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.
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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 2 / 29
Introduction and motivation 1©
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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 3 / 29
• 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©
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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 4 / 29
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.
Introduction and motivation 2©
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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 4 / 29
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©
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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 5 / 29
• 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−).
Running boundary conditions: trivial example 1©
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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 5 / 29
• 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−).
Running boundary conditions: trivial example 1©
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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 5 / 29
• 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−).
Running boundary conditions: trivial example 1©
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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 5 / 29
• 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−).
Running boundary conditions: trivial example 1©
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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 5 / 29
• 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−).
Running boundary conditions: trivial example 2©
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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 6 / 29
• 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→ ∞):
Running boundary conditions: trivial example 2©
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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 6 / 29
• 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=
ikik−ge−t = T (k; g(t))
where g(t) is the running coupling constant given by
g(t) = ge−t , −∞ < t < ∞.
• Corresponding running boundary conditions are:
ψ(0+) = ψ(0−),
• UV limit t→ ∞ (g→ 0):
• IR limit t→−∞ (g→ ∞):
Running boundary conditions: trivial example 2©
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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 6 / 29
• 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=
ikik−ge−t = T (k; g(t))
where g(t) is the running coupling constant given by
g(t) = ge−t , −∞ < t < ∞.
• Corresponding running boundary conditions are:
ψ(0+) = ψ(0−),
ψ ′(0+)−ψ ′(0−) = g(t)(
ψ(0+)+ψ(0−))
.
• UV limit t→ ∞ (g→ 0):
• IR limit t→−∞ (g→ ∞):
Running boundary conditions: trivial example 2©
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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 6 / 29
• 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=
ikik−ge−t = T (k; g(t))
where g(t) is the running coupling constant given by
g(t) = ge−t , −∞ < t < ∞.
• Corresponding running boundary conditions are:
ψ(0+) = ψ(0−),
ψ ′(0+)−ψ ′(0−) = g(t)(
ψ(0+)+ψ(0−))
.
• UV limit t→ ∞ (g→ 0):
• IR limit t→−∞ (g→ ∞):
Running boundary conditions: trivial example 2©
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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 6 / 29
• 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=
ikik−ge−t = T (k; g(t))
where g(t) is the running coupling constant given by
g(t) = ge−t , −∞ < t < ∞.
• Corresponding running boundary conditions are:
ψ(0+) = ψ(0−),
ψ ′(0+)−ψ ′(0−) = 0.
• UV limit t→ ∞ (g→ 0): Free boundary conditions.
• IR limit t→−∞ (g→ ∞):
Running boundary conditions: trivial example 2©
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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 6 / 29
• 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=
ikik−ge−t = T (k; g(t))
where g(t) is the running coupling constant given by
g(t) = ge−t , −∞ < t < ∞.
• Corresponding running boundary conditions are:
ψ(0+) = ψ(0−),
1g(t)
(
ψ ′(0+)−ψ ′(0−))
= ψ(0+)+ψ(0−).
• UV limit t→ ∞ (g→ 0): Free boundary conditions.
• IR limit t→−∞ (g→ ∞):
Running boundary conditions: trivial example 2©
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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 6 / 29
• 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=
ikik−ge−t = T (k; g(t))
where g(t) is the running coupling constant given by
g(t) = ge−t , −∞ < t < ∞.
• Corresponding running boundary conditions are:
ψ(0+) = ψ(0−),
0 = ψ(0+)+ψ(0−).
• UV limit t→ ∞ (g→ 0): Free boundary conditions.
• IR limit t→−∞ (g→ ∞): Dirichlet boundary conditions.
Running boundary conditions: trivial example 3©
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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 7 / 29
•
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.
Running boundary conditions: trivial example 3©
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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 7 / 29
• 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
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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 8 / 29
• 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• 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
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 9 / 29
Introduction
U(2) Family of Boundary Conditions
Exact RG Flow of Boundary Conditions
Generalization to Quantum Wire Junctions
Summary and Perspective
U(2) Family of BoundaryConditions
Introduction
U(2) Family of BoundaryConditions• U(2) family of boundaryconditions• S-matrix• Boundary bound states• Phase diagram
Exact RG Flow ofBoundary Conditions
Generalization to QuantumWire Junctions
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 10 / 29
U(2) family of boundary conditions 1©
Introduction
U(2) Family of BoundaryConditions• U(2) family of boundaryconditions• S-matrix• Boundary bound states• Phase diagram
Exact RG Flow ofBoundary Conditions
Generalization to QuantumWire Junctions
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 11 / 29
• 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.
U(2) family of boundary conditions 2©
Introduction
U(2) Family of BoundaryConditions• U(2) family of boundaryconditions• S-matrix• Boundary bound states• Phase diagram
Exact RG Flow ofBoundary Conditions
Generalization to QuantumWire Junctions
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 12 / 29
• Derivation of U(2) family of boundary conditions:
j(0+) = j(0−)
U(2) family of boundary conditions 2©
Introduction
U(2) Family of BoundaryConditions• U(2) family of boundaryconditions• S-matrix• Boundary bound states• Phase diagram
Exact RG Flow ofBoundary Conditions
Generalization to QuantumWire Junctions
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 12 / 29
• Derivation of U(2) family of boundary conditions:
j(0+) = j(0−)
⇔ ψ ′∗(0+)ψ(0+)−ψ∗(0+)ψ ′(0+) = ψ ′∗(0−)ψ(0−)−ψ∗(0−)ψ ′(0−)
U(2) family of boundary conditions 2©
Introduction
U(2) Family of BoundaryConditions• U(2) family of boundaryconditions• S-matrix• Boundary bound states• Phase diagram
Exact RG Flow ofBoundary Conditions
Generalization to QuantumWire Junctions
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 12 / 29
• Derivation of U(2) family of boundary conditions:
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−)
)
U(2) family of boundary conditions 2©
Introduction
U(2) Family of BoundaryConditions• U(2) family of boundaryconditions• S-matrix• Boundary bound states• Phase diagram
Exact RG Flow ofBoundary Conditions
Generalization to QuantumWire Junctions
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 12 / 29
• Derivation of U(2) family of boundary conditions:
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)
U(2) family of boundary conditions 2©
Introduction
U(2) Family of BoundaryConditions• U(2) family of boundaryconditions• S-matrix• Boundary bound states• Phase diagram
Exact RG Flow ofBoundary Conditions
Generalization to QuantumWire Junctions
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 12 / 29
• Derivation of U(2) family of boundary conditions:
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))
U(2) family of boundary conditions 3©
Introduction
U(2) Family of BoundaryConditions• U(2) family of boundaryconditions• S-matrix• Boundary bound states• Phase diagram
Exact RG Flow ofBoundary Conditions
Generalization to QuantumWire Junctions
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 13 / 29
• 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
U(2) Family of BoundaryConditions• U(2) family of boundaryconditions• S-matrix• Boundary bound states• Phase diagram
Exact RG Flow ofBoundary Conditions
Generalization to QuantumWire Junctions
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 14 / 29
• 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
U(2) Family of BoundaryConditions• U(2) family of boundaryconditions• S-matrix• Boundary bound states• Phase diagram
Exact RG Flow ofBoundary Conditions
Generalization to QuantumWire Junctions
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 15 / 29
• 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
U(2) Family of BoundaryConditions• U(2) family of boundaryconditions• S-matrix• Boundary bound states• Phase diagram
Exact RG Flow ofBoundary Conditions
Generalization to QuantumWire Junctions
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 16 / 29
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
U(2) Family of BoundaryConditions
Exact RG Flow ofBoundary Conditions• Exact RG flow ofboundary conditions• Symmetry classification
Generalization to QuantumWire Junctions
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 17 / 29
Exact RG flow of boundary conditions 1©
Introduction
U(2) Family of BoundaryConditions
Exact RG Flow ofBoundary Conditions• Exact RG flow ofboundary conditions• Symmetry classification
Generalization to QuantumWire Junctions
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 18 / 29
• 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.
Exact RG flow of boundary conditions 2©
Introduction
U(2) Family of BoundaryConditions
Exact RG Flow ofBoundary Conditions• Exact RG flow ofboundary conditions• Symmetry classification
Generalization to QuantumWire Junctions
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 19 / 29
• 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.
Exact RG flow of boundary conditions 3©
Introduction
U(2) Family of BoundaryConditions
Exact RG Flow ofBoundary Conditions• Exact RG flow ofboundary conditions• Symmetry classification
Generalization to QuantumWire Junctions
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 20 / 29
• 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}
Exact RG flow of boundary conditions 4©
Introduction
U(2) Family of BoundaryConditions
Exact RG Flow ofBoundary Conditions• Exact RG flow ofboundary conditions• Symmetry classification
Generalization to QuantumWire Junctions
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 21 / 29
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
U(2) Family of BoundaryConditions
Exact RG Flow ofBoundary Conditions• Exact RG flow ofboundary conditions• Symmetry classification
Generalization to QuantumWire Junctions
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 22 / 29
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+)
Generalization to QuantumWire Junctions
Introduction
U(2) Family of BoundaryConditions
Exact RG Flow ofBoundary Conditions
Generalization to QuantumWire Junctions• Quantum wire junctions• Example: Exact RG flowof Y-junction
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 23 / 29
Quantum wire junctions
Introduction
U(2) Family of BoundaryConditions
Exact RG Flow ofBoundary Conditions
Generalization to QuantumWire Junctions• Quantum wire junctions• Example: Exact RG flowof Y-junction
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 24 / 29
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
U(2) Family of BoundaryConditions
Exact RG Flow ofBoundary Conditions
Generalization to QuantumWire Junctions• Quantum wire junctions• Example: Exact RG flowof Y-junction
Summary and Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 25 / 29
• 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 1 relevant direction
Fixed point with 2 relevant directions
Arrows indicate the directions toward the infrared.
• Rich phase structure
• There exist 23 = 8 fixed points
Summary and Perspective
Introduction
U(2) Family of BoundaryConditions
Exact RG Flow ofBoundary Conditions
Generalization to QuantumWire Junctions
Summary and Perspective• Summary• Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 26 / 29
Summary
Introduction
U(2) Family of BoundaryConditions
Exact RG Flow ofBoundary Conditions
Generalization to QuantumWire Junctions
Summary and Perspective• Summary• Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 27 / 29
• 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)
Fixed point with 1 relevant direction
• 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
U(2) Family of BoundaryConditions
Exact RG Flow ofBoundary Conditions
Generalization to QuantumWire Junctions
Summary and Perspective• Summary• Perspective
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 28 / 29
• Generalization to spinning particle.
• Generalization to higher-dimensions (2D and 3D).
• Generalization to quantum field theory (very challenging).
• Physical applications (mandatory).
Workshop on Low Dimensional Quantum Systems @ HRI, Allahabad 29 / 29
Thank you for your attention.