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arXiv:1509.04180 · Physical review. B./Physical review. B

Bounds for low-energy spectral properties of center-of-mass conserving positive two-body interactions

Amila Weerasinghe, Tahereh Mazaheri, Alexander Seidel

Published Apr 19, 2016• 7 citations
arXivHTMLPDF

Abstract

We study the low-energy spectral properties of positive center-of-mass conserving two-body Hamiltonians as they arise in models of fractional quantum Hall states. Starting from the observation that positive many-body Hamiltonians must have ground-state energies that increase monotonously in particle number, we explore what general additional constraints can be obtained for two-body interactions with ``center-of-mass conservation'' symmetry, both in the presence and absence of particle-hole symmetry. We find general bounds that constrain the evolution of the ground-state energy with particle number, and in particular, constrain the chemical potential at $T=0$. Special attention is given to Hamiltonians with zero modes, in which case similar bounds on the first excited state are also obtained, using a duality property. In this case, in particular, an upper bound on the charge gap is also obtained. We further comment on center of mass and relative decomposition in disk geometry within the framework of second quantization.

DOI
10.1103/physrevb.93.155135

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We study the low-energy spectral properties of positive center-of-mass conserving two-body Hamiltonians as they arise in models of fractional quantum Hall states. Starting from the observation that positive many-body Hamiltonians must have ground-state energies that increase monotonously in particle number, we explore what general additional constraints can be obtained for two-body interactions with ``center-of-mass conservation'' symmetry, both in the presence and absence of particle-hole symmetry. We find general bounds that constrain the evolution of the ground-state energy with particle number, and in particular, constrain the chemical potential at T=0T=0T=0. Special attention is given to Hamiltonians with zero modes, in which case similar bounds on the first excited state are also obtained, using a duality property. In this case, in particular, an upper bound on the charge gap is also obtained. We further comment on center of mass and relative decomposition in disk geometry within the framework of second quantization.
The fractional quantum Hall (FQH) effect plays a prominent role in the study of topological phases of matter and of strongly correlated electron systems in general. FQH systems have been demonstrated to show many interesting novel properties such as fractional charges, and are believed to harbor even more intriguing phenomena such as fractional statistics. However, there remain many interesting questions to be addressed in this regime. The work reported in this thesis aims to push the envelope of our understanding of the low-energy properties of FQH states using microscopic principles. In the first part of the thesis, we present a systematic perturbative approach to study excitations in the thin cylinder/torus limit of the quantum Hall states. The approach is applied to the Haldane-Rezayi and Gaffnian quantum Hall states, which are both expected to have gapless excitations in the usual two-dimensional thermodynamic limit. For the Haldane-Rezayi state, we confirm that gapless excitations are present also in the ``one-dimensional'' thermodynamic limit of an infinite thin cylinder, in agreement with earlier considerations based on the wave functions alone. In contrast, we identify the lowest excitations of the Gaffnian state in the thin cylinder limit, and conclude that they are gapped, using a combination of perturbative and numerical means. We discuss possible scenarios for the cross-over between the two-dimensional and the one-dimensional thermodynamic limit in this case. In the second part of the thesis, we study the low energy spectral properties of positive center-of-mass conserving two-body Hamiltonians as they arise in models of FQH states. Starting from the observation that positive many-body Hamiltonians must have ground state energies that increase monotonously in particle number, we explore what general additional constraints can be obtained for two-body interactions with "center-of-mass conservation" symmetry, both in the presence and absence of particle-hole symmetry. We find general bounds that constrain the evolution of the ground state energy with particle number, and in particular constrain the chemical potential at T=0. Special attention is given to Hamiltonians with zero modes, in which case similar bounds on the first excited state are also obtained, using a duality property. In this case, in particular an upper bound on the charge gap is also obtained. We further comment on center-of-mass and relative-decomposition in disk geometry within the framework of second quantization.
We study the center-of-mass observable in one-dimensional many-body systems with translation-invariant interactions and extend the harmonic-rigidity mechanism from the one-body setting to an interacting many-body problem. We prove a sharp upper bound on the ground-state center-of-mass fluctuation in terms of the active spectral gap associated with the center-of-mass probe, and show that this bound does not require any positivity assumption on the ground state. In the positivity class, we characterize the equality case completely. Exact saturation occurs if and only if the external one-body traps are harmonic with a common frequency, while the interaction may remain arbitrary within the translation-invariant class. We also identify a natural rigidity defect measuring deviation from the harmonic extremal situation and prove quantitative near-saturation estimates controlling both the variance deficit and the spectral weight outside the first active shell. In this way, the paper establishes harmonic confinement as the unique static extremizer for the rigid interacting center-of-mass mode at fixed active gap.
We analyze many-body entanglement in interacting fermionic systems by using the MMM-body reduced density matrix. We demonstrate that if a particle number conserving fermionic Hamiltonian contains only up to MMM-body interaction terms, then its NNN-particle ground state cannot be maximally MMM-body entangled. As a key step in the proof, we show that the energy expectation value of a maximally MMM-body mixed state is equal to the spectral mean of the Hamiltonian on the corresponding NNN-particle subspace. We further demonstrate that the many-body entanglement structure of a ground state can place quantitative constraint on the interaction strength of its parent Hamiltonian. We illustrate the theorem and its implications in Hubbard and extended SYK models. Going beyond ground states, we analyze entanglement generation under unitary dynamics from Slater-determinant initial states in these models. We determine early-time growth and estimate entanglement saturation times. Finally, we derive explicit symmetry-refined saturation upper bounds for MMM-body entanglement in the presence of an Abelian symmetry.
For the numerical simulation of the fractional quantum Hall (FQH) effects on a finite disk, the rotational symmetry is the only symmetry that is used in diagonalizing the Hamiltonian. In this work, we propose a method of using the weak translational symmetry for the center of mass of the many-body system. With this approach, the bulk properties, such as the energy gap and the magneto-roton excitation are consistent with those in the closed manifolds like the sphere and torus. As an application, we consider the FQH phase and its phase transition in the fast rotated dipolar fermions. We thus demonstrate the disk geometry having versatility in analyzing the bulk properties beside the usual edge physics.
In this Letter, we investigate a class of Hamiltonians which, in addition to the usual center-of-mass momentum conservation, also have center-of-mass position conservation. We find that, regardless of the particle statistics, the energy spectrum is at least q-fold degenerate when the filling factor is p/q, where and are coprime integers. Interestingly, the simplest Hamiltonian respecting this type of symmetry encapsulates two prominent examples of novel states of matter, namely, the fractional quantum Hall liquid and the quantum dimer liquid. We discuss the relevance of this class of Hamiltonian to the search for featureless Mott insulators.
Past studies of fractional quantum Hall systems have found that the charge gap dominates the neutral gap for all relevant parameter choices. We report a wide-ranging proof that this domination is in fact a universal property of any Hamiltonian that satisfies a few simple structural properties: translation-invariance, charge conservation, dipole conservation, and a fractionally filled ground state. The result applies to both fermions and bosons. Our main tool is a new mathematical scheme, the gap comparison method, which provides a sequence of inequalities that relate the spectral gaps in successive particle number sectors. Our finding sheds new light on dipole conservation's profound effects on many-body physics.