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International Symposium on Lattice Field Theory (LATTICE)

Symmetry Breaking in an Extended-O(2) Model

Leon Hostetler, Ryo Sakai, Jin Zhang, Judah Unmuth-Yockey, Alexei Bazavov, Yannick Meurice

Published Dec 23, 2022• 1 citation
Publisher

Abstract

Motivated by attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we consider an extended-O(2) model that differs from the ordinary O(2) model by an explicit symmetry breaking term. Its coupling allows to smoothly interpolate between the O(2) model (zero coupling) and a $q$-state clock model (infinite coupling). In the latter case, a $q$-state clock model can also be defined for non-integer values of $q$. Thus, such a limit can also be considered as an analytic continuation of an ordinary $q$-state clock model to non-integer $q$. The phase diagram of the extended-O(2) model in the infinite coupling limit was established in our previous work, where it was shown that for non-integer $q$, there is a second-order phase transition at low temperature and a crossover at high temperature. In this work, we investigate the model at finite values of the coupling using Monte Carlo and tensor methods. The results may be relevant for configurable Rydberg-atom arrays.

DOI
10.22323/1.430.0014

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arXiv:2105.10450•Physical review. D/Physical review. D.•Sep 14, 2021•16 citations
Motivated by attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we consider an extended-O(2) model that differs from the ordinary O(2) model by an explicit symmetry breaking term. Its coupling allows to smoothly interpolate between the O(2) model (zero coupling) and a qqq-state clock model (infinite coupling). In the latter case, a qqq-state clock model can also be defined for non-integer values of qqq. Thus, such a limit can also be considered as an analytic continuation of an ordinary qqq-state clock model to non-integer qqq. The phase diagram of the extended-O(2) model in the infinite coupling limit was established in our previous work, where it was shown that for non-integer qqq, there is a second-order phase transition at low temperature and a crossover at high temperature. In this work, we investigate the model at finite values of the coupling using Monte Carlo and tensor methods. The results may be relevant for configurable Rydberg-atom arrays.
Motivated by attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we consider an extended-O(2) model that differs from the ordinary O(2) model by an explicit symmetry breaking term. Its coupling allows to smoothly interpolate between the O(2) model (zero coupling) and a qqq-state clock model (infinite coupling). In the latter case, a qqq-state clock model can also be defined for non-integer values of qqq. Thus, such a limit can also be considered as an analytic continuation of an ordinary qqq-state clock model to non-integer qqq. The phase diagram of the extended-O(2) model in the infinite coupling limit was established in our previous work, where it was shown that for non-integer qqq, there is a second-order phase transition at low temperature and a crossover at high temperature. In this work, we investigate the model at finite values of the coupling using Monte Carlo and tensor methods. The results may be relevant for configurable Rydberg-atom arrays.
Motivated by attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we study an extended-O(2) model in two dimensions that differs from the ordinary O(2) model by the addition of an explicit symmetry breaking term −hqcos⁡(qφ)-h_q\cos(qφ)−hq​cos(qφ). Its coupling hqh_qhq​ allows to smoothly interpolate between the O(2) model (hq=0h_q=0hq​=0) and a qqq-state clock model (hq→∞h_q\rightarrow\inftyhq​→∞). In the latter case, a qqq-state clock model can also be defined for noninteger values of qqq. Thus, such a limit can also be considered as an analytic continuation of an ordinary qqq-state clock model to noninteger qqq. In previous work, we established the phase diagram for noninteger qqq in the infinite coupling limit (hq→∞h_q\rightarrow\inftyhq​→∞). We showed that there is a second-order phase transition at low temperature and a crossover at high temperature. In this work, we seek to establish the phase diagram at finite values of the coupling using Monte Carlo and tensor methods. We show that for noninteger qqq, the second-order phase transition at low temperature and crossover at high temperature persist to finite coupling. For integer q=2,3,4q=2,3,4q=2,3,4, we know there is a second-order phase transition at infinite coupling (i.e. the well-known clock models). At finite coupling, we find that the critical exponents for q=3,4q=3,4q=3,4 vary with the coupling, and for q=4q=4q=4 the transition may turn into a BKT transition at small coupling. We comment on the similarities and differences of the phase diagrams with those of quantum simulators of the Abelian-Higgs model based on ladder-shaped arrays of Rydberg atoms.
Motivated by attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we study an extended-O(2) model in two dimensions that differs from the ordinary O(2) model by the addition of an explicit symmetry breaking term −hqcos(qφ) . Its coupling hq allows to smoothly interpolate between the O(2) model ( hq=0 ) and a q -state clock model ( hq→∞ ). In the latter case, a q -state clock model can also be defined for noninteger values of q . Thus, such a limit can also be considered as an analytic continuation of an ordinary q -state clock model to noninteger q . In previous work, we established the phase diagram for noninteger q in the infinite coupling limit ( hq→∞ ). We showed that there is a second-order phase transition at low temperature and a crossover at high temperature. In this work, we seek to establish the phase diagram at finite values of the coupling using Monte Carlo and tensor methods. We show that for noninteger q , the second-order phase transition at low temperature and crossover at high temperature persist to finite coupling. For integer q=2 , 3, 4, we know there is a second-order phase transition at infinite coupling (i.e. the well-known clock models). At finite coupling, we find that the critical exponents for q=3 , 4 vary with the coupling, and for q=4 the transition may turn into a Berezinskii-Kosterlitz-Thouless transition at small coupling. We comment on the similarities and differences of the phase diagrams with those of quantum simulators of the Abelian-Higgs model based on ladder-shaped arrays of Rydberg atoms. Published by the American Physical Society 2024
Motivated by attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we study an extended-O(2) model that differs from the ordinary O(2) model by the addition of an explicit symmetry breaking term. Its coupling allows to smoothly interpolate between the O(2) model (zero coupling) and a qqq-state clock model (infinite coupling). In the latter case, a qqq-state clock model can also be defined for noninteger values of qqq. Thus, such a limit can also be considered as an analytic continuation of an ordinary qqq-state clock model to noninteger qqq. In previous work, we established the phase diagram of the model in the infinite coupling limit. We showed that for noninteger qqq, there is a second-order phase transition at low temperature and a crossover at high temperature. In this work, we establish the phase diagram at finite values of the coupling using Monte Carlo and tensor methods. We show that for noninteger qqq, the second-order phase transition at low temperature and crossover at high temperature persist to finite coupling. For integer q=2,3,4q=2,3,4q=2,3,4, there is a second-order phase transition at infinite coupling (i.e. the clock models). At intermediate coupling, there are second-order phase transitions, but the critical exponents vary with the coupling. At small coupling, the second-order transition for q=4q=4q=4 may turn into a BKT transition.
The qqq-state clock model is a classical spin model that corresponds to the Ising model when q=2q=2q=2 and to the XYXYXY model when q→∞q\to\inftyq→∞. The integer-qqq clock model has been studied extensively and has been shown to have a single phase transition when q=2q=2q=2,333,444 and two phase transitions when q>4q>4q>4.We define an extended qqq-state clock model that reduces to the ordinary qqq-state clock model when qqq is an integer and otherwise is a continuous interpolation of the clock model to noninteger qqq. We investigate this class of clock models in 2D using Monte Carlo (MC) and tensor renormalization group (TRG) methods, and we find that the model with noninteger qqq has a crossover and a second-order phase transition. We also define an extended-O(2)O(2)O(2) model (with a parameter γγγ) that reduces to the XYXYXY model when γ=0γ=0γ=0 and to the extended qqq-state clock model when γ→∞γ\to\inftyγ→∞, and we begin to outline the phase diagram of this model. These models with noninteger qqq serve as a testbed to study symmetry breaking in situations corresponding to quantum simulators where experimental parameters can be tuned continuously.
Motivated by recent attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we define an extended-O(2) model by adding a $\ensuremath{\gamma}\mathrm{cos}(q\ensuremath{\varphi})$ term to the ordinary O(2) model with angular values restricted to a $2\ensuremath{\pi}$ interval. In the $\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, the model becomes an extended qqq-state clock model that reduces to the ordinary qqq-state clock model when qqq is an integer and otherwise is a continuation of the clock model for noninteger qqq. By shifting the $2\ensuremath{\pi}$ integration interval, the number of angles selected can change discontinuously and two cases need to be considered. What we call case 1 has one more angle than what we call case 2. We investigate this class of clock models in two space-time dimensions using Monte Carlo and tensor renormalization group methods. Both the specific heat and the magnetic susceptibility show a double-peak structure for fractional qqq. In case 1, the small-$\ensuremath{\beta}$ peak is associated with a crossover, and the large-$\ensuremath{\beta}$ peak is associated with an Ising critical point, while both peaks are crossovers in case 2. When qqq is close to an integer by an amount $\mathrm{\ensuremath{\Delta}}q$ and the system is close to the small-$\ensuremath{\beta}$ Berezinskii-Kosterlitz-Thouless transition, the system has a magnetic susceptibility that scales as $\ensuremath{\sim}1/(\mathrm{\ensuremath{\Delta}}q{)}^{1\ensuremath{-}1/{\ensuremath{\delta}}^{\ensuremath{'}}}$ with ${\ensuremath{\delta}}^{\ensuremath{'}}$ estimates consistent with the magnetic critical exponent $\ensuremath{\delta}=15$. The crossover peak and the Ising critical point move to Berezinskii-Kosterlitz-Thouless transition points with the same power-law scaling. A phase diagram for this model in the $(\ensuremath{\beta},q)$ plane is sketched. These results are possibly relevant for configurable Rydberg-atom arrays where the interpolations among phases with discrete symmetries can be achieved by varying continuously the distances among atoms and the detuning frequency.