NTU, Apr. 9, 2010
From
Random Walks
Non-Lorentz Invariant to
Field Theories
Susanne Reffert IPMU
based on arXiv:0803.1927, 0903.0732, 0905.0301, 0908.4429 with Robbert Dijkgraaf and Domenico Orlando
NTU, Apr. 9, 2010
Overview talk (~ 2 years of work)
one underlying idea
many realizations
generalize the random walk and use it to make non-rel. (D+1)dim
QFTs from D-dim ones
NTU, Apr. 9, 2010
Historical Picture
quantum dimer
quantum crystal
random partitions/
XXZ spin chain
stochastic quantization
0d FT/ sQM free boson/
quantum Lifshitz model
Hořava-Lifshitz gravity
Feynman rules for q Lifshitz model
full spectrum of XXZ spin chain
NTU, Apr. 9, 2010
Outline
•
Motivation: Why non-Lorentz invariant field theories?•
Introduction: Stochastic Quantization•
Where we go from here•
Examples, examples!•
Quantum Dimer•
Quantum Crystal•
Quantum Lifshitz Model•
Hořava-Lifshitz gravity•
Summary and ConclusionsNTU, Apr. 9, 2010
Non-Lorentz invariant
field theories
NTU, Apr. 9, 2010
Non-Lorentz invariant field theories
Lorentz invariance may not be a fundamental symmetry (absent at high energies)
It should appear as an emergent symmetry at long distances.
Non-Lorentz inv. (non-relativistic) field theories have an anisotropic scaling between space and time:
t → λzt, x → λx
z - dynamical critical exponent non-relativistic z ≠ 1.
NTU, Apr. 9, 2010
Non-Lorentz invariant field theories
Non-Lorentz inv. quantum field theories turn up in a
number of contexts: ultracold atoms at unitarity, nucleon scattering in some channels, a universality class of quantum critical behavior.
Solid state physics: (smectic phase of) liquid crystals Quantum Lifshitz model: (2+1)d, z=2
Hořava-Lifshitz gravity: candidate quantum field theory of gravity, (3+1)d, z=3
AdS/CFT: gravity duals of non-relativistic QFTs
Ardonne, Fendley, Fradkin
Hořava
Balasubramanian, McGreevy; Kachru, Liu, Mulligan
NTU, Apr. 9, 2010
Stochastic Quantization
NTU, Apr. 9, 2010
Stochastic Quantization
Stochastic variable X - range of values x and probability distribution P(x), describes random fluctuations of heat reservoir BG Stochastic process - process depending on time and a stochastic variable
Markov process - stoch. process in which P(t) only depends on its state at t-δt
Simplest Markov process: Brownian motion
Particle of mass m moving in liquid with friction coeff. α
m d
dt�v = −α�v(t) + �η(t)
stochastic force vector (collisions)
NTU, Apr. 9, 2010
Stochastic Quantization
Go from D-dim. to (D+1)-dim. field theory.
Time evolution of the quantized field is governed by a stochastic differential equation.
Stochastic quantization is a quantization scheme for quantizing a D-dim Euclidean field theory.
Introduce a new, fictitious time direction. Parisi, Wu
d
dtφ(x, t) = −δScl[φ]
δφ + η(x, t)
Langevin equation:
�η(x, t)� = 0 , �η(x1, t1)η(x2, t2)� = 2 δ(t1 − t2)δd(x1 − x2) . η(x, t) white Gaussian noise:
NTU, Apr. 9, 2010
Stochastic Quantization
φη(x, t)
Solve Langevin equation with initial condition →
In the limit of (thermal equilibrium), the d-dim.
Green’s functions are recovered.
t → ∞
tlim→∞�φη(x1, t) . . . φη(xk, t)�η = �φ(x1) . . . φ(xk)�
In general: expect supersymmetric, but non-Lorentz invariant quantum theory in (D+1) dimensions.
Supersymmetric in new t-direction!
�φη(x1, t1) . . . φη(xk, tk)�η =
� Dη exp�
−12 �
ddxdt η2(x, t)�
φη(x1, t1) . . . φη(xk, tk)
� Dη exp�
−12 �
ddx dt η2(x, t)�
Parisi, Sourlas
NTU, Apr. 9, 2010
Stochastic Quantization
Alternative approach: Fokker-Planck equation
In the limit of (thermal equilibrium), t → ∞
∂
∂tP [φ, t] =
�
ddx δ
δφ(x, t)
� δScl
δφ(x, t) + δ δφ(x, t)
�
P [φ, t]
The FP equation is a so-called Master equation
tlim→∞ P [φ, t] = Peq[φ] = e−Scl[φ]
� Dφ e−Scl[φ]
H[φ] = −1 2
δ2
δφ2 + 1 8
�δScl δφ
�2
− 1 4
δ2Scl δφ2
Hamiltonian formulation:
ψ0[φ] = e−Scl[φ]/2
Zero energy ground state
NTU, Apr. 9, 2010
Where we go from here
NTU, Apr. 9, 2010
What we are doing in the following is not stochastic quantization.
Where we go from here
We do not go to the limit of infinite time but take the new time direction seriously.
We consider the supersymmetric theory in (D+1) dimensions and its time evolution.
This resulting theory is not topological.
I am using the approach of stochastic quantization as a
factory that produces a variety of non-relativistic quantum field theories.
NTU, Apr. 9, 2010
Examples, examples!
NTU, Apr. 9, 2010
The quantum dimer
NTU, Apr. 9, 2010
Dimer Model
Graph G, nodes can be colored black and white such that
back nodes are only joined by edges to white nodes and vice versa: bipartite.
Let M be a subset of the set E of edges of G. M is called a matching, if its elements are links and no two of them are adjacent.
Link which joins a black and a white vertex: dimer.
If every vertex of G is saturated under M: perfect matching.
NTU, Apr. 9, 2010
Dimer model: Statistical mechanics of a system of random perfect matchings. In the simplest case: number of close packed dimer configurations (perfect matchings)?
Dimer Model
1 plaquette flip Elementary move: plaquette flip
NTU, Apr. 9, 2010
Quantum Dimer
Rokhsar, Kivelson
Resonant valence bond model (candidate for high temperature superconductivity)
Stochastic system. Dynamics obey master equation:
d
dt Pα(t) = �
β β�=α
(WαβPβ(t) − WβαPα(t))
Pα(0) = 1
Z e−gH(α) Z = �
α
e−gH(α)
WβαPα(0) = WαβPβ(0).
Unique stationary distribution:
Satisfies the detailed balance condition
Detailed balance implies the Markov property
class. Hamiltonian
NTU, Apr. 9, 2010
Quantum Dimer
Zero energy ground state: equal weighted sum over all perfect matchings.
plaquette flips counting of flippable plaquettes
kinetic term potential term
| ground � = �
α
| α �
Perfect matchings form basis for Hilbert space
NTU, Apr. 9, 2010
The quantum crystal
NTU, Apr. 9, 2010
The melting crystal revisited
One-to-one correspondence between perfect matchings on the hexagonal lattice and configurations of melting crystal via rhombus tiling.
NTU, Apr. 9, 2010
Statistical system of a melting crystal corner.
The melting crystal revisited
Obeys crystal melting rules.
NTU, Apr. 9, 2010
=1+ q+
...
q +
3q +
2q +
2q +
2Zcr = !
3d partitions
q#boxes =
"∞ n=1
1
(1 − qn)n q = e−1/T
Partition function is given by the MacMahon function:
The melting crystal revisited
NTU, Apr. 9, 2010
Correspondence to topological string A-model.
The partition function on C corresponds to the one of the melting crystal corner.
3
q = e−gs
Iqbal, Nekrasov, Okounkov, Pandharipande, Reshetikhin,Vafa
3d partitions quantum Kähler structures, localize to toric blow-ups of C3
Combinatorial interpretation for the topological vertex.
Aganagic, Klemm, Marino, Vafa
The melting crystal revisited
Generalization to all toric varieties.
Mozgovoy, Reineke; Ooguri , Yamazaki
NTU, Apr. 9, 2010
Quantum crystal melting
Quantize the statistical mechanical system. Stochastic system.
Hilbert space is spanned by classical configurations. Construct the quantum Hamiltonian such, that the ground state
reproduces the classical steady state distribution (Boltzmann).
Take the state graph:
The Hamiltonian is the q-analog of the Laplacian of the state graph!
NTU, Apr. 9, 2010
!ground|ground" = !
α
q#boxes = Ztop
|ground! = !
α
qN (α)/2|α!
potential
H|ground! = 0 H = −J
!
"
|!"#"| + |""#!| − " √
q|""#"| + 1
√q |!"#!|
#
Quantum crystal melting
adding a box
removing a box kinetic
counting
NTU, Apr. 9, 2010
Discrete model: random walk on state graph (box configurations are minima of the classical action).
Analogous to random movement of fields induced by Gaussian noise in stochastic quantization.
Quantum crystal melting
•
Markov property underlies both•
both obey a master equation (SQ: Fokker Planck)•
ground states and stationary distributions are the sameWhy are stochastic quantization and the quantization for the discrete models the same?
NTU, Apr. 9, 2010
The free boson
NTU, Apr. 9, 2010
Free Boson
Scld[ϕ] = κ 2
� ddx �
∂iϕ(xi) ∂iϕ(xi)�
i = 1, 2, . . . , d .
Consider the stochastic quantization of a free boson.
∂tϕ(t, xi) = κ
2 ∂i∂iϕ(t, xi) + η(t, xi) .
Langevin equation:
Z =
�
Dη e−12 R dt ddx η(t,xi)2 .
The partition function is defined by
Z =
�
Dϕ det
� δη δϕ
�����
η=∂tϕ(t,xi)+κ ∂i∂iϕ
e−12 R dt ddx(∂tϕ−κ2 ∂i∂iϕ)2
η ϕ
Change integration variable from to
NTU, Apr. 9, 2010
Free Boson
det
� δη δϕ
�����
η=∂tϕ(t,xi)+κ ∂i∂iϕ
= �
DψD ¯ψ e−R dt ddx ¯ψ(t,xi)(∂t−κ2 ∂i∂i)ψ(t,xi)
Sqd+1[ϕ, ψ, ¯ψ] = −
�
dt ddx � 1
2 (∂tϕ)2 + κ2 8
�∂i∂iϕ�2
− ¯ψ �
∂t − κ
2 ∂i∂i� ψ
�
Introduce fermionic fields to express the Jacobian:
Read off (d+1)-dim. action
Φ(t, xi, θ, ¯θ) = ϕ(t, xi) + ¯θψ(t, xi) + ¯ψ(t, xi)θ + ¯θθF (t, xi)
Introduce superfield for manifestly supersymmetric formulation:
Grassmann variables bosonic auxiliary field
NTU, Apr. 9, 2010
Free Boson
superpotential
Q = ¯ψ �
ıΠϕ − κ
2 ∂i∂iϕ�
, Q = ψ �
−ıΠϕ − κ
2 ∂i∂iϕ� {Q, Q} = 2H
Introduce supercharges
Πϕ = −ı δ δϕ
| Ψ0� = e−κ4 R ddx ∂iϕ∂iϕ| 0�
Q| Ψ0� = Q| Ψ0 � = 0
Bosonic ground state
Annihilated by supercharges
NTU, Apr. 9, 2010
Free Boson
super quantum Lifshitz model
Sqd+1[ϕ, ψ, ¯ψ] = −
� dt ddx � 1
2 (∂tϕ)2 + κ2 8
�∂i∂iϕ�2
− ¯ψ �
∂t − κ
2 ∂i∂i� ψ
�
Ardonne, Fendley, Fradkin
The quantum Lifshitz model describes a quantum critical point (where a continuous phase transition happens at T=0, driven by quantum fluctuations)
The super quantum Lifshitz model is better behaved than the bosonic quantum Lifshitz model and allows generalization to the interacting case.
NTU, Apr. 9, 2010
Feynman rules for the interacting theory
Application: perturbative solution for interacting model
W [φ] =
�
dx� 1
2∂iφ ∂iφ + 1
2m2φ2 + 1
3g2φ3
�
∂ φ(t, x)
∂t = −δW
δφ + η(t, x)
Langevin equation:
Instead of path integral, solve Langevin equation.
Action:
S[φ, ψ, ¯ψ] =
�
dt dx� 1
2 ˙φ2 + (∂i∂iφ)2 + m2∂iφ ∂iφ + g2φ ∂iφ ∂iφ + +m4φ2 + g2m2φ3 + g4φ4 + fermions
�
NTU, Apr. 9, 2010
Feynman rules for the interacting theory
φ(s, k) =
� ∞
0
dt �
dx�
e−ık·x−stφ(t, x)�
Consider the Fourier-Laplace transform:
Ω2 = �
k2 + m2�
Retarded Green’s function is solution of
s G(s, k) = −Ω2G(s, k) + 1
Substituting G and the Fourier-Laplace transformed field back into the Langevin eq., we get an integral equation:
φ(s, k) = G(s, k)η(s, k) − g2G(s, k) �
d[k1, k2, s1, s2]
×
�δ(k − k1 − k2)
s − s1 − s2 φ(s1, k1)φ(s2, k2)
�
+ G(s, k) φ0(k)
NTU, Apr. 9, 2010
Feynman rules for the interacting theory
Solve perturbatively in via Feynman diagramsg2
G(s, k)
η(s, k)
element Fourier-Laplace Fourier
s
k G(s, k) = 1
s + Ω2 G(ω, k) = 1
ıω + Ω2
s k
s� k�
2(2π)2 δ(k + k�)
(s + Ω2) (s� + Ω2) (s + s�)
(2π)2+1 δ(k + k�)δ(ω + ω�) ω2 + Ω4
s1 k1
s2 k2
s3 k3
g2δ(k1 − k2 − k3)
s1 − s2 − s3 g
2δ(k1 − k2 − k3)δ(ω1 − ω2 − ω3)
Table 1: Feynman rules for the cubic theory the field φ can be expanded as
φ = + + + . . . (3.26)
The Feynman rules for this model are summarized in Table 1. Note that even if the action in Eq. (3.21) has two cubic and a quartic interaction, the Feynman diagrams obtained from the Langevin equation only have one cubic vertex. This might at first seem surprising, but is again a simplifying consequence of the detailed balance condition.
3.3 Examples
3.3.1 Three–point function
As a first example, let us consider the three–point function �φ(s1, k1)φ(s2, k2)φ(s3, k3)� at tree level. Expanding the field as in Eq. (3.26), we find that at tree level the function is given by the sum of three contributions:
�φ(s1, k1)φ(s2, k2)φ(s3, k3)� = s1
k1
s2 k2
s3 k3
+ s1
k1
s2 k2
s3 k3
+ s1
k1
s2 k2
s3 k3
(3.27)
Using the Feynman rules in Table 1, it is immediate to find that each diagram gives a contri- bution of
−
� dsadsb 4π2
�
4g2
(si+Ω2i)�sj+Ω2j��
sa+Ω2j�
(sj+sa)(sk+Ω2k)(sb+Ω2k)(sk+sb)(si−sa−sb)
�
, (3.28)
9 φ(s, k) = G(s, k)η(s, k) − g2G(s, k) �
d[k1, k2, s1, s2]
×
�δ(k − k1 − k2)
s − s1 − s2 φ(s1, k1)φ(s2, k2)
�
+ G(s, k) φ0(k)
NTU, Apr. 9, 2010
Feynman rules for the interacting theory
Feynman rules:
element Fourier-Laplace Fourier
s
k G(s, k) = 1
s +Ω2 G(ω, k) = 1
ıω +Ω2 s
k
s� k�
2(2π)2δ(k + k�)
(s +Ω2) (s� + Ω2) (s +s�)
(2π)2+1 δ(k +k�)δ(ω +ω�) ω2 +Ω4
s1 k1
s2 k2 s3 k3
g2δ(k1 − k2 −k3)
s1 −s2 − s3 g2δ(k1 − k2 −k3)δ(ω1 −ω2 − ω3)
Table 1: Feynman rules for the cubic theory
the field φ can be expanded as
φ = + + + . . . (3.26)
The Feynman rules for this model are summarized in Table 1. Note that even if the action in Eq. (3.21) has two cubic and a quartic interaction, the Feynman diagrams obtained from the Langevin equation only have one cubic vertex. This might at first seem surprising, but is again a simplifying consequence of the detailed balance condition.
3.3 Examples
3.3.1 Three–point function
As a first example, let us consider the three–point function �φ(s1, k1)φ(s2, k2)φ(s3, k3)� at tree level. Expanding the field as in Eq. (3.26), we find that at tree level the function is given by the sum of three contributions:
�φ(s1, k1)φ(s2, k2)φ(s3, k3)� = s1
k1
s2 k2
s3 k3
+ s1
k1
s2 k2
s3 k3
+ s1
k1
s2 k2
s3 k3
(3.27)
Using the Feynman rules in Table 1, it is immediate to find that each diagram gives a contri- bution of
−
� dsadsb 4π2
�
4g2
(si+Ω2i)�sj+Ω2j��
sa+Ω2j�
(sj+sa)(sk+Ω2k)(sb+Ω2k)(sk+sb)(si−sa−sb)
�
, (3.28)
9
