The variational principle, relativistic quantum ﬁeld theory, …...• Due to the variational principle trying to ﬁnd the “lowest” ground-state energy. • To lowest order,

Tobias J. Osbornewith Jens Eisert, Ignacio Cirac, Jutho Haegeman, Henri

Verschelde, and Frank Verstraete

The variational principle, relativistic quantum field theory, and holographic

quantum states

http://tjoresearchnotes.wordpress.com

arXiv:1005.1268arXiv:1006.2409



Outline

Outline

• Feynman on the variational procedure and relativistic quantum field theory

Outline


• Holographic quantum states

Outline



• Topological effects

Outline




• UV sensitivity

Outline




• UV sensitivity

• Results

Feynman on the var. principle and RQFT

“... it is no damn good at all!”

R. P. Feynman, Proceedings of the International Workshop on Variational Calculations in Quantum Field

Theory. Wangerooge, West Germany. 1-4 Sept. 1987

Problem 1: UV sensitivity

“... but just a little shift in the behavior of the high frequencies is more important in minimizing the total energy.”

Problem 2: only gaussian trial states

“The second problem that we have in field theory is that we only have Gaussian trial states so far and we need what I like to call extensive wave functions.”

Non gaussian variational classes

• Since 1987 there have been many developments.

• Many non-gaussian trial states developed in the setting of low-dimensional quantum spin systems.

• Now we understand that physics of low-dimensional quantum spin systems can be captured by matrix product states.

Case study: matrix product states

For n distinguishable spin-1/2 particles the matrix product state (MPS) representation is given by:

|ψ =1

j1,...,jn=0

tr(Aj1Aj2 · · ·Ajn)|j1j2 · · · jn

Here the matrices are D by D matrices, where D is the so-called auxiliary or bond dimension.

Ajk

MPSMPS correctly capture the physics of naturally occurring quantum states in strongly interacting one-dimensional quantum spin systems.

Example:

H =n−1

j=1

hj

where

hj = I1···j−1 ⊗ h ⊗ Ij+2···n

MPS: equilibriumTheorem (Hastings). If the spectral gap is

then the (assumed) unique ground state

of H can be well-approximated by an MPS

with

Matthew B. Hastings, JSTAT, P08024 (2007)

∆

|Ω

|Ω

D ∼ poly(n, 1/, e1/∆)

MPS: non-equilibriumTheorem (TJO). If

and

then

can be well-approximated by an MPS with

Tobias J. Osborne, Phys. Rev. Lett. 97, 157202 (2006)

|t| ≤ O(log(n))

|ψ(t) = e itH|ψ(0)

|ψ(t)

D ∼ poly(n, 1/)|ψ(t)

MPS via sequential preparation

• One key physical insight obtained has been that all MPS can be sequentially generated by interacting an auxiliary system B with each spin j in the system.

















Proposal: variational field states via

sequential preparation

ProposalInput: (1) quantum field A in empty vacuum;(2) auxiliary system B in some state

Procedure: discretise A and interact A and B according to:

where and take the limit n = L → 0

H = K ⊗ I+√

n

j=1

δ(t − j)(iR ⊗ a†j + h.c.)

In words:

• Discretise the quantum field A into n pieces, modelled as harmonic oscillators

• Introduce an ancillary system B which evolves according to some internal dynamics

• At time t = j instantaneously interact A and B.

• Take limit n goes to infinity.

Continuous sequential preparation

• We now have a continuous sequential generation procedure

















• We let while holding

• We define where aj is the annihilation operator for the jth mode

• In the limit we obtain commutation relation

The continuum limit

→ 0 n = L

[ψ(x),ψ†(y)] = δ(x − y)

ψj = aj/√

Continuum limit cont.• Refer to the continuum of oscillators now

as “A”. This is a bosonic quantum field.

• The propagator for the overall system is

U(L) = T e−i L0 (K⊗I+iR⊗ψ†(s)−iR⊗ψ(s))ds

When acting on the field vacuum the propagator becomes

U(L)(IB ⊗ |Ω) = T e L0 (Q⊗I+R⊗ψ†(s))ds(IB ⊗ |Ω)

where Q(s) = −iK −1

2R†(s)R(s)

cMPS

• Resulting state of A alone, given by

are called continuum matrix product states (cMPS) - (introduced in arXiv:1002.1824)

• Variational parameters are the 2D2 functions:

σA = trBU(L)(|ΩAΩ|⊗ ρB)U†(L)

[Q(s)]lm [R(s)]lmand

• If we choose O = identity then we can compute by first tracing out the field A.

• The aux. system B undergoes, throughout sequential preparation process, dynamics generated by

• This is a Lindblad equation

A useful device

I

dρ

dx= −i [K, ρ]−

1

2R†R, ρ+ RρR†

Expectation values

• Therefore we can compute field expectation values by integrating Lindblad equation with additional insertions of R at the locations of and

• Derivatives of field operators can be dealt with similarly

• For a D-dimensional aux. system this is a finite computation.

ψ(x)

Expectation values can be calculated by studying B alone

A holographic principle for cMPS

Holography

Holography

• Suppose a quantum field is in a cMPS

Holography

• Suppose a quantum field is in a cMPS

• Any expectation value can be computed by studying the dynamics of a “boundary” dissipative theory of a lower geometric dimension.

Higher dimensions

!

"

!

!

Topological effectsB

Ar r

What is known

What is known

• cMPS yield a good non-gaussian variational class for non-relativistic theories: compares extremely well with analytic results for, eg., Lieb-Liniger model.

Thus we have addressed Feynman’s

gaussian criticism

The other problem: UV sensitivity

UV sensitivity• Due to the variational principle trying to find the

“lowest” ground-state energy.

• To lowest order, the ground state of a relativistic quantum field consists of the zero-point oscillations of all energy scales.

• Dominated by the infinite availability of high frequencies.

• In contrast, quantities of physical interest are related to the low-frequency modes.

UV sensitivity• In interacting theories where low and high

frequencies are coupled, the variational principle will exploit the addition of extra variational parameters to better describe the UV degrees of freedom at the expense of the relatively tiny contributions from low frequencies.

• This can lead to the paradoxical situation where the addition of variational parameters provides a worse approximation to physical quantities.

Dirac fermions

• For Dirac fermions in (1+1) dimensions the hamiltonian density is

• Our variational class is then defined by

|χ = T e +∞−∞ dx Q⊗I+

α Rα⊗ψ

†α(x)|ωB|ΩA

hD = −i

2ψ†(x)σy

dψ

dx(x) + h.c.+mψ†(x)σz ψ(x),

Where is the empty Dirac sea.|Ω

Cutoff properties

• Momentum distribution of a generic member of our variational class:

• We have for

• The region where this decay sets in is for

• This defines a soft momentum cutoff.

χ|ψ†α(k)ψβ(k )|χ = δ(k − k )nα,β(k)

nα,β(k) ≤ O(k−4) k →∞

k Q⊗ I+ I⊗Q+ Rα ⊗ Rα

∞ = Λ

Cutoffs

• Thus, by constraining the norm of the defining variational parameters

by, eg., lagrange multiplier, we can regulate the theory

Q Rαand

Dirac sea picture

k-4 decay

momentum k

low frequencies

high frequencies

Q ! c QR!! c R

!

mom

entu

m o

ccup

atio

nof

neg

ativ

e en

ergy

leve

ls … … ……

0

1

0 Λ 0 Λ' = cΛ! ∞Λ

Applying the variational procedure

• Pick some D, eg. D=2.

• Pick some initial values for

• Optimise energy of model, with lagrange multiplier imposing boundedness of norms.

• Plot momentum occupation as measure of success.

Q Rαand

Dirac fermions

D =

3D

= 1

2

D =

4

0

0.5

1.0

D =

2

0

0.5

1.0

k/Λ0 0.5 1.0 1.5 2.0

k / Λ0 0.5 1.0 1.5 2.0

n++

n--

|n+-|

Gross-Neveu model

• Shares many features with QCD, including, asymptotic freedom and spontaneous breaking of chiral symmetry.

• Hamiltonian density (N flavours):

hGN = −i

2ψ†aσ

y dψadx+ h.c.−

g2

2: (ψ†aσ

z ψa)2 :,

Gross-Neveu!(Λ

)"/

Λ

0.01

0.1

1

!-1(Λ) = [(N - 1)g(Λ)2]-1

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7

exactafitb

D = 6D = 8

D = 10D = 16

Expectation value of σ = χ|ψ†σz ψ|χ

Conclusions & outlook


• Introduced a variational class of quantum field states via sequential preparation



• Demonstrated holographic property




• Countered Feynman’s objections





• Cutting off bosonic theories





• Cutting off bosonic theories

• Many(!!) generalisations possible

Documents

The variational principle, relativistic quantum ﬁeld theory, …...• Due to the variational principle trying to ﬁnd the “lowest” ground-state energy. • To lowest order,