Lattice hard-core bosons: series expansions
6.2 Numerical linked cluster expansion
over the years, to support exotic phases and new physics for spin and boson models. The exploration of their possible phases and properties is an active area of research. A hotly pursued technique for the quantum simulation of lattice models is through ultra-cold atoms in optical lattices. Indeed there has been much progress in the manipulation of interactions and modelling of physical systems using ultra-cold gases. A wide range of condensed matter systems like Ising spin systems [124], Bose-Hubbard models [77], Fermi gases [125], to name a few, have been simulated using ultra-cold gases; an equally wide range of condensed matter phases continue to be uncovered in these ultra-cold systems. The hard-core boson model (in the absence of nearest neighbour interactions V) has indeed been realized in an optical lattice with 87Rb atoms [126]. Moreover, an optical kagome lattice has been realized recently by the overlay of two commensurate triangular optical lattices [127]. Our work in this chapter, where we primarily investigate thermodynamical and critical properties of interacting hard-core bosons on the kagome lattice, will therefore be relevant for experimental realizations of interacting hard-core bosons on such optical kagome lattices. We make some comparisons of the kagome model with the properties of the simpler one dimensional lattice. Our computations will employ the thermodynamic Bethe ansatz, statistics of the 8-vertex model, transfer matrix methods, and numerical linked cluster expansions.
6.2 Numerical linked cluster expansion
As explained in chapter 2 and Ref. [9], the series expansion can be done via bond, site or cell based expansions. In the present work, we work mainly with site and cell expansions; for the kagome lattice, a cell is a simple triangular graph. In our notation, ann-site/cell numerical linked cluster expansion (NLCE) refers tonbeing the maximum number of sites/triangles in the list of finite clusters. And unless otherwise specified, angular averages are taken in the grand canonical ensemble.
Additionally, as explained in the aforementioned chapter and reference, we need to extrapolate the series of results for finite clusters to the result for an infinitely large cluster: this essentially increases the temperature range of convergence. Here we use Wynn’s epsilon algorithm [26] and a numerically more stable version of it known as the Bordering method [27]; these were explained in detail in chapter 2 and Appendix C. Curiously, and perhaps satisfactorily, both bordering algorithms give the same result as the last Wynn approximants. Unless otherwise specified, we use 4 Wynn cycles for the 10-cells expansion, 2 Wynn cycles for 7-cells expansion and 6 Wynn cycles for the 14-sites expansion for interacting hard-core bosons on the kagome lattice.
The procedure is as follows: each of the partial sumssiin (2.39) correspond to the calculation of some thermodynamic quantity Pvia the basic linked cluster expansion equation (2.2) at order (explained in 2.1.2) i. Thus we obtain a sequence of partial sums for a range ofivalues; this is then extrapolated using Wynn’s algorithm and Brezinski’s bordering techniques described in section 2.4.3 to push down the temperature range of convergence. The desired observable P(c) appearing in (2.3) for every finite cluster is computed by determining all the eigenvalues of the Hamiltonian in question on the clusterc using the exact diagonalization technique outlined in section 2.3.
6.2.1 Thermodynamic quantities
In this chapter most of our analysis will be carried out within the grand canonical ensemble i.e. the chemical potential µis a free variable and our system will access states with all possible occupation densities for every finite cluster. This will enable us, via an inversion of variables through the Legendre transformation, to later inspect the system properties at a fixed density, which is relevant for bosonic systems in cold atom experiments. The partition function in the grand canonical ensemble at inverse
temperatureβ=1/T for the clustercis given by Z(c)=X
l
exph
−β
l(c)−µn(c)l i
, (6.1)
where the states in the cluster are labelled byl, the eigenvalue byl(c), and the number of particles in the statelbyn(c)l . The observableP(c) is computed as
P(c)= P
lPlexph
−β
l(c)−µn(c)l i
Z(c) . (6.2)
Eigenvalues and eigenvectors of three simple clusters are computed and indicated in table 6.1 for the interacting hard-core boson model at particular values ofV/tandµ/tfor illustrative purposes.
Table 6.1: Eigenvalues and eigenvectors for three clusters for all particle fillings atV/t = 0.5,µ/V =0, t =1.
Numbers within the ket indicate the number of bosons on each site.
Clusterc Eigenvaluel(c) Eigenvector 0
0
|0i
|1i
– 0
-1 1 0.5
|0,0i
0.707|0,1i+0.707|1,0i 0.707|0,1i −0.707|1,0i
|1,1i
– – 0
-1.414 0 1.414 -1.186 0.5 1.686 1
|0,0,0i
−0.5|1,0,0i −0.5|0,0,1i −0.707|0,1,0i
−0.707|1,0,0i+0.707|0,0,1i
−0.707|0,1,0i+0.5|1,0,0i+0.5|0,0,1i 0.454|1,1,0i+0.454|0,1,1i+0.766|1,0,1i
−0.707|1,1,0i+0.707|0,1,1i
−0.542|1,1,0i −0.542|0,1,1i+0.643|1,0,1i
|1,1,1i
Particle density and compressibility
Having determined the partition function in (6.1), we may use equation (6.2) to compute other thermo-dynamic quantities. The easiest is the particle density
ρ = 1
β
∂logZ
∂µ
= 1 Z
X
l
nlexp (−β(l−µnl)). (6.3)
6.2 Numerical linked cluster expansion
We need not perform the differentiation in the first line of the above formula because within linked cluster expansions, the second line of the above formula may be readily evaluated. This may be per-formed for a wide range of chemical potentials so as to obtain the region in phase space where the system has all possible filling factors for every temperature range of interest. From the particle density, we may evaluate the compressibility of the system i.e. how easily the density of the system can be changed via a small change in the chemical potential. In a phase where a finite particle gap exists, the compressibility is therefore zero; else it is finite. The compressibility we calculate is taken to be
K =
∂ρ
∂µ
T
= β
hN2i − hN i2
, (6.4)
whereN is the operatorP
inˆimeasuring the number of particles in the system, and the angular averages are taken in the grand canonical ensemble as in (6.1). Again within linked cluster expansions, we may directly compute the compressibility using the second line of (6.4). We will henceforth denote the mean square fluctuations, appearing in (6.4), of an operatorOash∆O2i.
Energy and entropy
The internal energy of the system is the expectation value of the Hamiltonian without the chemical potential terms
hH i ≡ hH˜ +µN i= 1 Z
X
l
lexp (−β(l−µnl)). (6.5) The entropy per site of the system, in units of the Boltzmann constantkB, is given at temperatureT =1/β by
S = −∂F
∂T
= logZ+βhH i, (6.6)
whereF=−TlogZis the free energy of the system.
Specific heat and Grüneisen parameter
The constant volume specific heat at any chemical potentialµand temperatureT values may be determ-ined from the entropy and particle number by the formula [128]
Cv(µ,T)=T
∂S
∂T
µ
− T
∂S
∂µ
2
T
K . (6.7)
The equation (6.7) may be modified to a form more amenable to our calculations as follows, culminating in (6.12). From (6.6), we get
T
∂S
∂T
µ
=h∆H2i, (6.8)
and
T
∂S
∂µ
T
=
∂hH i˜
∂µ
T
−µ
∂ρ
∂µ
T
, (6.9)
wherehH i˜ was defined in (6.5). Using (6.4), (6.8) and (6.9) in (6.7), we get
Cv(µ,T)= h∆H˜2i T2 − 1
T
∂hH i˜
∂µ
2
T
∂ρ
∂µ
T
, (6.10)
which is equation 11 in Ref. [9]. This may be further simplified by using
∂S
∂µ
T
=
∂ρ
∂T
µ
[128] and
∂ρ
∂T
µ
= h∆(H N)i
T2 ; (6.11)
the latter may be obtained by differentiating (6.3) with respect to temperature T. This leads to the simplified formula for the specific heat
Cv(µ,T)= 1 T2
h∆H2i −h∆(H N)i2 h∆N2i
. (6.12)
The distinct advantage of using formula (6.12) is that it avoids numerical differentiation of thermody-namic quantities as in (6.7) or (6.10), thus doing away with that much bit of numerical noise inextricably associated with the procedure of numerical differentiation. A related quantity of interest, also obtainable from the entropy, is the Grüneisen parameter, which is defined by [129]
Γ(T,r)=−1 T
∂S
∂µ
T
∂S
∂T
µ
(6.13)
whereris some parameter that measures the deviation from a critical point in the model; in our case,r= µ−µc
µc
, with the subscriptcindicating a critical value. Proceeding as before, (6.13) may be simplified so that differentiation is avoided to give
Γ(T,r)=−h∆(H N)i
h∆H2i . (6.14)
The authors of [129] argue that the Grüneisen parameter will scale with temperatureT as
Γ(T,0)∝T
−
1 νz
(6.15)