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arXiv:2001.07877 · Physical review. B./Physical review. B

Probing XY phase transitions in a Josephson junction array with tunable frustration

R. Cosmic, Kohei Kawabata, Yuto Ashida, Hiroki Ikegami, Shunsuke Furukawa, Pranay Patil, Jacob M. Taylor, Yasunobu Nakamura

Published Sep 10, 2020• 12 citations
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Abstract

The seminal theoretical works of Berezinskii, Kosterlitz, and Thouless presented a paradigm for phase transitions in condensed matter that are driven by topological excitations. These transitions have been extensively studied in the context of two-dimensional XY models---coupled compasses---and have generated interest in the context of quantum simulation. Here, we use a circuit quantum-electrodynamics architecture to study the critical behavior of engineered XY models through their dynamical response. In particular, we examine not only the unfrustrated case but also the fully frustrated case which leads to enhanced degeneracy associated with the spin rotational $[\mathrm{U}(1)]$ and discrete chiral (${Z}_{2}$) symmetries. The nature of the transition in the frustrated case has posed a challenge for theoretical studies while direct experimental probes remain elusive. Here we identify the transition temperatures for both the unfrustrated and fully frustrated XY models by probing a Josephson junction array close to equilibrium using weak microwave excitations and measuring the temperature dependence of the effective damping obtained from the complex reflection coefficient. We argue that our probing technique is primarily sensitive to the dynamics of the $\mathrm{U}(1)$ part.

DOI
10.1103/physrevb.102.094509

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The seminal theoretical works of Berezinskii, Kosterlitz, and Thouless presented a paradigm for phase transitions in condensed matter that are driven by topological excitations. These transitions have been extensively studied in the context of two-dimensional XY models---coupled compasses---and have generated interest in the context of quantum simulation. Here, we use a circuit quantum-electrodynamics architecture to study the critical behavior of engineered XY models through their dynamical response. In particular, we examine not only the unfrustrated case but also the fully frustrated case which leads to enhanced degeneracy associated with the spin rotational [U(1)][\mathrm{U}(1)][U(1)] and discrete chiral (Z2{Z}_{2}Z2​) symmetries. The nature of the transition in the frustrated case has posed a challenge for theoretical studies while direct experimental probes remain elusive. Here we identify the transition temperatures for both the unfrustrated and fully frustrated XY models by probing a Josephson junction array close to equilibrium using weak microwave excitations and measuring the temperature dependence of the effective damping obtained from the complex reflection coefficient. We argue that our probing technique is primarily sensitive to the dynamics of the U(1)\mathrm{U}(1)U(1) part.
Superconductivity provides a canonical example of a quantum phase of matter. When superconducting islands are connected by Josephson junctions in a lattice, the low-temperature state of the system can map to the celebrated XY model and its associated universality classes. This has been used to experimentally implement realizations of Mott insulator and Berezinskii-Kosterlitz-Thouless transitions to vortex dynamics analogous to those in type-II superconductors. When an external magnetic field is added, the effective spins of the XY model become frustrated, leading to the formation of topological defects (vortices). Here, we observe the many-body dynamics of such an array, including frustration, via its coupling to a superconducting microwave cavity. We take the design of the transmon qubit, but replace the single junction between two antenna pads with the complete array. This allows us to probe the system at 10 mK with minimal self-heating by using weak coherent states at the single (microwave) photon level to probe the resonance frequency of the cavity. We observe signatures of an ordered vortex lattice at rational flux fillings of the array.
A general framework is proposed to solve the two-dimensional (2D) fully frustrated XY model for the Josephson junction arrays in a perpendicular magnetic field. The essential idea is to encode the ground-state local rules induced by frustrations in the local tensors of the partition function. The partition function is then expressed in terms of a product of a 1D transfer matrix operator, whose eigenequation can be solved by an algorithm of matrix product states rigorously. The singularity of the entanglement entropy for the 1D quantum analog provides a stringent criterion to distinguish various phase transitions without identifying a priori order parameter. Two very close phase transitions are determined at ${T}_{c1}\ensuremath{\approx}0.4459$ and ${T}_{c2}\ensuremath{\approx}0.4532$, respectively. The former corresponds to a Berezinskii-Kosterlitz-Thouless phase transition describing the phase coherence of XY spins, and the latter is an Ising-like continuous phase transition below which a chirality order with spontaneously broken Z2{Z}_{2}Z2​ symmetry is established.
We study the phase diagram of the two-dimensional fully frustrated XY model (FFXY) and of two related models, a lattice discretization of the Landau-Ginzburg-Wilson Hamiltonian for the critical modes of the FFXY model, and a coupled Ising-XY model. We present Monte Carlo simulations on square lattices L × L , L ≲ 10 3 . We show that the low-temperature phase of these models is controlled by the same line of Gaussian fixed points as in the standard XY model. We find that, if a model undergoes a unique transition by varying temperature, then the transition is of first order. In the opposite case we observe two very close transitions: a transition associated with the spin degrees of freedom and, as temperature increases, a transition where chiral modes become critical. If they are continuous, they belong to the Kosterlitz-Thouless and to the Ising universality class, respectively. Ising and Kosterlitz-Thouless behavior is observed only after a preasymptotic regime, which is universal to some extent. In the chiral case, the approach is nonmonotonic for most observables, and there is a wide region in which finite-size scaling is controlled by an effective exponent ν eff ≈ 0.8. This explains the result ν ≈ 0.8 of many previous studies using smaller lattices.
We investigate the finite-temperature phase diagram of the classical ${J}_{1}\ensuremath{-}{J}_{2}$ XY model on a square lattice using a tensor network approach designed for frustrated spin systems. This model, characterized by competing nearest-neighbor and next-to-nearest-neighbor interactions, exhibits a complex interplay between U(1)U(1)U(1) and Z2{Z}_{2}Z2​ symmetries. Our study reveals an emergent intermediate phase around ${J}_{2}/{J}_{1}\ensuremath{\sim}0.505$, which is characterized by a Z2{Z}_{2}Z2​ long-range stripe order without phase coherence in the XY spins. The intermediate phase features two well-separated phase transitions: a higher-temperature Ising transition and a lower-temperature Berezinskii-Kosterlitz-Thouless (BKT) transition. The relative separation between these transitions is significantly larger than previously reported, enabling a clearer investigation of their distinct thermodynamic properties. For 0.5<J2/J1<0.5010.5<{J}_{2}/{J}_{1}<0.5010.5<J2​/J1​<0.501, two transitions merge into a single first-order phase transition, a phenomenon that cannot be explained solely by mapping to the Ising-XY model. As ${J}_{2}/{J}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$, the transition evolves continuously into the BKT universality class. These findings advance the understanding of the mechanisms driving phase transitions in frustrated spin systems, and they suggest potential experimental realizations in platforms such as ultracold atoms, Josephson junction arrays, and optical lattices.
We study the out-of-equilibrium dynamics of the fully frustrated XY model. At equilibrium, this model undergoes two phase transitions at two very close temperatures: a Kosterlitz–Thouless topological transition and a second-order phase transition between a paramagnetic phase and a low-temperature phase where the chiralities of the lattice plaquettes are antiferromagnetically ordered. We compute by means of Monte Carlo simulations two-time spin–spin and chirality–chirality autocorrelation and response functions. From the dynamics of the spin waves in the low-temperature phase, we extract the temperature-dependent exponent η. We provide evidence for logarithmic corrections above the Kosterlitz–Thouless temperature and interpret them as a manifestation of free topological defects. Our estimates of the autocorrelation exponent and the fluctuation-dissipation ratio differ from the XY values, while η( T KT ) lies at the boundary of the error bar. Indications for logarithmic corrections at the second-order critical temperature are presented. However, the coupling between angles and chiralities is still strong and explains why the autocorrelation exponent and fluctuation-dissipation ratio are far from the Ising values and seem stable.