumann fixed point (UVstable fixed point)

U = S = I

ψ ′(0+) = 0 = ψ ′(0−)

• Dirichlet fixed point (IRstable fixed point)

U = S =−I

ψ(0+) = 0 = ψ(0−)

• (0,π) fixed point

U = S =

(

cosϕ e−iθ sinϕeiθ sinϕ −cosϕ

)

ψ(0−) = eiθ tan ϕ2 ψ(0+), ψ ′(0−) = eiθ cot ϕ

2 ψ ′(0+)

Symmetry classification [cf. Cheon et al. (2000)]

Introduction






Marginal parameters (θ & ϕ) do not flow against RG. However, theycan be restricted by symmetry.

• Parity invariance⇒ θ = 0 & ϕ =± π2 :

{

ψ(0−) = ψ(0+)

ψ ′(0−) = ψ ′(0+)or

{

ψ(0−) =−ψ(0+)

ψ ′(0−) =−ψ ′(0+)

• Time-reversal invariance⇒ θ = 0:{

ψ(0−) = tan ϕ2 ψ(0+)

ψ ′(0−) = cot ϕ2 ψ ′(0+)

• PT-symmetry⇒ ϕ = π2 :

{

ψ(0−) = eiθ ψ(0+)

ψ ′(0−) = eiθ ψ ′(0+)

http://arxiv.org/abs/quant-ph/0008123


Introduction



Generalization to QuantumWire Junctions• Quantum wire junctions• Example: Exact RG flowof Y-junction



Quantum wire junctions

Introduction






0 jx1

x2

x j

xN

• Junction of N quantum wires

• Boundary condition at the junction point

(1−U)

ψ(01)...

ψ(0N)

− iL0(1+U)

ψ ′(01)...

ψ ′(0N)

=~0, U ∈U(N)

• Exact β -functions

β (α j(t)) =−sin α j(t), j = 1, · · · ,N

(α j: jth eigenphase of U)

Example: Exact RG flow of Y-junction

Introduction






• Exact RG flow of boundary conditions for N = 3 (Y-junction)

α1

α2

α3

0

π2π

π2π

Dirichlet fixed point

Neumann fixed point


Fixed point with 2 relevant directions

Arrows indicate the directions toward the infrared.

• Rich phase structure

• There exist 23 = 8 fixed points


Introduction




Summary and Perspective• Summary• Perspective


Summary

Introduction






• We derived the exact RG flow of U(2) family of boundaryconditions in the framework of one-particle quantum mechanics:

α+

α−

0 π 2π

π

2π

Dirichlet BC (IR stable fixed point)

Neumann BC (UV stable fixed point)


• There are 3 distinct universality classes of short-range interactions:

(1) If UV theory lies on the critical point (α+,α−) = (0,0), itremains on the Neumann fixed point.

(2) If UV theory lies on the critical line α+ = 0 (α− = 0), it flowsinto the (0,π)-fixed point ((π,0)-fixed point).

(3) All other short-range interactions flow into the Dirichlet fixedpoint in the long-wavelength limit.

Perspective

Introduction






• Generalization to spinning particle.

• Generalization to higher-dimensions (2D and 3D).

• Generalization to quantum field theory (very challenging).

• Physical applications (mandatory).


Thank you for your attention.

renormalization group flow of point defects in one dimensioncmworksh/ldqs/talks/satoshi.pdf ·...

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