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Quantum fluctuations in quantum lattice-systems with continuous symmetry 1

Tsutomu Momoi

Published Jan 1, 1996• 19 citations
OpenAlex

Abstract

We discuss conditions for the absence of spontaneous breakdown of continuous symmetries in quantum lattice systems at T = 0. Our analysis is based on Pitaevskii and Stringari’s idea that the uncertainty relation can be employed to show quantum fluctuations. For the one-dimensional systems, it is shown that the ground state is invariant under the continuous transformation if a certain uniform susceptibility is finite. For the two- and three-dimensional systems, it is shown that truncated correlation functions cannot decay any more rapidly than |r | −d+1 whenever the continuous symmetry is spontaneously broken. Both of these phenomena occur owing to quantum fluctuations. Our theorems cover a wide class of quantum lattice-systems having not-too-long-range interactions.

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We discuss conditions for the absence of spontaneous breakdown of continuous symmetries in quantum lattice systems at T = 0. Our analysis is based on Pitaevskii and Stringari’s idea that the uncertainty relation can be employed to show quantum fluctuations. For the one-dimensional systems, it is shown that the ground state is invariant under the continuous transformation if a certain uniform susceptibility is finite. For the two- and three-dimensional systems, it is shown that truncated correlation functions cannot decay any more rapidly than |r | −d+1 whenever the continuous symmetry is spontaneously broken. Both of these phenomena occur owing to quantum fluctuations. Our theorems cover a wide class of quantum lattice-systems having not-too-long-range interactions.
We consider a quantum many-body system on a lattice which exhibits a spontaneous symmetry breaking in its infinite volume ground states, but in which the corresponding order operator does not commute with the Hamiltonian. Typical examples are the Heisenberg antiferromagnet with a Néel order, and the Hubbard model with a (superconducting) off-diagonal long range order. In the corresponding finite system, the symmetry breaking is usually “obscured ” by “quantum fluctuation ” and one gets a symmetric ground state with a long range order. In such a situation, Horsch and von der Linden proved that the finite system has a low-lying eigenstate whose excitation energy is not more than of order N −1, where N denotes the number of the sites in the lattice. Here we study the situation where the broken symmetry is a continuous one. For a particular set of states (which are orthogonal to the ground state and with each other), we prove bounds for their energy expectation values. The bounds establish that there exist ever increasing numbers of low-lying eigenstates whose excitation energies are bounded by a constant times N −1. A crucial feature of the particular low-lying
We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with short-range interactions on a wide class of lattices. We prove that, if two observables anticommute with each other at large distance, then the nonvanishing spectral gap implies exponential decay of the corresponding correlation. When two observables commute with each other at large distance, the connected correlation function decays exponentially under the gap assumption. If the observables behave as a vector under the U(1) rotation of a global symmetry of the system, we use previous results on the large distance decay of the correlation function to show the stronger statement that the correlation function itself, rather than just the connected correlation function, decays exponentially under the gap assumption on a lattice with a certain self-similarity in (fractal) dimensions D < 2. In particular, if the system is translationally invariant in one of the spatial directions, then this self-similarity condition is automatically satisfied. We also treat systems with long-range, power-law decaying interactions. Spectral Gap and Exponential Decay of Correlations 2 1
The Kosterlitz–Thouless transition is studied from the representation of the systems’s ground state wavefunctions in terms of matrix product states for a quantum system on an infinite-size lattice in one spatial dimension. It is found that, in the critical regime for a one-dimensional quantum lattice system with continuous symmetry, the newly developed infinite matrix product state algorithm automatically leads to infinite degenerate ground states, due to the finiteness of the truncation dimension. This results in pseudo continuous symmetry spontaneous breakdown, which allows the introduction of a pseudo-order parameter that must be scaled down to zero, in order to be consistent with the Mermin–Wegner theorem. We also show that the ground state fidelity per lattice site exhibits a catastrophe point , thus resolving a controversy regarding whether or not the ground state fidelity is able to detect the Kosterlitz–Thouless transition.
It is widely believed that mean-field theory is exact for a wide range of classical long-range interacting systems. Is this also true once quantum fluctuations have been accounted for? As a test case we study the Hamiltonian mean-field (HMF) model for a system of bosons which is predicted (according to mean-field theory) to undergo a second-order quantum phase transition at zero temperature. The ordered phase is characterized by a spontaneously broken O(2) symmetry, which, despite occurring in a one-dimensional model, is not ruled out by the Mermin-Wagner theorem due to the presence of long-range interactions. Nevertheless, a spontaneously broken symmetry implies gapless Goldstone modes whose large fluctuations can restore broken symmetries. In this work we study the influence of quantum fluctuations by projecting the Hamiltonian onto the continuous subspace of symmetry-breaking mean-field states. We find that the energetic cost of gradients in the center-of-mass wave function inhibits the breaking of the O(2) symmetry, but that the energetic cost is very small, scaling as O(1/N^{2}). Nevertheless, for any finite N, no matter how large, this implies that the ground state has a restored O(2) symmetry. Implications for the finite-temperature phases, as well as the classical limit, of the HMF model are discussed.
We establish multiple interrelated, fundamental results in quantum many-body systems that can have long-range interactions. For a sufficiently long quantum spin chain, we first show that if the multispin interactions in the Hamiltonian decay fast enough as their ranges increase and the Hamiltonian is gapped, then the ground states satisfy the entanglement area law, even if there is a ground-state degeneracy due to a spontaneously broken discrete symmetry. This area law also holds for certain excited states. Second, if such a long-range interacting Hamiltonian has an anomalous symmetry, then the Lieb-Schultz-Mattis theorem applies, i.e., the Hamiltonian cannot have a unique gapped symmetric ground state. If the Hamiltonian contains only 2-spin interactions, then these results hold when the interactions decay faster than 1/r21/{r}^{2}1/r2, with rrr the distance between the two interacting spins. Third, we show that pure states with an anomalous symmetry, which may not be a ground state of any natural Hamiltonian, must be long-range entangled. The symmetries we consider include on-site internal symmetries combined with lattice translation symmetries, and they can also extend to purely internal but non-on-site symmetries. Moreover, these internal symmetries can be discrete or continuous. We explore the applications of these results through various examples.