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# Holographic Renormalization of a Massless Scalar Field in a Lifshitz background

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on 29 September 2010#### Transcript of Holographic Renormalization of a Massless Scalar Field in a Lifshitz background

Holographic Renormalization

Massless Scalar Field

Lifshitz background of a in a Wout Merbis Thesis Supervisor:

Dr. Marika Taylor University of Amsterdam Institute for Theoretical Physics AdS/CFT correspondence Gauge Theory String Theory

Conformally Invariant Field Theory

D dimensional

Operators N parallel D3-branes

String Theory on spacetime

D+1 dimensional

Fields Operator sourced by On-shell SUGRA partition function with field 1 2 3 4 Duals for Non-Relativistic Theories Consider field theories with

dynamical scale invariance: Why Would We Look at Dynamical Scaling? What Would Dynamical Scaling Imply? Dual of the Dynamically Scaled Field Theory Holographic Renormalization Method Asymptotic Expansion One-Point Correlation Function Higher-Point Correlation Functions Regularization and Renormalization Massless Scalar in a Lifshitz background Asymptotic Expansion Regularization and Renormalization One-Point correlation functions Two-Point functions Analytic Solutions Numerical Solutions = Matching of Symmetries on both sides This duality is a STRONG-WEAK duality Condensed Matter Applications

Poincaré invariance broken Non-Relativistic limit Adding Structure to AdS/CFT Finite Temperature Boundary Theory Schwarschild-AdS Black Hole Reissner-Nordström-AdS

Black Hole Finite Chemical Potential on Boundary Given a finite temperature, one can continue to calculate other thermodynamic quantities like: Partition Function

Free energy

Entropy (Bekenstein-Hawking entropy of BH corresponds to field theory entropy) More structure can be added by adding and/or deforming fields in the bulk spacetime, which correspond to field theory operators. Fermions at unitarity

non-Fermi metals near heavy electron critical points

Optimally doped cuprates Extending the AdS/CFT-correspondence to include dynamically scaled systems furthers our dictionary for holographic methods. Full conformal invariance is broken Anistropic scale invariance Lifshitz algebra: Schrödinger algebra: (include Galilean transformations) Look for metrics with isometries of Lifshitz algebra

Lifshitz metric: Physics of strongly coupled anisotropic field theory (in large N limit) is described by classical gravity on Lifshitz background Following AdS/CFT logic: Write any field as asymptotic expansion in radial coordinate Introduce logarithmic term when two coefficients coincide Take the asymptotic locally AdS metric (in Fefferman-Graham coordinates): In general, there are two independent solutions to the field equations: Write down diverging terms of on-shell action Write down the counterterm action Subtract away the infinities: Obtain the renormalized action

on the boundary Recall that the correlation function of a CFT operator is defined by the functional derivative of the supergravity action wrt the source Now we take the subtracted action, living on the regulated boundary ...and afterwards take the limit to the boundary Explicit calculation gives: i.e. the one-point function is related to the solution undetermined by near-boundary analysis, plus some local function of the source Higher-point correlation functions can be obtained by differentiating the one-point function wrt the source Lifshitz metric: Minimally coupled action Write the scalar field as an asymptotic expansion around its two solutions Assumption We assume the scalar field does not couple to gravity, so the field equations decouple and we only have to consider the free massless scalar field equation Solve field equation to first order in This gives the conformal dimension as the roots of: which reduces to the correct relation for AdS when Proceed to solve field equations order for order in

We get iterative solutions for the coefficients of the expansion: With these conformal dimensions we get an

expansion for and when Go through the steps of regularization and renormalization outlined earlier and when we get a logarithmic term in the counterterm action We get the 1-point correlation function Where: NOTE: one can always remove

by adding local finite counterterms The one-point correlation function depends on the coefficient of the expansion undetermined by the near-boundary analysis, So: we need the solution to the bulk field equations to obtain two-point correlation functions Then we can determine

to the required order in We need the two-point function: To obtain an expression for

we need to solve the field equation: Where we've Fourier transformed, and Now the 2-point function is a function of the scale invariant variable Analytic solutions for

can now be found for specific values of d and z i.e. when where U is the confluent hypergeometric function of the second kind Then: expand solution around

to find the coefficient The two point function then reads: Similar calculations for other integer values of d and z can be found in my thesis When z is not an integer, we need to solve the field equations numerically First step is to integrate out the field equations, allowing z and to vary, Also, the solution must be regular in the interior, so our boundary condition for the integration are: For the two-point function we need to extract

from We can do this by numerically calculating all the coefficients in the expansion of using: These coefficients are calculated very high order to obtain high accuracy Accuracy The accuracy is determined by Error in integrating out

Order of the largest coefficient in the expansion

Presence of higher order logarithmic terms By comparing to analytic results when present and by estimating the error made in the integration routine we get an accuracy of order Relative difference between numerical and analytical results for d=3 Conclusion We obtained renormalized two-point correlation functions of a Massless scalar in a Lifshitz background. SO? This furthers our understanding of gravitational duals for non-relativistic field theories. A tool which is useful for many strongly coupled condensed matter theories. of course, this is only a small step... ...but a lot of small steps... ...make a big one [Maldacena, 1997, GKPW, 1998] [Witten, 1998] [Kachru, Liu, Mulligan, 2008] [Skenderis review, 2002] And: The two-point function is: [Son, 2008]

Full transcriptMassless Scalar Field

Lifshitz background of a in a Wout Merbis Thesis Supervisor:

Dr. Marika Taylor University of Amsterdam Institute for Theoretical Physics AdS/CFT correspondence Gauge Theory String Theory

Conformally Invariant Field Theory

D dimensional

Operators N parallel D3-branes

String Theory on spacetime

D+1 dimensional

Fields Operator sourced by On-shell SUGRA partition function with field 1 2 3 4 Duals for Non-Relativistic Theories Consider field theories with

dynamical scale invariance: Why Would We Look at Dynamical Scaling? What Would Dynamical Scaling Imply? Dual of the Dynamically Scaled Field Theory Holographic Renormalization Method Asymptotic Expansion One-Point Correlation Function Higher-Point Correlation Functions Regularization and Renormalization Massless Scalar in a Lifshitz background Asymptotic Expansion Regularization and Renormalization One-Point correlation functions Two-Point functions Analytic Solutions Numerical Solutions = Matching of Symmetries on both sides This duality is a STRONG-WEAK duality Condensed Matter Applications

Poincaré invariance broken Non-Relativistic limit Adding Structure to AdS/CFT Finite Temperature Boundary Theory Schwarschild-AdS Black Hole Reissner-Nordström-AdS

Black Hole Finite Chemical Potential on Boundary Given a finite temperature, one can continue to calculate other thermodynamic quantities like: Partition Function

Free energy

Entropy (Bekenstein-Hawking entropy of BH corresponds to field theory entropy) More structure can be added by adding and/or deforming fields in the bulk spacetime, which correspond to field theory operators. Fermions at unitarity

non-Fermi metals near heavy electron critical points

Optimally doped cuprates Extending the AdS/CFT-correspondence to include dynamically scaled systems furthers our dictionary for holographic methods. Full conformal invariance is broken Anistropic scale invariance Lifshitz algebra: Schrödinger algebra: (include Galilean transformations) Look for metrics with isometries of Lifshitz algebra

Lifshitz metric: Physics of strongly coupled anisotropic field theory (in large N limit) is described by classical gravity on Lifshitz background Following AdS/CFT logic: Write any field as asymptotic expansion in radial coordinate Introduce logarithmic term when two coefficients coincide Take the asymptotic locally AdS metric (in Fefferman-Graham coordinates): In general, there are two independent solutions to the field equations: Write down diverging terms of on-shell action Write down the counterterm action Subtract away the infinities: Obtain the renormalized action

on the boundary Recall that the correlation function of a CFT operator is defined by the functional derivative of the supergravity action wrt the source Now we take the subtracted action, living on the regulated boundary ...and afterwards take the limit to the boundary Explicit calculation gives: i.e. the one-point function is related to the solution undetermined by near-boundary analysis, plus some local function of the source Higher-point correlation functions can be obtained by differentiating the one-point function wrt the source Lifshitz metric: Minimally coupled action Write the scalar field as an asymptotic expansion around its two solutions Assumption We assume the scalar field does not couple to gravity, so the field equations decouple and we only have to consider the free massless scalar field equation Solve field equation to first order in This gives the conformal dimension as the roots of: which reduces to the correct relation for AdS when Proceed to solve field equations order for order in

We get iterative solutions for the coefficients of the expansion: With these conformal dimensions we get an

expansion for and when Go through the steps of regularization and renormalization outlined earlier and when we get a logarithmic term in the counterterm action We get the 1-point correlation function Where: NOTE: one can always remove

by adding local finite counterterms The one-point correlation function depends on the coefficient of the expansion undetermined by the near-boundary analysis, So: we need the solution to the bulk field equations to obtain two-point correlation functions Then we can determine

to the required order in We need the two-point function: To obtain an expression for

we need to solve the field equation: Where we've Fourier transformed, and Now the 2-point function is a function of the scale invariant variable Analytic solutions for

can now be found for specific values of d and z i.e. when where U is the confluent hypergeometric function of the second kind Then: expand solution around

to find the coefficient The two point function then reads: Similar calculations for other integer values of d and z can be found in my thesis When z is not an integer, we need to solve the field equations numerically First step is to integrate out the field equations, allowing z and to vary, Also, the solution must be regular in the interior, so our boundary condition for the integration are: For the two-point function we need to extract

from We can do this by numerically calculating all the coefficients in the expansion of using: These coefficients are calculated very high order to obtain high accuracy Accuracy The accuracy is determined by Error in integrating out

Order of the largest coefficient in the expansion

Presence of higher order logarithmic terms By comparing to analytic results when present and by estimating the error made in the integration routine we get an accuracy of order Relative difference between numerical and analytical results for d=3 Conclusion We obtained renormalized two-point correlation functions of a Massless scalar in a Lifshitz background. SO? This furthers our understanding of gravitational duals for non-relativistic field theories. A tool which is useful for many strongly coupled condensed matter theories. of course, this is only a small step... ...but a lot of small steps... ...make a big one [Maldacena, 1997, GKPW, 1998] [Witten, 1998] [Kachru, Liu, Mulligan, 2008] [Skenderis review, 2002] And: The two-point function is: [Son, 2008]