JS
J. Soto Garcia
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Quantum Kibble-Zurek mechanism
The role of boundary conditions, endpoints, and kink types
Quantum phase transitions are characterized by the universal scaling laws in the critical region surrounding the transitions. This universality is also manifested in the critical real-time dynamics through the quantum Kibble- Zurek mechanism. In recent experiments on a Rydberg atom quantum simulator, the Kibble-Zurek mechanism was used to probe the nature of quantum phase transitions. In this paper, we analyze the caveats associated with this method and develop strategies to improve its accuracy. Focusing on two minimal models—transverse-field Ising and quantum three-state Potts, both in one dimension—we study the effect of boundary conditions, the location of the endpoints, and some subtleties in the definition of the kink operators. In particular, we show that the critical scaling of the most intuitive types of kinks is extremely sensitive to the correct choice of endpoint, while more advanced types of kinks exhibit remarkably robust universal scaling. Furthermore, we show that when kinks are tracked over the entire chain, fixed boundary conditions improve the accuracy of the scaling. Surprisingly, the Kibble-Zurek critical scaling appears to be equally accurate whether the fixed boundary conditions are chosen to be symmetric or antisymmetric. We also show that the density of kinks extracted in the central part of long chains obeys the predicted universal scaling for all types of boundary conditions. Finally, we test our kink definition for the Ising transition on the period-2 phase of the Rydberg model and show that it is more robust against the endpoint than the standard definition.
...
Quantum phase transitions are characterized by the universal scaling laws in the critical region surrounding the transitions. This universality is also manifested in the critical real-time dynamics through the quantum Kibble- Zurek mechanism. In recent experiments on a Rydberg atom quantum simulator, the Kibble-Zurek mechanism was used to probe the nature of quantum phase transitions. In this paper, we analyze the caveats associated with this method and develop strategies to improve its accuracy. Focusing on two minimal models—transverse-field Ising and quantum three-state Potts, both in one dimension—we study the effect of boundary conditions, the location of the endpoints, and some subtleties in the definition of the kink operators. In particular, we show that the critical scaling of the most intuitive types of kinks is extremely sensitive to the correct choice of endpoint, while more advanced types of kinks exhibit remarkably robust universal scaling. Furthermore, we show that when kinks are tracked over the entire chain, fixed boundary conditions improve the accuracy of the scaling. Surprisingly, the Kibble-Zurek critical scaling appears to be equally accurate whether the fixed boundary conditions are chosen to be symmetric or antisymmetric. We also show that the density of kinks extracted in the central part of long chains obeys the predicted universal scaling for all types of boundary conditions. Finally, we test our kink definition for the Ising transition on the period-2 phase of the Rydberg model and show that it is more robust against the endpoint than the standard definition.
Chains of Rydberg atoms have emerged as a powerful platform for exploring low-dimensional quantum physics. This success originates from the precise control of lattice geometries provided by optical tweezers, which allows access to a wide range of synthetic quantum phases. Experiments on one-dimensional arrays have stimulated tremendous progress in understanding quantum phase transitions into crystalline phases. However, the finite width of tweezers introduces small variations in interatomic distances, leading to quenched disorder in the interactions. In this Letter, we numerically study how such disorder alters the nature of two critical regimes observed in experiments. First, following experimental protocols, we analyze Kibble-Zurek dynamics and find a crossover from the clean Ising transition to the infinite-randomness fixed point as system size and disorder strength increase. Second, we show that the floating phase—an incommensurate Luttinger liquid phase emerging at stronger interactions—is localized by the disorder, yet preserves short-range incommensurate correlations with the same leading wave vector. Our results clearly reveal an additional conceptual challenge in understanding critical phenomena using Rydberg-based quantum simulators.
...
Chains of Rydberg atoms have emerged as a powerful platform for exploring low-dimensional quantum physics. This success originates from the precise control of lattice geometries provided by optical tweezers, which allows access to a wide range of synthetic quantum phases. Experiments on one-dimensional arrays have stimulated tremendous progress in understanding quantum phase transitions into crystalline phases. However, the finite width of tweezers introduces small variations in interatomic distances, leading to quenched disorder in the interactions. In this Letter, we numerically study how such disorder alters the nature of two critical regimes observed in experiments. First, following experimental protocols, we analyze Kibble-Zurek dynamics and find a crossover from the clean Ising transition to the infinite-randomness fixed point as system size and disorder strength increase. Second, we show that the floating phase—an incommensurate Luttinger liquid phase emerging at stronger interactions—is localized by the disorder, yet preserves short-range incommensurate correlations with the same leading wave vector. Our results clearly reveal an additional conceptual challenge in understanding critical phenomena using Rydberg-based quantum simulators.
Rydberg atoms trapped in optical tweezers have emerged as a powerful platform for exploring the physics of low-dimensional systems. Thanks to the precise control over the tweezers location, interatomic interactions, and couplings to external fields, these platforms have revealed a wide variety of critical phenomena.
This thesis numerically explores the quantum phases and phase transitions that can be engineered in quasi one-dimensional arrays of Rydberg atoms. It employs a combination of equilibrium and non-equilibrium approaches within the matrix product state framework.
We first studied the quantum Kibble–Zurek (KZ) mechanism in minimal one-dimensional models, showing how boundary conditions and quench endpoints affect scaling behavior. A refined definition of kinks was introduced, yielding robust results even away from the classical Hamiltonian. Building on this, we incorporated a blockade model into the time-evolving block decimation (TEBD) algorithm and combined KZ with finite time scaling (FTS) to study Rydberg chains, establishing a protocol suitable for experiments. This approach distinguished the Kosterlitz–Thouless transition from the Huse–Fisher universality class, providing an alternative way to experimentally detect the floating phase, and allowed the precise location of conformal points as well as the extraction of critical exponents.
Motivated by optical tweezer imperfections, we then examined the effect of disorder on the critical phenomena observed in these chains. We found that disorder alters the nature of quantum criticality, driving the system toward the infinite-disorder universality class, and visibly changing the nature of the KZ scaling. It also localizes the algebraic correlations of the floating phase, destroying Friedel oscillations in the bulk of the chain and saturating the entanglement entropy at a finite value.
Finally, we extended the analysis to two-leg Rydberg ladders, where the phase diagram revealed new phases with resonant Rydberg states between the upper and lower chains. This geometry produced novel critical phenomena, including the merging of two Ising transitions into the Ashkin–Teller universality class and the more exotic merging of an Ising and Pokrovsky–Talapov transition.
The findings presented here provide a theoretical guidance for investigating critical dynamics, the impact of disorder, and the emergence of exotic critical phenomena in both one and quasi-two-dimensional Rydberg atom arrays.
...
This thesis numerically explores the quantum phases and phase transitions that can be engineered in quasi one-dimensional arrays of Rydberg atoms. It employs a combination of equilibrium and non-equilibrium approaches within the matrix product state framework.
We first studied the quantum Kibble–Zurek (KZ) mechanism in minimal one-dimensional models, showing how boundary conditions and quench endpoints affect scaling behavior. A refined definition of kinks was introduced, yielding robust results even away from the classical Hamiltonian. Building on this, we incorporated a blockade model into the time-evolving block decimation (TEBD) algorithm and combined KZ with finite time scaling (FTS) to study Rydberg chains, establishing a protocol suitable for experiments. This approach distinguished the Kosterlitz–Thouless transition from the Huse–Fisher universality class, providing an alternative way to experimentally detect the floating phase, and allowed the precise location of conformal points as well as the extraction of critical exponents.
Motivated by optical tweezer imperfections, we then examined the effect of disorder on the critical phenomena observed in these chains. We found that disorder alters the nature of quantum criticality, driving the system toward the infinite-disorder universality class, and visibly changing the nature of the KZ scaling. It also localizes the algebraic correlations of the floating phase, destroying Friedel oscillations in the bulk of the chain and saturating the entanglement entropy at a finite value.
Finally, we extended the analysis to two-leg Rydberg ladders, where the phase diagram revealed new phases with resonant Rydberg states between the upper and lower chains. This geometry produced novel critical phenomena, including the merging of two Ising transitions into the Ashkin–Teller universality class and the more exotic merging of an Ising and Pokrovsky–Talapov transition.
The findings presented here provide a theoretical guidance for investigating critical dynamics, the impact of disorder, and the emergence of exotic critical phenomena in both one and quasi-two-dimensional Rydberg atom arrays.
...
Rydberg atoms trapped in optical tweezers have emerged as a powerful platform for exploring the physics of low-dimensional systems. Thanks to the precise control over the tweezers location, interatomic interactions, and couplings to external fields, these platforms have revealed a wide variety of critical phenomena.
This thesis numerically explores the quantum phases and phase transitions that can be engineered in quasi one-dimensional arrays of Rydberg atoms. It employs a combination of equilibrium and non-equilibrium approaches within the matrix product state framework.
We first studied the quantum Kibble–Zurek (KZ) mechanism in minimal one-dimensional models, showing how boundary conditions and quench endpoints affect scaling behavior. A refined definition of kinks was introduced, yielding robust results even away from the classical Hamiltonian. Building on this, we incorporated a blockade model into the time-evolving block decimation (TEBD) algorithm and combined KZ with finite time scaling (FTS) to study Rydberg chains, establishing a protocol suitable for experiments. This approach distinguished the Kosterlitz–Thouless transition from the Huse–Fisher universality class, providing an alternative way to experimentally detect the floating phase, and allowed the precise location of conformal points as well as the extraction of critical exponents.
Motivated by optical tweezer imperfections, we then examined the effect of disorder on the critical phenomena observed in these chains. We found that disorder alters the nature of quantum criticality, driving the system toward the infinite-disorder universality class, and visibly changing the nature of the KZ scaling. It also localizes the algebraic correlations of the floating phase, destroying Friedel oscillations in the bulk of the chain and saturating the entanglement entropy at a finite value.
Finally, we extended the analysis to two-leg Rydberg ladders, where the phase diagram revealed new phases with resonant Rydberg states between the upper and lower chains. This geometry produced novel critical phenomena, including the merging of two Ising transitions into the Ashkin–Teller universality class and the more exotic merging of an Ising and Pokrovsky–Talapov transition.
The findings presented here provide a theoretical guidance for investigating critical dynamics, the impact of disorder, and the emergence of exotic critical phenomena in both one and quasi-two-dimensional Rydberg atom arrays.
This thesis numerically explores the quantum phases and phase transitions that can be engineered in quasi one-dimensional arrays of Rydberg atoms. It employs a combination of equilibrium and non-equilibrium approaches within the matrix product state framework.
We first studied the quantum Kibble–Zurek (KZ) mechanism in minimal one-dimensional models, showing how boundary conditions and quench endpoints affect scaling behavior. A refined definition of kinks was introduced, yielding robust results even away from the classical Hamiltonian. Building on this, we incorporated a blockade model into the time-evolving block decimation (TEBD) algorithm and combined KZ with finite time scaling (FTS) to study Rydberg chains, establishing a protocol suitable for experiments. This approach distinguished the Kosterlitz–Thouless transition from the Huse–Fisher universality class, providing an alternative way to experimentally detect the floating phase, and allowed the precise location of conformal points as well as the extraction of critical exponents.
Motivated by optical tweezer imperfections, we then examined the effect of disorder on the critical phenomena observed in these chains. We found that disorder alters the nature of quantum criticality, driving the system toward the infinite-disorder universality class, and visibly changing the nature of the KZ scaling. It also localizes the algebraic correlations of the floating phase, destroying Friedel oscillations in the bulk of the chain and saturating the entanglement entropy at a finite value.
Finally, we extended the analysis to two-leg Rydberg ladders, where the phase diagram revealed new phases with resonant Rydberg states between the upper and lower chains. This geometry produced novel critical phenomena, including the merging of two Ising transitions into the Ashkin–Teller universality class and the more exotic merging of an Ising and Pokrovsky–Talapov transition.
The findings presented here provide a theoretical guidance for investigating critical dynamics, the impact of disorder, and the emergence of exotic critical phenomena in both one and quasi-two-dimensional Rydberg atom arrays.
Experiments on chains of Rydberg atoms appear as a playground to study quantum phase transitions in 1D. As a natural extension, we report a quantitative ground-state phase diagram of Rydberg atoms arranged in a two-leg ladder interacting via van der Waals potential. We address this problem numerically, using the density matrix renormalization group algorithm. Our results suggest that, quite remarkably, Zk crystalline phases, with the exception of the checkerboard phase, appear in pairs characterized by the same pattern of occupied rungs but distinguishable by a spontaneously broken Z2 symmetry between the two legs of the ladder. Within each pair, the two phases are separated by a continuous transition in the Ising universality class, which eventually fuses with the Zk transition, whose nature depends on k. According to our results, the transition into the Z2 - Z2 phase changes its nature multiple times, including an Ashkin-Teller transition that is surprisingly stable over an extended interval, followed by the Z4-chiral transition, and finally in a two step-process mediated melting via the floating phase. The transition into the Z3 phase with resonant states on the rungs belongs to the three-state Potts universality class at the commensurate point, to the Z3-chiral Huse-Fisher universality class away from it, and eventually it is through an intermediate floating phase. The Ising transition between Z3 and Z3 - Z2 phases, entering the floating phase, opens the possibility to realize lattice supersymmetry in Rydberg quantum simulators.
...
Experiments on chains of Rydberg atoms appear as a playground to study quantum phase transitions in 1D. As a natural extension, we report a quantitative ground-state phase diagram of Rydberg atoms arranged in a two-leg ladder interacting via van der Waals potential. We address this problem numerically, using the density matrix renormalization group algorithm. Our results suggest that, quite remarkably, Zk crystalline phases, with the exception of the checkerboard phase, appear in pairs characterized by the same pattern of occupied rungs but distinguishable by a spontaneously broken Z2 symmetry between the two legs of the ladder. Within each pair, the two phases are separated by a continuous transition in the Ising universality class, which eventually fuses with the Zk transition, whose nature depends on k. According to our results, the transition into the Z2 - Z2 phase changes its nature multiple times, including an Ashkin-Teller transition that is surprisingly stable over an extended interval, followed by the Z4-chiral transition, and finally in a two step-process mediated melting via the floating phase. The transition into the Z3 phase with resonant states on the rungs belongs to the three-state Potts universality class at the commensurate point, to the Z3-chiral Huse-Fisher universality class away from it, and eventually it is through an intermediate floating phase. The Ising transition between Z3 and Z3 - Z2 phases, entering the floating phase, opens the possibility to realize lattice supersymmetry in Rydberg quantum simulators.
The experimental realization of the quantum Kibble-Zurek mechanism in arrays of trapped Rydberg atoms has brought the problem of commensurate-incommensurate transition back into the focus of active research. Relying on equilibrium simulations of finite intervals, direct chiral transitions at the boundary of the period-3 and period-4 phases have been predicted. Here, we study how these chiral transitions can be diagnosed experimentally with critical dynamics. We demonstrate that chiral transitions can be distinguished from the floating phases by comparing Kibble-Zurek dynamics on arrays with different numbers of atoms. Furthermore, by sweeping in the opposite direction and keeping track of the order parameter, we identify the location of conformal points. Finally, combining forward and backward sweeps, we extract all critical exponents characterizing the transition.
...
The experimental realization of the quantum Kibble-Zurek mechanism in arrays of trapped Rydberg atoms has brought the problem of commensurate-incommensurate transition back into the focus of active research. Relying on equilibrium simulations of finite intervals, direct chiral transitions at the boundary of the period-3 and period-4 phases have been predicted. Here, we study how these chiral transitions can be diagnosed experimentally with critical dynamics. We demonstrate that chiral transitions can be distinguished from the floating phases by comparing Kibble-Zurek dynamics on arrays with different numbers of atoms. Furthermore, by sweeping in the opposite direction and keeping track of the order parameter, we identify the location of conformal points. Finally, combining forward and backward sweeps, we extract all critical exponents characterizing the transition.