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Monte Carlo Hamiltonian H.Jirari,H.Kr¨o ger ∗D´e partement de Physique,Universit´e Laval,Qu´e bec,Qu´e bec G1K 7P4,Canada X.Q.Luo CCAST(World Laboratory),P.O.Box 8730,Beijing 100080,China Department of Physics,Zhongshan University,Guangzhou 510275,China †Center for Computational Physics,School of Physics Science and Engineering,Zhongshan University,Guangzhou 510275,China K.J.M.Moriarty ‡Department of Mathematics,Statistics and Computational Science,Dalhousie University,Halifax,Nova Scotia B3H 3J5,Canada February 1,2008Abstract We construct an effective Hamiltonian via Monte Carlo from a given action.This Hamilto-nian describes physics in the low energy regime.We test it by computing spectrum,wave functions and thermodynamical obrvables (average energy and specific heat)for the free
system and the harmonic oscillator.The method is shown to work also for other local potentials.
PACS index:o3.65.-w,05.10.Ln
1Motivation
The motivation for constructing a Monte Carlo Hamiltonian comes from different direc-tions.
(i)The renormalization group`a la Kadanoff-Wilson[1]aims to construct a renormalized Hamiltonian,which describes physics at a critical point,bad on the assumption of scale invariance.Such Hamiltonian is suppod to have much less degrees of freedom than the original Hamiltonian.A recent further development of tho ideas is White’s density matrix renormalization group technique[2].Our goal is similar to the above in the n that we aim at an effective Hamiltonian,which has”less”degrees of freedom than the”original”Hamiltonian.But it differs in describing physics in the low energy domain instead of doing so at the critical point.
(ii)When one tries to solvefield theory in the Hamiltonian formulation,the standard way to proceed is by constructing a Fock space,parametrized by some high momentum cut-offand some occupation number cut-off(Tamm-Dancoffapproximation).When increasing tho parameters,which means increasing the upper bound of the energy,then typically the density of states increas in an exponential manner,which renders the system beyond any control.In contrast to that,the Monte Carlo
Hamiltonian is governed by a”small”number of low-energy degrees of freedom and the spectral density decreas with increasing energy.
(iii)The enormous success of latticefield theory over the last quarter of the century is certainly due to the fact that the Monte Carlo method with importance sampling is an excellent technique to solve high dimensional(and even”infinite”dimensional)integrals. Conventionally,one computes a transition amplitude of an operator and evaluates it nu-merically via Monte Metropolis algorithm[3]),
<O>= [dx]O[x]exp(−1
[dx]exp(−1
N c C O[C].(1)
S[x]). Here C stands for a path configuration drawn from the distribution P[x]=1
鱼香肉丝汁怎么调
¯h
The vitue of the Monte Carlo method lies in the property of yielding very good numerical
1
results. E.g.,solving afield theory model on a lattice of size204and measuring the obrvable from a number of configurations N c in the order of a few hundred typically yields results with statistical errors in the order of a few percent.In this way it has been possible to determine low lying baryon and meson mass quite precily[4].
On the other hand,one can express a transition amplitude in imaginary time via the Hamiltonian
<x fi,T|x in,0>=<x fi|e−HT/¯h|x in>
=
∞
n=1<x fi|E n>e−E n T/¯h<E n|x in>
空间清理≈<x fi|e−H ef T/¯h|x in>
=
N
ν=1<x fi|E effν>e−E ef fνT/¯h<E effν|x in>.(2)
In the last two lines we have approximated the Hamiltonian H by an effective Hamiltonian H eff,which has less degrees of ,it has only N eigenstates.The idea of the Monte Carlo Hamiltonian is that an effective Hamiltonian can be found via u of Monte Carlo,such that transition amplitudes become afinite sum over N eigenstates,where N is in the order of magnitude of N he number of equilibrium configurations,sufficient to cloly approximate the path integral of Eq.(1).
One might ask:What is the virtue of such a Hamiltonian?A list of physics problems, where progress has been slow with conventional methods including standard lattice tech-niques,and where such a Hamiltonian might bring progress are the following topics:
-Non-perturbative computation of cross ctions and decay amplitudes in many-body systems[5].
-Low-lying but excited states of the hadronic spectrum and the related question of quan-tum chaos in such a system.
-Hadron wave functions and the related question of hadron structure functions,in particu-lar for small x B and Q2.The Hamiltonian formulation is suited to compute wave functions, which is quite difficult in the Lagrangian lattice formulation.
-Finite temperature and in particularfinite density in baryonic matter.This is crucial
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for the quark-gluon plasma pha transition,the physics of neutron stars and cosmology. The Hamiltonian formulation is suited to compute the mean value of the energy(average energy).This is difficult to compute in the Lagrangian lattice formulation where one usu-ally computes the expectation value of the action.Finite density QED and QCD in the Lagrangian lattice formulation is hampered by the notorious complex action problem.
-Atomic physics:study of spectra and the question of quantum chaos.
-Condend matter physics:study of spin systems(computation of dynamical structure factors),and high T c superconductivity models(arch for electron pair attraction at very small energy).In the following we will outline how to construct such a Hamiltonian.
2Construction of H eff
In contrast to the statistical mechanics concept of the transfer matrix,which describes the time-evolution(we consider imaginary time)when advancing the system by a small discrete time step∆t=a
0and from which one can infer the Hamiltonian(a0→0),here we consider transition amplitudes<ψ|e−HT/¯h|φ>corresponding to afinite,long time T (T>>a0),for the purpo to reconstruct the spectrum in somefinite low energy domain. Let us start from a complete orthonormal basis of Hilbert states|e i>,i=1,2,3···and consider the matrix elements for a givenfixed N
M ij(T)=<e i|e−HT/¯h|e j>,i,j∈1,···,N.(3)
Under the assumption that H is Hermitian,M(T)is a positive,Hermitian matrix.Ele-mentary linear algebra implies that there is a unitary matrix U and a real,diagonal matrix D such that
M(T)=U†D(T)U.(4) On the other hand,projecting H onto the the subspace S N generated by thefirst N states of the basis|e i>,and using the eigenreprentation of such Hamiltonian,one has
狼王梦的读后感M ij(T)=
N
k=1<e i|E eff k>e−E ef f k T/¯h<E eff k|e j>,(5) 3
and we can identify
U†ik=<e i|E eff
k
>,D k(T)=e−E ef f k T/¯h.(6) Let us assume for the moment that the matrix elements M ij(T),i,j=1,···,N would be known.Then algebraic diagonalization of the matrix M(T)yields eigenvalues D k(T),k= 1,···,N,which by Eq.(6)gives the spectrum of energies,
E eff k =−
¯h
2
m˙x2+V(x) C.(10) 4
The Monte Carlo method with importance sampling is suited and conventionally applied to estimate a ratio of integrals,like in Eq.(1).Here we suggest to estimate the matrix elements M ij by splitting the action
S=S0+S V≡ T0dt1
x i+1x i dy x j+1x j dz [dx]exp[−S0[x]/¯h] y,T z,0,(12) where O≡exp[−S V/¯h]is treated as an obrvable.The ratio can be treated by standard Monte Carlo methods with importance sampling.The matrix elements M(0)ij,corresponding to the free action S0,are almost known analytically,
M(0)ij(T)= x i+1x i dy x j+1x j dz 2π¯h T exp −m
Z T r[He−βH]=−
∂log Z
∂T =k Bβ2
∂2log Z
m
