Resonant thermalization of periodically driven strongly correlated electrons

Resonant thermalization of periodically driven strongly correlated electrons

Abstract

We study the dynamics of the Fermi-Hubbard model driven by a time-periodic modulation of the interaction within nonequilibrium Dynamical Mean-Field Theory. For moderate interaction, we find clear evidence of thermalization to a genuine infinite-temperature state with no residual oscillations. Quite differently, in the strongly correlated regime, we find a quasi-stationary extremely long-lived state with oscillations synchronized with the drive (Floquet prethermalization). Remarkably, the nature of this state dramatically changes upon tuning the drive frequency. In particular, we show the existence of a critical frequency at which the system rapidly thermalizes despite the large interaction. We characterize this resonant thermalization and provide an analytical understanding in terms of a break down of the periodic Schrieffer-Wolff transformation.

\externaldocument

supp_mat

Recent advances in the ability to tailor and control light-matter interaction on the ultra-fast time scale Orenstein (2012); Nicoletti and Cavalleri (2016); Först et al. (2011); Subedi et al. (2014) have brought increasing interest in the manipulation of quantum phases of matter with periodic driving fields. Notable achievements are light-induced superconductivity Fausti et al. (2011); Mitrano et al. (2016), metal-to-insulator transition Lantz et al. (2017) and control of microscopic parameters such as the local interaction in organic Mott insulators Singla et al. (2015) and the band gap in excitonic insulators Mor et al. (2017). Similar ideas are applied to ultra-cold atoms in optical lattices Bloch et al. (2008); ? where driving fields are used, for instance, to engineer topological states Eckardt (2017); ?.

From a theoretical perspective, periodically driven, or Floquet, quantum systems are a long-standing subject of studies ranging from dynamical localization Dunlap and Kenkre (1986) and quantum dissipation Grifoni and Hänggi (1998) to quantum chaos Casati and Chirikov (1995) and, more recently, isolated quantum many-body systems Moessner and Sondhi (2017). Other topics of active research include drive-induced topological states Oka and Aoki (2009); Lindner et al. (2010) and artificial gauge fields Goldman and Dalibard (2014); driven electron-phonon coupling Knap et al. (2016); Babadi et al. (2017); Murakami et al. (2017); and integrable systems Lange et al. (2017), correlated electrons Tsuji et al. (2008); ?; Tsuji et al. (2011); Coulthard et al. (2017); Mazza and Georges (2017) or topological systems Dehghani et al. (2014); ?; Dehghani and Mitra (2015); ? in presence of dissipation.

In absence of integrability and of many-body localization, isolated out-of-equilibrium quantum many-body systems are expected to show thermalization of local observables at long times Rigol et al. (2007). Driven systems, which lack time translational invariance, are therefore brought to thermalize to a featureless infinite-temperature state consistent with maximum entropy and no energy conservation D’Alessio and Rigol (2014); Lazarides et al. (2014); Ponte et al. (2015); Genske and Rosch (2015). Yet, the transient dynamics may leave space to non-trivial extremely long-lived non-thermal states characterized by oscillations synchronized with the drive, a phenomenon known as Floquet prethermalization. This prethermal behavior can emerge in the high frequency limit Bukov et al. (2015a); Abanin et al. (2015); Kuwahara et al. (2016); Bukov et al. (2016a); Mori et al. (2016); Abanin et al. (2017); Machado et al. (2017) or be the consequence of a nearby integrable point in the system parameter space. In this case, as recently observed for weakly Bukov et al. (2015b); Canovi et al. (2016); Weidinger and Knap (2016) and strongly Seetharam et al. (2017); Herrmann et al. (2017) interacting systems, there are many quasi-integrals of motion that prevent thermalization except at very long times, similarly to what happens after a quantum quench Moeckel and Kehrein (2008). However, many intriguing questions remain wide open especially concerning the intermediate coupling and frequency regimes, where the most remarkable phenomena are expected to occur.

In this Letter we consider the Fermi-Hubbard model as paradigmatic example of strongly correlated electrons. The system is subject to a time-periodic modulation of the electron interaction, but it is otherwise isolated from any external reservoir. Starting from a thermal equilibrium state, we use non-equilibrium Dynamical Mean-Field Theory (DMFT) Aoki et al. (2014) to calculate the time evolution induced by the drive. First, we explicitly show that at moderate interaction the system thermalizes to the infinite-temperature state. Then, we turn to the regime of large interaction and find a long-lived prethermal state synchronized with the drive, except for a critical, resonant frequency where we find thermalization and a behavior reminiscent of a dynamical transition Eckstein et al. (2009); Schiró and Fabrizio (2010); Tsuji et al. (2013). A periodic Schrieffer-Wolff transformation shows that the Floquet prethermalization is due to the quasi-conservation of double occupancy at large interaction, with the resonant thermalization emerging in correspondence of a break down of such an expansion.

The system is governed by the following Hamiltonian:

(1)

where is the periodically driven interaction and is the hopping, which is such that the bare density of states reads (Bethe lattice). We take as unit of energy, frequency and inverse of time (). In these units the bare band-width is and the critical point of the Mott transition in DMFT is at and at an inverse temperature . We consider a thermal initial density matrix with and we fix the drive amplitude (cf. Supp. Mat. 1 Sec. LABEL:amplitude). For all times the interaction remains repulsive and the system stays half-filled () and particle-hole symmetric.

To calculate the time evolution induced by the drive we use nonequilibrium DMFT Aoki et al. (2014), which consists in mapping the lattice model described by Eq. (1) onto a quantum impurity problem with the following action:

(2)

where is the action associated to the local term in Eq. (1), is the three branch Keldysh contour Stefanucci and van Leeuwen (2013) and is the hybridization between the impurity and a nonequilibrium bath, which is self-consistently determined from the impurity Green function . Within the DMFT mapping, the impurity Green function coincides with the local lattice Green function and from it we can calculate various quantities directly in the thermodynamic limit, such as the double occupancy and the kinetic energy . The computation of the impurity Green function is a challenging task and, despite recent progresses Cohen et al. (2015); Antipov et al. (2017); Profumo et al. (2015), an efficient and numerically exact approach is still lacking. Here we resort to the non-crossing approximation Bickers (1987); Nordlander et al. (1999); Rüegg et al. (2013); Eckstein and Werner (2010); Eckstein et al. (2010); Eckstein and Werner (2011); ?; Strand et al. (2015); Golež et al. (2015) which consists in a first order self-consistent hybridization expansion and which we implement through a Dyson equation for the impurity atomic-state propagator (cf. Supp. Mat. Note1 () Sec. LABEL:nca_oca). For moderate interaction, we benchmark the results with the next-order one-crossing approximation (cf. Supp. Mat. Note1 () Sec. LABEL:benchmark).

We start by discussing the results for moderate average interaction (Fig. 1). The double occupancy shows fast oscillations with frequency comparable to the one of the drive  superimposed to a slower but exponential relaxation. Quite interestingly, after the initial transient and despite the continuous driving, the oscillations get fully damped and the double occupancy reaches the value  independently of the frequency. This is the value of a maximally disordered state and as such signals the thermalization to infinite temperature. With an exponential fit we can extract the thermalization time  which is minimum for and diverges for large frequency.

Figure 1: Thermalization to infinite temperature (). Top panel: Double occupancy for various drive frequencies  shows oscillations (shade) on top of an exponential relaxation (solid line). Right inset: Thermalization time . Bottom panels: Averaged spectral function , occupation function and distribution function for  show the evolution from the out-of-equilibrium state at  to the infinite-temperature thermal state at .

Thermalization is confirmed by the evolution of the Green function and in particular of the retarded component and the lesser component . In a thermal state these functions depend only on the difference  and their Fourier transform is related by the fluctuation-dissipation theorem. Out-of-equilibrium one can perform a Fourier transform with respect to  at fixed  Eckstein and Kollar (2008) and obtain the spectral function and the occupation function . As a consequence of the time-dependent interaction, these functions have oscillations in with period and are even negative for some . To extract meaningful information about the thermalization, which happens on times , we average and over a few periods and obtain positive and (cf. Supp. Mat. Note1 () Sec. LABEL:spec_func). The distribution function provides a simple indicator for thermalization since in the thermal state it equals the Fermi-Dirac distribution. In this case at early times we observe a non-thermal distribution with a pseudo-periodic structure in  with period . This feature is related to the so-called Floquet subbands characteristic of periodically driven systems Tsuji et al. (2009). Then, at later times we observe a remarkably flat distribution – clearly the only one to be at the same time thermal and pseudo-periodic. This establishes that the fluctuation-dissipation relation is satisfied with infinite temperature and therefore confirms thermalization.

Figure 2: Floquet prethermalization and resonant thermalization (). Left panel: Double occupancy  for various drive frequencies  shows oscillations (shade) on top of an exponential relaxation (solid line). Central panel: same for kinetic energy. Inset of left panel: Fourier transform . Right panels: Top left: stationary value with thermal value and initial value for reference. Bottom left: same for kinetic energy. Top right: estimated thermalization time . Bottom right: weight of the peak . Dotted lines mark the resonant frequency . For we see Floquet prethermalization with , and peaked at . For not only and but also the sharp minimum of and the vanishing of signal the resonant thermalization.

We now turn to the strong coupling regime at large average interaction (Fig. 2). The transition from moderate interaction appears to be rather smooth (cf. Supp. Mat. Note1 () Sec. LABEL:amplitude), however for large interaction, in contrast with above, we find qualitative differences as a function of the drive frequency. As a first indication, while for short times also in this case local observables oscillate on top of an exponential relaxation, now the stationary value depends on frequency. As we detail in the following, this signals the existence of different dynamical regimes. In particular, we find thermalization and damping of the oscillations only for a critical frequency, which we estimate to be , while for the other frequencies we observe a long-lived prethermal state.

For frequency below the double occupancy and the kinetic energy oscillate around an average which relaxes exponentially to a non-thermal plateau after the initial transient. While for moderate interaction these oscillations damp out, here they persist with constant amplitude. We calculate the Fourier transform where  is the prethermalization time when the plateau is attained and  is the maximum simulation time. The peaks of  at multiples of  demonstrate the synchronization of the oscillations with the drive. This, together with the non-thermal value of the plateau, are the distinctive features of a Floquet prethermal state in which the system appears to be trapped for times longer than numerically accessible. Since the plateau has a slight linear positive slope, we can extrapolate it to intercept the thermal value  and in this way estimate a thermalization time  which turns out to be orders of magnitude larger than at moderate interaction.

For frequency above we find a very similar prethermalization regime until, for , we observe a sharp threshold behavior. This value corresponds to the maximum energy for single-particle excitations above which the system appears to be unable to absorb energy and local observables are almost constant and equal to their initial equilibrium values. Accordingly, the thermalization time grows exponentially with frequency, in agreement with rigorous bounds Abanin et al. (2015, 2017). We remark also that, for a range of frequencies, the kinetic energy becomes positive, which is characteristic of a highly nonequilibrium state with population inversion. While a similar phenomenon is observed in other Floquet systems Tsuji et al. (2011), here it cannot be ascribed to an effective change of sign of the interaction, since this would also cause the double occupancy to increase above .

The above picture radically changes for the critical frequency where fast thermalization is found despite the large interaction. Here we observe an exponential relaxation of the double occupancy and of the kinetic energy to the thermal values, together with a full damping of oscillations. At this specific frequency the Floquet prethermal state is therefore melted away and the system is able to relax to the infinite-temperature thermal state. We name this phenomenon resonant thermalization since for the periodic modulation of the interaction is resonant with the energy  of doublon excitations, i.e. excitations that change the double occupancy. This resonant condition allows the absorption of energy from the drive and the creation of doublons, which are otherwise suppressed by the large average interaction through a well-known bottleneck mechanism Rosch et al. (2008); Sensarma et al. (2010). Remarkably, the behavior of the system around is strongly reminiscent of a dynamical transition Eckstein et al. (2009); Schiró and Fabrizio (2010); Tsuji et al. (2013). This is clearly seen in the estimated thermalization time  which has a sharp minimum for , as well as from the peak at of the Fourier transform . The weight of this peak goes to zero for with singular behavior, indicating the breakdown of synchronization and the approach to the stationary thermal value.

The above results are corroborated by the evolution of the spectral, occupation and distribution functions (Fig. 3). After the initial transient, these functions reach a stationary state independent of . This confirms that the plateau of the local observables corresponds to a true steady state of the system. For the distribution function is clearly non-thermal and pseudo--periodic, as also found for the non-thermal transient at moderate interaction. On the opposite, for the critical frequency we find a remarkably flat distribution which confirms the thermalization at infinite temperature. Interestingly, for , corresponding to positive kinetic energy, we indeed find a population inversion, as it is clear from the shift towards high energy of and the change of slope of with respect to .

Figure 3: Averaged spectral function , occupation function  and  for  and . Prethermalization for and thermalization for . For the population inversion is clear from the shift of towards higher energy and the change of slop of with respect to . Dotted lines mark the approximate middle of the Hubbard bands.

To gain an analytical insight into the Floquet prethermalization and the resonant thermalization we use a Floquet Schrieffer-Wolff transformation Schrieffer and Wolff (1966); Bukov et al. (2016b); Wysokiński and Fabrizio (2017). This conveniently describes the strong coupling regime, where doublon excitations are suppressed because of the large average interaction, thus preventing the system from absorbing energy unless the frequency of the drive is resonant with the doublon energy. In practice, we introduce a time-periodic unitary which eliminates perturbatively in the terms that do not conserve the double occupancy in the transformed Hamiltonian . This is obtained with an ansatz where is a periodic function determined imposing the vanishing of the commutator up to terms of a given order in , and where we decompose the kinetic energy in terms that do not change (), increase (), or decrease () the double occupancy (cf. Supp. Mat. Note1 () Sec. LABEL:s_w). For generic drive frequency the transformation is well behaved and at first order in we find:

(3)

where . Eq. (3) captures the Floquet prethermal state at long times multiples of (stroboscopic evolution) when the double occupancy is synchronized with the drive and oscillates around a frequency-dependent non-thermal value. However, for the critical value and its submultiples, the function develops a singularity and the transformation breaks down. This suggests that, at these frequencies, the Floquet prethermal state is unstable towards thermalization through non-perturbative processes in , as captured by DMFT. Calculations at large interaction and drive amplitude clearly show the resonant thermalization for frequencies and (cf. Supp. Mat. Note1 () Sec. LABEL:resonant).

The results we have presented here have a potential impact on various experiments, ranging from ultra-cold atoms in driven optical lattices, where one should observe a sudden increase of the heating rate Reitter et al. (2017) at ; to photo-excited organic Mott insulators Singla et al. (2015), where one should observe a sudden filling of the gap in the transient optical conductivity. We also envisage further theoretical study, in particular on the effect of non-local correlations in realistic lattices, which are likely to affect the lifetime of the prethermal plateau. Advances in the solution of the impurity problem would also be important, as they would permit further investigations of the transition between moderate and large interaction and the access to initial states at lower temperature.

In conclusion, to study periodically driven strongly correlated electrons, we have considered the Fermi-Hubbard model with time-periodic interaction. Within nonequilibrium DMFT we have calculated the evolution of local observables and of the local Green function, which provide evidence for thermalization or prethermalization. We have showed the existence of three dynamical regimes: (i) Thermalization to infinite temperature at moderate interaction, as expected for generic isolated quantum many-body systems; (ii) Floquet prethermalization at large interaction, characterized by oscillations of local observables around a non-thermal plateau and a stationary non-thermal distribution function; (iii) Resonant thermalization at large interaction for an isolated critical frequency , where local observables relax exponentially to the infinite-temperature thermal value, together with a damping of oscillations and a flat distribution function. We have then developed a periodic Schrieffer-Wolff transformation which captures the qualitative features of the Floquet prethermal state and whose breakdown for indicates the non-perturbative nature of the resonant thermalization phenomenon.

Acknowledgements.
This work is supported by the FP7/ERC, under Grant Agreement No. 278472-MottMetals. MS acknowledges support from a grant “Investissements d’Avenir” from LabEx PALM (ANR-10-LABX-0039-PALM) and from the CNRS through the PICS-USA-14750.

Footnotes

  1. See Supplemental Materials at …for additional data with different interaction and drive amplitude, for details on the non-crossing and one-crossing approximations, the spectral analysis and the Schrieffer-Wolff transformation.

References

  1. J. Orenstein, Physics Today 65, 44 (2012).
  2. D. Nicoletti and A. Cavalleri, Advances in Optics and Photonics 8, 401 (2016).
  3. M. Först, C. Manzoni, S. Kaiser, Y. Tomioka, Y. Tokura, R. Merlin,  and A. Cavalleri, Nature Physics 7, 854 (2011).
  4. A. Subedi, A. Cavalleri,  and A. Georges, Physical Review B 89, 220301 (2014).
  5. D. Fausti, R. I. Tobey, N. Dean, S. Kaiser, A. Dienst, M. C. Hoffmann, S. Pyon, T. Takayama, H. Takagi,  and A. Cavalleri, Science 331, 189 (2011).
  6. M. Mitrano, A. Cantaluppi, D. Nicoletti, S. Kaiser, A. Perucchi, S. Lupi, P. Di Pietro, D. Pontiroli, M. Riccò, S. R. Clark, D. Jaksch,  and A. Cavalleri, Nature 530, 461 (2016).
  7. G. Lantz, B. Mansart, D. Grieger, D. Boschetto, N. Nilforoushan, E. Papalazarou, N. Moisan, L. Perfetti, V. L. R. Jacques, D. Le Bolloc’h, C. Laulhé, S. Ravy, J.-P. Rueff, T. E. Glover, M. P. Hertlein, Z. Hussain, S. Song, M. Chollet, M. Fabrizio,  and M. Marsi, Nature Communications 8, 13917 (2017).
  8. R. Singla, G. Cotugno, S. Kaiser, M. Först, M. Mitrano, H. Y. Liu, A. Cartella, C. Manzoni, H. Okamoto, T. Hasegawa, S. R. Clark, D. Jaksch,  and A. Cavalleri, Physical Review Letters 115, 187401 (2015).
  9. S. Mor, M. Herzog, D. Golež, P. Werner, M. Eckstein, N. Katayama, M. Nohara, H. Takagi, T. Mizokawa, C. Monney,  and J. Stähler, Physical Review Letters 119, 086401 (2017).
  10. I. Bloch, J. Dalibard,  and W. Zwerger, Reviews of Modern Physics 80, 885 (2008).
  11. I. Bloch, J. Dalibard,  and S. Nascimbène, Nature Physics 8, 267 (2012).
  12. A. Eckardt, Reviews of Modern Physics 89, 011004 (2017).
  13. G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif,  and T. Esslinger, Nature 515, 237 (2014).
  14. D. H. Dunlap and V. M. Kenkre, Physical Review B 34, 3625 (1986).
  15. M. Grifoni and P. Hänggi, Physics Reports 304, 229 (1998).
  16. G. Casati and B. V. Chirikov, Quantum chaos: Between order and disorder (Cambridge University Press, 1995).
  17. R. Moessner and S. L. Sondhi, Nature Physics 13, 424 (2017).
  18. T. Oka and H. Aoki, Physical Review B 79, 081406 (2009).
  19. N. H. Lindner, G. Refael,  and V. Galitski, Nature Physics 7, 490 (2010).
  20. N. Goldman and J. Dalibard, Physical Review X 4, 031027 (2014).
  21. M. Knap, M. Babadi, G. Refael, I. Martin,  and E. Demler, Physical Review B 94, 214504 (2016).
  22. M. Babadi, M. Knap, I. Martin, G. Refael,  and E. Demler, Physical Review B 96, 014512 (2017).
  23. Y. Murakami, N. Tsuji, M. Eckstein,  and P. Werner, Physical Review B 96, 045125 (2017).
  24. F. Lange, Z. Lenarčič,  and A. Rosch, Nature Communications 8, 15767 (2017).
  25. N. Tsuji, T. Oka,  and H. Aoki, Physical Review B 78, 235124 (2008).
  26. N. Tsuji, T. Oka,  and H. Aoki, Physical Review Letters 103, 047403 (2009).
  27. N. Tsuji, T. Oka, P. Werner,  and H. Aoki, Physical Review Letters 106, 236401 (2011).
  28. J. R. Coulthard, S. R. Clark, S. Al-Assam, A. Cavalleri,  and D. Jaksch, Physical Review B 96, 085104 (2017).
  29. G. Mazza and A. Georges, Physical Review B 96, 064515 (2017).
  30. H. Dehghani, T. Oka,  and A. Mitra, Physical Review B 90, 195429 (2014).
  31. H. Dehghani, T. Oka,  and A. Mitra, Physical Review B 91, 155422 (2015).
  32. H. Dehghani and A. Mitra, Physical Review B 92, 165111 (2015).
  33. H. Dehghani and A. Mitra, Physical Review B 93, 245416 (2016).
  34. M. Rigol, V. Dunjko,  and M. Olshanii, Nature 452, 854 (2007).
  35. L. D’Alessio and M. Rigol, Physical Review X 4, 041048 (2014).
  36. A. Lazarides, A. Das,  and R. Moessner, Physical Review Letters 112, 150401 (2014).
  37. P. Ponte, A. Chandran, Z. Papić,  and D. A. Abanin, Annals of Physics 353, 196 (2015).
  38. M. Genske and A. Rosch, Physical Review A 92, 062108 (2015).
  39. M. Bukov, L. D’Alessio,  and A. Polkovnikov, Advances in Physics 64, 139 (2015a).
  40. D. A. Abanin, W. De Roeck,  and F. Huveneers, Physical Review Letters 115, 256803 (2015).
  41. T. Kuwahara, T. Mori,  and K. Saito, Annals of Physics 367, 96 (2016).
  42. M. Bukov, M. Heyl, D. A. Huse,  and A. Polkovnikov, Physical Review B 93, 155132 (2016a).
  43. T. Mori, T. Kuwahara,  and K. Saito, Physical Review Letters 116, 120401 (2016).
  44. D. A. Abanin, W. De Roeck, W. W. Ho,  and F. Huveneers, Physical Review B 95, 014112 (2017).
  45. F. Machado, G. D. Meyer, D. V. Else, C. Nayak,  and N. Y. Yao, arXiv:1708.01620  (2017).
  46. M. Bukov, S. Gopalakrishnan, M. Knap,  and E. Demler, Physical Review Letters 115, 205301 (2015b).
  47. E. Canovi, M. Kollar,  and M. Eckstein, Physical Review E 93, 012130 (2016).
  48. S. A. Weidinger and M. Knap, Scientific Reports 7, 45382 (2016).
  49. K. I. Seetharam, P. Titum, M. Kolodrubetz,  and G. Refael, arXiv:1710.09843  (2017).
  50. A. Herrmann, Y. Murakami, M. Eckstein,  and P. Werner, arXiv:1711.07241 , 1 (2017).
  51. M. Moeckel and S. Kehrein, Physical Review Letters 100, 175702 (2008).
  52. H. Aoki, N. Tsuji, M. Eckstein, M. Kollar, T. Oka,  and P. Werner, Reviews of Modern Physics 86, 779 (2014).
  53. M. Eckstein, M. Kollar,  and P. Werner, Physical Review Letters 103, 056403 (2009).
  54. M. Schiró and M. Fabrizio, Physical Review Letters 105, 076401 (2010).
  55. N. Tsuji, M. Eckstein,  and P. Werner, Physical Review Letters 110, 136404 (2013).
  56. See Supplemental Materials at …for additional data with different interaction and drive amplitude, for details on the non-crossing and one-crossing approximations, the spectral analysis and the Schrieffer-Wolff transformation.
  57. G. Stefanucci and R. van Leeuwen, Nonequilibrium Many-Body Theory of Quantum Systems (Cambridge University Press, Cambridge, 2013).
  58. G. Cohen, E. Gull, D. R. Reichman,  and A. J. Millis, Physical Review Letters 115, 266802 (2015).
  59. A. E. Antipov, Q. Dong, J. Kleinhenz, G. Cohen,  and E. Gull, Physical Review B 95, 085144 (2017).
  60. R. E. V. Profumo, C. Groth, L. Messio, O. Parcollet,  and X. Waintal, Physical Review B 91, 245154 (2015).
  61. N. E. Bickers, Reviews of Modern Physics 59, 845 (1987).
  62. P. Nordlander, M. Pustilnik, Y. Meir, N. S. Wingreen,  and D. C. Langreth, Physical Review Letters 83, 808 (1999).
  63. A. Rüegg, E. Gull, G. A. Fiete,  and A. J. Millis, Physical Review B 87, 075124 (2013).
  64. M. Eckstein and P. Werner, Physical Review B 82, 115115 (2010).
  65. M. Eckstein, T. Oka,  and P. Werner, Physical Review Letters 105, 146404 (2010).
  66. M. Eckstein and P. Werner, Physical Review B 84, 035122 (2011).
  67. M. Eckstein and P. Werner, Physical Review Letters 110, 126401 (2013).
  68. H. U. Strand, M. Eckstein,  and P. Werner, Physical Review X 5, 011038 (2015).
  69. D. Golež, M. Eckstein,  and P. Werner, Physical Review B 92, 195123 (2015).
  70. M. Eckstein and M. Kollar, Physical Review B 78, 205119 (2008).
  71. A. Rosch, D. Rasch, B. Binz,  and M. Vojta, Physical Review Letters 101, 265301 (2008).
  72. R. Sensarma, D. Pekker, E. Altman, E. Demler, N. Strohmaier, D. Greif, R. Jördens, L. Tarruell, H. Moritz,  and T. Esslinger, Physical Review B 82, 224302 (2010).
  73. J. R. Schrieffer and P. A. Wolff, Physical Review 149, 491 (1966).
  74. M. Bukov, M. Kolodrubetz,  and A. Polkovnikov, Physical Review Letters 116, 125301 (2016b).
  75. M. M. Wysokiński and M. Fabrizio, Physical Review B 96, 201115 (2017).
  76. M. Reitter, J. Näger, K. Wintersperger, C. Sträter, I. Bloch, A. Eckardt,  and U. Schneider, Physical Review Letters 119, 200402 (2017).
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
127070
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description