Hardcore Boson Model

Introduction

The hardcore boson model is what remains of the Bose-Hubbard Model in the limit $U \to \infty$ : bosons still hop between lattice sites, but the on-site repulsion is now so strong that double occupancy is forbidden outright, rather than merely costly. Each site can only be empty or singly occupied — exactly the same simplification the t-J Model makes to the fermionic Hubbard model, but applied to bosons instead of fermions.

The model, or more precisely its exact equivalence to a spin-1/2 system described below, was introduced by Matsubara and Matsuda (1956) as a lattice model for the superfluid transition in liquid helium — one of the earliest examples of mapping a genuine interacting quantum many-body problem onto an exactly equivalent spin model.

Because double occupancy is forbidden, the hardcore boson operators satisfy modified commutation relations:

[b_i, b_j^\dagger] = \delta_{ij} (1 - 2 b_i^\dagger b_i), \qquad (b_i^\dagger)^2 = (b_i)^2 = 0,

where $(b_i^\dagger)^2 = 0$ directly enforces the constraint: creating a second boson on an already-occupied site simply gives zero. Operators on different sites still commute with one another exactly as ordinary bosonic operators do — only the on-site algebra is modified. The Hamiltonian is

H = -t \sum_{\langle i,j \rangle} \left( b_i^\dagger b_j + \text{h.c.} \right) + V \sum_{\langle i,j \rangle} n_i n_j - \mu \sum_i n_i,

where $t$ is the hopping amplitude, $V$ is a nearest-neighbor interaction between bosons, $n_i = b_i^\dagger b_i$ is the (now strictly 0-or-1) occupation of site $i$ , and $\mu$ is the chemical potential. This is exactly the running example used to introduce the ALPS model XML format on the Site Basis and Hamiltonian Descriptions pages.

Physics of the model

An exact mapping to spin-1/2: the Matsubara-Matsuda transformation. Because a hardcore boson site has exactly two states — empty or occupied — it is algebraically identical to a spin-1/2 degree of freedom, via

b_i^\dagger = S_i^+, \qquad b_i = S_i^-, \qquad n_i = S_i^z + \tfrac{1}{2}.

Substituting this into the Hamiltonian above turns the hopping term into the transverse (XY) exchange $-2t \sum_{\langle i,j \rangle} (S_i^x S_j^x + S_i^y S_j^y)$ and the interaction term into the Ising exchange $V \sum_{\langle i,j \rangle} S_i^z S_j^z$ , plus a field term absorbing $\mu$ — in other words, exactly the anisotropic (XXZ) Heisenberg Model, in a field set by the boson density. Crucially, unlike the Jordan-Wigner transformation used for fermions on the Spinless Fermion Model page, this mapping needs no nonlocal string: bosons on different sites already commute, so the Matsubara-Matsuda transformation is completely local and works in any number of spatial dimensions, not just one.

Superfluidity and magnetic order are the same phenomenon. This equivalence turns the hardcore boson model’s phase diagram directly into the XXZ model’s phase diagram from the Heisenberg Model page. Superfluid order — a nonzero, phase-coherent expectation value $\langle b_i \rangle$ — is exactly transverse (XY) magnetic order, $\langle S_i^x \rangle, \langle S_i^y \rangle \neq 0$ . A commensurate, checkerboard-ordered Mott/charge-density-wave insulator — bosons sitting preferentially on one sublattice — is exactly Néel order along $z$ . The superfluid-insulator quantum phase transition of the boson language and the order-disorder quantum phase transition of the spin language are, on this exact mapping, the very same transition seen through two different sets of variables.

A real material realization: field-induced magnon condensation. This equivalence is not just formal. In real quantum magnets with a spin gap — a unique, non-degenerate ground state separated from all magnetic excitations by an energy gap — a strong-enough external magnetic field can close that gap and drive the system into a magnetically ordered state. In the hardcore boson language, the magnetic field plays the role of the chemical potential $\mu$ , the gapped, unmagnetized state is the vacuum (or Mott insulator) of bosons, and the field-induced ordered state is literally a Bose-Einstein condensate of magnons — this description was used successfully by Nikuni, Oshikawa, Oosawa, and Tanaka (2000) to explain the field-induced ordering observed experimentally in the quantum magnet TlCuCl $_3$ .

Phenomena

Superfluidity: at low density or weak interaction $V$ , hardcore bosons delocalize into a phase-coherent superfluid — equivalently, transverse (XY) magnetic order in the spin language.
Mott/charge-density-wave insulation: at commensurate filling (e.g. half filling on a bipartite lattice) and large $V$ , bosons localize into a checkerboard pattern — equivalently, Néel order in the spin language.
Quantum phase transitions: tuning density or $V/t$ drives the same superfluid-insulator transition already discussed for the Bose-Hubbard Model, now exactly mappable onto the magnetic ordering transition of an anisotropic spin-1/2 model.
Magnon Bose-Einstein condensation: field-induced ordering transitions in real gapped quantum magnets are described quantitatively as a Bose-Einstein condensation of hardcore-boson magnons (see above).

Methods

Because the model maps exactly onto a spin-1/2 Hamiltonian, every method applicable to the Heisenberg Model applies here as well, with no sign problem in any dimension:

Method	Strengths	Limitations	Applications
Worm algorithm / SSE — see Worm Algorithm / SSE	Sign-problem-free at any filling and in any dimension	None specific to this model	Superfluid density, phase diagrams, finite-temperature properties
Loop algorithm — see Quantum Monte Carlo	Highly efficient at exactly half filling ( $\mu = 0$ ), where the mapped spin model has no field	Away from half filling, the chemical potential becomes a field term that the loop algorithm handles poorly; use the worm/SSE codes instead	Half-filled (particle-hole symmetric) phase diagrams
ED — see sparsediag / fulldiag	Exact results for small systems; the hardcore constraint is built into the local Hilbert space at no extra cost	Limited to small systems	Small-system benchmarks
DMRG — see Density Matrix Renormalization Group	Highly accurate for 1D chains and ladders	Less efficient for genuinely 2D/3D systems	Ground states of 1D hardcore-boson chains

There is no dedicated ALPS tutorial for this model by name, but it is the worked example used throughout the ALPS Model Definitions pages, and the Bose-Hubbard Model tutorials (MC-05, DWA-01, DWA-02) illustrate the same worm and directed-worm methods on the closely related finite- $U$ model.

For an overview of the other models in ALPS, see Models in ALPS.

Anderson Impurity Model Bose-Hubbard Model