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Effect of Particle Statistics and Frustration on Ground-State Energy

文星 聶

Published Mar 24, 2014• 0 citations
Publisher

Abstract

We study the connections among particle statistics, frustration, and groundstate energy, in many-particle systems.Statistics of identical particles is one of the most fundamental concepts, which pervades all of quantum mechanics.A consequence of the particle statistics appears in the ground-state energy.We compare the ground-state energies of bosons and fermions with the same Hamiltonian.In noninteracting systems, the influence of particle statistics on the ground-state energy is trivial: the ground-state energy of noninteracting bosons is lower than that of free fermions because of Bose-Einstein condensation (BEC) and Pauli exclusion principle.The relation that bosons have a lower ground-state energy than fermions is described as natural inequality in this thesis.However, the comparison of the ground-state energies of bosons and fermions is not trivial in the presence of interaction, because the simple argument based on the perfect BEC breaks down.In a system of interacting bosons, it is in fact already a nontrivial question whether the BEC actually takes place.In strongly correlated systems, the influence of particle statistics on the ground-state energy is still a relatively unexplored area.We have found a sufficient condition for the natural inequality to hold for spinless and spinful cases respectively, without relying on the occurrence of BEC.That is, if all the hopping amplitudes are nonnegative, the ground-state energy of hard-core bosons is still lower than that of fermions.The same argument implies that, once we relax the condition of nonnegative hopping amplitudes, it is possible to reverse the inequality so that the ground-state energy of bosons could be higher than that of fermions.By relaxing the condition, we indeed have found several concrete examples in which the ground-state energy of hard-core bosons is proved to be higher than that of fermions.Many of the examples are even proved rigorously in the thermodynamic limit.Our study leads to a novel physical understanding of the effects of particle statistics, in terms of frustration in quantal phase.This is more general than the picture based on the perfect BEC, and is indeed applicable to systems with interaction.We can map a quantum many-particle problem to a single-particle problem on a fictitious lattice in higher dimensions.When all the hopping amplitudes are nonnegative and the particles are bosons, the corresponding single-particle problem also has only nonnegative hopping amplitudes.In such a case, there is no frustration in the quantal phase of the wavefunction.On the other hand, Fermi statistics of the original particles gives an effective magnetic flux in the corre-sponding single-particle problem.This implies a frustration in the phase of the wavefunction, which can be regarded as "statistical frustration": it is induced by the Fermi statistics and it leads to a destructive interference among propagation along different paths.In the presence of a non-vanishing flux in the original many-body problem, we observe that there is also a magnetic flux in the corresponding single-particle problem, inducing a frustration among quantal phases, which we name hopping frustration.In the original many-particle problem, the statistical frustration appears rather differently from the hopping frustration.However, upon mapping to the single-particle problem on the fictitious lattice, both hopping frustration and statistical frustration are represented by non-vanishing flux in the fictitious lattice.This provides a unified understanding of frustration.Based on the unified understanding, we find the mechanism as to why the ground-state energy of hard-core bosons can be higher than that of fermions.For many-body bosonic systems, in which there is no statistical frustration, introduction of hopping frustration will not decrease the ground-state energy.This is known as Simon's universal diamagnetism of bosons.On the other hand, in many-body fermionic systems, where statistical frustration already exists, hopping frustration if introduced, is expected to compete with statistical frustration and sometimes partially cancel with each other, resulting in the reversed inequality between the ground-state energies of the hard-core bosons and fermions.

DOI
10.15083/00006601

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We study the connections among particle statistics, frustration, and groundstate energy, in many-particle systems.Statistics of identical particles is one of the most fundamental concepts, which pervades all of quantum mechanics.A consequence of the particle statistics appears in the ground-state energy.We compare the ground-state energies of bosons and fermions with the same Hamiltonian.In noninteracting systems, the influence of particle statistics on the ground-state energy is trivial: the ground-state energy of noninteracting bosons is lower than that of free fermions because of Bose-Einstein condensation (BEC) and Pauli exclusion principle.The relation that bosons have a lower ground-state energy than fermions is described as natural inequality in this thesis.However, the comparison of the ground-state energies of bosons and fermions is not trivial in the presence of interaction, because the simple argument based on the perfect BEC breaks down.In a system of interacting bosons, it is in fact already a nontrivial question whether the BEC actually takes place.In strongly correlated systems, the influence of particle statistics on the ground-state energy is still a relatively unexplored area.We have found a sufficient condition for the natural inequality to hold for spinless and spinful cases respectively, without relying on the occurrence of BEC.That is, if all the hopping amplitudes are nonnegative, the ground-state energy of hard-core bosons is still lower than that of fermions.The same argument implies that, once we relax the condition of nonnegative hopping amplitudes, it is possible to reverse the inequality so that the ground-state energy of bosons could be higher than that of fermions.By relaxing the condition, we indeed have found several concrete examples in which the ground-state energy of hard-core bosons is proved to be higher than that of fermions.Many of the examples are even proved rigorously in the thermodynamic limit.Our study leads to a novel physical understanding of the effects of particle statistics, in terms of frustration in quantal phase.This is more general than the picture based on the perfect BEC, and is indeed applicable to systems with interaction.We can map a quantum many-particle problem to a single-particle problem on a fictitious lattice in higher dimensions.When all the hopping amplitudes are nonnegative and the particles are bosons, the corresponding single-particle problem also has only nonnegative hopping amplitudes.In such a case, there is no frustration in the quantal phase of the wavefunction.On the other hand, Fermi statistics of the original particles gives an effective magnetic flux in the corre-sponding single-particle problem.This implies a frustration in the phase of the wavefunction, which can be regarded as "statistical frustration": it is induced by the Fermi statistics and it leads to a destructive interference among propagation along different paths.In the presence of a non-vanishing flux in the original many-body problem, we observe that there is also a magnetic flux in the corresponding single-particle problem, inducing a frustration among quantal phases, which we name hopping frustration.In the original many-particle problem, the statistical frustration appears rather differently from the hopping frustration.However, upon mapping to the single-particle problem on the fictitious lattice, both hopping frustration and statistical frustration are represented by non-vanishing flux in the fictitious lattice.This provides a unified understanding of frustration.Based on the unified understanding, we find the mechanism as to why the ground-state energy of hard-core bosons can be higher than that of fermions.For many-body bosonic systems, in which there is no statistical frustration, introduction of hopping frustration will not decrease the ground-state energy.This is known as Simon's universal diamagnetism of bosons.On the other hand, in many-body fermionic systems, where statistical frustration already exists, hopping frustration if introduced, is expected to compete with statistical frustration and sometimes partially cancel with each other, resulting in the reversed inequality between the ground-state energies of the hard-core bosons and fermions.
We study the connections among particle statistics, frustration, and ground-state energy in quantum many-particle systems. In the absence of interaction, the influence of particle statistics on the ground-state energy is trivial: the ground-state energy of noninteracting bosons is lower than that of free fermions because of Bose-Einstein condensation and Pauli exclusion principle. In the presence of hard-core or other interaction, however, the comparison is not trivial. Nevertheless, the ground-state energy of hard-core bosons is proved to be lower than that of spinless fermions, if all the hopping amplitudes are non-negative. The condition can be understood as the absence of frustration among hoppings. By mapping the many-body Hamiltonian to a tight-binding model on a fictitious lattice, we show that the Fermi statistics of the original particles introduces an effective magnetic flux in the fictitious lattice. The latter can be effectively regarded as a kind of frustration since it leads to a destructive interference among different paths along which a single particle is propagating. If we introduce hopping frustration, the hopping frustration is expected to compete with the effective frustration due to the Fermi statistics, leading to the possibility that the ground-state energy of hard-core bosons can be higher than that of fermions. We present several examples, in which the ground-state energy of hard-core bosons is proved to be higher than that of fermions due to the hopping frustration. The basic ideas were reported in a previous paper [W.-X. Nie, H. Katsura, and M. Oshikawa, Phys. Rev. Lett. 111, 100402 (2013)]; more details and several extensions, including one to the spinful case, are discussed in this paper.
We compare the ground-state energies of bosons and fermions with the same form of the Hamiltonian. If both are noninteracting, the ground-state energy of bosons is always lower, owing to Bose-Einstein condensation. However, the comparison is nontrivial when bosons do interact. We first prove that, when the hopping is unfrustrated (all the hopping amplitudes are non-negative), hard-core bosons still must have a lower ground-state energy than fermions. If the hopping is frustrated, bosons can have a higher ground-state energy than fermions. We prove rigorously that this inversion indeed occurs in several examples.
In this work we investigate themes related to many-body systems in which multiple ground states are accessible, a condition known as frustration. Frustration can arise in a number of contexts, and we consider the consequences of this situation with some examples from condensed-matter physics. In some magnetic materials interactions between spins are such that no single spin configuration provides a unique ground state. In the class of frustrated magnets where the number of ground states is extensive, thermal fluctuations are strong even at temperatures significantly below the interaction strength. At such temperatures spins are highly correlated, and small perturbations may have profound consequences. In this thesis we provide an example of this. By considering classical n-component spins with nearest-neigbour exchange on a frustrated octahedral lattice we show that – in the limit where exchange interactions are large – the system is in a disordered, correlated phase where correlations have the form of a dipole field. This is termed a Coulomb phase. From this phase we induce an ordering transition, lifting the degeneracy with weak, additional short-range interactions. By studying the transition in the solvable limit of n → ∞, we discover that the transition has identical thermodynamics to that of a magnetic system interacting through long-range, dipolar forces. Finally, we provide a more apposite characterisation of the transition, where the high-temperature side of the transition is described through the fluctuations of solenoidal fields, and the ordering corresponds to a condensation of these fields. In a separate part of the thesis, we investigate the influence of disorder on frustrated lattices. We study a two-dimensional tight-binding model with nearest-neighbour hopping and on-site disorder. Restricting the allowed states to being those from the low-lying manifold of ground states, the disorder feeds through to act as effective disorder in the hopping terms, which decay algebraically with distance. The quasi-long range nature of this effective hopping leads to a situation in which the resultant single-particle eigenstates are critical, and we probe their behaviour numerically with a transfer matrix calculation.
This thesis encompasses a number of problems related to the number fluctuations from the ground state of ideal particles in different statistical ensembles. In the microcanonical ensemble most of these problems may be solved using number theory. Given an energy E, the well-known problem of finding the number of ways of distributing N bosons over the excited levels of a one-dimensional harmonic spectrum, for instance, is equivalent to the number of restricted partitions of E. As a result, the number fluctuation from the ground state in the microcanonical ensemble for this system may be found analytically. When the particles are fermions instead of bosons, however, it is difficult to calculate the exact ground state number fluctuation because the fermionic ground state consists of many levels. By breaking up the energy spectrum into particle and hole sectors, and mapping the problem onto the classic number partitioning theory, we formulate a method of calculating the particle number fluctuation from the ground state in the microcanonical ensemble for fermions. The same quantity is calculated for particles interacting via an inverse-square pairwise interaction in one dimension. In the canonical ensemble, an analytical formula for the ground state number fluctuation is obtained by using the mapping of this system onto a system of noninteracting particles obeying the Haldane-Wu exclusion statistics. In the microcanonical ensemble, however, the result can be obtained only for a limited set of values of the interacting strength parameter. Usually, for a discrete set of a mean-field single-article quantum spectrum and in the micro canonical ensemble, there are many combinations of exciting particles from the ground state. The spectrum given by the logarithms of the prime number sequence, however, is a counterexample to this rule. Here, as a consequence of the fundamental theorem of arithmetic, there is a one-to-one correspondence between the microstate and the macrostate, resulting in the vanishing of number fluctuation for all excitations. The use of the canonical or grand canonical ensembles, on the other hand, gives a substantial number fluctuation from the ground state. For a related spectrum, that given by the logarithms of an integer n, the microcanonical number fluctuation is non-zero but the application of the other ensembles is still not valid. These two spectra are examples of systems where canonical and grand canonical ensembles averagings yield answers different from the microcanonical result. Some models in physics may be used to obtain formulae known in the theory of number partition. For the same problem of N ideal bosons in a one-dimensional harmonic oscillator potential mentioned earlier, it is well known that the asymptotic (N --> ∞) density of states is identical to the Hardy-Ramanujan formula for the number of partitions of an integer n. The same statistical mechanics technique for the density of states of bosons in a power-law spectrum yields the partitioning formula for the number of partitions of n into a sum of 8th powers of a set of integers. By considering only the particle sector of the fermionic spectrum, a formula for the number of distinct partitions of n is obtained. For the s = 1 case and for finite N, the Erdos-Lehner formula for the restricted partitions, and a new formula for the distinct and restricted partitions are derived. As a diversion, we discuss the microcanonical entropy which may be uniquely defined in terms of the macrostate, or equivalently the many-body degeneracy of the state, at a given energy. The many-body degeneracy factor, however, is exceedingly difficult to calculate in general. It is thus desirable to find a different way to calculate the micro canonical entropy. It has been recently suggested that the microcanonical entropy may be accurately reproduced by including a logarithmic correction to the canonical entropy. This claim is readily tested using some of the models mentioned above, where the many-body degeneracy may be determined exactly. In addition, we also consider a system of N distinguishable particles in a d-dimensional harmonic energy spectrum. In this case the many-body degeneracy factor can be obtained analytically in a closed form.
We have used the stochastic series expansion quantum Monte Carlo method to study interacting hard-core bosons on the square lattice with pair-hopping processes supplementing the standard single-particle hopping. Such pair hopping arises in effective models for frustrated quantum magnets. Our goal is to investigate the effects of the pair-hopping process on the commonly observed superfluid, insulating (Mott), and supersolid ground-state phases in the standard hard-core boson model with various interaction terms. The model is specifically motivated by the observation of finite dispersion of two-magnon bound states in neutron-diffraction experiments SrCu2(BO3)2{\mathrm{SrCu}}_{2}{({\mathrm{BO}}_{3})}_{2}SrCu2​(BO3​)2​. Our results show that the pair hopping has different effects on Mott phases at different filling fractions, ``melting'' them at different critical pair-hopping amplitudes. Thus, it appears that pair hopping may play an important role in determining which out of a potentially large number of Mott phases (stabilized by details of the charge-diagonal interaction terms) actually survive the totality of quantum fluctuations present.
We investigate the consequences of correlated hopping on the ground state properties of hard-core bosons on a square lattice as revealed by extensive exact diagonalizations and quantum Monte Carlo simulations. While for noninteracting hard-core bosons the effective attraction induced by the correlated hopping leads to phase separation at low density, we show that a modest nearest-neighbor repulsion suppresses phase separation, leading to a remarkable low-density pairing phase with no single particle Bose-Einstein condensation but long-range two-particle correlations, signaling a condensation of pairs. We also explain why the unusual properties of the pairing phase are a real challenge for standard one-worm quantum Monte Carlo simulations.