Effective Action Approach to Quantum Phase Transitions in Bosonic Lattices MASSACHUSETTS INSTTIJE OF TECHNOLOGY by JUL 0 7 2009 Barry J Bradlyn LIBRARIES Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of BACHELOR OF SCIENCE at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2009 @ Massachusetts Institute of Technology 2009. All rights reserved ARCHIVES Author .ent of Physics May 4, 2009 Certified by. Roman Jackiw Professor, Department of Physics Thesis Supervisor Accepted by. Pritchard Senior Thesis Coordinator, Department of Physics Effective Action Approach to Quantum Phase Transitions in Bosonic Lattices by Barry J Bradlyn Submitted to the Department of Physics on May 4, 2009, in partial fulfillment of the requirements for the degree of BACHELOR OF SCIENCE Abstract In this thesis, I develop a new, field-theoretic method for describing the quantum phase transition between Mott insulating and superfluid states observed in bosonic optical lattices.
I begin by adding to the Hamiltonian of interest a symmetry breaking source term. Using time-dependent perturbation theory, I then expand the grand- canonical free energy as a double power series in both the tunneling and the source term. From here, an order parameter field is introduced, and the underlying effective action is derived via a Legendre transformation. After determining the Ginzburg- Landau expansion of the effective action to first order in the tunneling term.
expres- sions for the Mott insulator-superfluid phase boundary, condensate density, average particle number, and compressibility are derived and analyzed in detail. Additionally, excitation spectra in the ordered phase are found by considering both longitudinal and transverse variations of the order parameter. Finally, these results are applied to the concrete case of the Bose-Hubbard Hamiltonian on a three dimensional cubic lat- tice, and compared with the corresponding results from mean-field theory. Although both approaches yield the same Mott insulator - superfluid phase boundary to first order in the tunneling, the predictions of the effective action theory turn out to be superior to the mean-field results deeper into the superfluid phase.
Thesis Supervisor: Roman Jackiw Title: Professor, Department of Physics Acknowledgments I would foremost like to thank Ednilson Santos and Dr. Axel Pelster for the collab- oration which led to the work presented in this thesis. Additionally, I would like to thank Prof. Roman Jackiw for his advice and supervision.
Furthermore, I would like to thank Hagen Kleinert, Flavio Nogueira, and Matthias Ohliger for fruitful discus- sions and suggestions. Finally, I acknowledge financial support from both the German Academic Exchange Service (DAAD) and the German Research Foundation (DFG) within the Collaborative Research Center SBF/TR 12 Symmetries and Universality in Mesoscopic Systems.1 Derivation of the Bose-Hubbard Model .2 Ginzburg-Landau Theory. 13 2 The Grand-Canonical Free Energy 15 3 The Effective Action 23 3.1 Physical Quantities in the Static Case .2 Physical Quantities in the Dynamic Case. 31 4 An Application: The Bose-Hubbard Hamiltonian 37 4.1 Effective Action Predictions in the Static Case .2 Effective Action Predictions in the Dynamic Case.
44 5 Summary and Conclusion 49 Chapter 1 Introduction Recent developments in the field of dilute ultracold quantum gasses [1, 2, 3, 4, 5] have led to the experimental investigation of atoms in periodic potentials [6]. They are a fascinating new generation of many-particle quantum systems as they allow for the study of a variety of solid-state phenomena under perfectly controlled conditions [4, 6, 7,8. For instance, bosonic lattice systems show a quantum phase transition for varying lattice depths. In deep lattices the tunneling between lattice sites is suppressed, and a Mott insulating state forms with a fixed number of bosons residing on each lattice site.
For shallow lattices, however, the dominance of inter- site tunneling allows for bosons to spread coherently over the whole lattice, forming a superfluid. The occurrence of such a quantum phase transition between a Mott insu- lator and a superfluid is observable, for instance, in tinme-of-flight absorption pictures taken after switching off the lattice potential. They inmage momentum distributions integrated along one axis, and therefore by Heisenberg's unIertainty principle give information about the corresponding spatial distributions. the localization of atoms in the Mott phase results in diffuse absorption pictures.
while the delocalized superfluid phase gives rise to Bragg-like interference patterns. The theoretical analysis of this quantum phase transition is usually based on the Bose-Hubbard model Hamiltonian [13, 14, 15, 16], HBH = [U&UaQ&(,- 1) - fta 1- t >aa, (1.1) where i, and i are bosonic annihilation and creation operators, p is the chemical potential, and (i, j) signifies a sum over nearest neighbor sites i and j. Additionally, U parameterizes the on-site interaction energy between two atoms at a given site, and t characterizes the kinetic energy, in this case given by the tunneling of an atom between two neighboring lattice sites. The quartic on-site coupling term, however, makes an exact diagonalization of (1.
Thus, while Monte-Carlo simulations have proven fruitful for obtaining numerical results [17, 18, 19], analytic descriptions of bosonic lattices near the quantum phase boundary have so far been typically limited to imean-field [13, 15] or strong-coupling approximations [20, 21]. Currently, the most precise analytic result for the whole Mott insulator-superfluid phase diagram in a three dimensional cubic lattice at zero temperature is found in Ref. Therein, a Landau expansion for an effective potential with a spatially and temporally global order parameter is derived. In this thesis, we generalize the results of Ref.
[22] by allowing for a spatially and temporally varying order parameter, thus determining a Ginzburg-Landau expansion for the effective action. This allows us to obtain an approximate analytic description of bosonic lattice systelms near the quantum phase 1)oundarv. This work is based primarily upon my previous paper, Ref [23] To this end we proceed as follows. The remainder of this Chapter will present a (derivation of the Bose-Hubbard model, and an overview of the Ginzburg-Landau theory of phase transitions.
we consider a very general type of Hainiltonian consisting of an arbitrary on-site interaction and an arbitrary tunneling terin. of which the Bose-Hubbard Hamiltonian is a special case. and determine the g]l amd-( ainonical free energy to first order in the tiuneling term. This tunneling laproximlation is motivated by the fact that in three diimensions, the Mott insulator- supeliflid quantum phase transition is observed to occur for small values of t/U (note that the tunneling expansion is related to the landom-walk expansion of Refs.
Next, in Chapter 3 we introduce an order parameter field and derive a Ginzburg-Landau expansion of the effective action, allowing for the computation of physical quantities near the phase boundary in both the Mott insulator and the superfluid phase.2 present predictions of our effective-action theory for both static homogeneous and spatio-temporally varying order parameter fields, including expressions for the particle density, the compressibility, the superfluid density, and the excitation spectra. Finally, in Chapter 4, we specify our results to the Bose-Hubbard Hamiltonian, and compare them to the predictions of the standard mean-field theory. Although both approaches yield the same approximation for the location of the phase boundary, our effective action approach turns out to be superior to the mean-field theory for the following reasons. First, we demonstrate that the effective action approach leads to qualitatively better results deeper into the superfluid phase.
Secondly, in contrast to the mean-field approximation, the effective action approach can be systematically extended to higher orders in the tunneling parameter in order to quantitatively improve the results, as has already been demonstrated for the case of the effective potential in Ref.1 Derivation of the Bose-Hubbard Model We can see how Eq.1) comes about by examining the setup of optical lattice experiments. bosons are confined using pairs of counter-propegating lasers aligned along all three spatial axes [4, 5. This establishes standing waves in all three dimensions, creating a cubic array of electric field intensity maxima. Due to the Stark effect.
the b)osonic atoms are attracted to these intensity nmaxima. as there electronic energy will be lower there due to level splitting. the effect of the laser beams is to establish a potential energy term in the Hamiltonian of the form attice( ) = V sill 2 ( 7 X)_ .2) 11 where a = 1, 2, 3 is the spatial direction, and a is the laser wavelength. Additionally, we suppose that there is a hard-core repulsive interaction due to the presence of two atoms at the same point, parameterized by the coupling constant g.
Putting these terms together, we can construct the second-quantized Hamiltonian for the lattice system as H J= d3/t(5) + Vlattice(;) +g t(>)$(5) ,p(S), (1.3) where g(Y) and /i (5) are the annihilation and creation operators for bosons at posi- tion Z. They satisfy the canonical commutation relations Since the potential Vattice(Y) is periodic in 5, we can identify each intensity maximum as a lattice site i and, via Bloch's theorem, switch to a basis of Wannier functions [27] w,,(5) localized about i which satisfy fd w ( )w3( = 6b, ) (1. the bosonic field operators become V() - Ziwz(X) ,t(X) lZw'(.6) where i' and 6, are bosonic creation and annihilation operators at site i obeying the standard coninutation relations [az &j = [at!at 0=.7) For our analysis, the specific form of the Wannier functions is no important, but they can be found numerically if one isso inclined [28]. Up to now.
our treatment has been exact. we sul)stitute the Wannier-function decomposition of the field operators into Eq.3), and make the approximation that the Wannier functions are strongly localized enough that all but nearest neighbor overlap integrals can be neglected. With the definitions t =- - d"3xw*() Vlattice(x) w () (1.9) we recover the Bose-Hubbard Hamiltonian (1.2 Ginzburg-Landau Theory Our main tool for analyzing phase transitions in the Bose-Hubbard model will be the Ginzburg-Landau theory [29, 30]. The primary tenant of this framework is that different phases are characterized by different symmetry properties.
In the case of the insulator-superfluid phase transition we will be examining, the relevant symmetry is the breaking of global U(1) phase symmetry , -- ee' in the superfluid ground state [30]. Clearly, from Eq.1), this is a symmetry of the Hamiltonian, so that the breaking of it by the ground state of the system is referred to as spontaneous symmetry breaking. The fundamental quantity of the Ginzburg-Landau theory is the order parameter, which quantifies the extent of the symmetry breaking. The order parameter should be 0 in the symmetric (in this case.
the insulating state), and acquires a nonzero value at the phase boundary. Because we are interested in symmetry breaking of the creation and annihilation operator phases. we choose as our order parameter field ( = K, 0(7) (1.10) This quantity must acquire a nonzero value if the phase symmetry shown above is broken. To quantify the properties of the phase transition, a thermodynamic potential must be found which depends on the order parameter field.
For us, this role will be served by the effective action F. Since the Hamiltonian does not break the phase symmetry, we expect the effective action to depend only on even powers of ,b. The standard Ginzburg-Landau theory proceeds by expanding the thermodynamic poten- tial to fourth order in the order parameter. Since in equilibrium all thermodynamic potentials are stationary (i.
their differentials are zero when the appropriate state variables are held fixed) [29], we can find the equilibrium value of the order parameter by setting the first derivative of F equal to 0. From there, many other thermodynamic quantities may be determined. _~~~ ___ I*~LIX---L ~__rX^~__ Chapter 2 The Grand-Canonical Free Energy We consider bosons on a background lattice with lattice sites denoted by i. Suppose they are described by a Hamiltonian of the form ft = o + ft, (2.1) which depends on bosonic creation and annihilation operators o, and a,.
We assume that Ht is a sum of local terms each diagonal in the occupation number basis, i.2) so its energy eigenvalues are given by E n, = f(nj) (2.