# Bose condensation of direct excitons in an off-resonant cavity at elevated temperatures

###### Abstract

We theoretically predict the possibility to observe a macroscopically coherent state of two-dimensional direct excitons at high temperatures, thanks to the dramatic increase of their lifetime in the cavities designed to inhibit exciton recombination. For a single GaAs quantum well embedded in a photonic layered heterostructure with subwavelength period, we show the exciton radiative decay to be strongly suppressed. Quantum hydrodynamic approach is used to study the Berezinskii-Kosterlitz-Thouless crossover in a finite exciton system with intermediate densities. As the system is cooled down below the estimated critical temperatures, the drastic growth of the correlation length is accompanied by a manyfold increase of the photoluminescence intensity.

Introduction.—Despite long-standing theoretical predictions keldysh (); rice (), the experimental observation of a macroscopically coherent state of excitons — bound pairs of electrons and holes in a semiconductor — for decades remained a challenging task and a subject of heated discussions exc2 (); exc1 (); exc3 (); butov1 (); timofeev1 (); jetpl0840329 (); butov2 (). Exciton Bose-Einstein condensation (BEC), once realized, could provide a plethora of beautiful observable phenomena with excitons, such as stimulated backscattering and multi-photon coherence jetpl0740288 (), topological effects prb091161413 (), supersolidity prb082014508 (), ballistic transport prb088195309 (), spin vortices prb089035302 () and currents prl110246403 (), etc. One of the major obstacles on this way, together with the inhomogeneities and excess of free carriers kukushkin (), is the relatively high exciton radiative recombination rate which hinders effective thermalization. Therefore attempts to experimentally achieve excitonic BEC were mostly focused on electronically engineered systems utilizing indirect excitons (IX) in coupled quantum wells (CQWs) under the influence of electric field exc1 (); exc3 (); butov1 (); timofeev1 (); butov2 (), which allows lifetimes longer than the characteristic timescales of relaxation. Compared to IX, direct excitons are more tightly bound and thus can have higher densities and higher critical temperatures of BEC. However, this comes with much shorter lifetimes that prevent their effective cooling, and they recombine before reaching the condensed state.

While electronic engineering as described above was the most fruitful approach so far elongating the excitons’ lifetime and leading to coherence, there is yet another way to control their radiative properties, which utilizes photonic engineering. As well known from cavity quantum electrodynamics, if an excited light source is embedded into a photonic material environment, its recombination can be both greatly enhanced purcell () or inhibited kleppner (). It can be vastly employed in devices which performance is limited by spontaneous emission, such as low-threshold lasers, heterojunction transistors, single photon emitters, etc. For example, ways to experimentally control the spontaneous emission rate have been successfully demonstrated for quantum dots (QDs) in laterally structured microcavities bayer (), as well as quantum wells (QWs) and QDs in two-dimensional photonic crystals noda ().

In this paper, we show the possibility to suppress direct exciton recombination in a single QW using the inhibited spontaneous emission effect, by embedding it into an off-resonant cavity. Relying on our theoretical calculations, we propose two specific GaAs-based geometries for experimental realization, provide optical properties of these structures, and predict exciton lifetimes up to tens of nanoseconds and more. A hydrodynamic quantum field theory joined with the Bogoliubov description is then employed to investigate the Berezinskii-Kosterlitz-Thouless (BKT) crossover in the gas of direct excitons in the regime of intermediate correlations (at elevated densities), allowing to estimate the critical temperatures of condensation and calculate the change in intensity of photoluminescence (PL) along normal direction as the system is cooled down below the crossover temperature.

Optical properties.—Considering a single GaAs QW embedded in a periodic photonic heterostructure, we seek the gap opening up in the electromagnetic density of states. In contrast to distributed Bragg reflectors, we propose a short-period (subwavelength) metallic/dielectric structure, so that the exciton recombination frequency appears well inside this bandgap. Using metal is essential to enhance the contrast between the refractive indices of layers and suppress coupling to the in-plane guided photon modes. We assume a spatially-separated cw pump (see e.g. pump ()) at a frequency outside the gap which allows optically generated QW excitons to readily relax to lowest-energy states while moving to the central region of the sample. To obtain the field distribution inside the medium and optimize the layers widths and their number, we numerically solve the Maxwell problem for electric field with the dielectric constant , where denotes the three-dimensional Laplacian, is the layers growth direction, is the frequency of field, and k and r are the in-plane wavevector and radius-vector, respectively.

In our work we consider two specific geometries. The first realization is based on a 8-nm GaAs QW embedded in a 52 nm AlGaAs (off-resonant) cavity layer sandwiched between the periodic structure with 4.5 pairs of Ag/SiO layers (15/60 nm thickness, respectively), and is depicted schematically in Fig. 1. 8-nm QWs are well studied in works on IX, so we relate to the data of Ref. prb059001625 () with regard to direct exciton recombination line eV and exciton lifetime without cavity ps (recalculated for a single QW). Intensities as a function of are shown in Fig. 1, for the frequency corresponding to the exciton recombination and a higher frequency suggested for the pump ( eV), first being suppressed by a factor of , while the latter lies in the region of the cavity resonance (see also Fig. 2(a)) and has a maximum in the region of the QW.

Fig. 2 summarizes the optical properties of the proposed structure in terms of lifetime. Spectral dependence of the emission rate in the normal direction is shown in Fig. 2(a), and Fig. 2(b) provides inverse lifetimes of excitons with an in-plane wavevector (i.e. emission rate into different angles at ). For , one can deduce the lifetime of excitons in their ground state ns. For excitons with , lifetime grows, and parasitic optical recombination into the in-plane (guided) photon modes is suppressed. The reason to use metallic layers in the structure was to eliminate those guided modes which inevitably appeared when the structure was composed of dielectric layers. Fig. 2(c) shows the dependence of the lifetime on the number of layers in the structure. The optimal layer number provides the lifetimes long enough for thermalization, but not too long to avoid non-radiative recombination which happens on s-timescales kukushkin1 ().

The second geometry we suggest is based on rapidly developing technology allowing to selectively remove substrate and bond thin layers (up to monolayers) of semiconductors. In order to elevate the exciton density and hence the BEC critical temperature, we consider an ultra narrow (2–4 monolayers) single GaAs QW embedded in a 40 nm AlGaAs layer sandwiched between the heterostructure with 4.5 alternate layers of 20/30 nm thick Ag/SiO. In this ultra-narrow case, fluctuations of the energy bandgap due to fluctuations of Al and Ga concentrations in the AlGaAs barrier lead to a strong disorder. This can be overcome by e.g. placing on both sides of the QW (i) thin AlAs layers prb045011403 () or (ii) short-period superlattices (GaAs)/(AlAs) () representing a continuous medium for carriers prb050014416 (). For this geometry, the recombination energy is estimated as eV and exciton lifetime without cavity ps. The lifetime of ground-state excitons in the suggested structure deduced from the Maxwell equations as described above is equal 45 ps (for details, see the Supplemental material). However, for such an ultra-narrow QW the effect of dimensionality allows fourfold increase of the exciton density as compared to wider QWs prb042008928 (), so the density can be taken as high as cm, whereas for the 8-nm QW the lower estimate is cm prb045011403 ().

The exciton gas at those densities is in the regime of intermediate correlations and cannot be readily described by the mean field approximation. To achieve a better analytical description, we combine the use of the Bogoliubov theory with quantum hydrodynamic approach. It can be shown numerically tbp () that this combination provides good accuracy even when correlations are not weak.

Quantum hydrodynamic description.—We are interested in describing exciton BEC and superfluid phase. While in macroscopic 2D system BEC is forbidden hohenberg (), and only superfluid BKT transition takes place jpc006001181 (), in mesoscopic systems BEC can exist due slow decrease of the density matrix with temperature kane (). Therefore we will consider finite but large 2D system of the size , where the disappearance of BEC happens as the BKT crossover, and describe the behavior of the equilibrium, one-body density matrix in the long-wavelength (hydrodynamic) limit, i.e. at large (of the order of ). The resulting expression has the form:

(1) |

where is the exciton mass, is their temperature, is the area of the quantization, is the superfluid density renormalized by vortex pairs as compared to the uniform superfluid density . Account of free vortices in the system is taken according to Kosterlitz jpc007001046 () by introducing the factor ( denotes the distance between free vortices prl040000783 ()). The Bogoliubov spectrum of excitations is given by:

(2) |

where contains contributions from two-, three- and many-body interactions in the hydrodynamic Hamiltonian (we refer to the Supplemental material for detailed analytical derivation). The constant factor in (1) is equal to the total exciton density , and is defined consistently with the ultraviolet cutoff at short distances that cannot be calculated analytically within the hydrodynamic description Popov () and has to be introduced under the sum in (1) by hand. Note that for a single-component uniform superfluid in the limit of low temperatures () and weak correlations the expression (1) is accurate tbp ().

In order to estimate the effective interaction in (2), one needs to consider a series of ladder diagrams which are dependent on the chemical potential due to the logarithmic divergence of the integrals at the lower limit in the case of small densities () SLY (). Following Mora and Castin prl102180404 (), we expand the energy functional up to the third order in terms of a small density-dependent parameter , extrapolating the result of prl102180404 () to the crossover regime (i.e. intermediate correlations), basing on the comparison of the expansion coefficients with numerical simulations tbp (). Interaction of direct excitons is short-ranged LJ (), so that is weakly dependent on momenta. Hence we assume , being the compressibility of the system. We obtain

(3) |

where is defined by the transcendental equation , is the 2D wavevector-dependent exciton scattering length pra081013612 () and is Euler’s constant. The numerical constant is evaluated as . Note that defining , one needs to take a non-zero scattering wavevector tbp ().

The superfluid density in (1) is renormalized by the presence of vortex pairs with separations and can be obtained from the problem of ‘‘dielectric’’ screening of the static supercurrent jpc006001181 (), as follows: . Here is the effective scale-dependent ‘‘dielectric constant’’, , is the local superfluid density, and , denoting the healing length (for details, see the Supplemental material). Then, for an infinite 2D system the BKT transition temperature equals prl039001201 (). However, for a large finite system of the size , the BKT crossover temperature is given by

(4) |

where the denominator is an analytical fit to the numerical calculation pla366000487 () with the parameters , , and valid for the 2D model. From (4), one obtains the distance between free vortices:

(5) |

The local superfluid density in Eqs. (4)–(5) is calculated with the use of the Landau formula

(6) |

containing the spectrum of excitations , where is the spin index and is the spin degeneracy factor.

Finally, the results obtained above allow us to evaluate the asymptotic of the one-body density matrix (1), and the values of the true superfluid density and the condensate density as

(7) |

Fig. 3 (a) shows the results of calculations for the densities (7) depending on temperature for the narrow QW realization, revealing K. For 8 nm GaAs QW shown in Fig. 1, the results are plotted in the inset of Fig. 3 (a) with K. Dependence of the critical temperature (4) on the number of particles at a fixed density is shown in Fig. 3 (b) and (c) for 8-nm QW and narrow QW, respectively.

Accounting for spin.—In the results displayed in Fig. 3 (a), excitons are treated as spinless particles with the spectrum given by (2). This can be justified by employing the Zeeman effect. In order to analyze quantitatively at which magnetic fields one can neglect spin, we solve the BKT transition problem in magnetic field to evaluate the spectrum in dilute and low-temperature limit. In this case, the lowest branch is given by , while the higher branches have the form . Here are the Zeeman shifts in which all –factors are taken to be equal 1, so that at ( in GaAs). As one can see from the simulation results shown in Fig. 3 (b) for the 8-nm QW, even for moderate fields the depletion of the superfluid component is low: is lowered less than by at T, and less than by at T. This underlines the consistency of our spinless approximation. In the ultra-narrow QW, the depletion of the superfluid component by magnetic field is more pronounced but still moderate (see Fig. 3 (c)).

We complete our analysis by estimating the change in PL intensity in the region of the BKT crossover. The exciton lifetime is given by

(8) |

where is equal to the radiative lifetime of exciton with the momentum p (see Fig. 2(b)). The factor indicates that the system is not fully thermalized at large scales: in thermal equilibrium . During thermalization, drops after the appearance of quasi-condensed phase. In particular, as one can see in Fig. 4(a), in the absence of quasi-condensate (), is approximately two orders of magnitude longer than when coherence length m. Fig. 4(b) displays sharp increase of the coherence length in thermalized system at the crossover and the corresponding manyfold growth of the emission intensity from the structure: as compared to intensity at , it is 45 and 82 times higher below for 8-nm QW and ultra-narrow QW, respectively.

It should be mentioned that we assume the QWs to be of a high quality with low inhomogeneities, e.g. same as used in Ref. butov2 (). However, when disorder is taken into account, critical temperatures are still estimated to be 4 K and higher (see the Supplemental material).

In conclusion, we proposed the method to increase the lifetime of direct excitons in single GaAs QWs by employing photonic engineering, and predict their transition to the superfluid phase and condensation at temperatures from 4.8 K to 19 K, depending on the geometry. For comparison, these values are well above K demonstrated in CQWs butov2 (), and are not lower than the temperatures theoretically estimated for IX superfluidity in spatially-separated MoS layers butov3 (). We would like to note that for a transition metal dichalcogenide (TMD) monolayer embedded in an off-resonant cavity, our theory predicts as high as 85 K. However, while the optical recombination of excitons would be suppressed, Auger processes being dominant in TMDs glazov () present the main obstacle for exciton relaxation.

###### Acknowledgements.

Acknowledgements.—The authors are grateful to Oleg Kotov for discussions. N.S.V. acknowledges the financial support by the Russian Foundation for Basic Research, according to the research Project No. 16–32–60066 mol_a_dk, and the Council of the President of the Russian Federation for Support of Young Scientists and Scientific Schools (Project No. MK–201.2017.2). Yu.E.L. is supported by the Program of Basic Research of the High School of Economy.## References

- (1) L. V. Keldysh, A. N. Kozlov, Sov. Phys. JETP 27, 521–528 (1968); Yu. E. Lozovik, V. I. Yudson, Sov. Phys. JETP 44, 389–397 (1976).
- (2) X. Zhu, P. B. Littlewood, M. S. Hybertsen, and T. M. Rice, Phys. Rev. Lett. 74, 1633 (1995).
- (3) J. L. Lin and J. P. Wolfe, Phys. Rev. Lett. 71, 1222 (1993).
- (4) T. Fukuzawa, E. E. Mendez, and J. M. Hong, Phys. Rev. Lett. 64, 3066 (1990).
- (5) L. V. Butov and A. I. Filin, Phys. Rev. B 58, 1980 (1998).
- (6) L. V. Butov, C. W. Lai, A. L. Ivanov, A. C. Gossard, and D. S. Chemla, Nature 417, 47 (2002).
- (7) A. V. Larionov, V. B. Timofeev, P. A. Ni, S. V. Dubonos, I. Hvam, K. Soerensen, JETP Lett. 75, 570 (2002); V. B. Timofeev, Phys. Usp. 48, 295-306 (2005).
- (8) A. V. Gorbunov, V. B. Timofeev, JETP Lett. 84, 329 (2006); M. Alloing, M. Beian, M. Lewenstein, D. Fuster, Y. González, L. González, R. Combescot, M. Combescot, and F. Dubin, Europhys. Lett. 107, 10012 (2014).
- (9) A. A. High, J. R. Leonard, A. T. Hammack, M. M. Fogler, L. V. Butov, A. V. Kavokin, K. L. Campman, and A. C. Gossard, Nature 483, 584 (2012).
- (10) Yu. E. Lozovik and I. V. Ovchinnikov, JETP Letters 74, 288 (2001); Yu. E. Lozovik and I. V. Ovchinnikov, Phys. Rev. B 66, 075124 (2002).
- (11) C.-E. Bardyn, T. Karzig, G. Refael, and T. C. H. Liew, Phys. Rev. B 91, 161413(R) (2015).
- (12) I. L. Kurbakov, Yu. E. Lozovik, G. E. Astrakharchik, and J. Boronat, Phys. Rev. B 82, 014508 (2010).
- (13) A. V. Kavokin, M. Vladimirova, B. Jouault, T. C. H. Liew, J. R. Leonard, L. V. Butov, Phys. Rev. B 88, 195309 (2013).
- (14) H. Sigurdsson, T. C. H. Liew, O. Kyriienko, and I. A. Shelykh, Phys. Rev. B 89, 035302 (2014).
- (15) A. A. High, A. T. Hammack, J. R. Leonard, Sen Yang, L.V. Butov, T. Ostatnicky, M. Vladimirova, A. V. Kavokin, T. C. H. Liew, K. L. Campman, and A. C. Gossard, Phys. Rev. Lett. 110, 246403 (2013).
- (16) V. V. Soloviev, I. V. Kukushkin, J. Smet, K. von Klitzing, W. Dietsche, JETP Letters 83, 553 (2006).
- (17) E. M. Purcell, Phys. Rev. 69, 681 (1946).
- (18) D. Kleppner, Phys. Rev. Lett. 47, 233 (1981); E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
- (19) Y. Yamamoto, S. Machida, G. Björk, Phys. Rev. A 44, 657 (1991); M. Bayer, F. Weidner, A. Larionov, A. McDonald, A. Forchel, T. L. Reinecke, Phys. Rev. Lett. 86, 3168 (2001).
- (20) M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, S. Noda, Science 308, 1296 (2005); A. Kress, F. Hofbauer, N. Reinelt, M. Kaniber, H. J. Krenner, R. Meyer, G. Böhm, and J. J. Finley, Phys. Rev. B 71, 241304 (2005); D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vučković, Phys. Rev. Lett. 95, 013904 (2005).
- (21) A. T. Hammack, M. Griswold, L.V. Butov, L. E. Smallwood, A. L. Ivanov, and A. C. Gossard, Phys. Rev. Lett. 96, 227402 (2006).
- (22) L. V. Butov, A. Imamoglu, A. V. Mintsev, K. L. Campman and A. C. Gossard, Phys. Rev. B 59, 1625 (1999).
- (23) V. V. Soloviev, I. V. Kukushkin, J. Smet, K. von Klitzing, W. Dietsche, JETP Letters 84, 222 (2006).
- (24) R. Eccleston, B. F. Feuerbacher, J. Kuhl, W. W. Rühle, and K. Ploog, Phys. Rev. B 45, 11403 (1992).
- (25) S.-F. Ren, J.-B. Xia, H.-X. Han, and Z.-P. Wang, Phys. Rev. B 50, 14416 (1994).
- (26) 2D exciton Bohr radius is known to be 2 times smaller than in 3D, its binding energy and are 4 times higher. Therefore for an ultra-narrow QW, densities can be assumed four times as high as in a very wide (27-nm) QW prb045011403 (), where and are almost the same as in bulk. See L. C. Andreani, A. Pasquarello, Phys. Rev. B 42, 8928 (1990).
- (27) N.S. Voronova, I.L. Kurbakov, A.S. Pliashechnik, and Yu.E. Lozovik (to be published).
- (28) P. C. Hohenberg, Phys. Rev. 158, 383 (1967).
- (29) J. M. Kosterlitz, D. J. Thouless, J. Phys. C 6, 1181 (1973).
- (30) J. W. Kane, L. P. Kadanoff, Phys. Rev. 155, 80 (1967).
- (31) J. M. Kosterlitz, J. Phys. C 7, 1046 (1974).
- (32) V. Ambegaokar, B. I. Halperin, D. R. Nelson, and E. D. Siggia, Phys. Rev. Lett. 40, 783 (1978).
- (33) V. N. Popov, Functional Integrals in Quantum Field Theory and Statistical Physics (Reidel, Dordrecht, 1983).
- (34) Yu. E. Lozovik, V. I. Yudson, Physica A 93, 493 (1978).
- (35) C. Mora and Y. Castin, Phys. Rev. Lett. 102, 180404 (2009).
- (36) Precisely, we describe bare interaction of excitons by Lennard-Jones potential , with eV, nm in 8-nm QW and eV, nm in the ultra-narrow QW. The system properties appear to be weakly dependent on .
- (37) G. E. Astrakharchik, J. Boronat, I. L. Kurbakov, Yu. E. Lozovik, and F. Mazzanti, Phys. Rev. A 81, 013612 (2010).
- (38) D. R. Nelson and J. M. Kosterlitz, Phys. Rev. Lett. 39, 1201 (1977).
- (39) Yu. E. Lozovik, I. L. Kurbakov, and M. Willander, Phys. Lett. A 366, 487 (2007).
- (40) M. M. Fogler, L. V. Butov, and K. S. Novoselov, Nat. Comm. 5, 4555 (2014).
- (41) M. Kulig, J. Zipfel, P. Nagler, S. Blanter, C. Schüller, T. Korn, N. Paradiso, M. M. Glazov, and A. Chernikov, Phys. Rev. Lett. 120, 207401 (2018).

Bose condensation of direct excitons in an off-resonant cavity at elevated temperatures — Supplemental Material

In this supplemental material, we provide additional figures and details of theoretical calculations, and address the effects of disorder.

## Appendix A Maxwell description and Ultra narrow quantum well

In order to obtain field distribution inside the structure, we solve the Maxwell problem for the 3D vector of electric field E as given in the main text. Given the in-plane translational symmetry of the structure (shown in Fig. 1 of the main text for 8-nm QW and in Fig. 5(a) of this Supplemental material for the ultra-narrow QW), we obtain , where is the layers growth direction, is the frequency of the field, and k and r are the 2D in-plane wavevector and radius-vector, respectively. As a result, we are solving the equation

(9) |

with the complex dielectric constant of the structure corresponding to the given frequency. In particular, for silver layers we used the following data: at eV (as shown in Fig. 1 of the main text); at eV; at eV (as shown in Fig. 5(a)); and at eV. Comparing the resulting profiles of the electric field intensity for different k (corresponding to the angle of emission) to in vacuum, we obtain suppression of the field. Given lifetimes of the QW exciton without cavity (70 ps for 8-nm QW prb059001625 () and 10 ps for ultra-narrow QW (estimate)) and the suppression of field intensity, we calculate the lifetimes of excitons with different inside the structure. Results of the calculations for the 8-nm QW are shown in the main text. For the ultra-narrow QW, the corresponding results are given in Fig. 5 (a), (c), (d), and (e). Fig. 5(b) schematically shows one of the suggested ways to overcome disorder due to fluctuations of Al and Ga atoms concentrations in the AlGaAs barrier, by placing AlAs monolayers by both sides of the ultra-narrow QW prb045011403 (). The disorder caused by the fluctuations of QW widths is suppressed when the AlGaAs (or GaAs) heterointerface is capped by exact number of monolayers.

## Appendix B Quantum hydrodynamic description with Bogoliubov spectrum

Describing coherence in the system, we are interested in the behavior of the equilibrium, one-body density matrix in the long-wavelength (hydrodynamic) limit, i.e. at large . Here is the exciton field operator and the brackets denote the averaging over the thermal equilibrium state of a 2D uniform exciton system with the constant number of particles. To find the asymptotic behavior of , we substitute the field operator as

(10) |

where and are the superfluid density and phase operators, respectively, being the uniform superfluid density and the density fluctuations operator.

To avoid the explicit use of the phase operator which is not well defined rmp040000411 (), in our calculations we will instead work with the operator of superfluid velocity . Splitting it into longitudinal (phonon) and transverse (vortical) parts as , we assume and treat vortical effects by introducing appropriate renormalization. In particular, we take account of free vortices in the system (if those are present) in the form of Kosterlitz renormalization jpc007001046 (), while the renormalization by vortex pairs can be accounted for inside () prb024002526 (). The resulting expression that allows to explicitly evaluate in the long-wavelength limit is as follows:

(12) |

Here denotes the distance between free vortices prl040000783 () (in the case of no free vortices ), and the constant factor in front of the exponent depends on the choice of the ultraviolet (UV) cutoff and will be defined below. Note that the integral in (12) does not depend on the choice of the contour connecting the points and r prb024002526 ().

Making use of the Fourier transform, one has

(13) |

where , is the area of quantization, and we took into account .

To calculate the average in (12), we substitute (10) into the grand canonical Hamiltonian

(14) |

where is the exciton mass, is their chemical potential, is the exciton-exciton two-particle interaction, and denote three-body and other many-body interaction operators. After transformations, one can separate the harmonic part of the hydrodynamic Hamiltonian (14) as (see prb049001205 (); prb049012938 ()), where contains all anharmonic terms and

(15) |

In (15), denotes the Fourier transforms of the density fluctuations operator , and contains contributions from two-, three- and many-body interactions in the hydrodynamic Hamiltonian (14). The transformation

(16) |

brings (15) to the diagonalized form , where is the annihilation operator of a collective phonon with the momentum p satisfying bosonic commutation relations , , and is the Bogoliubov spectrum of excitations:

(17) |

Substituting (16) into (13) and (12), and taking into account and , where is the temperature of excitons, one gets the result

(18) |

It should be pointed out that the hydrodynamic quantum field theory approach described above is valid only in the long-wavelength limit . In other words, the accuracy of the result (18) is limited by the choice of the cutoff at short distances. Hence our approach involves two phenomenological parameters, the constant factor that is contained in (12) and the UV-cutoff factor :

(19) |

that cannot be calculated analytically within the hydrodynamic description Popov () and has to be introduced under the sum in (18) by hand. These two parameters are to be defined consistently with each other. Choosing in the form

(20) |

leads to the constant in (12) being equal to the full exciton density . Thus we obtain the long-wavelength asymptotic for the one-body density matrix which appears in the main text:

(21) |

## Appendix C Dielectric screening and superfluid density renormalization

As the renormalization due to free vortices is taken into account according to Kosterlitz jpc007001046 () by introducing the factor (see (12), (21)), the superfluid density renormalized only by the presence of vortex pairs and is obtained from the problem of ‘‘dielectric’’ screening of the static supercurrent jpc006001181 (). The effective scale-dependent ‘‘dielectric constant’’ of this screening for 2D model satisfies the equation

(22) |

In (22), , is the local superfluid density, and , denoting the separation between vortices and the microscopic phonon length scale of the order of the vortex core radius, i.e. the healing length (for a realistic system of 2D excitons, ). When (i.e. ), the interaction of a pair of vortices separated by is unaffected by any other pair, which provides the boundary condition for (22): .

For one-body density matrix between the points and r, all pairs with do not take part in the renormalization of the superfluid density prb024002526 (). On the other hand, if is larger than the distance between free vortices , the vortex pair itself should be considered as two free vortices. Hence the global superfluid density in (21) is renormalized only by the pairs with separation and depends on as

(23) |

while the true superfluid density which is correspondent to the screening on the scales of the order of the system size , is given by

(24) |

For an infinite system the BKT transition temperature is given by prl039001201 () (where ). However, for a large finite system of the size the BKT crossover temperature is obtained from the numerical analysis of Eqs. (22), (24) in the assumption pla366000487 ():

(25) |

The denominator in (25) is an analytical fit to the numerical calculation pla366000487 () with the parameters , , and valid for the 2D model. Since the BKT crossover corresponds to the appearance of free vortices in the system, for the system of the size it occurs when . This allows to obtains the distance between free vortices from (25) (see Eq. (5) of the main text).

## Appendix D Disorder

While in this work we assume QWs to be of a high quality with low inhomogeneities, e.g. same as reported in Refs. prb059001625 (); butov2 (), for completeness of our analysis we estimate the effects of disorder. In order to do so, we take local superfluid density as given by Eq. (6) of the main text and substract from the quantity

(26) |

that represents the depletion of the superfluid component by disorder prb049012938 (). Here and are the width and the amplitude of the random field fluctuations, respectively:

For medium quality of the QW, we take m and assume the amplitude to correspond to the fluctuations of the QW width by one monolayer nm (where is the reduced mass of an electron and a hole, and the effective QW width takes into account the tunneling of carriers under the AlGaAs barrier, nm). Fig. 6 shows the calculations result for the QWs of the width in the range from =11 nm to 18 nm. For wider QWs, the effect of disorder is negligible. For more narrow QWs such as considered in our manuscript, the high quality of the structures is essential. However even when disorder is taken into account, the critical temperature is estimated to be 4 K and higher.

## References

- (42) L. V. Butov, A. Imamoglu, A. V. Mintsev, K. L. Campman and A. C. Gossard, Phys. Rev. B 59, 1625 (1999).
- (43) R. Eccleston, B. F. Feuerbacher, J. Kuhl, W. W. Rühle, and K. Ploog, Phys. Rev. B 45, 11403 (1992).
- (44) J. W. Kane and L. P. Kadanoff, Phys. Rev. 155, 80 (1967).
- (45) R. Kubo, J. Phys. Soc. Japan 17, 1100 (1962).
- (46) P. Carruthers and M. M. Nieto, Rev. Mod. Phys. 40, 411 (1968).
- (47) J. M. Kosterlitz, J. Phys. C 7, 1046 (1974).
- (48) P. Minnhagen and G. G. Warren, Phys. Rev. B 24, 2526 (1981).
- (49) V. Ambegaokar, B. I. Halperin, D. R. Nelson, and E. D. Siggia, Phys. Rev. Lett. 40, 783 (1978).
- (50) H.-F. Meng, Phys. Rev. B 49, 1205 (1994).
- (51) S. Giorgini, L. Pitaevskii, and S. Stringari, Phys. Rev. B 49, 12938 (1994).
- (52) V. N. Popov, Functional Integrals in Quantum Field Theory and Statistical Physics (Reidel, Dordrecht, 1983).
- (53) J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973).
- (54) D. R. Nelson and J. M. Kosterlitz, Phys. Rev. Lett. 39, 1201 (1977).
- (55) Yu. E. Lozovik, I. L. Kurbakov, and M. Willander, Phys. Lett. A 366, 487 (2007).
- (56) A. A. High, J. R. Leonard, A. T. Hammack, M. M. Fogler, L. V. Butov, A. V. Kavokin, K. L. Campman, and A. C. Gossard, Nature 483, 584 (2012).