Mapping between Hamiltonians with attractive and repulsive potentials on a lattice
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p Mapping between Hamiltonians with attractive and repulsivepotentials on a lattice
Yogesh N. Joglekar
Department of Physics, Indiana University Purdue UniversityIndianapolis (IUPUI), Indianapolis, Indiana 46202, USA (Dated: October 15, 2018)
Abstract
Through a simple and exact analytical derivation, we show that for a particle on a lattice, thereis a one-to-one correspondence between the spectra in the presence of an attractive potential ˆ V and its repulsive counterpart − ˆ V . For a Hermitian potential, this result implies that the number oflocalized states is the same in both, attractive and repulsive, cases although these states occur above(below) the band-continnum for the repulsive (attractive) case. For a PT -symmetric potential thatis odd under parity, our result implies that in the PT -unbroken phase, the energy eigenvalues aresymmetric around zero, and that the corresponding eigenfunctions are closely related to each other. ntroduction: The energy spectrum of a quantum particle in an attractive potential V ( r ),in general, consists of discrete eigenvalues for which the eigenfunctions are localized in realspace, and continuum eigenvalues with non square-integrable eigenfunctions. The energyspectrum for the corresponding repulsive potential − V ( r ) has only continuum eigenval-ues [1, 2]. This situation changes dramatically when the particle is confined to a lattice or,equivalently, is exposed to a periodic potential. Indeed, repulsively bound two-atom stateshave been explored in detail since their experimental discovery in optical lattices [3, 4] andcontinue to be a source of ongoing work [5] in the context of the Bose-Hubbard model [6, 7].We note that in the Bose-Hubbard model, the interaction between the two atoms is short-ranged and is tuned via the Feschback resonance [3]. However, to our knowledge, theproperties of single-particle states localized in the vicinity of a generic repulsive potential(defined below) have not been studied. In another area, localized states in parity + time-reversal ( PT ) symmetric one-dimensional lattice models, too, have been explored in recentyears. These explorations have focused on the PT -symmetry breaking in the presence ofattractive (real) on-site potentials with random PT -symmetric complex parts [11].In this note, through a simple but exact derivation, we show that for a single particle ona lattice, there is a one-to-one correspondence between its energy spectrum in the presenceof an attractive potential and the repulsive counterpart, and that the corresponding eigen-functions have identical probability distributions. For PT -symmetric potentials that areodd under parity (and hence time-reversal), we show that if the PT -symmetry is unbroken,the energy spectrum must be symmetric around zero. One-dimensional Model:
Let us start with the Hamiltonian for a particle on a one-dimensional lattice with only nearest-neighbor hopping energy
J > H = − J X i (cid:16) c † i c i +1 + c † i +1 c i (cid:17) (1)where c † i and c i are creation and annihilation operators at site i respectively. The externalpotential is given by ˆ V = P j V j c † j c j . We define the potential to be attractive provided P j V j < | ψ α i = P j f α,j | j i be an eigenstate of theHamiltonian ˆ H + = ˆ H + ˆ V with energy E α where | j i denotes a single-particle state localizedat site j . The coefficients f α,j obey the recursion relation − J [ f α,j +1 + f α,j − ] + V j f α,j = E α f α,j . (2)2e now consider the staggered wavefunction | φ α i = P j f α,j ( − j | j i . Using Eq.(2) it isstraightforward to show that the staggered wavefunction satisfies the following equationˆ H | φ α i = (cid:16) − E α + ˆ V (cid:17) | φ α i . (3)Thus, it is an eigenfunction of the conjugate Hamiltonian ˆ H − = ˆ H − ˆ V with eigenvalue − E α . When ˆ V = 0, the energy spectrum is given by ǫ k = − J cos( ka ) and representsthe well-known continuum band from − J to 2 J where a is the lattice spacing. In thistrivial case, indeed the eigenfunction | ψ k i = P j sin( kj ) | j i and its staggered counterpart | φ k i = P j sin [( π − k ) j ] | j i have energies ± ǫ k respectively.Our result shows that if an attractive external potential ˆ V has n bound states belowits continuum with energies E m ( m = 1 , . . . , n ), then the corresponding repulsive potential − ˆ V must have an equal number of bound states above its continuum with energies − E m .Since the staggered wavefunction | φ α i varies over the lattice length-scale a , it is energeticallyexpensive and ill-defined in the continuum limit a →
0. Physically, in the continuum limit,the absence of lattice-site scattering centers makes it impossible for a particle to localizenear the repulsive potential. However, on a lattice, the probability distributions for the twostates - a localized bound state | ψ α i with energy E α ≤ − J in an attractive potential and thelocalized bound state | φ α i with energy − E α ≥ +2 J in the repulsive potential - are identical.As a concrete example, we numerically obtain the spectrum for a lattice with N = 29 sitesand a quadratic potential that vanishes at the ends, V m = Λ( m − N − m ) /N , where m = 1 , . . . , N , N = ( N + 1) / V N = Λ. Figure 1 shows theground state wavefunction ψ Gm for the attractive case, Λ /J = − .
5, (left panel) along withthe highest-energy state wavefunction φ m for the repulsive case, Λ /J = +0 . φ m = ( − m +1 ψ G,m . Two-particle Case:
We can generalize this result in a straightforward manner to treat in-terparticle interaction ˆ U = P ij U i − j ˆ n i ˆ n j where the on-site number operator is given byˆ n i = c † i c i . In the two-particle sector, the recursion relation satisfied by the relative-coordinatewavefunction is given by [6, 7] − J K (cid:2) ψ Kα,m +1 + ψ Kα,m − (cid:3) + U ( r m ) ψ Kα,m = E Kα ψ Kα,m . (4)Here − π/a ≤ K ≤ π/a is the lattice momentum associated with the center-of-mass of thetwo particles, J K = J cos( Ka ) is the effective hopping energy, r m = am = a ( i − j ) is the3
10 20 3000.10.20.30.4 m (position along the lattice) G r ound − s t a t e w a v e f un c t i on ψ G , m S t agge r ed w a v e f un c t i on φ m FIG. 1. (color online) (a) The left panel shows the dimensionless ground-state wavefunction ψ G,m for an attractive quadratic potential V m = Λ( m − N − m ) /N where N = 29 = (2 N + 1) isthe lattice size and Λ /J = − .
5. As expected for a quadratic potential ground-state, ψ G,m is aGaussian with width x = a ( N t/ | Λ | ) / ∼ .
61. (b) The right panel shows the dimensionlesshighest-energy state wavefunction φ m for its repulsive counterpart with Λ /J = +0 .
5. We see thatthe φ m is indeed the staggered version of the ground-state wavefunction φ G,m . distance between the two particles on the lattice located at sites i and j , and U ( r m ) isthe real-space interaction between the two particles. Note that for a non-local interparticleinteraction U ( r m ), multiple bound-state ψ Kα solutions are generic, although, in the contextof the Bose-Hubbard model, only one [3] or two [6] have been discussed. If ψ Kα is aneigenfunction of the Hamiltonian ˆ H + ˆ U with energy E Kα , Eq. 4 implies that the staggeredwavefunction φ Kα defined by φ Kα ( r m ) = ( − m ψ Kα ( r m ) is an eigenfunction of the conjugateHamiltonian ˆ H − ˆ U with energy − E Kα .Two-particle bound states in the presence of on-site and nearest-neighbor repulsivedensity-density interactions on a lattice have been extensively investigated [3, 5, 6]. Ourderivation shows that they are a generic feature of any density-density interaction on a lat-tice, and this result is true for square lattices in higher dimensions. Note that the quantumstatistics of the particles only constrains the relative wavefunction ψ Kα ( r m ) to be odd (spin-less fermions) or even (bosons or spin-singlet fermions) under parity; however, it does not4ffect the one-to-one correspondence between the spectra for the two Hamiltonians ˆ H ± ˆ U .Thus, two-atom bound-states with attractive and repulsive interactions in optical lattices(bosons) [3], the donor and acceptor impurity levels in semiconductors (fermions) [8], aswell as the localized phonon modes (collective bosonic excitation) [9, 10] around a soft orstiff impurity can all be thought of as manifestations of the correspondence between spectrafor ˆ H + and H − . PT Symmetric Potential:
The mapping between the two Hamiltonians ˆ H + and ˆ H − is validindependent of the properties of the potential ˆ V including its Hermiticity; the on-site po-tential elements V j may be complex. However, for a PT -symmetric potential that is oddunder parity (and hence, time reversal), V ∗ j = − V j = V − j , it follows that ˆ H ∗ + = ˆ H − where* denotes complex conjugation. Therefore, it follows from ˆ H + | ψ α i = E α | ψ α i that the wave-function | ψ ∗ α i = P j f ∗ α,j | j i is an eigenstate of the conjugate Hamiltonian ˆ H − with eigenvalue+ E ∗ α . In the continuum limit, it has been shown that a wide class of such potentials, in-cluding V ( x ) = ix and V ( x ) = i sin n +1 ( x ) have purely real energy spectra [12, 13]. Ifthe PT -symmetry is unbroken, E ∗ α = E α , then it follows that ˆ H − | φ α i = − E α | φ α i andˆ H − | ψ ∗ α i = + E α | ψ ∗ α i .This explicit construction of wavefunctions with equal and opposite energies implies thatfor any arbitrary PT -symmetric potential that is odd under parity, if the PT symmetryis not broken, the energy spectrum must be symmetric around zero. It also shows thatthe corresponding wavefunctions in the two cases have components that are simply related:[+ E α , f ∗ α,j ] ↔ [ − E α , f α,j ( − j ]. As an example, we consider the simplest “finite lattice”with 2 points. (Our result is equally applicable to a finite lattice.) The Hamiltonian in thiscase is given by ˆ H − = − J ˆ σ x + iγ ˆ σ z where ( σ x , σ z ) are the Pauli matrices in the site-indexspace [14] and a real γ ensures that the potential is odd under parity as well as time-reversal.The eigenvalues in this case are given by E ± = ± p J − γ . Thus the PT -symmetry in thiscase is not broken as long as γ ≤ J . The corresponding (unnormalized) eigenfunctions [15]are given by [14] |±i = ± e ∓ iθ (5)where θ = arctan( γ/ p J − γ ) is real when γ ≤ J . Therefore, in the PT -unbroken phase,the eigenvectors for positive and negative energies indeed are related by f − ,j = ( − j f ∗ + ,j where j = 0 ,
1. 5 onclusion:
Our result, through a one-to-one mapping between attractive and repulsivepotentials on a lattice, shows that localized states in repulsive potentials are ubiquitous.These states can be explored via local measurements. In contrast to the bound-states withenergies below the continuum band, these localized states with energies above the continuumband will decay into the continnum states. They may thus provide a useful spectroscopictool in optical lattices as well as engineered electronic materials with a small bandwidth. [1] L.D. Landau and E.M. Lifschitz,
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