Lifshitz metric with hyperscaling violation
from string/M theory
Shibaji Roy
Saha Institute of Nuclear Physics
December 16, 2012
(SINP)
December 16, 2012
1 / 23
Based on:
P. Dey and S. Roy, “Lifshitz-like space-time from intersecting
branes in string/M theory,” JHEP 1206, 129 (2012)
[arXiv:1203.5381 [hep-th]].
P. Dey and S. Roy, “Intersecting D-branes and Lifshitz-like
space-time,” Phys. Rev. D 86, 066009 (2012) [arXiv:1204.4858
[hep-th]].
P. Dey and S. Roy, “Holographic entanglement entropy of the near
horizon 1/4 BPS F-Dp bound states,” arXiv:1208.1820 [hep-th].
P. Dey and S. Roy, “Lifshitz metric with hyperscaling violation
from NS5-Dp states in string theory,” arXiv:1209.1049 [hep-th].
(SINP)
December 16, 2012
2 / 23
Plan of the talk
Introduction and motivation
Construction of F-Dp solution
Near horizon limit and Lifshitz + hyperscaling
violating solutions
Phase structures
Other solutions
Conclusion
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December 16, 2012
3 / 23
Introduction and Motivation:
What is a Lifshitz metric?
ds2Lif
dt2
= − 2z +
u
Pd
i=1 (dx
u2
i )2
+
du2
u2
where 0 < u < ∞ and d is the spatial dimensions of the boundary
u → 0. u is the radial coordinate.
The metric has a scaling symmetry t → λz t, xi → λxi , u → λu
Metric is non-relativistic as t and xi scale differently for z 6= 1.
z = 1 is the relativistic limit which has AdSd+2 form with
SO(d + 1, 2) symmetry. (z is the dynamical critical exponent)
It also has time and space translation symmetry, spatial rotation
symmetry, and P , T symmetry. Particle conservation is not a
symmetry ⇒ particle production possible.
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December 16, 2012
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Intro...
Lifshitz metric can be obtained from an effective gravity action which
is Einstein gravity with negative cosmological constant and a 1-form
and a 2-form gauge field [Kachru-Liu-Mulligan].
Why Lifshitz system?
Non-relativistic systems are of general interests in condensed
matter physics.
Lifshitz symmetry arises in some strongly correlated electron
systems near critical points.
Multicritical points in some magnetic materials, liquid crystals or
near quantum critical point of high Tc cuprate superconductors.
Lifshitz space-times represent the gravity duals of such condensed
matter systems.
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December 16, 2012
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Intro...
Can it be obtained from some fundamental theory like
string theory? (not so easy!)
The effective action of Kachru-Liu-Mulligan has been tried to
embed in string theory (“bottom-up” approach) by several groups.
There has been attempts to obtain it by deforming the known
string theory solutions in a more direct way (“top-down”
approach).
Some references:
Hartnoll-Polchinski-Silverstein-Tong, Balasubramanian-Narayan, Donos-Gauntlett,
Gregory-Parameswaran-Tasinato-Zvala, Cassani-Faedo, Halmagyi-Petrini-Zaffaroni,
Gouteraux-Kiritsis, Narayan, Chemissany-Hartlong, Singh, Blaaback-Danielsson-van
Riet
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December 16, 2012
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Intro...
What is hyperscaling and its violation?
Behavior of systems near a phase transition is characterized by
critical exponents organizing the systems into universality classes.
Critical exponents satisfy certain relations known as hyperscaling
relations.
One such relation is (Josephson Identity): dν = 2 − α
where the specific heat C ∼ τ −α , and the correlation length ξ ∼ τ −ν
with τ = (T − Tc )/Tc is the reduced temperature and d is the spatial
dimension.
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December 16, 2012
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Intro...
Violation of hyperscaling means Josephson Identity is violated by a
hyperscaling violation exponent θ by
(d − θ)ν = 2 − α
So, θ has the effect of reducing the spatial dimensions.
Hyperscaling violation was first observed in random field Ising model
(RFIM) [Imry-Ma, Fisher]
How does hyperscaling violation shows its effect in the
gravity dual?
Hyperscaling is the property that the dimensionful quantities scale by
their natural length dimensions. So, for example, if the only length
scale is the correlation length ξ, the entropy should scale as S ∼ ξ d
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December 16, 2012
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Intro...
Dynamical properties of a system is governed by the dynamical
critical exponent z, where the characteristic time scale of the
theory goes as ζ ≡ tchar ∼ ξ z .
Since this is the natural scale in the dynamical theory the entropy
scales as S ∼ ξ d = ζ d/z .
With temperature the entropy therefore scales as S ∼ T d/z if the
hyperscaling is intact.
If the hyperscaling is violated the entropy scales as S ∼ T (d−θ)/z .
Now it can be easily seen that this temp. behavior of the entropy can
not be obtained from Lifshitz metric. For Lifshitz metric at finite
temp. S ∼ Area ∼ u−d
h , where uh is the horizon radius. Since u → λu,
−d/z ∼ T d/z (as t is inverse temp.)
t → λz t, so, entropy S ∼ u−d
h ∼t
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December 16, 2012
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Intro...
Not difficult to see – Correct T behavior of S with hyperscaling
violation can be obtained if the metric is modified with an overall
factor u2α with α, a constant to be determined.
The entropy will be modifed as
d(α−1)
dα
S ∼ Area ∼ u−d
h uh = uh
d(α−1)
Since uh
scales as td(α−1)/z , so, the entropy will scale as
d(1−α)/z
T
. Equating that with T (d−θ)/z , we get α = θ/d and so,
ds2new = u2θ/d ds2Lif
Such solutions have been obtained from Einstein-Maxwell-Dilaton
action and also modifying some string theory solutions. We will obtain
such solutions directly from certain string/M theory solutions.
(SINP)
December 16, 2012
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Construction of F-Dp solution
We will look for BPS solutions.
t and xi (for i = 1, . . . d) should scale differently.
So, dt2 and (dxi )2 should be multiplied by harmonic functions
with different powers and then take near horizon limit.
Not possible for single branes like Dp-branes, F-strings,
NS5-branes etc.
Possible for bound states of branes consisting of composites of
more than one type of branes.
1/2 BPS, non-threshold bound states (F-Dp, D(p − 2)-Dp, NS5-Dp
etc.) do not produce the right structure. Here the near horizon (or
decoupling) limit produce NCOS, NCYM, ODp theories (due to
the interacting nature of the constituent branes).
Next we look at 1/4 BPS, threshold bound states.
(SINP)
December 16, 2012
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F-Dp solution
In particular we construct some new F-Dp bound states. These are
1/4 BPS and threshold. [There are 1/2 BPS, non-threshold F-Dp bound
states (well-known [Lu-SR]) constructed from (p, q) string solution of Schwarz
and applying T-duality in the transverse directions of the strings. They give
NCOS theory in the decoupling limit.]
The steps of constructions are (from known D1/D5 soln of type IIB):
T5
T4
D1(5)/D5(12345) −→ D0/D4(1234) −→ D1(4)/D3(123)
T3
S
D1(4)/D3(123) −→ F(4)/D3(123) −→ F(4)/D2(12)
{z
}
|
{z
}
|
F/D3
T2
F/D2
T1
F(4)/D2(12) −→ F(4)/D1(1) −→ F(4)/D0
|
{z
}
| {z }
F/D1
F/D0
T6
T5
F(4)/D3(123) −→ F(4)/D4(1235) −→ F(4)/D5(12356)
{z
}
{z
}
|
|
F/D4
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F/D5
December 16, 2012
12 / 23
F-Dp solution
Using various duality rules the F-Dp (for 0 ≤ p ≤ 5) solution of type
IIB/A string theory can be written in a compact form as (in a suitable
coordinate)
ds2
1
=
H22 [−H1−1 H2−1 dt2 + H2−1
p
X
(dxi )2 +H1−1 (dxp+1 )2 +dr 2 + r 2 dΩ27−p ]
| {z }
i=1
F strings
| {z }
Dp branes
3−p
2
e2φ
=
B[2]
=
H2
H1
1 − H1−1 dt ∧ dxp+1 ,
A[p+1] =
1 − H2−1 dt ∧ dx1 ∧ · · · ∧ dxp
Here H1,2 = 1 + Q1,2 /r 6−p are the two harmonic functions with
Q1,2 , the charges of F-strings and Dp-branes respectively.
Dilaton φ is not constant in general.
In the metric dt2 and dx2 terms are multiplied with different
powers of harmonic functions and can lead to Lifshitz-like
solutions.
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December 16, 2012
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Near horizon limit and Lifshitz + hyperscaling violating
solution
As F-Dp solutions have horizon at r = 0, near horizon limit means
we take r → 0, then the harmonic functions take the form
H ≈ Q1,2 /r 6−p and the solutions simplify accordingly.
We further make a coordinate transformation r → 1/r.
Then define a new coordinate u by u2 = r 4−p for p 6= 4.
p = 4 case needs to be discussed separately.
With these, F-Dp solutions take the form,
ds
2
=
1
2
Q2 u
2−p
4−p
"
−
dt2
Q1 Q2 u
4(5−p)
4−p
+
Pp
i 2
i (dx )
2
Q2 u
(dxp+1 )2
4
du2
+
+
+ dΩ27−p
2
2
Q1 u
(4 − p) u2
#
3−p
2φ
=
B[2]
=
e
(6−p)(1−p)
Q2 2
u (4−p)
Q1
1
dt ∧ dxp+1 ,
−
2(6−p)
Q1 u 4−p
(SINP)
1
A[p+1] = −
Q2 u
2(6−p)
4−p
dt ∧ dx1 ∧ · · · ∧ dxp
December 16, 2012
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Near horizon limit...
We next compactify the metric on S7−p and write the resultant metric
in the Einstein frame to get,
ds2p+3
2
p+1
= Q1
p(4−p)−(p−2)
Q2
2[
u (4−p)(p+1)
]

−
dt2
Q1 Q2 u
4(5−p)
4−p
+
Pp
i 2
i (dx )
Q 2 u2

(dxp+1 )2
4
du2 
+
+
Q 1 u2
(4 − p)2 u2
If we rescale the coordinates to absorb the constant Q1 , Q2 , 4/(4 − p)2
etc, we precisely get Lifshitz metric with hyperscaling violation from
string theory solution. To find the various exponents we note that
under the rescaling
t → λ2(5−p)/(4−p) t ≡ λz t,
x1,...,(p+1) → λx1,...,(p+1) ,
u → λu
the metric is not invariant but changes as,
dsp+3 → λ
(SINP)
p(4−p)−(p−2)
(4−p)(p+1)
dsp+3 ≡ λθ/d dsp+3
December 16, 2012
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Near horizon limit...
We thus find the dynamical critical exponent z, hyperscaling
violation exponent θ and the spatial dimensions d of the boundary
theory (obtained from F-Dp soln) to be,
z=
2(5 − p)
,
4−p
θ =p−
p−2
,
4−p
d = p+1
Also under the scaling the other fields transform as,
φ → φ+
(6 − p)(1 − p)
log λ,
2(4 − p)
2−p
B[2] → λ 4−p B[2] ,
A[p+1] → λ
2−(p−2)2
4−p
A[p+1]
Since the exponents are obtained from genuine gravity solutions it
can be easily checked that they automatically satisfy Null Energy
Condition [Dong-Harrison-Kachru-Torroba-Wang]
(d − θ)(d(z − 1) − θ) ≥ 0,
(SINP)
(z − 1)(d + z − θ) ≥ 0
December 16, 2012
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Near horizon limit...
z=
2(5 − p)
4−p
,
θ = p−
p−2
4−p
,
d= p+1
From the above values it can be checked that for p = 2, θ = d − 1, one can
calculate the holographic entanglement entropy (HEE) in this case [Dey-Roy]
and it has a logarithmic violation of area law, indicating that the dual theory
represents compressible states with fermi surface. [Huijse-Sachdev-Swingle],
[Ogawa-Takayanagi-Ugajin]
For p = 3, θ < d − 1 and for p = 5, θ > d and in both cases HEE gives
[Dey-Roy] the usual area law. [Bombelli-Koul-Lee-Sorkin], [Srednicki]
For p = 0, 1, d − 1 < θ < d and HEE [Dey-Roy] has new area law violation (in
between linear and logarithmic), indicating that dual theories have new phases.
[Dong-Harrison-Kachru-Torroba-Wang]
As we mentioned p = 4 case has to be treated separately. Instead of defining u
coordinate if we keep the original r coordinate it is easy to see that the near
horizon limit does not lead to Lifshitz-like space-time, but gives an AdS2 × S3
× E5 space-time.
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December 16, 2012
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Phase structures
We noted before that the dilaton of the near horizon geometry of
the F-Dp solutions we constructed are not constant
(3−p)/2
/Q1 )u(6−p)(1−p)/(4−p) ] in general except for p = 1.
[e2φ = (Q2
As the radial variable u (which is the RG flow parameter in the
boundary) changes, the effective string coupling eφ also changes.
However, for the gravity description to remain valid eφ must
remain small. This gives some restrictions on u.
Another restriction on u comes from the curvature of the metric,
which also should remain small.
For large eφ we have to lift the solution to eleven dimensions (for
type IIA) or go to the S-dual frame (for type IIB).
The phase structures have to be analysed on case by case basis. We
will discuss F-D2 (type IIA) solution only. The other cases are similar.
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December 16, 2012
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Phase structures...
Near horizon F-D2 Soln
The metric and the dilaton in this case have the forms,
1
2
"
dt2
−
+
Q1 Q2 u6
2
=
Q2
e2φ
=
Q22
Q1 u2
ds
P2
i=1 (dx
Q2 u2
i 2
)
(dx3 )2
4 du2
+
+
+ dΩ25
2
Q1 u
9 u2
#
1
This gives Lifshitz metric with hyperscaling violation with the
parameters (z, θ, d) = (3, 2, 3). The gravity description is valid for
1/4
1/2
1/4
1/2
u ≫ Q2 /Q1 with Q2 ≫ 1. When u ≤ Q2 /Q1 , eφ is large and we
have to uplift the solution to M-theory. The uplifted solution takes the
form
1
3
1
3
2
=
Q1 Q2 u
A[3]
=
−
ds
2
3
"
dt2
−
+
Q1 Q2 u6
1
dt ∧ dx3 ∧ dx11 ,
Q1 u4
(SINP)
P2
i=1 (dx
Q2 u2
i 2
)
(dx3 )2 + (dx11 )2
du2
+
+
+ dΩ25
Q1 u2
u2
A′[3] = −
#
1
dt ∧ dx1 ∧ dx2
Q2 u4
December 16, 2012
19 / 23
Phase structures...
2
1
1
2
ds
=
Q13 Q23 u 3
A[3]
=
−
1
Q1 u4
"
−
dt2
Q1 Q2 u6
3
dt ∧ dx ∧ dx
11
+
,
P2
i 2
(dx3 )2 + (dx11 )2
du2
2
i=1 (dx )
+
+
+ dΩ5
2
2
Q2 u
Q1 u
u2
′
A[3] = −
1
Q2 u4
1
dt ∧ dx ∧ dx
#
2
The above configuration represents near horizon limit of two
intersecting M2-branes. It also gives Lifshitz metric with hyperscaling
violation with (z, θ, d) = (3, 3, 4). This gravity description is valid for
√
√
1/4
1/2
u ≫ 1/ Q1 Q2 , i.e., in the range 1/ Q1 Q2 ≪ u ≤ Q2 /Q1 . [Note that
here u → ∞ ⇒ going to IR and u → 0 ⇒ going to UV]
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December 16, 2012
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Other solutions
In the following we tabulate some other solutions of string theory
which also give Lifshitz metric with hyperscaling violations:
(SINP)
Type
z
θ
d
NS5-Dp
1≤p≤6
0
9−p
7−p
F-Dp
0 ≤ p ≤ 5, p 6= 4
D0-D4
D1-D3
D2-D2′
D2-D6
D3-D5
D4-D4′
2(5−p)
4−p
p−
p−2
4−p
p+1
4
2
4
0
6
4
December 16, 2012
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Conclusion
We have constructed some new string/M theory solutions which,
in the near horizon limit and after compactification gives Lifshitz
metric (with hyperscaling violation.) We found some delocalized
F-D1 solution which in the near horizon limit gives Lifshitz metric
(without hyperscaling violation).
They might be the gravity dual of some condensed matter system
near quantum critical point.
We have mainly looked at intersecting 1/4 BPS solutions involving
two types of branes, namely, F-strings/D-branes, two D-branes,
NS5/D-branes. There may be more solutions involving waves and
KK-monopoles giving rise to Lifshitz metric + hyperscaling
violation.
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December 16, 2012
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Conclusion...
There may be more solutions involving three or more branes
having less susy giving rise to this geometry (some of these
solutions are discussed in [Kim, arXiv:1210.0540]).
Also, there are non-supersymmetric solutions of string theory
which can give Lifshitz-like geometry.
To find the exhaustive list one has to look at those solutions.
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December 16, 2012
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