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arXiv:1510.02108 · Physical review. B./Physical review. B

Kaleidoscope of quantum phases in a long-range interacting spin-1 chain

Zhexuan Gong, Mohammad F. Maghrebi, A. Hu, Michael Foss‐Feig, Philip Richerme, Christopher R. Monroe, Alexey Vyacheslavovich Gorshkov

Published May 11, 2016• 84 citations
arXivHTMLPDF

Abstract

Motivated directly by recent trapped-ion quantum simulation experiments, we carry out a comprehensive study of the phase diagram of a spin-1 chain with XXZ-type interactions that decay as $1/{r}^{\ensuremath{\alpha}}$, using a combination of finite and infinite-size DMRG calculations, spin-wave analysis, and field theory. In the absence of long-range interactions, varying the spin-coupling anisotropy leads to four distinct and well-studied phases: a ferromagnetic Ising phase, a disordered XY phase, a topological Haldane phase, and an antiferromagnetic Ising phase. If long-range interactions are antiferromagnetic and thus frustrated, we find primarily a quantitative change of the phase boundaries. On the other hand, ferromagnetic (nonfrustrated) long-range interactions qualitatively impact the entire phase diagram. Importantly, for $\ensuremath{\alpha}\ensuremath{\lesssim}3$, long-range interactions destroy the Haldane phase, break the conformal symmetry of the XY phase, give rise to a new phase that spontaneously breaks a $U(1)$ continuous symmetry, and introduce a possibly exotic tricritical point with no direct parallel in short-range interacting spin chains. Importantly, we show that the main signatures of all five phases found could be observed experimentally in the near future.

DOI
10.1103/physrevb.93.205115

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Motivated directly by recent trapped-ion quantum simulation experiments, we carry out a comprehensive study of the phase diagram of a spin-1 chain with XXZ-type interactions that decay as 1/r\ensuremathα1/{r}^{\ensuremath{\alpha}}1/r\ensuremathα, using a combination of finite and infinite-size DMRG calculations, spin-wave analysis, and field theory. In the absence of long-range interactions, varying the spin-coupling anisotropy leads to four distinct and well-studied phases: a ferromagnetic Ising phase, a disordered XY phase, a topological Haldane phase, and an antiferromagnetic Ising phase. If long-range interactions are antiferromagnetic and thus frustrated, we find primarily a quantitative change of the phase boundaries. On the other hand, ferromagnetic (nonfrustrated) long-range interactions qualitatively impact the entire phase diagram. Importantly, for \ensuremathα\ensuremath≲3\ensuremath{\alpha}\ensuremath{\lesssim}3\ensuremathα\ensuremath≲3, long-range interactions destroy the Haldane phase, break the conformal symmetry of the XY phase, give rise to a new phase that spontaneously breaks a U(1)U(1)U(1) continuous symmetry, and introduce a possibly exotic tricritical point with no direct parallel in short-range interacting spin chains. Importantly, we show that the main signatures of all five phases found could be observed experimentally in the near future.
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.
Topological phases of matter are primarily studied in systems with short-range interactions. In nature, however, nonrelativistic quantum systems often exhibit long-range interactions. Under what conditions topological phases survive such interactions, and how they are modified when they do, is largely unknown. By studying the symmetry-protected topological phase of an antiferromagnetic spin-1 chain with $1/{r}^{\ensuremath{\alpha}}$ interactions, we show that two very different outcomes are possible, depending on whether or not the interactions are frustrated. While unfrustrated long-range interactions can destroy the topological phase for $\ensuremath{\alpha}\ensuremath{\lesssim}3$, the topological phase survives frustrated interactions for all $\ensuremath{\alpha}>0$. Our conclusions are based on strikingly consistent results from large-scale matrix-product-state simulations and effective-field-theory calculations, and we expect them to hold for more general interacting spin systems. The models we study can be naturally realized in trapped-ion quantum simulators, opening the prospect for experimental investigation of the issues confronted here.
The phase diagram of the quantum spin-12\frac{1}{2}21​ antiferromagnetic ${J}_{1}^{\phantom{\rule{0.16em}{0ex}}}\text{\ensuremath{-}}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ XXZ chain was obtained by Haldane using bosonization techniques [Haldane, Phys. Rev. B 25, 4925 (1982); Phys. Rev. B 26, 5257 (1982)]. It supports three distinct phases for $0\ensuremath{\le}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}/{J}_{1}^{\phantom{\rule{0.16em}{0ex}}}<\frac{1}{2}$, i.e., a gapless algebraic spin-liquid phase, a gapped long-range ordered Neel phase, and a gapped long-range ordered dimer phase. Even though the Neel and dimer phases are not related hierarchically by a pattern of symmetry breaking, it was shown that they meet along a line of quantum critical points with a U(1) symmetry and central charge c=1c=1c=1. Here, we extend the analysis made by Haldane on the quantum spin-12\frac{1}{2}21​ antiferromagnetic ${J}_{1}^{\phantom{\rule{0.16em}{0ex}}}\text{\ensuremath{-}}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ XYZ chain using both bosonization and numerical techniques. We show that there are three Neel phases and the dimer phase that are separated from each other by six planes of phase boundaries realizing Gaussian criticality when $0\ensuremath{\le}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}/{J}_{1}^{\phantom{\rule{0.16em}{0ex}}}<\frac{1}{2}$. We also show that each long-range ordered phase harbors topological point defects (domain walls) that are dual to those across the phase boundary in that a defect in one ordered phase locally binds the other type of order around its core. By using the bosonization approach, we identify the critical theory that describes simultaneous proliferation of these dual point defects, and show that it supports an emergent U(1) symmetry that originates from the discrete symmetries of the XYZ model. To confirm this numerically, we perform exact diagonalization and density-matrix renormalization-group calculations and show that the critical theory is characterized by the central charge c=1c=1c=1 with critical exponents that are consistent with those obtained from the bosonization approach. Furthermore, we generalize the field-theoretic description of direct continuous phase transition to higher dimensions, especially in d=3d=3d=3, by using a nonlinear sigma model (NLSM) with a topological term. In particular, we propose the $\ensuremath{\pi}$-flux phase on the cubic lattice with local quartic interactions as a platform for direct continuous phase transition and deconfined criticality. We discuss possible phase diagrams for the $\ensuremath{\pi}$-flux phase on the cubic lattice with these quartic interactions from the renormalization flow of NLSMs.
Abstract Applying the (infinite) density-matrix renormalisation group technique, we explore the effect of an explicit dimerisation on the ground-state phase diagram of the spin-1 XXZ chain with single-ion anisotropy D. We demonstrate that the Haldane phase between large-D and antiferromagnetic phases survives up to a critical dimerisation only. As a further new characteristic the dimerisation induces a direct continuous Ising quantum phase transition between the large-D and antiferromagnetic phases with central charge c=1/2c=1/2c=1/2 c = 1 / 2 , which terminates at a critical end-point where c=7/10c=7/10c=7/10 c = 7 / 10 . Calculating the critical exponents of the order parameter, neutral gap and spin–spin-correlation function, we find β=1/8\beta =1/8β=1/8 β = 1 / 8 (1/24), ν=1\nu =1ν=1 ν = 1 (5/9), and η=1/4\eta =1/4η=1/4 η = 1 / 4 (3/20), respectively, which proves the Ising (tricritical Ising) universality class in accordance with field-theoretical predictions.
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.