Elementary excitations of ordered quantum magnets are harmonic spin waves. Yet, when ordering breaks a continuous symmetry, excitations at zero momentum cannot really be treated as harmonic, because they correspond to the Goldstone mode that restores the symmetry in finite-size systems. Here, the authors show that zero-momentum excitations can be treated in a fully nonlinear way in terms of a macroscopic spin variable (the rotor): its spectrum reconstructs the so-called Anderson tower of states, a hallmark of continuous-symmetry breaking systems. The rotor variable can be considered as nearly separate from finite-momentum spin waves, leading to a quantitative reconstruction of the low-energy excitation spectrum of finite-size systems.
Elementary excitations of ordered quantum magnets are harmonic spin waves. Yet, when ordering breaks a continuous symmetry, excitations at zero momentum cannot really be treated as harmonic, because they correspond to the Goldstone mode that restores the symmetry in finite-size systems. Here, the authors show that zero-momentum excitations can be treated in a fully nonlinear way in terms of a macroscopic spin variable (the rotor): its spectrum reconstructs the so-called Anderson tower of states, a hallmark of continuous-symmetry breaking systems. The rotor variable can be considered as nearly separate from finite-momentum spin waves, leading to a quantitative reconstruction of the low-energy excitation spectrum of finite-size systems.
The nonequilibrium dynamics of quantum spin models is a most challenging topic, due to the exponentiality of Hilbert space, and it is central to the understanding of the many-body entangled states that can be generated by state-of-the-art quantum simulators. A particularly important class of evolutions is the one governed by U(1)-symmetric Hamiltonians, initialized in a state that breaks the U(1) symmetry-the paradigmatic example being the evolution of the so-called one-axis-twisting (OAT) model, featuring infinite-range interactions between spins. In this Letter, we show that the dynamics of the OAT model can be closely reproduced by systems with power-law-decaying interactions, thanks to an effective separation between the zero-momentum degrees of freedom, associated with the so-called Anderson tower of states, and reconstructing an OAT model, as well as finite-momentum ones, associated with spin-wave excitations. This mechanism explains quantitatively the recent numerical observation of spin squeezing and Schrödinger cat-state generation in the dynamics of dipolar Hamiltonians, and it paves the way for the extension of this observation to a much larger class of models of immediate relevance for quantum simulations.
In frustrated magnetic systems with a subextensive number of classical ground states, quantum zero-point fluctuations can select a unique long-range ordered state, a celebrated phenomenon referred to as order by quantum disorder (ObQD). For frustrated spin-1/2 models, unbiased numerical methods able to expose ObQD are necessary. We show that ObQD can be identified from exact diagonalization calculations through an analysis akin to the Anderson tower of states associated with spontaneous symmetry breaking. By defining an effective quantum rotor model, we describe the competition between ObQD-induced localization of the rotor and its tunneling between symmetry-related ground states, identifying the crossover length scale from the finite-size regime where the rotor is delocalized to the infinite system-size limit where it becomes localized. This rotor model relates the characteristic splittings in the exact diagonalization energy spectrum to the ObQD selection energy scale, providing an estimate that can be compared to spin wave calculations. We demonstrate the general applicability of this approach in one-, two-, and three-dimensional frustrated spin models that exhibit ObQD.
Spin squeezing---a central resource for quantum metrology---can be generated via the nonlinear, entangling evolution of an initially factorized spin state. Here we show that robust (i.e., persistent) squeezing dynamics is generated by a very large class of S=1/2 spin Hamiltonians with axial symmetry, in relationship with the existence of a peculiar structure of the low-lying Hamiltonian eigenstates---the so-called Anderson's tower of states. Such states are fundamentally related to the appearance of spontaneous symmetry breaking in quantum systems; and, for models with sufficiently high connectivity, they are parametrically close to the eigenstates of a planar rotor (Dicke states), in that they feature an anomalously large value of the total angular momentum. Our central insight is that starting from a coherent spin state, a generic U(1)-symmetric Hamiltonian featuring the Anderson's tower of states generates the same squeezing evolution at short times as the one governed by the paradigmatic one-axis-twisting (or planar-rotor) model of squeezing dynamics. The full squeezing evolution of the planar-rotor model is seemingly reproduced for interactions decaying with distance r as ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ when $\ensuremath{\alpha}<5d/3$ in d dimensions. Our results connect quantum simulation with quantum metrology by unveiling the squeezing power of a large variety of Hamiltonian dynamics that are currently implemented by different quantum simulation platforms.
This chapter turns to the models obtained by the quantum–classical mapping QC on the D -dimensional, N -component, classical ferromagnets in (3.3) with N ≥ 2. These are the O( N ) quantum rotor models in d = D − 1 dimensions, originally written down in (1.23). The quantum Ising model studied previously had a discrete Z 2 symmetry. An important new ingredient in the rotor models will be the presence of a continuous symmetry: The physics is invariant under a uniform, global O( N ) transformation on the orientation of the rotors, which is broken in the magnetically ordered state. We will introduce the important concept of the spin stiffness , which characterizes the rigidity of the ordered state and determines the dispersion spectrum of the low energy “spin-wave” excitations. Apart from this, much of the technology and the physical ideas introduced earlier for the d = 1 Ising chain will generalize straightforwardly, although we will no longer be able to obtain exact results for crossover functions. The characterization of the physics in terms of three regions separated by smooth crossovers, the high- T and the two low- T regions on either side of the quantum critical point, will continue to be extremely useful and will again be the basis of our discussion. Because we will consider models in spatial dimensions d > 1, it will be possible to have a thermodynamic phase transition at a nonzero temperature. We shall be particularly interested in the interplay between the critical singularities of the finite-temperature transition and those of the quantum critical point.
Universal spin squeezing from the tower of states of U(1)-symmetric spin Hamiltonians
Spin squeezing -- a central resource for quantum metrology -- results from the non-linear, entangling evolution of an initially factorized spin state. Here we show that universal squeezing dynamics is generated by a very large class of S=1/2 spin Hamiltonians with axial symmetry, in relationship with the existence of a peculiar structure of the low-lying Hamiltonian eigenstates -- the so-called Anderson's tower of states. Such states are fundamentally related to the appearance of spontaneous symmetry breaking in quantum systems, and they are parametrically close to the eigenstates of a planar rotor (Dicke states), in that they feature an anomalously large value of the total angular momentum. We show that, starting from a coherent spin state, a generic U(1)-symmetric Hamiltonian featuring the Anderson's tower of states generates the same squeezing evolution at short times as the one governed by the paradigmatic one-axis-twisting (or planar rotor) model of squeezing dynamics. The full squeezing evolution is seemingly reproduced for interactions decaying with distance r as r−α when α<5d/3 in d dimensions. Our results connect quantum simulation with quantum metrology by unveiling the squeezing power of a large variety of Hamiltonian dynamics that are currently implemented by different quantum simulation platforms.
In magnetically ordered systems, the breaking of SU(2) symmetry in the thermodynamic limit is associated with the appearance of a special type of low-lying excitations in finite-size energy spectra, the so-called tower of states (TOS). In the present work, we numerically demonstrate that there is a correspondence between the SU(2) tower of states and the lower part of the ground-state entanglement spectrum (ES). Using state-of-the-art density matrix renormalization group (DMRG) calculations, we examine the ES of the 2D antiferromagnetic J1-J2 Heisenberg model on both the triangular and kagome lattice. At large ferromagnetic J2, the model exhibits a magnetically ordered ground state. Correspondingly, its ES contains a family of low-lying levels that are reminiscent of the energy tower of states. Their behavior (level counting, finite-size scaling in the thermodynamic limit) sharply reflects TOS features, and is characterized in terms of an effective entanglement Hamiltonian that we provide. At large system sizes, TOS levels are divided from the rest by an entanglement gap. Our analysis suggests that (TOS) entanglement spectroscopy provides an alternative tool for detecting and characterizing SU(2)-broken phases using DMRG.