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.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.

arXiv:1510.01325Physical Review LettersJul 11, 2017128 citations

Elementary quantum gates between long-distance qubits mediated by a resonator

Ming-Cui Li, Aixi Chen

Quantum Information ProcessingSep 28, 20201 citation

Continuous symmetry breaking in a trapped-ion spin chain

Lei Feng, Or Katz, Casey Haack +5 more

Abstract One-dimensional systems exhibiting a continuous symmetry can host quantum phases of matter with true long-range order only in the presence of sufficiently long-range interactions. In most physical systems, however, the interactions are short-ranged, hindering the emergence of such phases in one dimension. Here we use a one-dimensional trapped-ion quantum simulator to prepare states with long-range spin order that extends over the system size of up to $23$ spins and is characteristic of the continuous symmetry-breaking phase of matter. Our preparation relies on simultaneous control over an array of tightly focused individual-addressing laser beams, generating long-range spin-spin interactions. We also observe a disordered phase with frustrated correlations. We further study the phases at different ranges of interaction and the out-of-equilibrium response to symmetry-breaking perturbations. This work opens an avenue to study new quantum phases and out-of-equilibrium dynamics in low-dimensional systems.Abstract One-dimensional systems exhibiting a continuous symmetry can host quantum phases of matter with true long-range order only in the presence of sufficiently long-range interactions. In most physical systems, however, the interactions are short-ranged, hindering the emergence of such phases in one dimension. Here we use a one-dimensional trapped-ion quantum simulator to prepare states with long-range spin order that extends over the system size of up to $23$ spins and is characteristic of the continuous symmetry-breaking phase of matter. Our preparation relies on simultaneous control over an array of tightly focused individual-addressing laser beams, generating long-range spin-spin interactions. We also observe a disordered phase with frustrated correlations. We further study the phases at different ranges of interaction and the out-of-equilibrium response to symmetry-breaking perturbations. This work opens an avenue to study new quantum phases and out-of-equilibrium dynamics in low-dimensional systems.

arXiv:2211.01275NatureNov 15, 202350 citations

Quantum Gates with a Selective Interaction

E. Solano, França M. Santos, P. Milman

Lecture notes in physicsJan 1, 20012 citations

Quantum Gates and Circuits

Belal Ehsan Baaquie, L. C. Kwek

Springer eBooksJan 1, 20233 citations

Algebraic Relaxation of Strong-to-Weak Symmetry-Breaking Order via Charge-Sector Broadening

Zi-Xi Fang, Kai Zhang

Strong-to-weak spontaneous symmetry breaking (SW-SSB) is a distinctive phenomenon of mixed quantum states, characterized by nonlinear long-range order associated with strong symmetry. In realistic open quantum systems, however, strong symmetry is macroscopically fragile and can be explicitly reduced to weak symmetry through local charge exchange with the environment. In this paper, we investigate how the SW-SSB order decays under such strong-symmetry-breaking Liouvillian dynamics. For discrete $\mathbb{Z}_2$ symmetry, a local symmetry-breaking perturbation drives an exponential decay of the Rényi-1 long-range correlator, revealing the fragility of discrete SW-SSB order. By contrast, for continuous $U(1)$ symmetry, local charge exchange broadens the distribution over symmetry-charge sectors and leads to parametrically slower algebraic relaxation. We establish a rigorous algebraic lower bound on the connected Rényi-1 long-range correlation, demonstrating the robustness of this slow relaxation. In the thermodynamic limit, we further uncover a two-stage dynamical scaling: $\tilde{R}_1\sim t^{-1}$ at early times and $\tilde{R}_1\sim t^{-1/z}$ at long times, with $z$ set by the underlying hydrodynamic dynamics. As a numerical verification, we consider fermionic chains with $z=2$ and $z=4$ , which are excellent agreement with the theoretical predictions.Strong-to-weak spontaneous symmetry breaking (SW-SSB) is a distinctive phenomenon of mixed quantum states, characterized by nonlinear long-range order associated with strong symmetry. In realistic open quantum systems, however, strong symmetry is macroscopically fragile and can be explicitly reduced to weak symmetry through local charge exchange with the environment. In this paper, we investigate how the SW-SSB order decays under such strong-symmetry-breaking Liouvillian dynamics. For discrete $\mathbb{Z}_2$ symmetry, a local symmetry-breaking perturbation drives an exponential decay of the Rényi-1 long-range correlator, revealing the fragility of discrete SW-SSB order. By contrast, for continuous $U(1)$ symmetry, local charge exchange broadens the distribution over symmetry-charge sectors and leads to parametrically slower algebraic relaxation. We establish a rigorous algebraic lower bound on the connected Rényi-1 long-range correlation, demonstrating the robustness of this slow relaxation. In the thermodynamic limit, we further uncover a two-stage dynamical scaling: $\tilde{R}_1\sim t^{-1}$ at early times and $\tilde{R}_1\sim t^{-1/z}$ at long times, with $z$ set by the underlying hydrodynamic dynamics. As a numerical verification, we consider fermionic chains with $z=2$ and $z=4$, which are excellent agreement with the theoretical predictions.

cond-mat.mes-hallCross-listedarXiv:2610.00474Oct 2, 20260 citations

23

spins and is characteristic of the continuous symmetry-breaking phase of matter. Our preparation relies on simultaneous control over an array of tightly focused individual-addressing laser beams, generating long-range spin-spin interactions. We also observe a disordered phase with frustrated correlations. We further study the phases at different ranges of interaction and the out-of-equilibrium response to symmetry-breaking perturbations. This work opens an avenue to study new quantum phases and out-of-equilibrium dynamics in low-dimensional systems.

\mathbb{Z}_2

symmetry, a local symmetry-breaking perturbation drives an exponential decay of the Rényi-1 long-range correlator, revealing the fragility of discrete SW-SSB order. By contrast, for continuous

U(1)

symmetry, local charge exchange broadens the distribution over symmetry-charge sectors and leads to parametrically slower algebraic relaxation. We establish a rigorous algebraic lower bound on the connected Rényi-1 long-range correlation, demonstrating the robustness of this slow relaxation. In the thermodynamic limit, we further uncover a two-stage dynamical scaling:

\tilde{R}_1\sim t^{-1}

at early times and

\tilde{R}_1\sim t^{-1/z}

at long times, with

z

set by the underlying hydrodynamic dynamics. As a numerical verification, we consider fermionic chains with

z=2

and

z=4

, which are excellent agreement with the theoretical predictions.