Re-entrance was not predicted for the 1D BHM until more sophisticated and accurate approximations were applied. In particular it was first revealed by using 12th order SCP expansions 17. After this discovery further calculations with DMRG 15 and QMC methods 19, 20 convincingly established the phenomenon. Their original results are replotted in Fig. 1b showing the most accurate, and essentially exact, result for the 1D BHM phase diagram. A clear consensus from these calculations is that the MI lobe in 1D and for small filling factor n ∼ 1, has a triangular pointed shape which is quite different from the rounded shape revealed in the MF approximation. Moreover the tip of the lobe, also highlighted in Fig. 1b from the DMRG calculation 15, occurs at a much larger value of t than MF predicts owing to the KT nature of its transition in 1D. Roughly speaking re-entrance occurs due to a combination of this elongation of the tip with the kinetic energy driven tendency for all density contours to eventually have a negative slope. With this view it is therefore also natural that re-entrance only occurs in 1D because in higher dimension the transition at the tip of the MI lobe is not a KT critical point and instead occurs at values of t increasingly close to the MF prediction. Ultimately these two effects cause the hole boundary to become concave before it intersects with the particle boundary. This then makes it possible to traverse, by increasing t alone, a series of phases transitions MI–SF–MI–SF at constant µ in the region just above the tip. That re-entrance occurs in this region, where the interplay between hopping and interactions are most significant, illustrates how strong correlations are key to its appearance. We note also that increasing the filling n causes bosonic enhancement of the kinetic energy and consequently the MI lobe shrinks as ≈1/n. Since the tip then occurs at much smaller values of t re-entrance disappears for n > 1. For this reason we focus entirely on n = 1 filling in this work.
The sophisticated methods employed to determine this structure however, obscure the underlying physical intuition behind re-entrance owing to their complexity. Our main aim here is to capture re-entrance within the simplest possible approximation in order to gain further insight into the origin of this remarkable phenomenon. We now examine in turn some common approaches.
3.1 Multi-site mean-field decoupling
The MF decoupling approach is often a highly successful means of determining the qualitative structure of the phase diagram for quantum lattice systems. For the BHM the most effective strategy is to decouple the hopping term via the factorization 21, 4
where denotes the ground state average. Applying this factorization leaves the interaction term untouched but reduces the BHM to a sum of identical single site Hamiltonians
The rounded MI lobes predicted by the conventional single-site MF was shown earlier in Fig. 1a. We now examine the results produced by a multi-site generalization of this approach 24, 25. Specifically, we apply the same decoupling approximation to the hopping terms connecting the boundaries of each L-site blocks as shown schematically in Fig. 2a. A virtue of this approach is that the decoupled systems now have a 1D chain geometry and so the approximation departs from the effective infinite dimensional geometry of the single-site scheme. The MF decoupling is equivalent to assuming a product ansatz
over all L-site blocks , with where is the ground state of the cluster Hamiltonian
Figure 2. (online color at: www.pss-b.com) (a) A block of L sites is represented together with its nearest neighbors from the two contiguous blocks. The hopping between the two sites at the end of the central block with the sites outside of it are approximated using a homogeneous SF order parameter which is determined self-consistently 24, 25. In doing so, the Hamiltonian of the central block is decoupled from its neighbor sites rendering the system numerically solvable. (b) Several MI lobes are shown which were computed using mean-field decoupling on the hopping between adjacent 1D blocks of L sites. The size of the block vary from the conventional L = 1 up to L = 8 site clusters. The solid grey lines are the boundaries from a 12th order SCP expansion which are included as a reference.
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Here the boundary MF is enforcing a homogeneous order parameter whose value is determined self-consistently as before. The MI and SF phase are again signalled by Φ = 0 and Φ ≠ 0, respectively. By partitioning the system in to clusters of L > 1 sites our description loses its full translational symmetry, but in turn can describe MI and SF phases by more complex states with quantum correlations, such as particle–hole fluctuations, present within each block.
As seen in Fig. 2b, we find that the extension to a multi-site MF approach has only a marginal influence on both the shape and size of the predicted MI lobes. Going to large chains does extend the lobe to fractionally larger t, though the qualitative form of the MI lobes remain rounded similar to the single-site version. In principle as L → ∞ this approach will converge to an exact diagonalization of the BHM. However, the results for numerically accessible chains sizes up to L = 8 show this convergence is extremely slow and no re-entrant behavior is visible. This suggests that the symmetry breaking induced by the boundary MF decoupling of the hopping is dominant and washes out vital physics underpinning re-entrance.
3.2 Small system exact diagonalization
In Fig. 3 the resulting particle and hole boundaries for the MI lobe for increasing system sizes N are shown. An indication that these curves do not produce genuine phase boundaries is that the particle and hole boundaries do not intersect for any finite N and so predict a MI lobe of infinite size. However, a key feature, clearly visible for all system sizes examined, is that the hole boundary already shows concavity at moderate values of t characteristic of re-entrance. Moreover there is a visible tendency of the finite-size system boundaries to slowly converge to the expected MI lobe in the thermodynamic limit as predicted by the 12th order SCP expansion. This feature is the basis of finite-size scaling calculations 26, 29.
Figure 3. (online color at: www.pss-b.com) The sequence of particle and hole boundaries computed from small finite-sized clusters with periodic boundary conditions 26, 29. System sizes go from N = 2 to N = 11 sites as the outermost to innermost lines, respectively. The solid grey lines are the boundaries for a 12th order SCP expansion which are included as a reference.
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Both the finite-sized cluster and the multi-site MF approximations are based on an ansatz with a repeated cluster of variable size which breaks the underlying translational symmetry of the BHM Hamiltonian in Eq. (1). Yet the phase boundaries predicted for each of these methods are rather different. We see that the finite-sized cluster calculation produces systematic overestimations of the MI lobe for any cluster size while the multi-site MF approximation produces a systematic underestimation. The reason for this is that unlike MF decoupling the finite-sized cluster approach does not break the U(1) global symmetry of the BHM and is unable to mimic the quasi-long range order characteristic of a SF thereby favoring the MI phase. In contrast the multi-site MF approximation can break this symmetry favoring the SF phase. That a precursor to re-entrance is visible in finite-sized clusters demonstrates that a symmetry preserving description of kinetic energy is crucial to its emergence in an approximation. This conclusion is further enhanced by results obtained with more sophisticated approaches such as the variational cluster perturbation method 30. There the system is partitioned into finite-size clusters, like that used in the multi-site MF approximation, however hopping between the clusters is instead treated perturbatively. Crucially this allows the U(1) symmetry to be preserved in both the SF and MI phases. This phase diagram produced by this method then resemble the exact-diagonalization results here and also similarly predict re-entrance for any cluster size 30.
3.3 Real-space renormalization group
The exponential growth in the Hilbert space with N means that exact diagonalization cannot be feasibly pushed much beyond the sizes considered here already. For this reason we examine the predictions of a real-space RG approach for the BHM following the scheme originally introduced by Singh and Rokhsar 31 some time ago. In that work they focused on the canonical ensemble at unit-filling to analyse the KT transition point at the tip. Here for the first time we reformulate their scheme within the quasi-grand-canonical ensemble approach, just applied to the finite-sized clusters, to obtain an RG prediction of the complete phase diagram of the BHM. The RG approximation combines two crucial features of the MF and finite-sized clusters considered already. First, like finite-sized clusters it preserves the U(1) global symmetry of the BHM. Second, like the MF decoupling RG scheme curtails the exponential growth of the system's Hilbert space enabling the thermodynamic limit to be reached.
The real-space RG scheme applied is based on partitioning the 1D system into non-overlapping two site blocks. In contrast to the other approximations the RG scheme 31 simplifies the BHM by truncating the on-site Fock basis to the occupation states , and thereby excluding triple or higher occupancies. In this basis the kinetic contribution to the block Hamiltonian , describing hopping between adjacent sites within a block, is
where th and tp, specify the hopping amplitude for a hole or additional particle, respectively, while to is the amplitude for the creation or annihilation of a particle–hole pair. When constructing directly from the hopping terms in the BHM in Eq. (1) we have that reflecting the fact that the BHM is not particle–hole symmetric. The block Hamiltonian is then completed by adding the interaction and chemical potential terms, which in the truncated basis are diagonal on-site operators with diagonal matrix elements (0, 0, U) and (0, µ, 2µ), respectively 31.
The RG scheme begins by diagonalizing and identifying three eigenstates , , and which represent the lowest energy state with one, two and three bosons per block, respectively. The Fock state truncation enables these states to be readily determined as
with corresponding energies
where . Using this appropriately chosen reduced basis of states for a single block we can now interpret the block as a new super-site and formulate a renormalized block Hamiltonian spanning two such adjacent super-sites. In this way the RG scheme dramatically limits the degrees of freedom retained. The Hamiltonian has exactly the same type of terms as , once the replacements , etc. have been made, but possesses different matrix elements , , , U′, and µ′. The new on-site matrix elements are given by
Figure 4. (online color at: www.pss-b.com) (a) The real-space RG scheme adopted here 31. The Hamiltonian of a two site block is diagonalized and the lowest three eigenstates are kept. This forms an isometry mapping which renormalizes a block of two sites into a single super-site coupled to its neighbors with a different hopping, that is, t → t′. (b) This RG scheme is equivalent to variationally minimizing a TTN ansatz. Here each isometry is represented as a three leg tensor (triangle) composed as a hierachical network. (c) The MI lobe predicted by the real-space RG scheme is displayed, along with the green point marking the predicted KT critical point 31. The inset is a zoom of phase boundaries around the tip highlighting the observed re-entrance. The solid gray lines are again the boundaries from a 12th order SCP expansion included as a reference.
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