Supersymmetric Index for Small Black Holes

Chandramouli Chowdhury,1,2 Ashoke Sen,2 P Shanmugapriya,3 Amitabh Virmani3

1 Mathematical Sciences and STAG Research Centre, University of Southampton,
Highfield, Southampton SO17 1BJ, United Kingdom 2 International Centre for
Theoretical Sciences, Bengaluru - 560089, India 3 Chennai Mathematical
Institute, Kelambakkam, Tamil Nadu, India 603103

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###### Abstract

Supersymmetric elementary string states in the compactified heterotic string
theory are described by small black holes that have zero area event horizon.
In this paper we compute the supersymmetric index of such elementary string
states using gravitational path integral. The dominant contribution to the
path integral comes from an Euclidean rotating black hole solution of the
supergravity theory with a finite area event horizon, but the logarithm of the
index, computed from the saddle point, vanishes. Nevertheless we show that the
solution is singular on certain subspaces of the horizon where higher
derivative corrections can be important, and once the higher derivative
corrections are taken into account the solution could yield a finite result
for the logarithm of the index whose form agrees with the microscopic results
up to an overall numerical constant. While the numerical constant is not
determined in our analysis, we show that it is independent of the details of
the compactification and even the number of non-compact dimensions, in
agreement with the microscopic results.

###### Contents

  1. 1 Introduction
  2. 2 Microscopic results for small black hole entropy
  3. 3 Strategy for macroscopic computation of small black hole entropy
  4. 4 The black hole solution
    1. 4.1 The solution
    2. 4.2 BPS limit and Euclidean continuation
    3. 4.3 Periodicity of the $\tau$ circle
    4. 4.4 $(D-1)$-dimensional base and the singular subspace
  5. 5 The geometry near the singularity
    1. 5.1 Coordinates centered at the singular surface
    2. 5.2 Large charge limit
  6. 6 Wald entropy and the index
  7. 7 Generalization to other compactifications

##  1 Introduction

The excitations of a string lead to an infinite tower of massive states with
large degeneracy. It has long been suspected that sufficiently heavy states of
the string can be identified as black holes and that the logarithm of the
degeneracy of such string states may provide an explanation of the black hole
entropy[1, 2, 3, 4, 5, 6]. However since the counting of states of the
elementary string can be carried out reliably at weak string coupling, while
the description of the system as a black hole holds at large string coupling,
quantitative comparison between the microscopic and macroscopic calculations
is difficult.

This problem can be overcome by considering a special class of elementary
string states that preserve some of the supersymmetries of the theory and
hence saturate the BPS bound[7]. For such states there are general arguments
that the log of the degeneracy computed at weak coupling remains the same at
strong coupling and hence can be compared with the entropy of the black hole
carrying the same charges. One finds however that the corresponding black hole
has vanishing area of the event horizon and hence the entropy vanishes. It was
argued in [8, 9, 10] that once we take into account the higher derivative
corrections to the effective action, the black hole may acquire a finite
entropy. Using the symmetries of the theory and a scaling argument, one could
determine the dependence of the entropy on the charges up to an overall
numerical factor whose value depends on the details of the higher derivative
corrections[8]. The dependence of the entropy on the charges, obtained this
way, precisely matches with the known result from the microscopic counting[8,
9, 10]. The numerical constant was determined later in [11] under the
assumption that only certain four derivative terms contribute to the entropy
of the black hole.

There is however one caveat in this analysis. The analysis described above
relies on the degeneracy of BPS states being protected under quantum
corrections so that it takes the same value for weak and strong coupling.
However it is not the degeneracy, but a closely related quantity called the
supersymmetric index, that is protected this way. So we need to make an extra
assumption, namely that the degeneracy and the index should be equal both at
strong and weak coupling. In the weak coupling limit, where the states can be
described as excitations of the fundamental string, this is known to be the
case for compactification of heterotic string theory. However for type II
string compactification this fails – the index vanishes while the degeneracy
continues to be large. At strong coupling where the black hole description is
valid, there is no general argument showing the equality of degeneracy and the
index either in the heterotic string theory or in type II string theory. See
e.g. section 5 of [12] for some discussion on these issues.

The goal of this paper is to circumvent this problem by directly analyzing the
index of fundamental string states in heterotic string theory compactified on
$(10-D)$ dimensional torus and other manifolds. On the microscopic side the
computation is straightforward and the result has been known for many years.
On the macroscopic side we use a procedure for computing index developed
recently in [13], following earlier work on $AdS$ spaces[14]. This procedure
has been successful in reproducing correctly the leading result[13, 15] as
well as logarithmic correction to the logarithm of the index of black holes
with regular finite area event horizon[16, 17]. We use this method to
calculate the leading contribution to the logarithm of the index of small
black holes.

It turns out that the saddle point that contributes to the log of the index of
small black holes is not a small black hole with zero area event horizon but a
rotating Euclidean black hole with finite area event horizon. Nevetheless,
when one follows the algorithm for computing the index using this black hole
saddle point, one finds that in the two derivative theory the index vanishes,
just as the entropy of the small black holes vanished in the two derivative
theory. One finds however that despite having an event horizon of finite area,
this black hole solution is not completely smooth, – it has singularity on a
subspace of the horizon where higher derivative corrections may become
important. We carefully analyze the geometry near the singularity, and by
using the symmetries of the theory and a scaling argument very similar to the
ones used in [8], we can completely determine the corrected logarithm of the
index up to an overall numerical factor. This dependence of the index on the
charges found this way turns out to be in perfect agreement with the
microscopic results for the index. We furthermore show that the undetermined
numerical factor is independent of the dimension $D$ of the non-compact part
of the space-time even though the full black hole solution depends non-
trivially on $D$. This is also in agreement with the microscopic results.

The rest of the paper is organized as follows. In section 2 we review the
microscopic results for the index of fundamental string states. In section 3
we describe the general strategy that we shall use for the macroscopic
computation of the log of the index. In section 4 we describe the rotating
black hole solution that contributes to the index and show that the log of the
index computed from this solution vanishes in the two derivative theory. We
also show that the solution is singular on a subspace of the horizon. In
section 5 we zoom in on the region near the singularity and express the
solution in a rescaled coordinate system that is useful for studying the
solution near the singularities. In section 6 we present the argument based on
scaling and the symmetries that determines the higher derivative correction to
the log of the index up to an overall numerical constant. We show that the
dependence of this correction on the charges carried by the state is precisely
what is expected from the microscopic analysis. In section 7 we generalize
these results to other compactifications of heterotic string theory.

Since the paper is technical, for the benefit of the reader we would like to
point to the final results of our paper that are contained in section 6.
Eqs.(6.1)-(6.6) describe the solution near the singularity. These
configurations are universal, i.e. independent of the parameters of the
solution like the charges or the asymptotic values of the moduli, except for
two factors. First the string metric (6.1) contains a direct sum of a
universal four dimensional metric and a nearly flat $(D-4)$ dimensional metric
$b^{2}d\Omega^{D-4}$, whose only role in our analysis is to provide a factor
of the volume of the $(D-4)$ dimensional space, proportional to $b^{D-4}$, in
the evaluation of the action. The second source of non-universality is an
additive constant $-\ln(g_{s}^{-2}m_{0}b^{D-4})$ in the expression for the
dilaton field $\Phi$ as given in (6.6), which also provides an overall
multiplicative factor in the action but does not affect our analysis in any
other way. Expressing these non-universal factors in terms of the charges
carried by the black hole, we arrive at the result (6.12) which is our final
result for the logarithm of the index. Here $K$ is an unknown numerical factor
that can in principle be determined with a better understanding of the higher
derivative terms in the action, or, equivalently, a better understanding of
the world-sheet $\sigma$ model with target space determined by the solutions
(6.1)-(6.6).

##  2 Microscopic results for small black hole entropy

We consider heterotic string theory compactified on a $(10-D)$ dimensional
torus so that we have a theory in $D$ non-compact space-time directions. The
vacuum of this theory is characterized by the string coupling $g_{s}$ that can
be identified as the vacuum expectation value of $e^{\Phi/2}$, $\Phi$ being
the dilaton field and the vacuum expectation value of a $(36-2D)\times(36-2D)$
matrix valued scalar field $M$, satisfying

$MLM^{T}=L,\qquad M^{T}=M,\qquad L=\pmatrix{-I_{26-D}&0\cr 0&I_{10-D}}\,,$
(2.1)

where $I_{n}$ denotes $n\times n$ identity matrix. We shall denote by
$\Phi_{\infty}$ and $M_{\infty}$ the asymptotic values of the fields $\Phi$
and $M$.

The elementary string states in this theory carry $36-2D$ different charges,
which we can divide into a $(10-D)$ dimensional charge vector $Q_{R}$ and a
$(26-D)$ dimensional charge vector $Q_{L}$. If we denote by $Q$ the full
$(36-D)$ dimensional charge vector, then we can write

$Q_{R}={1\over 2}(M_{\infty}+L)Q,\qquad Q_{L}={1\over 2}(M_{\infty}-L)Q\,.$
(2.2)

We also define,

$Q_{R}^{2}\equiv{1\over 2}Q^{T}(M_{\infty}+L)Q,\qquad Q_{L}^{2}\equiv{1\over
2}Q^{T}(M_{\infty}-L)Q,\qquad Q^{2}\equiv Q^{T}LQ=Q_{R}^{2}-Q_{L}^{2}\,.$
(2.3)

Among the elementary string states there are a special class of states, known
as half BPS states, that are invariant under eight of the sixteen unbroken
supersymmetries of the theory. The mass of such a state, measured in the
string metric, is given by

$m_{BPS}=\sqrt{Q^{T}(M_{\infty}+L)Q}=\sqrt{2\,Q_{R}^{2}}\,.$ (2.4)

At this point we need to describe our convention. We shall work in the
$\alpha^{\prime}=1$ unit so that if we have a fundamental string carrying $n$
units of momentum and $w$ units of winding along an internal circle of radius
$R$, it has mass $wR+n/R$. In our convention $Q_{R}=(wR+n/R)/\sqrt{2}$ and
$Q_{L}=(wR-n/R)/\sqrt{2}$ for this state so that (2.4) holds and we have
$Q_{R}^{2}-Q_{L}^{2}=2nw$.

The microscopic entropy of such BPS states, defined as the log of the
degeneracy of the states, can be computed and yields the result (see e.g. [4,
8])

$S_{micro}=4\pi\sqrt{Q_{R}^{2}-Q_{L}^{2}\over 2}=2\sqrt{2}\pi\sqrt{Q^{2}}\,,$
(2.5)

in the limit of large charges. However this is not protected under quantum
corrections. The more relevant quantity for our analysis is an appropriate
index, known as the fourth helicity trace index, defined as

${1\over 4!}\,Tr_{Q}\left[e^{-\beta(H-m_{BPS})}(-1)^{F}(2J)^{4}\right]\,,$
(2.6)

where $Tr_{Q}$ denotes trace over all states carrying charge $Q$ and zero
momentum, $(-1)^{F}$ takes value 1 and $-1$ for bosonic and fermionic states
respectively, $H$ is the Hamiltonian and $J$ is the component of the angular
momentum carried by the state in some fixed two dimensional plane. $\beta$ is
an arbitrary parameter labelling the inverse temperature of the system but the
index is supposed to be independent of $\beta$ since only those states whose
$H$ eigenvalue is equal to $m_{BPS}$ contribute to the trace. For heterotic
string theory on $T^{10-D}$ this quantity can be shown to be equal to the
degeneracy in the weak coupling limit, and so the log of the index is given by
(2.5). However this index can also be shown to be protected under quantum
corrections and hence takes the same value at strong coupling.

Our goal is to try to reproduce (2.5) from macroscopic analysis, i.e. from the
analysis of the classical geometry describing a solution carrying these
charges.

##  3 Strategy for macroscopic computation of small black hole entropy

In any two derivative theory in $D$ space-time dimensions containing the
metric $g_{\mu\nu}$, 2-form field $B_{\mu\nu}$, $U(1)$ gauge fields
$\\{A_{\mu}^{(i)}\\}$ and scalar fields $\\{\phi_{\alpha}\\}$, there is a
scaling relation under which

$g_{\mu\nu}\to\lambda^{2}\,g_{\mu\nu},\quad
B_{\mu\nu}\to\lambda^{2}\,B_{\mu\nu},\quad
A^{(i)}_{\mu}\to\lambda\,A^{(i)}_{\mu},\quad\phi_{\alpha}\to\phi_{\alpha},$
(3.1)

and the action scales as $\lambda^{D-2}$. For a classical black hole solution
carrying mass $m$, charges $Q$ and angular momenta $J$, if we scale the fields
as in (3.1), the various quantities parametrizing the solution scale as

$m\to\lambda^{D-3}\,m,\quad Q\to\lambda^{D-3}\,Q,\quad
J\to\lambda^{D-2}\,J\,.$ (3.2)

Under this scaling the Bekenstein-Hawking entropy $S_{BH}$ of a regular black
hole scales in the same way as the action:

$S_{BH}\to\lambda^{D-2}\,S_{BH}\,.$ (3.3)

We can take the macroscopic limit by taking $\lambda$ to be large. Note that
under the same scaling (2.5) scales by a factor of $\lambda^{D-3}$, leading to
an apparent contradiction. This issue was resolved in [8] by noting that the
classical BPS black hole in the two derivative supergravity theory has zero
area event horizon and hence vanishing entropy and we need to take into
account higher derivative corrections to get the correct scaling of the
entropy. We shall carry out a similar analysis in this paper, but focussing
from the beginning on the computation of the index on the macroscopic side as
well.

We now outline the main steps in our analysis.

  1. 1.

Since we are interested in computing the index, we follow the strategy
described in [13], where we begin with an electrically charged black hole
carrying a charge vector $Q$ and angular momentum $J$ in a two dimensional
plane. The angular momentum $J$ is adjusted so that the chemical potential
$\Omega$ dual to this angular momentum, also called the angular velocity, is
given by

$\Omega=-2\pi\mathrm{i}/\beta\,,$ (3.4)

where $\beta$ is the inverse temperature of the black hole. The effect of this
chemical potential is to insert a factor of $e^{2\pi\mathrm{i}J}$ into the
path integral, which in turn plays the role of $(-1)^{F}$ that appears in
(2.6). Therefore this rotating black hole solution provides the dominant
saddle point in the computation of the index. Then the leading semiclassical
result $S_{macro}$ for the log of the index, computed from the gravitational
path integral, is given by

$S_{macro}=S_{wald}+2\pi\mathrm{i}J\,,$ (3.5)

where $S_{wald}$ is the Wald entropy of the black hole, which can also be
computed following the approach of Gibbons and Hawking where we first compute
the partition function by computing the path integral over all the fields and
then use the appropriate thermodynamic relations to extract the entropy[18].
We shall denote by $S_{wald}$ the expression for the black hole entropy
including the effect of higher derivative corrections, irrespective of which
way it is calculated. For two derivative theories this agrees with the
Bekenstein-Hawking entropy $S_{BH}$ of the black hole given by $A/(4G_{N})$
where $A$ is the horizon area and $G_{N}$ is the Newton’s gravitational
constant. The $2\pi\mathrm{i}J$ term in (3.5) is the effect of the chemical
potential that inserts the explicit factor of $e^{2\pi\mathrm{i}J}$ in the
path integral. Note that according to (2.6) we also need to insert a factor of
$(2J)^{4}$ into the path integral, but as discussed in [16, 17], this factor
just soaks up the fermion zero modes associated with broken supersymmetries in
the path integral and does not have any other effect.

  2. 2.

We then show that for this solution

$S_{BH}+2\pi\mathrm{i}J=0\,.$ (3.6)

If this had been non-zero, then according to (3.2), (3.3) this would scale as
$\lambda^{D-2}$. On the other hand (2.5) and (3.2) shows that $S_{micro}$
scales as $\lambda^{D-3}$. This would lead to an obvious contradiction which
is avoided by (3.6).

  3. 3.

We can now ask whether higher derivative corrections to (3.5) could resolve
the disagreement. Since $J$ is measured from the fields at infinity where the
higher derivative corrections can be ignored, we focus on $S_{wald}$. If the
horizon had been smooth, then the four derivative corrections to $S_{wald}$
would be of order $\lambda^{D-4}$ according to (3.1) since two extra
derivatives need to be contracted with a $g^{\mu\nu}$. This would be
inconsistent with (2.5) which scales as $\lambda^{D-3}$. We show that the
solution is singular near a subspace, and that in the neighbourhood of this
subspace we can choose a coordinate system in which the string frame metric
becomes independent of the charges and the dilaton takes the form

$e^{-\Phi}=\sqrt{Q_{R}^{2}-Q_{L}^{2}}\ F$ (3.7)

where $F$ is a function of the coordinates that does not depend on the
charges. Since $e^{\Phi/2}$ expectation value gives the string coupling, (3.7)
shows that for large charges, the string coupling at the horizon is small.
Hence we can restrict our analysis to tree level string theory. On the other
hand since the metric and $\partial_{\mu}\Phi$ are of order unity (having no
dependence on the charges) we see that the $\alpha^{\prime}$ corrections are
of order unity, and we need to take into account all order corrections in
$\alpha^{\prime}$ to compute the entropy. However these corrections are
completely universal, independent of the charges.

  4. 4.

This argument shows that if the higher derivative corrections generate a
finite entropy, then the dependence of the entropy on the charges comes solely
from the $\sqrt{Q_{R}^{2}-Q_{L}^{2}}$ factor in (3.7). This will appear as an
overall multiplicative factor in the expression for the on-shell action and
hence the entropy. So we can write

$S_{macro}=C\,\sqrt{Q_{R}^{2}-Q_{L}^{2}}\,,$ (3.8)

in agreement with (2.5). Here $C$ is an unknown numerical constant that
depends on the details of how the $\alpha^{\prime}$ corrections modify the
near horizon geometry. However $C$ is independent of $\Phi_{\infty}$ and
$M_{\infty}$. We also show that $C$ is independent of $D$, in agreement with
(2.5).

Finally we would like to remark that since the macroscopic computation of the
index will be done using the full space-time geometry, it implicitly uses the
grand canonical ensemble, i.e. instead of computing the trace over fixed
charge states as in (2.6) we sum over all charged states weighted by
$e^{-\beta\mu.Q}$ and then perform the Legendre transform of the logarithm of
the result with respect to the chemical potential $\mu$. Of course we shall
not see this being done explicitly, but the gravitational formula (3.5) for
the index that we shall use does this automatically. If the index is a smooth
function of the charges then the result of this computation differs from the
actual index by subleading logarithmic terms that we do not keep track of.
However if the index is not a smooth function of the charges, e.g. if it
switches sign as we move from one element of the charge lattice to its
neighbouring element, then the procedure described above can give results that
differ significantly from the actual index, – see e.g. [14] for similar issues
for AdS back holes. For heterotic string compactification we know that the
microscopic index is a smooth function of the charges for large charge. We
shall assume that this holds on the macroscopic side as well.

##  4 The black hole solution

We begin by writing down the two derivative action describing the massless
bosonic fields in heterotic string theory compactified on $T^{10-D}$ [19]:

$\displaystyle\displaystyle S$ $\displaystyle=$ $\displaystyle C_{D}\int
d^{D}x\sqrt{-\det
G}\,e^{-\Phi}\,\Big{[}R_{G}+G^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi+{1\over
8}G^{\mu\nu}\,\mathrm{Tr}(\partial_{\mu}ML\partial_{\nu}ML)$ (4.1)
$\displaystyle-{1\over
12}G^{\mu\mu^{\prime}}G^{\nu\nu^{\prime}}G^{\rho\rho^{\prime}}H_{\mu\nu\rho}H_{\mu^{\prime}\nu^{\prime}\rho^{\prime}}-G^{\mu\mu^{\prime}}G^{\nu\nu^{\prime}}F^{(j)}_{\mu\nu}\,(LML)_{jk}\,F^{(k)}_{\mu^{\prime}\nu^{\prime}}\Big{]}\,,$

where,

$F^{(j)}_{\mu\nu}=\partial_{\mu}A^{(j)}_{\nu}-\partial_{\nu}A^{(j)}_{\mu}\,,$
(4.2)
$H_{\mu\nu\rho}=\partial_{\mu}B_{\nu\rho}+2A_{\mu}^{(j)}L_{jk}F^{(k)}_{\nu\rho}+\partial_{\nu}B_{\rho\mu}+2A_{\nu}^{(j)}L_{jk}F^{(k)}_{\rho\mu}+\partial_{\rho}B_{\mu\nu}+2A_{\rho}^{(j)}L_{jk}F^{(k)}_{\mu\nu},$
(4.3)

and $R_{G}$ is the Ricci scalar associated with the metric $G_{\mu\nu}$. $M$
and $L$ have been introduced in (2.1). $C_{D}$ is an arbitrary $D$-dependent
constant, which can be absorbed into the dilaton field $\Phi$. This action is
invariant under an $O(26-D,10-D)$ symmetry under which

$M\to\Omega M\Omega^{T},\qquad
A_{\mu}^{(j)}\to\Omega_{jk}A_{\mu}^{(k)},\qquad\forall\ \Omega\ \hbox{such
that}\ \Omega^{T}L\,\Omega=L\,.$ (4.4)

Furthermore this symmetry is known to survive $\alpha^{\prime}$ correction to
the effective action, i.e. this is an exact symmetry of tree level string
theory[20, 21].

$G_{\mu\nu}$ appearing in (4.1) is the string frame metric. The canonical
Einstein frame metric $g_{\mu\nu}$ is related to this via the equation:

$g_{(can)\mu\nu}=g_{s}^{4/(D-2)}e^{-2\Phi/(D-2)}G_{\mu\nu}\,,$ (4.5)

where $g_{s}^{2}$ is the asymptotic value of $e^{\Phi}$. The $g_{s}$ dependent
factor is normally not included in the definition of the Einstein frame
metric, but we have included it here so that if $G_{\mu\nu}$ approaches
$\eta_{\mu\nu}$ asymptotically, so does $g_{\mu\nu}$. We shall always work
with the string frame metric unless mentioned otherwise.

###  4.1 The solution

Let $\alpha$ and $\gamma$ be two boost parameters. Let $\vec{n}$ be a $(26-D)$
dimensional unit vector and $\vec{p}$ be a $(10-D)$ dimensional unit vector.
The general non-extremal solution in $D$-dimensions with single rotation takes
the form [22]:

$\displaystyle\displaystyle ds^{2}$ $\displaystyle\equiv$ $\displaystyle
G_{\mu\nu}dx^{\mu}dx^{\nu}$ (4.6) $\displaystyle=$
$\displaystyle(\rho^{2}+a^{2}\cos^{2}\theta)\Big{\\{}-\Delta^{-1}(\rho^{2}+a^{2}\cos^{2}\theta-2m\rho^{5-D})dt^{2}+(\rho^{2}+a^{2}-2m\rho^{5-D})^{-1}d\rho^{2}+d\theta^{2}$
$\displaystyle+\Delta^{-1}\sin^{2}\theta[\Delta+a^{2}\sin^{2}\theta(\rho^{2}+a^{2}\cos^{2}\theta+2m\rho^{5-D}\cosh\alpha\cosh\gamma)]\,d\phi^{2}$
$\displaystyle-2\Delta^{-1}m\rho^{5-D}a\sin^{2}\theta(\cosh\alpha+\cosh\gamma)dtd\phi+\rho^{2}\cos^{2}\theta(\rho^{2}+a^{2}\cos^{2}\theta)^{-1}d\Omega^{D-4}\Big{\\}}\,,$

where,

$\Delta=(\rho^{2}+a^{2}\cos^{2}\theta)^{2}+2m\rho^{5-D}(\rho^{2}+a^{2}\cos^{2}\theta)(\cosh\alpha\cosh\gamma-1)+m^{2}\rho^{10-2D}(\cosh\alpha-\cosh\gamma)^{2}.$
(4.7)

$\\{t,\rho,\theta,\phi\\}$ are the four spacetime coordinates and
$d\Omega^{D-4}$ is the round metric on a $(D-4)$\- dimensional unit sphere.
The dilaton supporting the solution is:

$\Phi=2\ln g_{s}+{1\over
2}\ln{(\rho^{2}+a^{2}\cos^{2}\theta)^{2}\over\Delta}.$ (4.8)

The U(1) vector fields $A^{(j)}$ for $1\leq j\leq(26-D)$ are determined by the
unit vector $\vec{n}$:

$\displaystyle\displaystyle A^{(j)}_{t}$ $\displaystyle=$
$\displaystyle-{n^{(j)}\over\sqrt{2}}\Delta^{-1}m\rho^{5-D}\sinh\alpha\\{(\rho^{2}+a^{2}\cos^{2}\theta)\cosh\gamma+m\rho^{5-D}(\cosh\alpha-\cosh\gamma)\\},$
$\displaystyle A^{(j)}_{\phi}$ $\displaystyle=$
$\displaystyle{n^{(j)}\over\sqrt{2}}\Delta^{-1}m\rho^{5-D}a\sinh\alpha\sin^{2}\theta\\{\rho^{2}+a^{2}\cos^{2}\theta+m\rho^{5-D}\cosh\gamma(\cosh\alpha-\cosh\gamma)\\}\,,$

and the remaining vector fields $A^{(j)}$ for $j\geq(27-D)$ are determined by
the unit vector $\vec{p}$:

$\displaystyle\displaystyle A^{(j)}_{t}$ $\displaystyle=$
$\displaystyle-{p^{(j-26+D)}\over\sqrt{2}}\Delta^{-1}m\rho^{5-D}\sinh\gamma\\{(\rho^{2}+a^{2}\cos^{2}\theta)\cosh\alpha+m\rho^{5-D}(\cosh\gamma-\cosh\alpha)\\},$
$\displaystyle A^{(j)}_{\phi}$ $\displaystyle=$
$\displaystyle{p^{(j-26+D)}\over\sqrt{2}}\Delta^{-1}m\rho^{5-D}a\sinh\gamma\sin^{2}\theta\\{\rho^{2}+a^{2}\cos^{2}\theta+m\rho^{5-D}\cosh\alpha(\cosh\gamma-\cosh\alpha)\\}.$

The two-form field takes the form,

$B_{t\phi}=\Delta^{-1}m\rho^{5-D}a\sin^{2}\theta(\cosh\alpha-\cosh\gamma)\\{\rho^{2}+a^{2}\cos^{2}\theta+m\rho^{5-D}(\cosh\alpha\cosh\gamma-1)\\}\,,\\\
$

and the matrix of scalars takes the form,

$M=I_{36-2D}+\pmatrix{Pnn^{T}&Unp^{T}\cr Upn^{T}&Ppp^{T}}\,,$ (4.11)

where,

$\displaystyle\displaystyle P$ $\displaystyle=$ $\displaystyle
2\Delta^{-1}m^{2}\rho^{10-2D}\sinh^{2}\alpha\sinh^{2}\gamma\,,$ $\displaystyle
U$ $\displaystyle=$
$\displaystyle-2\Delta^{-1}m\rho^{5-D}\sinh\alpha\sinh\gamma\\{\rho^{2}+a^{2}\cos^{2}\theta+m\rho^{5-D}(\cosh\alpha\cosh\gamma-1)\\}\,.$
(4.12)

Since $P$ and $U$ fall off for large $\rho$, we see from (4.11) that
$M_{\infty}=I_{36-2D}$. However using the symmetry transformation (4.4) we can
generate any other $M_{\infty}$ from this solution, and hence setting
$M_{\infty}$ to $I_{36-2D}$ does not restrict the solution in any way.

###  4.2 BPS limit and Euclidean continuation

The BPS limit is obtained by taking $m\to 0$, $\gamma\to\infty$ keeping
$m_{0}\equiv m\cosh\gamma$ finite. The BPS solution is also written in [22].
Let us in addition consider the analytic continuation to Euclidean space,
together with a redefinition of the rotation parameter $a$ and the fields:

$\displaystyle t=\mathrm{i}\,\tau,$ $\displaystyle a=\mathrm{i}\,b,\qquad
J=\mathrm{i}J_{E},\qquad\Omega=-\mathrm{i}\,\Omega_{E},$ $\displaystyle
A^{(j)}=\mathrm{i}A_{(E)}^{(j)},$ $\displaystyle B=-B_{(E)}.$ (4.13)

The last analytic continuation is required for consistency with (4.3). This
gives the following Euclidean solution,

$\displaystyle\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle(\rho^{2}-b^{2}\cos^{2}\theta)\Big{\\{}\Delta^{-1}(\rho^{2}-b^{2}\cos^{2}\theta)d\tau^{2}+(\rho^{2}-b^{2})^{-1}d\rho^{2}+d\theta^{2}$
(4.14)
$\displaystyle+\Delta^{-1}\sin^{2}\theta[\Delta-b^{2}\sin^{2}\theta(\rho^{2}-b^{2}\cos^{2}\theta+2m_{0}\rho^{5-D}\cosh\alpha)]\,d\phi^{2}$
$\displaystyle+2\Delta^{-1}m_{0}\rho^{5-D}b\sin^{2}\theta d\tau
d\phi+\rho^{2}\cos^{2}\theta(\rho^{2}-b^{2}\cos^{2}\theta)^{-1}d\Omega^{D-4}\Big{\\}},$

where now

$\Delta=(\rho^{2}-b^{2}\cos^{2}\theta)^{2}+2m_{0}\rho^{5-D}\cosh\alpha\,(\rho^{2}-b^{2}\cos^{2}\theta)+m_{0}^{2}\rho^{10-2D}\,,$
(4.15) $\Phi=2\ln g_{s}+{1\over
2}\ln{(\rho^{2}-b^{2}\cos^{2}\theta)^{2}\over\Delta}.$ (4.16)

The gauge fields are given by

$\displaystyle\displaystyle A_{(E)}^{(j)}{}_{\tau}$ $\displaystyle=$
$\displaystyle-{n^{(j)}\over\sqrt{2}}\Delta^{-1}m_{0}\rho^{5-D}\sinh\alpha(\rho^{2}-b^{2}\cos^{2}\theta)\,,$
$\displaystyle A_{(E)}^{(j)}{}_{\phi}$ $\displaystyle=$
$\displaystyle-{n^{(j)}\over\sqrt{2}}\Delta^{-1}(m_{0})^{2}\rho^{10-2D}b\sinh\alpha\sin^{2}\theta,\qquad\hbox{for
$1\leq j\leq 26-D$}\,,$ (4.17)

and

$\displaystyle\displaystyle A_{(E)}^{(j)}{}_{\tau}$ $\displaystyle=$
$\displaystyle-{p^{(j-26+D)}\over\sqrt{2}}\Delta^{-1}m_{0}\rho^{5-D}\\{(\rho^{2}-b^{2}\cos^{2}\theta)\cosh\alpha+m_{0}\rho^{5-D}\\}\,,$
$\displaystyle A_{(E)}^{(j)}{}_{\phi}$ $\displaystyle=$
$\displaystyle{p^{(j-26+D)}\over\sqrt{2}}\Delta^{-1}m_{0}\rho^{5-D}b\sin^{2}\theta\\{\rho^{2}-b^{2}\cos^{2}\theta+m_{0}\rho^{5-D}\cosh\alpha\\},$
(4.18) $\displaystyle\hskip 72.26999pt\hbox{for $j\geq 27-D$}\,.$

The two form field takes the form,

$B_{(E)\tau\phi}=-\Delta^{-1}m_{0}\rho^{5-D}b\sin^{2}\theta\\{\rho^{2}-b^{2}\cos^{2}\theta+m_{0}\rho^{5-D}\cosh\alpha\\}\,,$
(4.19)

and the scalar fields are given by,

$M=I_{36-2D}+\pmatrix{Pnn^{T}&Unp^{T}\cr Upn^{T}&Ppp^{T}}\,,$ (4.20)

where

$\displaystyle\displaystyle P$ $\displaystyle=$ $\displaystyle
2\Delta^{-1}m_{0}^{2}\rho^{10-2D}\sinh^{2}\alpha,$ $\displaystyle U$
$\displaystyle=$
$\displaystyle-2\Delta^{-1}m_{0}\rho^{5-D}\sinh\alpha\\{\rho^{2}-b^{2}\cos^{2}\theta+m_{0}\rho^{5-D}\cosh\alpha\\}.$
(4.21)

Note that with the redefinitions given in (LABEL:eanalytic) the solution is
real. The horizon is at $\rho=b$ and the relevant region of space-time is
$\rho\geq b$. Also the solution is singular at $\rho=b$, $\cos^{2}\theta=1$.
This is easiest to see by examining the dilaton given in (4.16), which blows
up at these values.

We now compute the electric charges carried by the solution. For this we note
that for large $\rho$, we have

$F^{(j)}_{(E)\rho\tau}\simeq\partial_{\rho}A^{(j)}_{(E)\tau}=\cases{(D-3){n^{(j)}\over\sqrt{2}}m_{0}\,\rho^{2-D}\,\sinh\alpha\quad\hbox{for
$1\leq j\leq
26-D$}\cr(D-3){p^{(j-26+D)}\over\sqrt{2}}m_{0}\,\rho^{2-D}\,\cosh\alpha\quad\hbox{for
$27-D\leq j\leq 36-D$}}\,.$ (4.22)

We define the electric charges $Q^{(j)}$ as the integral of ${1\over 2}{\delta
S\over\delta F^{(j)}_{(E)\rho\tau}}$ over the $(D-2)$ dimensional sphere at a
large but fixed value of $\rho$.111We have used the factor of 2 in the
denominator to compensate for the fact that the gauge field kinetic term in
(4.1) does not have the usual $1/4$. We shall see that this normalization is
consistent with the convention of section 2. Using the fact that
$e^{-\Phi_{\infty}}=1/g_{s}^{2}$ and $M_{\infty}=I_{36-2D}$,

$Q=\sqrt{2}\,(D-3)\,C_{D}\,g_{s}^{-2}\,m_{0}\,\omega_{D-2}\,\pmatrix{n\sinh\alpha\cr
p\cosh\alpha}\equiv\pmatrix{Q_{L}\cr Q_{R}}\,,$ (4.23)

where $\omega_{k}$ represents the volume of a unit sphere in $k$ dimensions.
Since $n$ and $p$ are unit vectors, we get

$\sqrt{Q_{R}^{2}-Q_{L}^{2}}=\sqrt{Q^{T}LQ}=\sqrt{2}\,(D-3)\,C_{D}\,g_{s}^{-2}\,m_{0}\,\omega_{D-2}\,.$
(4.24)

We can also calculate the ADM mass of the black hole. For this we use (4.14),
(4.16) and (4.5) to write down the expression for the canonical Einstein
metric:

$\displaystyle\displaystyle ds_{(can)}^{2}$ $\displaystyle=$
$\displaystyle(\rho^{2}-b^{2}\cos^{2}\theta)\left({\rho^{2}-b^{2}\cos^{2}\theta\over\Delta^{1/2}}\right)^{-2/(D-2)}$
(4.25)
$\displaystyle\Big{\\{}\Delta^{-1}(\rho^{2}-b^{2}\cos^{2}\theta)d\tau^{2}+(\rho^{2}-b^{2})^{-1}d\rho^{2}+d\theta^{2}$
$\displaystyle+\Delta^{-1}\sin^{2}\theta[\Delta-b^{2}\sin^{2}\theta(\rho^{2}-b^{2}\cos^{2}\theta+2m_{0}\rho^{5-D}\cosh\alpha)]\,d\phi^{2}$
$\displaystyle+2\Delta^{-1}m_{0}\rho^{5-D}b\sin^{2}\theta d\tau
d\phi+\rho^{2}\cos^{2}\theta(\rho^{2}-b^{2}\cos^{2}\theta)^{-1}d\Omega^{D-4}\Big{\\}},$

Using this and eq.(14) of [23], we get

$m_{ADM}={(D-2)\,\omega_{D-2}\over 16\pi
G_{N}}\lim_{\rho\to\infty}\rho^{D-3}(1-g_{\tau\tau})={\omega_{D-2}\over 8\pi
G_{N}}\,(D-3)\,m_{0}\cosh\alpha\,,$ (4.26)

where the Newton’s constant $G_{N}$ for the action (4.1) is given by

$G_{N}={1\over 16\pi C_{D}\,g_{s}^{-2}}\,.$ (4.27)

This gives

$m_{ADM}=2\,\omega_{D-2}\,C_{D}\,g_{s}^{-2}\,(D-3)\,m_{0}\cosh\alpha=\sqrt{2\,Q_{R}^{2}}\,,$
(4.28)

in agreement with (2.4).

The overall normalization of the charges depends on the normalization of the
U(1) gauge fields appearing in (4.1). We have chosen the normalization so that
we get (4.28). We have also carefully kept track of all dependence on the
asymptotic moduli and the dimension $D$. In particular, if we use the
transformation (4.4) to generate a solution with some other $M_{\infty}$, the
dependence of $m_{ADM}$ on $M_{\infty}$ and the charge $Q$ will be captured by
the dependence of $Q_{R}^{2}$ on $M_{\infty}$ and $Q$ via (2.3), since (2.3)
is invariant under $O(26-D,10-D)$ transformation.

The area of the $\rho=b$ surface in the Einstein frame is

$A_{H}=m_{0}b\,\omega_{D-2}\,.$ (4.29)

The Bekenstein-Hawking entropy of the black hole is given by

$S_{BH}=\frac{A_{H}}{4G_{N}}={m_{0}b\,\omega_{D-2}\over 4G_{N}}\,.$ (4.30)

The Euclidean angular momentum $J_{E}$ of the black hole is,

$J_{E}=\frac{\omega_{D-2}}{8\pi G_{N}}m_{0}b.$ (4.31)

Clearly,

$S_{BH}-2\pi J_{E}=0.$ (4.32)

This reproduces (3.6) with the identification given in (LABEL:eanalytic).

###  4.3 Periodicity of the $\tau$ circle

To understand the periodicity of the $\tau$ circle, let us introduce

$\displaystyle\displaystyle\widetilde{\rho}$ $\displaystyle=\sqrt{\rho-b},$
$\displaystyle\phi$
$\displaystyle=\widetilde{\phi}-\frac{b^{D-4}}{m_{0}}\tau.$ (4.33)

With these coordinate changes, the relevant part of the metric near
$\widetilde{\rho}=0$ is

$\displaystyle\displaystyle ds^{2}$ $\displaystyle\simeq$ $\displaystyle
2b\sin^{2}\theta\left\\{\left(d\widetilde{\rho}^{2}+\widetilde{\rho}^{2}\frac{b^{2D-8}}{m_{0}^{2}}d\tau^{2}\right)\right.$
$\displaystyle\left.+\frac{b}{2}\left(d\theta^{2}+\frac{m_{0}^{2}}{m_{0}^{2}+2b^{D-3}m_{0}\cosh\alpha\sin^{2}\theta+b^{2D-6}\sin^{4}\theta}\sin^{2}\theta
d\widetilde{\phi}^{2}+\cot^{2}\theta d\Omega^{D-4}\right)\right\\}.$

In writing this metric, we have dropped the terms of order
$\widetilde{\rho}^{2}d\tau d\widetilde{\phi}$ and various other higher order
terms in the expansion in $\widetilde{\rho}$. Away from the singular subspace
on which $\sin\theta$ vanishes, the $(\widetilde{\rho},\tau)$ part of the
metric is smooth at $\widetilde{\rho}=0$ provided $\tau$ has periodicity

$(\tau,\widetilde{\phi})\equiv(\tau+\beta,\widetilde{\phi})\quad\mbox{where}\quad\beta=2\pi\frac{m_{0}}{b^{D-4}}\,.$
(4.35)

Using (4.3), this translates to

$(\tau,\phi)\equiv\left(\tau+\beta,\phi-{b^{D-4}\over m_{0}}\beta\right)\,.$
(4.36)

Of course $\phi$ itself has a periodic identification under shift by $2\pi$ in
order that the metric (4.40) is non-singular for $\rho>b$. From this we can
determine the angular velocity of Euclidean black hole to be

$\Omega_{E}=\frac{b^{D-4}}{m_{0}}.$ (4.37)

(4.35) and (4.37) give,

$\beta\Omega_{E}=2\pi,$ (4.38)

in agreement with (3.4).

###  4.4 $(D-1)$-dimensional base and the singular subspace

Remarkably, the string frame metric (4.14) takes the following form in general
dimensions222Ref. [22] also noted this with $\alpha=0$.

$ds^{2}=f(d\tau+\omega_{\phi}d\phi)^{2}+ds^{2}_{\mathrm{base}},$ (4.39)

where

$ds^{2}_{\mathrm{base}}=\left(\frac{\rho^{2}-b^{2}\cos^{2}\theta}{\rho^{2}-b^{2}}\right)d\rho^{2}+(\rho^{2}-b^{2}\cos^{2}\theta)d\theta^{2}+(\rho^{2}-b^{2})\sin^{2}\theta
d\phi^{2}+\rho^{2}\cos^{2}\theta d\Omega^{D-4},$ (4.40)
$f=\frac{(\rho^{2}-b^{2}\cos^{2}\theta)^{2}}{\Delta}=\left(1+\frac{1}{\rho^{D-5}}\frac{m_{0}e^{\alpha}}{\rho^{2}-b^{2}\cos^{2}\theta}\right)^{-1}\left(1+\frac{1}{\rho^{D-5}}\frac{m_{0}e^{-\alpha}}{\rho^{2}-b^{2}\cos^{2}\theta}\right)^{-1},$
(4.41)

and

$\omega_{\phi}=bm_{0}\frac{\rho^{5-D}}{\rho^{2}-b^{2}\cos^{2}\theta}\sin^{2}\theta.$
(4.42)

The base metric $ds^{2}_{\mathrm{base}}$ does not depend on the charges and is
in fact flat. To see this, let us introduce,

$\displaystyle\displaystyle y_{1}$ $\displaystyle=$
$\displaystyle\sqrt{\rho^{2}-b^{2}}\,\sin\theta\,\cos\phi,$ $\displaystyle
y_{2}$ $\displaystyle=$
$\displaystyle\sqrt{\rho^{2}-b^{2}}\,\sin\theta\,\sin\phi,$ $\displaystyle
y_{3}$ $\displaystyle=$ $\displaystyle\rho\,\cos\theta\cos\psi_{1}$
$\displaystyle y_{4}$ $\displaystyle=$
$\displaystyle\rho\cos\theta\sin\psi_{1}\cos\psi_{2}\,,$ $\displaystyle\cdot$
$\displaystyle\cdot$ $\displaystyle y_{D-2}$ $\displaystyle=$
$\displaystyle\rho\cos\theta\sin\psi_{1}\cdots\sin\psi_{D-5}\cos\psi_{D-4}\,,$
$\displaystyle y_{D-1}$ $\displaystyle=$
$\displaystyle\rho\cos\theta\sin\psi_{1}\cdots\sin\psi_{D-5}\sin\psi_{D-4}\,,$

where $\psi_{1},\ldots\psi_{D-4}$ are the angles labelling points on the
$(D-4)$ sphere. In this coordinate system the base metric takes the form:

$ds_{\rm base}^{2}=dy_{1}^{2}+dy_{2}^{2}+dy_{3}^{2}+\ldots+dy^{2}_{D-1}.$
(4.44)

Despite having a flat base, the metric is singular is at $\rho=b$,
$\cos^{2}\theta=1$ since $f$ vanishes and $\omega_{\phi}$ blows up there. To
determine the geometry of the singular surface, note from (4.4) that $y_{1}$
and $y_{2}$ vanish on the singular subspace and $y_{3},\cdots,y_{D-1}$ lie on
the surface of a $(D-4)$ dimensional sphere of radius $b$. The range of the
coordinate $\theta$ is $(0,\pi)$ for $D=4$ whereas for $D>4$ it is
$(0,\frac{\pi}{2})$. Therefore for $D>4$, we have a connected singular surface
at $\rho=b$, $\theta=0$ which is a $S^{D-4}$ sphere. In $D=4$ the singular
subspace is a pair of points corresponding to $\rho=b$, $\theta=0,\pi$. Since
$S^{0}$ consists of a pair of points, we can describe the singular surface as
$S^{D-4}$ for all $D$. This will be important later.

$f$ computed from (4.41) can be written as

$f^{-1}=\left(1+\frac{m_{0}e^{\alpha}}{2\,\rho^{D-4}}\left(\frac{1}{R_{+}}+\frac{1}{R_{-}}\right)\right)\left(1+\frac{m_{0}e^{-\alpha}}{2\,\rho^{D-4}}\left(\frac{1}{R_{+}}+\frac{1}{R_{-}}\right)\right),$
(4.45)

where

$R_{\pm}=\rho\pm b\cos\theta\,.$ (4.46)

In $D=4$, $R_{\pm}$ have the interpretation of the distance of a point from
the locations of the singularities at $\rho=b$, $\theta=0,\pi$. In this case
$f^{-1}$ is simply a product of two harmonic functions and the solution can be
written in a standard IWP form[15, 24].

##  5 The geometry near the singularity

Eq. (4.32) shows that in the two derivative approximation to the effective
action, the macroscopic index vanishes. According to the scaling argument
given between (3.6) and (3.7), higher derivative corrections to $S_{wald}$
from the non-singular regions cannot produce the answer known from the
microscopic counting. We shall now focus on the singular regions near the
horizon and explore whether higher derivative correction to $S_{wald}$ from
the singular region can produce the desired result.

###  5.1 Coordinates centered at the singular surface

In order to analyse the solution near the singular surface $\cos\theta=1$, we
introduce the following coordinates near the singular subspace. Define,

$\widetilde{R}=\rho-b\cos\theta,\qquad\cos\widetilde{\theta}={\rho\cos\theta-b\over\rho-b\cos\theta}.$
(5.1)

The inverse coordinate transformations are,

$\rho=\frac{1}{2}\left(\widetilde{R}+\sqrt{\widetilde{R}^{2}+4b\widetilde{R}\cos\widetilde{\theta}+4b^{2}}\right),\qquad\cos\theta=\frac{1}{2b}\left(-\widetilde{R}+\sqrt{\widetilde{R}^{2}+4b\widetilde{R}\cos\widetilde{\theta}+4b^{2}}\right).$
(5.2)

$\widetilde{\theta}$ has range $(0,\pi)$ for all $D$. These coordinates have
the important property that when $\widetilde{R}=0$, we have $\rho=b$ and
$\cos\theta=1$, i.e., we are at the singular surface. For $D=4$ the second
copy of the singular subspace is at $\widetilde{R}=2b$,
$\widetilde{\theta}=\pi$, but near that region it is more appropriate to
introduce new coordinate system by replacing $\cos\theta$ by $-\cos\theta$ in
(5.1).

###  5.2 Large charge limit

We now make a final coordinate transformation,

$\widetilde{\theta}=2\Theta,\qquad\widetilde{\phi}=\phi+\frac{\tau}{m_{0}b^{4-D}},\qquad\widetilde{\tau}=\frac{\tau}{m_{0}b^{4-D}},$
(5.3)

and take the limit $m_{0}\to\infty$ and $b\to\infty$ by taking

$m_{0}\sim\lambda^{D-3},\qquad b\sim\lambda,\qquad\lambda\to\infty\,,$ (5.4)

keeping fixed
$\widetilde{R},\widetilde{\theta},\widetilde{\tau},\widetilde{\phi},\Omega^{D-4}$.
The scaling of $b$ follows from (3.2) after noting that $J_{E}\propto m_{0}b$.
The limit gives

$\displaystyle\displaystyle\rho$ $\displaystyle\simeq$ $\displaystyle
b+\frac{\widetilde{R}}{2}(1+\cos\widetilde{\theta})$ (5.5)
$\displaystyle\cos\theta$ $\displaystyle\simeq$ $\displaystyle
1-\frac{\widetilde{R}}{2b}(1-\cos\widetilde{\theta})$ (5.6)
$\displaystyle\sin^{2}\theta$ $\displaystyle\simeq$
$\displaystyle\frac{\widetilde{R}}{b}(1-\cos\widetilde{\theta})$ (5.7)
$\displaystyle(\rho^{2}-b^{2}\cos^{2}\theta)$ $\displaystyle\simeq$
$\displaystyle 2b\widetilde{R}$ (5.8) $\displaystyle\Delta$
$\displaystyle\simeq$ $\displaystyle
m_{0}^{2}b^{10-2D}-b^{6-2D}m_{0}\widetilde{R}\left\\{(D-5)b^{3}m_{0}(1+\cos\widetilde{\theta})-4b^{D}\cosh\alpha\right\\}$
(5.9) $\displaystyle f$ $\displaystyle\simeq$
$\displaystyle\frac{4\widetilde{R}^{2}}{m_{0}^{2}b^{8-2D}}$ (5.10)
$\displaystyle\omega_{\phi}$ $\displaystyle\simeq$ $\displaystyle
m_{0}b^{4-D}\sin^{2}\frac{\widetilde{\theta}}{2}.$ (5.11)

We have kept terms in the expansion to the order that will be needed in our
computation of the solution up to terms that vanish for large $\lambda$.
Combining all this together and using (4.39), we arrive at a universal form
for the metric:

$ds^{2}\simeq
d\widetilde{R}^{2}+4\widetilde{R}^{2}d\Theta^{2}+4\widetilde{R}^{2}\sin^{2}\Theta
d\widetilde{\phi}^{2}+4\widetilde{R}^{2}\cos^{2}\Theta
d\widetilde{\tau}^{2}+b^{2}d\Omega^{D-4}.$ (5.12)

$\Theta$ has range $\left(0,\frac{\pi}{2}\right)$ and $\widetilde{\phi}$ and
$\widetilde{\tau}$ have periodicities $2\pi$. The metric is singular at
$\widetilde{R}=0$. Since $b$ is large, we can replace $b^{2}d\Omega^{D-4}$
with $\sum_{i=1}^{D-4}dx_{i}^{2}$ locally.

From this we see that near the singularity the metric has a universal form
independent of $D$. If on the other hand we take a spherically symmetric
solution by taking the rotation parameter $b$ to zero, the size of the $(D-4)$
dimensional sphere goes to zero and the nature of the singularity becomes
dependent on $D$[9].

We now turn to the matter fields in the large charge limit at fixed
$\widetilde{R}$, $\widetilde{\tau}$, $\widetilde{\phi}$, $\Theta$. In this
limit (4.2) gives

$\displaystyle\displaystyle P$ $\displaystyle\simeq$ $\displaystyle
2\sinh^{2}\alpha\,,$ $\displaystyle U$ $\displaystyle\simeq$ $\displaystyle\
-2\sinh\alpha\cosh\alpha\,.$ (5.13)

(4.20) now implies,

$M\simeq
I_{36-2D}+\pmatrix{2\sinh^{2}\alpha\,\,(nn^{T})&-2\sinh\alpha\cosh\alpha\,\,(np^{T})\cr-2\sinh\alpha\cosh\alpha\,\,(pn^{T})&2\sinh^{2}\alpha\,\,(pp^{T})}\,.$
(5.14)

For the gauge fields, we get, for $1\leq j\leq 26-D$,

$\displaystyle\displaystyle A_{(E)}^{(j)}{}_{\widetilde{\tau}}$
$\displaystyle\simeq$
$\displaystyle-\sqrt{2}n^{(j)}\cos^{2}\Theta\sinh\alpha\,\widetilde{R},$
$\displaystyle A_{(E)}^{(j)}{}_{\widetilde{\phi}}$ $\displaystyle\simeq$
$\displaystyle-\sqrt{2}n^{(j)}\sin^{2}\Theta\sinh\alpha\,\widetilde{R}\,,$
(5.15)

and for $j\geq 27-D$,

$\displaystyle\displaystyle A_{(E)}^{(j)}{}_{\widetilde{\tau}}$
$\displaystyle\simeq$
$\displaystyle-{p^{(j-26+D)}\over\sqrt{2}}b^{4-D}m_{0}+\sqrt{2}p^{(j-26+D)}\cos^{2}\Theta\cosh\alpha\,\widetilde{R},$
$\displaystyle A_{(E)}^{(j)}{}_{\widetilde{\phi}}$ $\displaystyle\simeq$
$\displaystyle\sqrt{2}p^{(j-26+D)}\sin^{2}\Theta\cosh\alpha\,\widetilde{R}\,.$
(5.16)

The constant additive term in the expression for
$A_{(E)}^{(j)}{}_{\widetilde{\tau}}$ contributes to the three form field
strength defined in (4.3), but can be dropped after computing this field
strength.

For the two form field, we shall directly write down the expression for the
gauge invariant three form field strength after taking into account the
contribution from the Chern-Simons term in (4.3). This takes the form:

$H_{(E)}\equiv{1\over 6}H_{(E)\mu\nu\rho}dx^{\mu}\wedge dx^{\nu}\wedge
dx^{\rho}\simeq 8\,\widetilde{R}^{2}\,\sin\Theta\,\cos\Theta\,d\Theta\wedge
d\widetilde{\phi}\wedge d\widetilde{\tau}\,,$ (5.17)

where

$H_{(E)\mu\nu\rho}\equiv\partial_{\mu}B_{(E)\nu\rho}+2A_{(E)\mu}^{(j)}L_{jk}F^{(k)}_{(E)\nu\rho}+\partial_{\nu}B_{(E)\rho\mu}+2A_{(E)\nu}^{(j)}L_{jk}F^{(k)}_{(E)\rho\mu}+\partial_{\rho}B_{(E)\mu\nu}+2A_{(E)\rho}^{(j)}L_{jk}F^{(k)}_{(E)\mu\nu}\,.$
(5.18)

Finally, the dilaton behaves as follows,

$\exp[-\Phi]\simeq g_{s}^{-2}\,\frac{b^{4-D}m_{0}}{2\widetilde{R}}.$ (5.19)

If we regard this near horizon geometry as a solution in four dimensional
heterotic string theory by regarding the coordinates along $\Omega^{D-4}$ as
describing an almost flat compact space, then the canonical metric in four
dimensions is given by $e^{-\Phi}G_{\mu\nu}$. This describes a flat four
dimensional metric, as can be seen by making a change of coordinates
$\widetilde{R}=\widetilde{r}^{2}$. The significance of this observation is not
clear to us.

Note that the matrix valued scalar field $M$ and the gauge fields depend of
the parameter $\alpha$. The $\alpha$ dependence on the background $M$ can be
removed by doing a $O(26-D,10-D)$ transformation described in (4.4) by
choosing,

$\Omega=I_{36-2D}+\pmatrix{(\cosh\alpha-1)\,\,(nn^{T})&\sinh\alpha\,\,(np^{T})\cr\sinh\alpha\,\,(pn^{T})&(\cosh\alpha-1)\,\,(pp^{T})}\,.$
(5.20)

This gives

$M\simeq I_{36-2D}.$ (5.21)

The vector fields also change under this. We get, for $1\leq j\leq 26-D$,

$\displaystyle\displaystyle A{}{}_{(E)}^{(j)}{}_{\widetilde{\tau}}$
$\displaystyle\simeq$ $\displaystyle 0,$ $\displaystyle
A{}{}_{(E)}^{(j)}{}_{\widetilde{\phi}}{}$ $\displaystyle\simeq$ $\displaystyle
0\,,$ (5.22)

and for $j\geq 27-D$,

$\displaystyle\displaystyle A{}{}_{(E)}^{(j)}{}_{\widetilde{\tau}}$
$\displaystyle\simeq$
$\displaystyle\sqrt{2}p^{(j-26+D)}\widetilde{R}\cos^{2}\Theta,$ $\displaystyle
A{}{}_{(E)}^{(j)}{}_{\widetilde{\phi}}$ $\displaystyle\simeq$
$\displaystyle\sqrt{2}p^{(j-26+D)}\widetilde{R}\sin^{2}\Theta\,,$ (5.23)

where we have dropped the constant term in the expression for
$A{}{}_{(E)}^{(j)}{}_{\widetilde{\tau}}$. Since we no longer need to use the
gauge field for computing $H_{(E)}$, this is an allowed operation. The new
fields do not depend on the parameter $\alpha$. Furthermore using an
$O(10-D)\subset O(26-D,10-D)$ rotation we could bring the unit vector $p$ to
any standard form, e.g. having only the first component non-zero.

##  6 Wald entropy and the index

Let us summarize the solution in the large charge limit after the
$O(26-D,10-D)$ rotation (5.20). We have,

$\displaystyle\displaystyle ds^{2}$ $\displaystyle\simeq$ $\displaystyle
d\widetilde{R}^{2}+4\widetilde{R}^{2}d\Theta^{2}+4\widetilde{R}^{2}\sin^{2}\Theta
d\widetilde{\phi}^{2}+4\widetilde{R}^{2}\cos^{2}\Theta
d\widetilde{\tau}^{2}+b^{2}d\Omega^{D-4},$ (6.1) $\displaystyle\hskip
72.26999pt0\leq\Theta\leq{\pi\over
2},\qquad\widetilde{\phi}\equiv\widetilde{\phi}+2\pi,\qquad\widetilde{\tau}\equiv\widetilde{\tau}+2\pi\,,$
$M\simeq I_{36-2D}\,,$ (6.2) $\displaystyle\displaystyle
A{}{}_{(E)}^{(j)}{}_{\widetilde{\tau}}$ $\displaystyle\simeq$ $\displaystyle
0,$ $\displaystyle A{}{}_{(E)}^{(j)}{}_{\widetilde{\phi}}{}$
$\displaystyle\simeq$ $\displaystyle 0\,,\qquad\hbox{for $1\leq j\leq
26-D$}\,,$ (6.3) $\displaystyle\displaystyle
A{}{}_{(E)}^{(j)}{}_{\widetilde{\tau}}$ $\displaystyle\simeq$
$\displaystyle\sqrt{2}p^{(j-26+D)}\widetilde{R}\cos^{2}\Theta,$ $\displaystyle
A{}{}_{(E)}^{(j)}{}_{\widetilde{\phi}}$ $\displaystyle\simeq$
$\displaystyle\sqrt{2}p^{(j-26+D)}\widetilde{R}\sin^{2}\Theta\,,\qquad\hbox{for
$j\geq 27-D$}\,,$ (6.4) $H_{(E)}\simeq
8\,\widetilde{R}^{2}\,\sin\Theta\,\cos\Theta\,d\Theta\wedge
d\widetilde{\phi}\wedge d\widetilde{\tau}\,,$ (6.5)

and

$\exp[-\Phi]\simeq g_{s}^{-2}\,\frac{b^{4-D}m_{0}}{2\widetilde{R}}\,.$ (6.6)

We now note the following features of the solution:

  1. 1.

Since the original configuration described in section 4.2 solved the equations
of motion of the two derivative theory for all values of the parameters, and
the configuration given in (6.1)-(6.6) is obtained as a limit of the original
solution for large $m_{0}$, $b$, (6.1)-(6.6) should also satisfy the equations
of motion of the two derivative theory. We have checked this by explicit
computation, after replacing the $b^{2}d\Omega^{D-4}$ part of the metric by
flat metric. However since for finite $\widetilde{R}$ the curvature invariants
and other invariants associated with this solution are of order unity, we
expect the solution to be modified by higher derivative corrections. Such
corrections should fall off for large $\widetilde{R}$ where the invariants are
small.

  2. 2.

In the large charge limit, since $b\sim\lambda$ and $m_{0}\sim\lambda^{D-3}$,
$e^{-\Phi}$ given in (6.6) is of order $\lambda$. Since $e^{-\Phi}$ can be
identified as the inverse square of the string coupling locally, we see that
near the singularity the string coupling is small. Hence we can ignore string
loop corrections and focus only on the $\alpha^{\prime}$ correction to the
solution.

  3. 3.

Since $\alpha^{\prime}$ correction to the effective action does not destroy
the $O(26-D,10-D)$ symmetry, the $O(26-D,10-D)$ transformation that was used
to bring the solution to the form (6.1)-(6.6) is a valid symmetry and leaves
the $\alpha^{\prime}$ corrected formula for $S_{wald}$ invariant. In fact we
can use the $O(10-D)$ subgroup of $O(26-D,10-D)$ to bring the vector $p$ to
some canonical form, e.g. only with the first component non-zero. In this case
the solution is universal except for the $b^{2}d\Omega^{D-4}$ term in the
metric (6.1) and the overall factor of $g_{s}^{-2}\,b^{4-D}m_{0}$ in the
expression for $e^{-\Phi}$ in (6.6). In the limit of large $b$ the
$b^{2}d\Omega^{D-4}$ part of the metric describes almost locally flat $(D-4)$
dimensional space and the $\alpha^{\prime}$ corrections are suppressed by
inverse power of $\lambda$. At the leading order this part of the metric will
give an overall multiplicative factor of $b^{D-4}\omega_{D-4}$ to the action
and hence to $S_{wald}$ but has no effect on the $\alpha^{\prime}$ correction
to the rest of the field configuration. On the other hand from (4.1) we see
that the effect of the $g_{s}^{-2}\,b^{4-D}m_{0}$ in the expression for
$e^{-\Phi}$ is to multiply the action by a factor of
$g_{s}^{-2}\,b^{4-D}m_{0}$ but it also does not affect the $\alpha^{\prime}$
corrections to the solution since these corrections involve $\Phi$ only
through its derivatives. Therefore $\alpha^{\prime}$ corrections to the
solution take a universal form, even independent of $D$, and modify only the
$\widetilde{R}$ and $\Theta$ dependence of various fields in (6.1)-(6.6).

Taking into account the overall factor of $C_{D}$ in (4.1) we see that the net
$\alpha^{\prime}$ correction to $S_{wald}$, and hence to $S_{macro}$ according
to (3.5), (3.6), takes the form,

$S_{macro}=K\times C_{D}\times b^{D-4}\omega_{D-4}\times
g_{s}^{-2}\,b^{4-D}m_{0}=K\,C_{D}\,g_{s}^{-2}\,m_{0}\omega_{D-4}\,,$ (6.7)

where $K$ is a universal numerical constant that can in principle be computed
from $\alpha^{\prime}$ correction to the solution (6.1)-(6.6). In particular,
$K$ is independent of $D$.

We now recall from (4.24):

$\sqrt{Q_{R}^{2}-Q_{L}^{2}}=\sqrt{Q^{T}LQ}=\sqrt{2}\,(D-3)\,C_{D}\,g_{s}^{-2}\,m_{0}\,\omega_{D-2}\,.$
(6.8)

Using this we can express (6.7) as

$S_{macro}={K\over\sqrt{2}}{\omega_{D-4}\over(D-3)\omega_{D-2}}\,\sqrt{Q_{R}^{2}-Q_{L}^{2}}\,.$
(6.9)

Using the result

$\omega_{k}=2\pi^{(k+1)/2}/\Gamma[(k+1)/2]\,,$ (6.10)

we get

$\omega_{D-2}/\omega_{D-4}=2\pi/(D-3)\,.$ (6.11)

Substituting this into (6.9) we get

$S_{macro}={K\over 2\pi\sqrt{2}}\,\sqrt{Q_{R}^{2}-Q_{L}^{2}}\,.$ (6.12)

This would agree with (2.5) if $K=8\pi^{2}$. While we do not have a way to
calculate $K$ at present, the $D$-independence of (6.12) is consistent with
the $D$-independence of (2.5).

Note that the dependence on the parameter $b$, that controls the temperature
of the black hole via (4.35), has disappeared. This is consistent with the
fact that the index should be independent of the temperature. Note also that
the result is independent of the combination
$Q^{T}M_{\infty}Q=Q_{L}^{2}+Q_{R}^{2}$. This can be traced to the
$\alpha$-independence of the result, – the $O(26-D,10-D)$ transformation that
removes the $\alpha$ dependence of $M$ near the singularity also removes the
$\alpha$ dependence of the gauge fields. Such a dependence would have been
inconsistent with the microscopic results which do not depend on $M_{\infty}$.
This was also a feature in the analysis in [8].

##  7 Generalization to other compactifications

The above analysis can be easily generalized to other ${\cal N}=4$ or ${\cal
N}=2$ supersymmetric compactifications of heterotic string theory. For this we
use the observation of [10] that the black hole solution describing BPS
elementary string states can be embedded in a universal consistent truncation
of the supergravity theory used above. Furthermore the universality is
maintained to all orders in $\alpha^{\prime}$. Therefore the analysis given
above can be carried out without any further modification and we shall arrive
at the expression (6.12) for the macroscopic entropy with the same $K$. The
microscopic counting also remains unchanged, since the asymptotic growth of
BPS states is controlled by the central charge of the left-moving matter
sector of the world-sheet theory and this is always equal to 26. Note however
that in order to get BPS fundamental string in the first place, we must have
at least one internal circle on which the string can wrap, i.e. we cannot have
such states in ten dimensional heterotic string theory or the six dimensional
theory obtained by compactifying heterotic string theory on $K3$.

Finally we shall make a few remarks on the type II string compactification.
For definiteness let us consider the case of torus compactification. The
microscopic index for the fundamental string states vanishes in this case due
to the cancellation between the contributions from the bosonic and fermionic
states. On the macroscopic side the relevant part of the two derivative action
is very similar to the one we have analyzed here and identical arguments lead
to a form (6.12) for the logarithm of the index. Therefore for consistency the
coefficient $K$ must vanish for type II theory. This may be related to the
fact that higher derivative corrections are relatively rare in type II string
theory. For example the four derivative term, that was used in [11] to get a
finite result for $K$ in the heterotic string theory, is absent in type II
string theory. However to get a complete picture we need to learn how to
compute $K$ by taking into account the effect of all the $\alpha^{\prime}$
corrections to the effective action.

Acknowledgement: We would like to thank A H Anupam for discussions and
collaboration during the initial stages of this work. CC thanks David Turton
and Shasha Tyukov for discussions. AS thanks Sameer Murthy for discussions. AV
thanks David Chow for email correspondence. CC is supported by the STFC
consolidated grant (ST/X000583/1) “New Frontiers in Particle Physics,
Cosmology and Gravity”. A.S. is supported by ICTS-Infosys Madhava Chair
Professorship and the J. C. Bose fellowship of the Department of Science and
Technology.

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# Fingerprints of Majorana Fermions in High-Harmonic Spectroscopy

Adhip Pattanayak  Department of Physics, Indian Institute of Technology
Bombay, Powai, Mumbai 400076, India  Sumiran Pujari  Department of Physics,
Indian Institute of Technology Bombay, Powai, Mumbai 400076, India  Gopal
Dixit<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Bombay, Powai, Mumbai 400076, India<EMAIL_ADDRESS>1Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai
400076, India

###### Abstract

The observation of Majorana fermions as collective excitations in condensed-
matter systems is an ongoing quest, and several state-of-the-art experiments
have been performed in the last decade. As a potential avenue in this
direction, we simulate the high-harmonic spectrum of Kitaev’s superconducting
chain model that hosts Majorana edge modes in its topological phase, and find
their fingerprints in the spectral profiles. It is well-known that this system
exhibits a topological–trivial superconducting phase transition. We
demonstrate that high-harmonic spectroscopy is sensitive to the phase
transition in presence of open boundary conditions due to the presence or
absence of these edge modes. The population dynamics of the Majorana edge
modes are different from the bulk modes, which is the underlying reason for
the distinct harmonic profile of both the phases. On the contrary, in presence
of periodic boundary conditions with only bulk modes, high-harmonic
spectroscopy becomes insensitive to the phase transition with similar harmonic
profiles in both phases.

Emission of radiation at prominently higher integer multiple frequencies of
the incident laser frequency due to strong nonlinear interaction of intense
laser fields with matter is known as high-harmonic generation (HHG) 1; 2; 3.
The emitted radiation encodes the information about laser-driven sub-cycle
electron dynamics which forms the basis of HHG spectroscopy. In recent years,
HHG in solids has become a method of choice to probe various aspects of
condensed matter such as the observation of Bloch oscillations 4; 5, examining
the dynamics of the defects in solids 6; 7, band structure tomography 8; 9;
10, observation of the valley pseudospin 11; 12, imaging valence electrons 13;
14, realisation of petahertz currents in solids 15; 16, probing Berry phases
17; 18, and phase transitions driven by both light and topology 19; 20; 21;
22; 23; 24; 25; 26; 27; 28; 29.

In the field of strongly correlated matter, topological superconductivity and
associated phase transitions is presently one of the major topics 30; 31 as it
is associated with the emergence of the Majorana fermion – particles that are
their own antiparticles 32. The notion of the Majorana fermion is also of
great importance in nuclear and particle physics, apart from solid-state
physics 33. In a condensed matter context, Majorana fermions are emergent
quasiparticle excitations that have been proposed as the physical basis to
realize qubits that are robust against decoherence and fault-tolerant
topological quantum computation 34; 35; 36; 37. In this work, we propose a way
to probe the existence of Majorana fermions by probing the topological–trivial
superconducting phase transition using HHG spectroscopy.

Experimental observation of Majorana fermions is an ongoing quest, and several
experiments on semiconductor nanowires, magnetic chains and superconductors
have been reported 38; 39; 40; 41; 42; 43; 44; 45. Topological superconductors
are considered as a promising platform for Majorana fermions since the
original proposal of Kitaev. 46 Thus for this study, we choose the
paradigmatic one dimensional (1D) Kitaev chain of spinless fermions as a
suitable model system. The many-body Hamiltonian for this 1D superconducting
chain is

$\mathcal{H}=-\mu\sum_{j=1}^{L}c_{j}^{\dagger}c_{j}-t_{0}\sum_{j=1}^{L}\left(c_{j+1}^{\dagger}c_{j}+c_{j}^{\dagger}c_{j+1}\right)+\Delta\sum_{j=1}^{L}\left(c_{j}c_{j+1}+c_{j+1}^{\dagger}c_{j}^{\dagger}\right).$
(1)

Here, $\mu$ is the onsite energy, $t_{0}$ is a nearest neighbour hopping term,
and $\Delta$ is the superconducting pairing term (chosen to be purely real),
which creates or annihilates pairs of particles at neighbouring lattice sites.
$c_{j}^{\dagger}$ ($c_{j}$) is fermionic creation (annihilation) operator at
site $j$ and $L$ represents the number of lattice sites.

In the appropriate region of parameter space, the system described by Eq. (1)
can host a topological superconductor that leads to the emergence of Majorana
zero-energy edge modes (MZMs) 33; 43; 44. The boundary of the phase transition
from the topological to the trivial superconducting side depends on the
critical values of $\mu$, $t_{0}$ and $\Delta$ 30; 33. For example, for
$\Delta=t_{0}$, the system is in the trivial phase when
$\left|\mu/t_{0}\right|>2$ in the thermodynamic limit $L\rightarrow\infty$.
This is qualitatively similar to the limit
$\left|\mu/t_{0}\right|\rightarrow\infty$. For $\left|\mu/t_{0}\right|<2$, the
system behaves as a topological superconductor with boundary MZMs in presence
of edges. At $\left|\mu/t_{0}\right|=2$, the gap closes which causes the
boundary zero modes to hybridize and disappear 35.

To set the stage for the HHG study, we briefly recall this topological–trivial
phase structure and its association with the MZMs. Following Ref. 46, one
rewrites Eq. (1) in terms of Majorana operators
$c^{\dagger}_{j}=\frac{1}{2}(\gamma_{j,1}+i\gamma_{j,2})$ and
$c_{j}=\frac{1}{2}(\gamma_{j,1}-i\gamma_{j,2})$ to arrive at
$\mathcal{H}=\frac{i}{2}\Big{[}-\mu\sum_{j=1}^{L}\gamma_{j,1}\gamma_{j,2}+(-t_{0}+\Delta)\sum_{j=1}^{L}\gamma_{j,1}\gamma_{j+1,2}+(t_{0}+\Delta)\sum_{j=1}^{L}\gamma_{j,2}\gamma_{j+1,1}\Big{]}$
up to a constant. The system is in the trivial phase when two Majorana
fermions are bound on the same site as sketched in Fig 1(a). This is clearly
seen when $\mu$ is the dominant scale in $\mathcal{H}$ as expressed in terms
of the Majorana operators. On the other hand, two unpaired Majorana fermions
emerge at the boundary when Majorana fermions are bound in pairs on the
neighbouring sites as sketched in Fig 1(b) that corresponds with the system
being in the topological superconducting phase. This is easiest to see for
$t_{0}=\Delta$ and $\mu=0$. More generally in the topological phase, it is
understood that the two unpaired Majorana fermions (or MZMs) are exponentially
localised at each physical boundary 46. This notion of emergent unpaired
Majorana fermions at the boundary or edge, expectedly, gets lost when closed
(periodic) boundary condition (PBC) is employed as sketched in Fig 1(c). The
topological–trivial transition can be seen by tracking the lowest many-body
excitation energy as a function of $\left|\mu/t_{0}\right|$ for the open chain
as shown in Fig 1(d). On the topological side, there is an effectively zero
energy (fermionic) excitation which is composed from the unpaired Majorana
modes localised at the opposite edges as in Fig. 1(b), whereas on the trivial
side, such zero energy modes are lost.

Figure 1: Notion of Majorana fermions in one dimensional chain. Two phases of
the 1D superconducting Kitaev chain with open boundary: (a) the trivial phase
has Majorana fermions (blue spheres) bound in pairs located on the same site
of the physical lattice, represented by grey spheres. The other phase is (b)
the topological phase, where Majorana fermions are bound in pairs located on
the neighbouring sites leading to two unpaired Majorana fermions at both ends
of the chain, represented by the red spheres. (c) The two unpaired Majorana at
the ends of the chain again become bound in a pair when periodic or closed
boundary condition is employed. (d) Lowest many-body excitation energy for
$L=8$ and 128 as a function of $|\mu/t_{0}|$. Figure 2: High-harmonic spectrum
as a function of $\mu/t_{0}$ with different boundary conditions. High-harmonic
spectra on the two side of the phase transition of the 1D superconducting
Kitaev chain. The spectra for (a) the topological phase at $\mu/t_{0}=1.8$,
and (b) the trivial phase at $\mu/t_{0}=2.2$ when the open boundary condition
(OBC) is used. The spectra (c) and (d) are same as (a) and (b), respectively,
with periodic boundary condition (PBC) imposed on the Kitaev chain. $L=128$ is
used while simulating the harmonic spectra.

High-harmonic spectra for the 1D Kitaev chain with open boundary conditions
(OBC) and PBC are presented in Fig. 2. The left and right panels in the figure
correspond to topological and trivial phases, respectively, on either side of
the transition at $\mu/t_{0}=2.0$. To observe the sensitivity of HHG
spectroscopy on the two phases, we have chosen the values of $\mu/t_{0}$ close
to transition point. A gradual decrease in the harmonic intensity up to
15${}^{\textrm{th}}$ harmonics is visible when the system is in the
topological phase at $\mu/t_{0}=1.8$ [see Fig. 2(a)].

Figure 3: Temporal dynamics of Majorana zero modes in the presence of strong
laser pulse. Population dynamics during the laser pulse for (a) doubly
degenerate Majorana zero-energy modes (MZM1 and MZM2 in the legend) in the BdG
spectrum on the topological side at $\mu/t_{0}=1.8$ and (b) the corresponding
non-zero non-degenerate modes (which we have still called “MZM1” and “MZM2” to
avoid extra notation) on the trivial side at $\mu/t_{0}=2.2$ with OBC. The
population dynamics (c) and (d) are same as (a) and (b), respectively with PBC
where none of the modes are at zero energy due to the absence of MZMs.

However, the spectrum for the trivial phase at $\mu/t_{0}=2.2$ is
significantly different from the topological phase, as evident from the right
panel of the figure [see Fig. 2(b)]. In this case, the extent of the harmonics
is relatively broad, ranging from 3${}^{\textrm{rd}}$ to 15${}^{\textrm{th}}$
order. The intensity of the harmonics in the plateau region is comparable, and
the spectra extend up to the 17${}^{\textrm{th}}$ harmonic. In both the
phases, the harmonic spectra exhibit odd harmonics only as expected from the
inversion symmetry present in the 1D system. Note that energy eigen-spectra
are almost same at $\mu/t_{0}=1.8$ and 2.2 as evident from Fig. S1. The band
gap at $\mu/t_{0}=1.8$ is $\sim$0.109 eV (if MZM states at zero energy are
excluded), and at $\mu/t_{0}=2.2$ is $\sim$0.11 eV. In Fig. S7, we also
present a series of harmonic spectra for different values of $\mu/t_{0}$ (=
1.6, 1.7, 1.8, 1.9, 2.1, 2.2, 2.3, 2.4) across the transition through which
their differences on the two sides of the transition becomes evident.

Figure 4: High-harmonic spectrum with and without Majorana zero modes of the
1D superconducting Kitaev chain with open boundary condition (OBC). Harmonic
spectrum where for both occupied and unoccupied MZM states are not considered
for (a) topological phase $\mu/t_{0}=1.8$, and (b) trivial superconducting
phase $\mu/t_{0}=2.2$. Harmonic spectrum originating only from occupied MZM
state for (c) topological phase $\mu/t_{0}=1.8$, and (d) trivial
superconducting phase $\mu/t_{0}=2.2$.

The situation becomes very different when PBC is imposed on the 1D Kitaev
chain such that there are no boundaries. In this case, the harmonic spectra
corresponding to $\mu/t_{0}=1.8$ and 2.2 are very similar as reflected from
Figs. 2(c) and 2(d). The small differences between both the spectra arise due
to slight changes in the eigen-spectra at $\mu/t_{0}=1.8$ and 2.2. The
analysis of Fig. 2 concludes that the harmonic spectra are very similar when
PBC is employed, whereas markedly different for both the phases when OBC is
employed. This indicates that the MZMs are playing a distinct role in the
harmonic generation as seen by the contrast in the HHG spectra in both the
phases with different boundary conditions. In the following, we will
investigate how the MZMs are associated with the different harmonic spectra in
the two phases.

To elucidate the role of the MZMs in the harmonic generation, population
dynamics of the doubly degenerate MZMs in Bogoliubov-de Gennes (BdG) spectrum
on the topological side is presented in Fig. 3. We recall that the degeneracy
of the two zero modes is lifted in the trivial phase, whereby they lose their
Majorana character. When compared to these corresponding non-degenerate and
non-zero modes in the trivial phase, the population dynamics during the laser
pulse on the topological side at $\mu/t_{0}=1.8$ shows that the degenerate
Majorana states show drastically different behaviours [see Figs. 3(a) and
3(b)]. This noticeable difference in population dynamics in both the phases
owing to the topologically enforced zero mode character of the Majorana
excitations or lack thereof can be correlated with the different harmonic
spectra observed in both the phases as shown in Fig. 2.

In presence of PBC, when there are no MZMs present, the temporal evolution of
the corresponding modes are quite similar at $\mu/t_{0}=1.8$ and 2.2 [see
Figs. 3(c) and 3(d)]. This similarity in their temporal evolution indicates
that both the modes are contributing in the same way to the harmonic
generation at $\mu/t_{0}=1.8$ and 2.2; this correlates with almost the same
harmonic spectra for both the phases with PBC as shown in Figs. 2(c) and 2(d)
in contrast to the above discussion on the previous case with OBC. Moreover,
their behaviour is similar to the OBC case on the trivial side [Fig. 3(b)]
when these modes have become part of the bulk and not localized at the
boundary anymore. We recall again that the notion of the unpaired Majorana
fermions exists only on the topological side with OBC which is in line with
the preceding discussions. The present findings thus establish that the
harmonic spectra and the population dynamics of the MZMs are different for
both the phases for the 1D Kitaev chain with OBC. This concludes that high-
harmonic spectroscopy is sensitive to the presence of the MZMs. By probing the
superconducting phase transition at the phase boundary using HHG, one could
potentially probe the presence of the Majorana fermions. On the other hand,
the notion of unpaired Majorana fermion does not exist for the 1D Kitaev chain
with PBC. As a result, the harmonic spectra and the population dynamics are
similar for different values of $\mu/t_{0}$.

In order to highlight further the role of MZMs in the differences of the
harmonic profiles in presence of OBC, we also show the harmonic spectra
obtained by artificially suppressing the presence of MZMs in the eigen-
spectra. The harmonic spectra at $\mu/t_{0}=1.8$ and 2.2 with OBC are almost
same as evident from Figs. 4(a) and 4(b). The minor difference in the spectra
are originating from the minor difference in the eigen-spectra. Moreover, the
harmonic spectra solely originating from one occupied MZM at $\mu/t_{0}=1.8$
and corresponding non-zero mode at $\mu/t_{0}=2.2$ is shown in Figs. 4(c) and
4(d). As reflected from the figure, the spectra are drastically different from
each other. Results presented in Fig. 4 unequivocally establish that the
similarities and differences in the spectra under different situations are due
to the presence or absence of the MZM modes. Also, the system continues to
exhibit different harmonic profiles in the two phases (for OBC) for different
laser intensities (see Figs. S3 and S4) and wavelengths (see Figs. S5 and S6).

We now briefly comment on the domain of validity for the observations made
earlier. Keeping our choice of the laser parameters fixed, we find that a
variation of $\sim 11\%$ in the $t_{0}=\Delta$ scale does not change our
conclusions. Keeping the $t_{0}$ scale fixed, a variation of $~{}\sim\pm 8\%$
in $\Delta/t_{0}$ keeps conclusions unchanged as well. Beyond these
variations, the laser parameters have to be tuned accordingly to find the
suitable values. Thus, analogous simulations can be helpful in determining the
appropriate laser characteristics when applying HHG to different physical
systems. Finally, it is expected that the HHG obtained from the time evolution
of the many-body Hamiltonian in Eq. (1) and from that of the BdG-recasted
“one-particle” Hamiltonian should be the same in principle. We have confirmed
this expectation by explicit numerical simulations (results not shown here),
which is an independent check on the accuracy of our numerical time evolution
procedures.

In summary, our work demonstrates that HHG spectroscopy is sensitive to the
topological–trivial phase transition in 1D superconducting Kitaev chain. The
extent of the harmonic plateau and energy cutoff in the trivial phase are
distinguishable from the harmonic spectra in the topological phase. This
distinguishability is further consolidated by various harmonic spectra for
different intensities and wavelengths of driving laser as presented in
supplementary material. The population dynamics of the two Majorana states and
their contributions to the total harmonic spectra are very different for both
the phases, which we established to be correlated with the differences in the
HHG spectra of the two phases. This distinct signature may potentially point
to a new probe of Majorana modes in solids. A physical explanation for the
numerically observed differences in the population dynamics of MZMs and bulk
modes that were shown to underlie the HHG spectra is desirable for better
understanding. Finally, the feasibility of coupling existing realisations of
topological superconductivity with potential MZMs to _intense_ laser fields in
the laboratory is an open issue 47, and we hope our work stimulates attempts
in this direction.

##  I Methods

To probe the topological–trivial phase transition by HHG spectroscopy, we
imagine that the 1D Kitaev chain is exposed to an ultrashort, strong laser
pulse. For simulating this process, it is useful to recast the Hamiltonian,
given in Eq. (1), in the two-component Nambu-Bogoliubov-de Gennes (BdG)
formalism. 48 The Hamiltonian of Eq. (1) can be written as
$\mathcal{H}=\frac{1}{2}\;C^{\dagger}\;H_{\textrm{BdG}}\;C$ with
$C=(c_{1},...,...,c_{L},c^{\dagger}_{1},...,...,c^{\dagger}_{L})^{T}$ as a
column vector containing all fermionic creation and annihilation operations.
The one-particle matrix elements contained in $H_{\textrm{BdG}}$ can be
compactly written as

$\displaystyle H_{\textrm{BdG}}$ $\displaystyle=$
$\displaystyle-\mu\sigma_{z}\sum_{n}\left|n\right\rangle\left\langle n\right|$
(2)
$\displaystyle-\sum_{n}[(t_{0}\sigma_{z}-i\Delta\sigma_{y})\left|n\right\rangle\left\langle
n+1\right|+\textrm{h.c.}]$

by using Pauli matrices $\sigma$. $H_{\textrm{BdG}}$ thus has the dimension of
$2L\times 2L$, where $|n\rangle$ corresponds to the $n$-th site of the chain
and h.c. stands for the hermitian conjugate. Here, $\sigma$ operates on the
particle and hole indices of the BdG formalism.

In presence of the laser pulse, the hopping term now acquires a time-dependent
Peierls phase $\theta(t)$. The time derivative of $\theta(t)$ is directly
proportional to the applied electric field as
$E(t)=-\frac{1}{a_{0}}\frac{d\theta(t)}{dt}=-\frac{dA(t)}{dt}$ with $a_{0}$ as
the lattice constant and $A(t)$ as the time-dependent vector potential of the
driving laser pulse. Here, we are thus assuming a spatially constant field due
to the laser. This assumption corresponds to the limit when the wavelength of
the laser is much larger than the 1D system under probe which will govern our
choice of laser wavelength subsequently. Also, the value of $\Delta$ changes
as a function of time in the presence of laser (see Fig. S2). To simulate the
time-dependent changes in (real) $\Delta$, we have used the fact
$\Delta(t)\propto\langle c_{j}c_{j+1}\rangle(t)$ for the pairing term as time
progresses, e.g. see Ref. 49. Time-dependent Schrodinger equation for the
modes of the BdG-recasted Hamiltonian is numerically solved to simulate the
high-harmonic spectrum along with time-evolving $\Delta(t)$ – i.e., at each
(small) discrete time step, the BdG modes are first computed using the
Hamiltonian parameters from the previous time step, and then the pairing term
coefficient $\Delta$ is re-computed using these new set of BdG modes just
computed to supply the Hamiltonian parameters for the next time step. One
ensures the convergence of this procedure with respect to the size of the
discretized time steps which simulates the continuous time evolution.

The high-harmonic power spectrum is obtained by the modulus square of the
Fourier-transform of the total current
$j(t)=\sum_{k}\left\langle\psi_{k}(t)|\hat{J}(t)|\psi_{k}(t)\right\rangle$
where

$\hat{J}(t)=-iea_{0}t_{0}\sigma_{z}\sum_{n}[\mathrm{e}^{-i\theta(t)}\left|n\right\rangle\left\langle
n+1\right|-\textrm{h.c.}],$ (3)

and $|\psi_{k}(t)\rangle$ is the time-propagated state at time $t$. Half of
the states in the BdG spectrum make up the many-body ground state which is
considered as the initial state for the time evolution in order to obtain the
harmonic spectra. On the topological side, this corresponds to (lower) one of
the two zero-energy modes in the BdG spectrum being occupied.

In this work, $t_{0}$ = 0.26 eV and $a_{0}\sim$ 0.4 nm are considered for the
simulations. In what follows, we set $\Delta=t_{0}$ at the initial time
instant to describe our main observations. A linearly polarised laser pulse of
9.1 $\mu$m wavelength having ten optical cycles with sine squared envelope is
used. The polarisation of the driving pulse is along the 1D Kitaev chain and
has a peak amplitude of 30 MV/m. Choice of the laser parameters is motivated
from Ref. 50. Time-step of 0.1 atomic unit ($\sim$ 2.5 as) is used for time
propagation.

## Data Availability

Data that support the plots within this paper and other findings of this study
are available from the corresponding authors upon reasonable request.

## Code Availability

Code that support the findings of this study are available from the
corresponding authors on reasonable request.

## Acknowledgements

A. P. acknowledges fruitful discussion with Souvik Bandyopadhyay from IIT
Kanpur, and sandwich doctoral fellowship from Deutscher Akademischer
Austauschdienst (DAAD, reference no. 57440919). SP acknowledges financial
support from Science and Engineering Research Board (SERB) India
(SRG/2019/001419). G. D. acknowledges financial support from Science and
Engineering Research Board (SERB) India (Project No. ECR/2017/001460).

## Competing Interests

The authors declare no competing interests.

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Supplementary Material: Fingerprints of Majorana Fermions in High-Harmonic
Spectroscopy

Adhip Pattanayak1

Sumiran Pujari1 Gopal Dixit1

### Supplementary Note 1: Bogoliubov-de Gennes Energy Eigenspectrum.

Supplementary Fig. S1: Bogoliubov-de Gennes Energy eigenspectrum of system in
(a) topological phase $\mu/t_{0}=1.8$, and (b) trivial superconducting phase
$\mu/t_{0}=2.2$.

### Supplementary Note 2: Time-Variation in $\Delta$

$\Delta(t)/t_{0}$ varies with time in the presence of strong laser field. The
value of $\Delta(t)/t_{0}$ remains to its initial value approximately up to 3
optical cycles of the laser. A drop in the absolute value of $\Delta(t)/t_{0}$
is observed near the end of the third optical cycle. The value of
$\Delta(t)/t_{0}$ decreases approximately to 7% at the end the laser pulse.
This variation in $\Delta$ corresponds to a laser parameter with field
amplitude $3\times 10^{8}$ V/m, ten-optical cycles and frequency of 32.9 THz
(0.136 eV) and $sine$ squared envelope. We have performed similar numerical
calculation for different set of laser parameters to get time-dependent
$\Delta(t)$ for that particular set.

Supplementary Fig. S2: Temporal variation of $\Delta$ in the presence of the
ten-optical cycles, $sine$ squared envelope laser with field amplitude
$3\times 10^{8}$ V/m, and frequency of 32.9 THz (0.136 eV).

### Supplementary Note 3: High-Harmonic Spectra with different laser
intensities

Supplementary Fig. S3:  High-harmonic spectrum for (a) topological phase
$\mu/t_{0}=1.8$, and (b) trivial superconducting phase $\mu/t_{0}=2.2$ of the
1D superconducting Kitaev chain with open boundary condition. The spectra
correspond to a laser pulse with frequency 32.9 THz and amplitude $8\times
10^{7}$ V/m. Supplementary Fig. S4:  Same as in Fig. S3 except laser with
amplitude $8\times 10^{8}$ V/m.

### Supplementary Note 4: High-Harmonic Spectra with different laser
wavelengths

Supplementary Fig. S5:  High-harmonic spectrum for (a) topological phase
$\mu/t_{0}=1.8$, and (b) trivial superconducting phase $\mu/t_{0}=2.2$ of the
1D superconducting Kitaev chain with open boundary condition. The spectra
correspond to a laser pulse with frequency 26.6 THz and amplitude $8\times
10^{8}$ V/m. Supplementary Fig. S6:  Same as in Fig. S5 except laser with
frequency 36.2 THz.

### Supplementary Note 3: High-Harmonic Spectra

Supplementary Fig. S7:  High-harmonic spectrum for different values of
$\mu/t_{0}$ across the topological–trivial superconducting phase transition in
1D superconducting Kitaev chain with open boundary condition. The spectra
correspond to a laser pulse with frequency 32.9 THz and amplitude $8\times
10^{7}$ V/m.