Thermodynamics from Monte Carlo Hamiltonian

Abstract

We construct an effective low-energy Hamiltonian from the classical action via Monte Carlo with importance sampling. We use Monte Carlo (i) to compute matrix elements of the transition amplitude and (ii) to construct stochastically a basis. The MC Hamiltonian allows to obtain energies and wave functions of low-lying states. It allows also to compute thermodynamical observables in some temperature window (starting from temperature zero). We present examples from lattice field theory (Klein-Gordon model).

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arXiv:hep-lat/0202016v1 11 Feb 2002
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Thermodynamics from Monte Carlo
Hamiltonian
L.A. Carona
H. Kr¨ ogera∗
G. Melkonyana
X.Q. Luob
K.J.M. Moriartyc
aD´ epartement de Physique, Universit´ e Laval,
Qu´ ebec, Qu´ ebec G1K 7P4, Canada
bDepartment of Physics, Zhongshan University,
Guangzhou 510275, China
cDepartment of Mathematics, Statistics and Computer Science,
Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada
Abstract
We construct an effective low-energy Hamiltonian from the clas-
sical action via Monte Carlo with importance sampling. We use
Monte Carlo (i) to compute matrix elements of the transition am-
plitude and (ii) to construct stochastically a basis. The MC Hamil-
tonian allows to obtain energies and wave functions of low-lying
states. It allows also to compute thermodynamical observables in
some temperature window (starting from temperature zero). We
present examples from lattice field theory (Klein-Gordon model).
∗talk given by H. Kr¨ oger at QMC Trento 2001, email: hkroger@phy.ulaval.ca

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L.A. Caron, H. Kr¨ oger, G. Melkonyan, X.Q. Luo, K. Moriarty
1Introduction
The most interesting phenomena in physics occur in many-body systems.
The difficulty lies in solving models of such systems. A brilliant idea to do
this is the renormalisation group approach ` a la Kadanoff, Wilson, Migdal
and others. It suggests to compute critical phenomena by constructing
an effective (renormalized) Hamiltonian, by thinning out degrees of free-
dom in an appropriate manner. Here we suggest to adapt this idea for
the pupose to construct an effective Hamiltonian in a low-energy window.
The basic idea to thin out degrees of freedom is taken over from lattice
field theory, where infinite-dimensional integrals (path integrals) are suc-
cessfully simulated by a small number of representative configurations. In
analogy to the representative configurations of path integrals, we construct
a representative basis of Hilbert states via a Monte Carlo algorithm with
importance sampling. Furthermore we use the Monte Carlo algorithm to
compute matrix elements of the transition amplitude.
In standard Lagrangian lattice field theory, one is used to construct
a corresponding lattice Hamiltonian via the transfer matrix, which corre-
sponds to the transition amplitude for infinitesimal short time (when the
lattice spacing atgoes to zero). As a result, one finds a Hamilton operator
(on the lattice) which still has the character of a many-body Hamiltonian.
In contrast to that, we start out from the transition amplitude for a fi-
nite time (in imaginary time corresponding to some finite temperature).
We compute the transition matrix for all transitions, where the states are
taken from the finite stochastically generated basis. As a result, we find
an effective Hamiltonian, i.e. its spectrum and eigen states. It is evident
that such Hamiltonian covers only a window of physics, which in this case
is a temperature window, starting from temperature zero.
From the viewpoint of the enormous success of standard Lagrangian
lattice field theory, one may ask: Do we need such an effective Hamil-
tonian? To give an example: Spectrum and wave functions of excited
states. Wave functions in conjunction with the energy spectrum contain
more physical information than the energy spectrum alone. Although lat-
tice QCD simulations in the Lagrangian formulation give good estimates
of the hadron masses, one is yet far from a comprehensive understand-
ing of hadrons. Let us take as example a new type of hardrons made
of gluons, the so-called glueballs. Lattice QCD calculations [1] predict
the mass of the lightest glueball with quantum number JPC= 0++, to

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Thermodynamics from Monte Carlo Hamiltonian
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be 1650 ± 100MeV . Experimentally, there are at least two candidates:
f0(1500) and fJ(1710). The investigation of the glueball production and
decays can certainly provide additional important information for experi-
mental determination of a glueball. Therefore, it is important to be able to
compute the glueball wave function. The standard Lagrangian lattice for-
mulation has made slow progress also in some other areas, like on hadron
structure functions (not only moments) in particular at small Q2and at
small xB, as well as on scattering amplitudes.
2Effective Hamiltonian
Let us discuss the construction of the effective Hamiltonian in several
steps [2]. First, consider in quantum mechanics in 1-dim the motion of a
single particle of mass m under the influence of a local potential V (x). Its
classical action is given by
dtm
2˙ x2− V (x) .
Given the classical action, one can determine Q.M. transition amplitudes.
Like in Lagrangian lattice field theory, we use imaginary time in what
follows. We consider the transition amplitude for some finite time T (tin=
0, tfi= T) and for all combinations of positions xi,xj. Here xi, xj run
over a finite discrete set of points {x1,...,xN} located on the real axis.
Suppose these points are equidistantly distributed (spacing ∆x) over some
interval [−L,+L]. The transition amplitudes are given by the (Euclidean)
path integral,
?
where SE denotes the Euclidean action. Suppose for the moment that
those transition amplitudes were known. From that one can construct an
approximate, i.e. effective Hamiltonian. Note that the matrix M(T) =
[Mij(T)]N×Nis a real, positive and Hermitian matrix. It can be factorized
into a unitary matrix U and a real diagonal matrix D(T),
S =
?
(1)
Mij(T) =[dx]exp[−SE[x]/¯ h]
????
xi,T
xj,0, (2)
M(T) = U†D(T) U .(3)
Because the matrix M(T) can be expressed in terms of the full Hamilto-
nian H by
Mij(T) = ?xi|e−HT/¯ h|xj? , (4)

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L.A. Caron, H. Kr¨ oger, G. Melkonyan, X.Q. Luo, K. Moriarty
the matices U and D have the following meaning,
U†
Dk(T) = e−Eeff
ik=< xi|Eeff
k
>
k
T/¯ h, (5)
i.e., the k−th eigenvector |Eeff
identified with the k−th column of matrix U†and the energy eigenvalues
Eeff
k
of Heff can be identified with the logarithm of the diagonal matrix
elements of D(T). This yields an effective Hamiltonian,
k
> of the effective Hamiltonian Heffcan be
Heff=
N
?
k=1
|Eeff
k
> Eeff
k
< Eeff
k
|.(6)
Note that in the above we have been mathematically a bit sloppy. The
states |xi? are not Hilbert states. We have to replace |xi? by some ”lo-
calized” Hilbert state. This can be done by introducing box states. We
associate to each xisome box state bi, defined by
1/√∆xi if xi< x ≤ xi+1
0 else
bi(x) =
?
(7)
where ∆xi= xi+1−xi. When we use an equidistant distribution of xi, i.e.
∆xi= ∆x, we refer to the basis of box states as the regular basis. Below
we will consider a stochastic basis.
3 Matrix elements
We compute the matrix element Mij(T) directly from the action via Monte
Carlo with importance sampling. This is done by writing each transition
matrix element as a ratio of two path integrals. This is done by splitting
the action into a non-interacting part and an interacting part,
S = S0+ SV .(8)
This allows to express the transition amplitude by
Mij(T) = M(0)
ij(T)
?xi+1
xi
dy?xj+1
?xi+1
xj
dz?[dx] exp[−SV[x]/¯ h] exp[−S0[x]/¯ h]
dy?xj+1
???
y,T
z,0
xi
xj
dz?[dx] exp[−S0[x]/¯ h]
???
y,T
z,0
,
(9)

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Thermodynamics from Monte Carlo Hamiltonian
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Here e−S0/¯ his the weight factor and e−SV/¯ his the observable. M(0)
stands for the transition amplitude of the noninteracting system, which
is (almost) known analytically. For details see ref.[2]. Carrying out these
steps allows to construct an effective Hamiltonian, which reproduces well
low energy physics, provided that the box functions cover an interval large
enough (depending on the range of the potential) and the resolution ∆x
is small enough. This can be achieved with a small numerical effort for
a one-body system in 1 dimension. Our goal is, however, to solve many-
body systems. For such purpose, the above regular basis fails. What to
do in such systems is the subject of the following section.
ij(T)
4Stochastic basis
It is evident that the regular basis defined above becomes prohibitively
large when applied to a many-body system. For example, in a spin model
of a 1-dimensional chain of 30 atoms with spin 1/2, the Hilbert space
has the dimension D = 230= 1073741824. For such situations we wish to
construct a smaller basis which gives an effective Hamiltonian reproducing
well low-energy observables. Why should such a basis exist in the first
place? The heuristic argument is the Euclidean path integral, which, when
evaluated via Monte Carlo with importance sampling, gives a good answer
for the transition amplitude. In particular, this is possible by taking into
account a ”small” number of configurations (e.g. in the order of 100 -
1000). In a crude way the configurations correspond to basis functions.
Thus we expect that suitably chosen basis functions exist, the number of
which is in the order of 100 - 1000, which yields a satisfactory effective
low energy Hamiltonian.
How can we construct such a ”small” basis? As example consider a free
particle in D = 1 dimension. Recall: For the free system the transition
amplitude reads
GEucl(x,T;y,0) =
?
m
2π¯ hTexp[−m
2¯ hT(x − y)2].(10)
This is a positive function for all x, y, T. It can be interpreted as a
probability density. We put y = 0 and define a probability density P(x)
by
P(x) =
1
ZGEucl(x,T;0,0),