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arXiv:2103.04271 · Physical Review A

Long-range-ordered phase in a quantum Heisenberg chain with interactions beyond nearest neighbors

Zehan Li, Sayan Choudhury, W. Vincent Liu

Published Jul 6, 2021• 15 citations
arXivHTMLPDF

Abstract

Spin ensembles coupled to optical cavities provide a powerful platform for engineering synthetic quantum matter. Recently, we demonstrated that cavity-mediated infinite-range interactions can induce fast scrambling in a Heisenberg $\mathrm{XXZ}$ spin chain [Phys. Rev. Research 2, 043399 (2020)]. In this paper, we analyze the kaleidoscope of quantum phases that emerge in this system from the interplay of these interactions. Employing both analytical spin-wave theory as well as numerical density-matrix renormalization group calculations, we find that there is a large parameter regime where the continuous $U(1)$ symmetry of this model is spontaneously broken and the ground state of the system exhibits $\mathrm{XY}$ order. This kind of symmetry breaking and the consequent long-range order is forbidden for short-range interacting systems by the Mermin-Wagner theorem. Intriguingly, we find that the $\mathrm{XY}$ order can be induced by even an infinitesimally weak infinite-range interaction. Furthermore, we demonstrate that in the $U(1)$ symmetry-broken phase, the half-chain entanglement entropy violates the area law logarithmically. Finally, we discuss a proposal to verify our predictions in state-of-the-art quantum emulators.

DOI
10.1103/physreva.104.013303

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Spin ensembles coupled to optical cavities provide a powerful platform for engineering synthetic quantum matter. Recently, we demonstrated that cavity-mediated infinite-range interactions can induce fast scrambling in a Heisenberg XXZ\mathrm{XXZ}XXZ spin chain [Phys. Rev. Research 2, 043399 (2020)]. In this paper, we analyze the kaleidoscope of quantum phases that emerge in this system from the interplay of these interactions. Employing both analytical spin-wave theory as well as numerical density-matrix renormalization group calculations, we find that there is a large parameter regime where the continuous U(1)U(1)U(1) symmetry of this model is spontaneously broken and the ground state of the system exhibits XY\mathrm{XY}XY order. This kind of symmetry breaking and the consequent long-range order is forbidden for short-range interacting systems by the Mermin-Wagner theorem. Intriguingly, we find that the XY\mathrm{XY}XY order can be induced by even an infinitesimally weak infinite-range interaction. Furthermore, we demonstrate that in the U(1)U(1)U(1) symmetry-broken phase, the half-chain entanglement entropy violates the area law logarithmically. Finally, we discuss a proposal to verify our predictions in state-of-the-art quantum emulators.
We study quantum phase transitions in Heisenberg antiferromagnetic chains with a staggered power-law decaying long-range interactions. Employing the density-matrix renormalization group (DMRG) algorithm and the fidelity susceptibility as the criticality measure, we establish more accurate values of quantum critical points than the results obtained from the spin-wave approximation, quantum Monte Carlo, and DMRG in literatures. The deviation is especially evident for strong long-range interactions. We extend isotropic long-range interactions to the anisotropic cases and find that kaleidoscope of quantum phases emerge from the interplay of anisotropy of the long-range exchange interaction and symmetry breaking. We demonstrate nonfrustrating long-range interactions induce the true long-range order in Heisenberg antiferromagnetic chains with a continuous symmetry breaking, lifting the restrictions imposed by the Mermin-Wagner theorem.
Long-range interactions can fundamentally reshape the low-energy properties of low-dimensional quantum matter, altering both continuous symmetry breaking and topological phenomena. However, their impact on the quantum criticality separating these regimes remains poorly understood. We determine the ground-state phase diagram and critical properties of the spin-one Heisenberg chain with single-ion anisotropy and staggered antiferromagnetic power-law interactions, using matrix-product state (MPS) calculations complemented by high-order series expansions (pCUT+MC). Such long-range, non-frustrated interactions circumvent the Hohenberg-Mermin-Wagner theorem, thereby stabilizing continuous symmetry breaking (CSB) phases in direct competition with the Haldane phase. We map out the resulting phase diagram and analyze the entanglement entropy scaling behavior in the U(1) and SU(2) CSB phases, finding logarithmic corrections beyond the short-range, universal contributions expected from linearly dispersed Goldstone modes. We further characterize all critical boundaries through finite-size scaling of either the entanglement entropy or the staggered magnetization. In particular, the large-D-to-U(1)-CSB transition exhibits unconventional, continuously varying critical exponents as a function of the long-range decay exponent with a strong dependence on the imposed boundary conditions leading to distinct finite-size scalings for sufficiently long-range potentials. Remarkably, the Haldane-to-U(1)-CSB transition likewise displays unconventional quantum criticality with distinct continuously varying critical exponents. Our work positions this model as a target for near-term atomic platforms with tunable long-range couplings and exhibiting natural single-ion anisotropy, offering a minimal playground for exploring the interplay between long-range interactions, continuous symmetry breaking, and topology.
Continuous symmetry breaking (CSB) in low-dimensional systems, forbidden by the Mermin-Wagner theorem for short-range interactions, may take place in the presence of slowly decaying long-range interactions. Nevertheless, there is no stringent bound on how slowly interactions should decay to give rise to CSB in 1D quantum systems at zero temperature. Here, we study a long-range interacting spin chain with U(1) symmetry and power-law interactions V(r)∼1/r^{α}. Using a number of analytical and numerical techniques, we find CSB for α smaller than a critical exponent α_{c}(≤3) that depends on the microscopic parameters of the model. Furthermore, the transition from the gapless XY phase to the gapless CSB phase is mediated by the breaking of conformal and Lorentz symmetries due to long-range interactions, and is described by a universality class akin to, but distinct from, the Berezinskii-Kosterlitz-Thouless transition. Signatures of the CSB phase should be accessible in existing trapped-ion experiments.
In a recent paper Nahum argued on the basis of a renormalization-group analysis that, contrary to standard lore, ground states of 1D spin chains with short-range interactions can spontaneously break U(1) ``easy-plane'' spin rotation symmetry. True long-range order of (Sx,Sy)(S^x,S^y)(Sx,Sy) arises at the phase transition between two quasi-long-range-ordered phases. Here we present detailed numerical results for an anisotropic spin-1 chain model. We find a magnetization exponent β≈0.29β\approx0.29β≈0.29, consistent with the εεε-expansion prediction β≈0.28β\approx0.28β≈0.28, a divergence of the Luttinger parameter on approaching the transition from both sides with an exponent that we relate to Nahum's transition, and finite-size spectra consistent with a dynamical critical exponent z≈2z\approx2z≈2. Our results support the critical behavior predicted by Nahum's field theory.
We explored interaction cold atoms in new quantum regimes that have no prior analogue in condensed matter materials in three related main topics. First, motivated by recent advance in orbitally tuned Feshbach resonance experiments, we analyze the ground-state phase diagram and related low-energy excitation spectra of a high partial wave interacting Bose gas. Remarkably different from what was previously known in the p-wave case, the atomic superfluid is found to be momentum-independent in the present d-wave case. What is more, we study the quantum fluctuations in the condensates of a mixture of bosonic atoms and molecules with interspecies p-wave interaction. Our analysis shows that the quantum phase of coexisting atomic and molecular condensates is unstable at the mean-field level. The quantum Lee-Huang-Yang correction to the mean-field energy provides a remarkable mechanism to self-stabilize the phase. The correlated order spontaneously breaks a rich set of global U(1) gauge, atomic spin, spatial rotation and translation, and time-reversal symmetries. Second, we study the dynamics of a non-integrable spin chain model composed of two ingredients - a nearest neighbor Ising coupling, and an infinite range XX interaction. Unlike other fast scrambling many-body systems, this model is not known to be dual to a black hole. We demonstrate that our model exhibits fast scrambling for a wide parameter regime, accompanied by a fast growth of the entanglement entropy, as well as a swift change in the magnetization. Third, to extend the study of the highly chaotic many-body model, we analyze the ground state phases of the model with a nearest neighbor XXZ interaction and an infinite range XX interaction. By employing spin-wave theory, we find that there is a large parameter regime where the continuous U(1) symmetry of this model is spontaneously broken, which is not possible in the absence of the infinite range interaction by the Mermin-Wagner theorem. Furthermore, we demonstrate that in the U(1) symmetry broken phase, the half chain entanglement entropy violates the area law logarithmically. Our work demonstrates that the interplay of short and long range interactions can lead to novel quantum phases of matter.
Abstract The Mermin-Wagner theorem states that spontaneous continuous symmetry breaking is prohibited in systems with short-range interactions at spatial dimension D ≤ 2. For long-range interactions with a power-law form (1/ r α ), the theorem further forbids ferromagnetic or antiferromagnetic order at finite temperature when α ≥ 2 D . However, the situation for α ∈ (2, 4) at D = 2 is not covered by the theorem. To address this, we conduct large-scale quantum Monte Carlo simulations and field theoretical analysis. Our findings show spontaneous breaking of S U (2) symmetry in the ferromagnetic Heisenberg model with 1/ r α -form long-range interactions at D = 2. We determine critical exponents through finite-size analysis for α < 3 (above the upper critical dimension with Gaussian fixed point) and 3 ≤ α < 4 (below the upper critical dimension with non-Gaussian fixed point). These results reveal new critical behaviors in 2D long-range Heisenberg models, encouraging further experimental studies of quantum materials with long-range interactions beyond the Mermin-Wagner theorem’s scope.