376 research outputs found
Symplectic geometry and Hamiltonian flow of the renormalisation group equation
It is argued that renormalisation group flow can be interpreted as being a
Hamiltonian vector flow on a phase space which consists of the couplings of the
theory and their conjugate \lq\lq momenta", which are the vacuum expectation
values of the corresponding composite operators. The Hamiltonian is linear in
the conjugate variables and can be identified with the vacuum expectation value
of the trace of the energy-momentum operator. For theories with massive
couplings the identity operator plays a central role and its associated
coupling gives rise to a potential in the flow equations. The evolution of any
quantity , such as -point Green functions, under renormalisation group flow
can be obtained from its Poisson bracket with the Hamiltonian. Ward identities
can be represented as constants of the motion which act as symmetry generators
on the phase space via the Poisson bracket structure.Comment: 30 page
Potential Flow Of The Renormalisation Group In A Simple Two Component Model
The renormalisation group (RG) flow on the space of couplings of a simple
model with two couplings is examined. The model considered is that of a single
component scalar field with self interaction coupled, via Yukawa
coupling, to a fermion in flat four dimensional space. The RG flow on the two
dimensional space of couplings, , is shown to be derivable from a
potential to sixth order in the couplings, which requires two loop calculations
of the -functions. The identification of a potential requires the
introduction of a metric on and it is shown that the metric defined
by Zamalodchikov, in terms of two point correlation functions of composite
operators, gives potential flow to this order.Comment: 7 pages Typset in PlainTeX, C Version 3.14
Bose condensation and branes
When the cosmological constant is considered to be a thermodynamical variable
in black hole thermodynamics, analogous to a pressure, its conjugate variable
can be thought of as a thermodynamic volume for the black hole. In the AdS/CFT
correspondence this interpretation cannot be applied to the CFT on the boundary
but, from the point of view of the boundary gauge theory, varying the
cosmological constant in the bulk is equivalent to varying the number of colors
in the gauge theory. This interpretation is examined in the case of
, for SUSY Yang-Mills at large , and the
variable thermodynamically conjugate to , a chemical potential for color, is
determined. It is shown that the chemical potential in the high temperature
phase of the Yang-Mills theory is negative and decreases as temperature
increases, as expected. For spherical black holes in the bulk the chemical
potential approaches zero as the temperature is lowered below the Hawking-Page
temperature and changes sign at a temperature that is within one part in a
thousand of the temperature at which the heat capacity diverges.Comment: 9 pages, 1 figur
The intrinsic curvature of thermodynamic potentials for black holes with critical points
The geometry of thermodynamic state space is studied for asymptotically
anti-de Sitter black holes in D-dimensional space times. Convexity of
thermodynamic potentials and the analytic structure of the response functions
is analysed. The thermodynamic potentials can be used to define a metric on the
space of thermodynamic variables and two commonly used such metrics are the
Weinhold metric, derived from the internal energy, and the Ruppeiner metric,
derived from the entropy. The intrinsic curvature of these metrics is
calculated for charged and for rotating black holes and it is shown that the
curvature diverges when heat capacities diverge but, contrary to general
expectations, the singularities in the Ricci scalars do not reflect the
critical behaviour.
When a cosmological constant is included as a state space variable it can be
interpreted as a pressure and the thermodynamically conjugate variable as a
thermodynamic volume. The geometry of the resulting extended thermodynamic
state space is also studied, in the context of rotating black holes, and there
are curvature singularities when the heat capacity at constant angular velocity
diverges and when the black hole is incompressible. Again the critical
behaviour is not visible in the singularities of the thermodynamic Ricci
scalar.Comment: 35 pages, 3 figure
Thermodynamic stability of asymptotically anti-de Sitter rotating black holes in higher dimensions
Conditions for thermodynamic stability of asymptotically anti-de Sitter
rotating black holes in D-dimensions are determined. Local thermodynamic
stability requires not only positivity conditions on the specific heat and the
moment of inertia tensor but it is also necessary that the adiabatic
compressibility be positive. It is shown that, in the absence of a cosmological
constant, neither rotation nor charge is sufficient to ensure full local
thermodynamic stability of a black hole.
Thermodynamic stability properties of anti-de Sitter Myers-Perry black holes
are investigated for both singly spinning and multi-spinning black holes.
Simple expressions are obtained for the specific heat and moment of inertia
tensor in any dimension. An analytic expression is obtained for the boundary of
the region of parameter space in which such space-times are thermodynamically
stable.Comment: 30 pages, 3 figures. References added, minor typos corrected in v
The Modular and Renormalisation Groups in the Quantum Hall Effect
An analytic form for the crossover of the conductivity tensor between two
Hall plateaux, as a function of the external magnetic field, is proposed. The
form of the crossover is obtained from the action of a symmetry group, a
particular subgroup of the modular group, on the upper-half complex
conductivity plane, by assuming that the beta-function describing the crossover
is a holomorphic function of the conductivity. The group action also leads to a
selection rule, |p_1q_2-p_2q_1|=1, for allowed transitions between Hall
plateaux with filling factors p_1/q_1 and p_2/q_2, where q_1 and q_2 are odd.Comment: Talk presented at the Workshop on the Exact Renormalisation Group,
Faro, Portugal, 10-12th September, 1998. Typeset in Latex, 9 pages, 3 figure
Renormalisation Flow and Geodesics on the Moduli Space of Four Dimensional N=2 Supersymmetric Yang-Mills Theory
It is shown that the beta functions for four dimensional N=2 supersymmetric
Yang-Mills theory without matter give integral curves on the moduli space some
of which are geodesics of the natural metric on the moduli space. In particular
the flow lines which cross-over from from the weak coupling limit
(asymptotically free theory) to the singular points, representing the strong
coupling limit, are geodesics. A possible connection with irreversibility is
discussed.Comment: 9 pages, 1 figure, typeset in PlainTe
Geodesic Renormalisation Group Flow
It is shown that the renormalisation group flow in coupling constant space
can be interpreted in terms of a dynamical equation for the couplings analogous
to viscous fluid flow under the action of a potential. For free scalar field
theory the flow is geodesic in two dimensions, while for it is only
geodesic in certain limits, e.g. for vanishing external source. For the 1-D
Ising model the renormalisation flow is geodesic when the external magnetic
field vanishes.Comment: PlainTeX file, 11 pages, 5 figure
Compressibility of rotating black holes
Interpreting the cosmological constant as a pressure, whose thermodynamically
conjugate variable is a volume, modifies the first law of black hole
thermodynamics. Properties of the resulting thermodynamic volume are
investigated: the compressibility and the speed of sound of the black hole are
derived in the case of non-positive cosmological constant. The adiabatic
compressibility vanishes for a non-rotating black hole and is maximal in the
extremal case --- comparable with, but still less than, that of a cold neutron
star. A speed of sound is associated with the adiabatic compressibility,
which is is equal to for a non-rotating black hole and decreases as the
angular momentum is increased. An extremal black hole has
when the cosmological constant vanishes, and more generally is bounded
below by .Comment: 8 pages, 1 figure, uses revtex4, references added in v
- …