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## Appendix

Here we present the appendices to the main paper (The Quasar Feedback Survey:
characterising CO excitation in quasar host galaxies). Firstly we present the
tables of data used in the paper in appendix A. We then present all the
spectra in appendix B, for which the line profiles and flux estimates have
been determined. Finally, in appendix C we show contour plots showing the
extent of CO(1-0) in the ACA data for the 9 targets with detections, plotted
over optical images of the galaxies.

##  Appendix A Tables of data

We first present details of the CO observations including project id, beam
sizes, rms and observation times in Table 5. We also present tables showing
information about the spectra and line fits including the S/N of lines,
integrated fluxes, line widths, velocity offsets and luminosities in Tables 6
– 10. We also present the line profile information from the fits to the
ionised gas observations in Table 11.

Line | Source Name | Beam Size | rms, $\Delta v$ = 100 km $\rm s^{-1}$ | $\rm t_{obs}$ | Detection?
---|---|---|---|---|---
|  | (arcsec) | (mJy) | (minutes) |
CO(1-0) Atacama Compact Array | J0909+1052 | 12.7 | 1.8 | 43.8 | N
project id: 2019.2.00194.S | J0945+1737 | 11.9 | 1.7 | 31.2 | L
| J0958+1439 | 12.7 | 1.1 | 85.7 | D
| J1000+1242 | 12.8 | 1.3 | 92.7 | D
| J1016+0028 | 12.2 | 1.2 | 42.3 | L
| J1055+1102 | 11.7 | 1.1 | 44.3 | L
| J1100+0846 | 12.3 | 2.5 | 10.2 | D
| J1108+0659 | 12.8 | 1.1 | 35.3 | D
| J1114+1939 | 14.1 | 1.7 | 63.5 | N
| J1116+2200 | 13.1 | 1.7 | 51.4 | D
| J1222-0007 | 12.7 | 1.3 | 30.2 | N
| J1316+1753 | 12.9 | 1.6 | 21.2 | N
| J1518+1403 | 12.5 | 1.9 | 12.1 | D
CO(2-1) APEX id: E-0105.B-0713A-2020 | J0909+1052 | 31.6 | 26.5 | 44.7 | N
| J1016+0028 | 30.2 | 7.4 | 51.1 | N
| J1055+1102 | 31.0 | 15.6 | 19.1 | N
| J1108+0659 | 32.0 | 18.0 | 12.8 | D
| J1114+1939 | 32.5 | 9.9 | 31.3 | D
| J1116+2200 | 30.9 | 14.4 | 19.2 | D
| J1222-0007 | 31.7 | 18.7 | 19.2 | N
| J1430+1939 | 29.4 | 10.6 | 58.3 | D
| J1518+1403 | 30.8 | 6.8 | 128.8 | D
CO(2-1) APEX id: E-0100.B-0166B-2017 | J0945+1737 | 30.5 | 7.6 | 42.0 | D
| J0958+1439 | 30.0 | 6.0 | 71.9 | N
| J1000+1242 | 31.1 | 3.2 | 239.1 | D
| J1010+0612 | 29.8 | 12.0 | 23.0 | D
| J1010+1413 | 32.4 | 6.8 | 60.0 | D
| J1100+0846 | 29.8 | 18.9 | 27.5 | D
| J1316+1753 | 31.2 | 11.0 | 49.6 | L
| J1356+1026 | 30.4 | 24.0 | 35.8 | N
CO(3-2) APEX id: E-0105.B-0713B-2020 | J0909+1052 | 21.0 | 17.9 | 64.0 | N
| J0945+1737 | 20.4 | 14.1 | 46.5 | N
| J0958+1439 | 20.0 | 18.1 | 248.3 | L
| J1000+1242 | 20.7 | 10.6 | 60.1 | L
| J1010+0612 | 19.8 | 10.0 | 76.2 | D
| J1010+1413 | 21.6 | 9.6 | 53.8 | D
| J1016+0028 | 20.1 | 9.5 | 148.8 | N
| J1055+1102 | 20.7 | 15.4 | 15.9 | N
| J1100+0846 | 19.9 | 14.2 | 57.3 | D
| J1108+0659 | 21.3 | 24.2 | 32.0 | D
| J1114+1939 | 21.6 | 25.9 | 18.5 | N
| J1116+2200 | 20.6 | 17.8 | 63.5 | D
| J1222-0007 | 21.2 | 21.4 | 12.8 | N
| J1316+1753 | 20.8 | 16.0 | 64.9 | N
| J1356+1026 | 20.3 | 11.3 | 95.5 | N
| J1518+1403 | 20.6 | 6.9 | 52.6 | D
CO(3-2) ALMA project id: 2016.1.01535.S | J1430+1339 | 4.3 | 10.3 | 26.2 | D
CO(6-5) APEX id: E-0109.B-0710A-2022 | J0945+1737 | 10.2 | 82.1 | 128.7 | N
| J1010+0612 | 9.9 | 58.2 | 156.1 | N
| J1100+0846 | 9.9 | 49.0 | 111.3 | N
| J1430+1339 | 9.8 | 40.0 | 292.0 | L
CO(7-6) APEX id: E-0109.B-0710A-2022 | J1108+0659 | 9.1 | 118.0 | 96.4 | N
| J1116+2200 | 8.8 | 124.9 | 87.5 | N
| J1518+1403 | 8.8 | 68.3 | 245.2 | N
Table 5: Details of the CO observations including the CO line, telescope used and project id, the source name, the beam size, the rms, observing times ($\rm t_{obs}$) and whether we class a detection "D", low S/N detection "L" or a non-detection "N" based on criteria stated in Section B. Name | S/N | SCO(1-0) | CO(1-0) $V_{50}$ | CO(1-0) $W_{80}$ | $\rm L^{\prime}_{CO(1-0)}$
---|---|---|---|---|---
|  | (Jy km $\rm s^{-1}$ ) | (km $\rm s^{-1}$ ) | (km $\rm s^{-1}$ ) | (1x109 $\times$ K km $\rm s^{-1}$ pc2)
J0909+1052 | – | < 2.1 | – | – | < 2.8
J0945+1737 | 4.8 | 2.7 $\pm$ 0.7 | 243 $\pm$ 65 | 539 $\pm$ 167 | 2.1 $\pm$ 0.6
J0958+1439 | 8.6 | 2.6 $\pm$ 0.3 | -71 $\pm$ 13 | 215 $\pm$ 34 | 1.5 $\pm$ 0.2
J1000+1242 | 5.6 | 1.9 $\pm$ 0.6 | 259 $\pm$ 68 | 621 $\pm$ 246 | 1.8 $\pm$ 0.5
J1010+0612 | – | – | – | – | –
J1010+1413 | – | – | – | – | –
J1016+0028 (core) | 4.6 | 1.3 $\pm$ 0.4 | -116 $\pm$ 39 | 288 $\pm$ 101 | 0.8 $\pm$ 0.3
J1016+0028 North Lobe |  | 0.6 $\pm$ 0.1 | 611 $\pm$ 68 | 116 $\pm$ 23 | 0.4 $\pm$ 0.1
J1016+0028 South Lobe |  | 1.0 $\pm$ 0.2 | \- 365 $\pm$ 16 | 126 $\pm$ 24 | 0.6 $\pm$ 0.1
J1055+1102 | 3.1 | 1.0 $\pm$ 0.4 | -6 $\pm$ 74 | 381 $\pm$ 189 | 1.0 $\pm$ 0.4
J1100+0846 (Total) | 7.6 | 3.7 $\pm$ 0.3 | 33 $\pm$ 42 | 378 $\pm$ 41 | 1.7 $\pm$ 0.3
J1100+0846 (red) | 6.2 | 2.3 $\pm$ 0.4 | 79 $\pm$ 16 | 142 $\pm$ 41 | 1.1 $\pm$ 0.3
J1100+0846 (blue) | 3.8 | 1.4 $\pm$ 0.4 | -197 $\pm$ 15 | 204 $\pm$ 45 | 0.7 $\pm$ 0.2
J1108+0659 (Total) | 32.2 | 8.4 $\pm$ 0.5 | 88 $\pm$ 5 | 354 $\pm$ 40 | 13.4 $\pm$ 3.1
J1108+0659 (core) | 22.1 | 4.7 $\pm$ 1.3 | 112 $\pm$ 5 | 66 $\pm$ 8 | 7.5 $\pm$ 2.0
J1108+0659 (blue wing) | 14.2 | 3.7 $\pm$ 1.4 | -64 $\pm$ 68 | 151 $\pm$ 40 | 5.9 $\pm$ 2.3
J1114+1939 | – | < 2.0 | – | – | < 3.9
J1116+2200 | 12.4 | 7.4 $\pm$ 0.5 | 122 $\pm$ 17 | 578 $\pm$ 44 | 7.3 $\pm$ 0.5
J1222-0007 | – | < 2.1 | – | – | < 3.0
J1316+1753 | – | < 1.8 | – | – | < 2.4
J1356+1026 | – | – | – | – | –
J1430+1339 | – | – | – | – | –
J1518+1403 | 9.3 | 3.8 $\pm$ 0.5 | 47 $\pm$ 14 | 220 $\pm$ 36 | 1.2 $\pm$ 0.2
Table 6: Properties of the CO(1-0) line emission observed with the ALMA ACA. We present the integrated S/N, the integrated flux from the single with Gaussian line fit, the velocity offset of the peak compared to $v$ = 0 km $\rm s^{-1}$ (defined from the SDSS redshift), line width ($W_{80})$ and the line luminosity. Upper limits are 3 $\sigma$ upper limits based on the median rms in the spectra and using the average line profile from other transitions of the same target, or from others with CO(1-0) detections. Those with no data shown do not have the required data to present. Name | S/N | SCO(2-1) | CO(2-1) $V_{50}$ | CO(2-1) $W_{80}$ | $\rm L^{\prime}_{CO(2-1)}$
---|---|---|---|---|---
|  | (Jy km $\rm s^{-1}$ ) | (km $\rm s^{-1}$ ) | (km $\rm s^{-1}$ ) | (1x109 $\times$ K km $\rm s^{-1}$ pc2)
J0909+1052 | – | < 34.1 | – | – | < 12.6
J0945+1737 | 5.1 | 11.6 $\pm$ 2.4 | 25 $\pm$ 49 | 528 $\pm$ 125 | 2.3 $\pm$ 0.5
J0958+1439 | – | < 6.5 | – | – | < 1.0
J1000+1242 | 7.3 | 6.4 $\pm$ 1.2 | -64 $\pm$ 47 | 544 $\pm$ 119 | 1.7 $\pm$ 0.3
J1010+0612 | 4.2 | 12.3 $\pm$ 4.5 | 30 $\pm$ 74 | 453 $\pm$ 190 | 1.4 $\pm$ 0.5
J1010+1413 | 5.0 | 11.4 $\pm$ 2.4 | -10 $\pm$ 44 | 462 $\pm$ 112 | 5.5 $\pm$ 1.2
J1016+0028 (core) | 1.9 | < 6.6 | – | - | < 1.1
J1055+1102 | – | < 16.9 | – | – | < 5.9
J1100+0846 (Total) | 5.9 | 22.8 $\pm$ 3.4 | -18 $\pm$ 39 | 378 $\pm$ 41 | 2.7 $\pm$ 0.4
J1100+0846 (red) | 8.1 | 14.1 $\pm$ 2.6 | 79 $\pm$ 16 | 142 $\pm$ 41 | 1.7 $\pm$ 0.3
J1100+0846 (blue) | 5.0 | 8.8 $\pm$ 2.2 | -197 $\pm$ 15 | 204 $\pm$ 45 | 1.0 $\pm$ 0.3
J1108+0659 (Total) | 10.0 | 59.4 $\pm$ 15.6 | 9 $\pm$ 19 | 366 $\pm$ 112 | 24.0 $\pm$ 9.0
J1108+0659 (core) | 9.5 | 29.7 $\pm$ 13.2 | 35 $\pm$ 18 | 93 $\pm$ 26 | 12.0 $\pm$ 9.0
J1108+0659 (blue wing) | 6.4 | 29.8 $\pm$ 17.0 | -236 $\pm$ 185 | 277 $\pm$ 112 | 12.0 $\pm$ 7.0
J1114+1939 | 8.7 | 21.4 $\pm$ 6.6 | 280 $\pm$ 106 | 756 $\pm$ 270 | 10.3 $\pm$ 3.2
J1116+2200 | 6.3 | 26.4 $\pm$ 4.6 | 233 $\pm$ 41 | 517 $\pm$ 104 | 6.5 $\pm$ 1.1
J1222-0007 | – | < 21.9 | – | – | < 9.7
J1316+1753 | 3.7 | 11.1 $\pm$ 3.6 | -370 $\pm$ 43 | 293 $\pm$ 110 | 3.0 $\pm$ 1.0
J1356+1026 | – | < 30.9 | – | – | < 5.8
J1430+1339 | 5.7 | 21.2 $\pm$ 7.9 | 31 $\pm$ 106 | 631 $\pm$ 271 | 1.8 $\pm$ 0.7
J1518+1403 | 11.5 | 15.5 $\pm$ 1.8 | -14 $\pm$ 19 | 293 $\pm$ 49 | 3.6 $\pm$ 0.4
Table 7: Properties of the CO(2-1) line emission observed with APEX. We present the integrated S/N, the integrated flux from the single with Gaussian line fit, the velocity offset of the peak compared to $v$ = 0 km $\rm s^{-1}$ (defined from the SDSS redshift), line width ($W_{80})$ and the line luminosity. Upper limits are 3$\sigma$ upper limits based on the median rms in the spectra and using the average line profile from other transitions of the same target, or from others with CO(2-1) detections. Those with no data shown do not have the required data to present. Name | S/N | SCO(3-2) | CO(3-2) $V_{50}$ | CO(3-2) $W_{80}$ | $\rm L^{\prime}_{CO(3-2)}$
---|---|---|---|---|---
|  | (Jy km $\rm s^{-1}$) | (km $\rm s^{-1}$) | (km $\rm s^{-1}$) | (1x109 $\times$ K km $\rm s^{-1}$ pc2)
J0909+1052 | – | < 22.5 | – | – | < 3.3
J0945+1737 | 2.7 | < 19.2 | – | – | < 1.7
J0958+1439 | 4.3 | 13.3 $\pm$ 2.7 | 188 $\pm$ 59 | 578 $\pm$ 137 | 1.5 $\pm$ 0.4
J1000+1242 | 3.1 | 8.6 $\pm$ 3.0 | -38 $\pm$ 81 | 576 $\pm$ 207 | 1.2 $\pm$ 0.4
J1010+0612 | 5.5 | 10.3 $\pm$ 3.1 | 179 $\pm$ 36 | 264 $\pm$ 94 | 0.5 $\pm$ 0.2
J1010+1413 | 12.1 | 37.5 $\pm$ 3.4 | -83 $\pm$ 30 | 673 $\pm$ 77 | 8.1 $\pm$ 0.7
J1016+0028 (core) | 2.1 | < 7.0 | – | – | < 0.5
J1055+1102 | – | < 14.8 | – | – | < 1.7
J1100+0846 (Total) | 7.4 | 31.7 $\pm$ 3.7 | -21 $\pm$ 21 | 354 $\pm$ 36 | 1.7 $\pm$ 1.2
J1100+0846 (red) | 12.0 | 19.4 $\pm$ 2.7 | 58 $\pm$ 12 | 210 $\pm$ 36 | 1.0 $\pm$ 0.1
J1100+0846 (blue) | 7.6 | 12.3 $\pm$ 2.5 | -184 $\pm$ 13 | 167 $\pm$ 39 | 0.7 $\pm$ 0.1
J1108+0659 (Total) | 14.2 | 118.8 $\pm$ 20.5 | 55 $\pm$ 17 | 529 $\pm$ 141 | 21.0 $\pm$ 4.0
J1108+0659 (core) | 14.7 | 71.3 $\pm$ 13.9 | 142 $\pm$ 17 | 276 $\pm$ 41 | 12.7 $\pm$ 2.5
J1108+0659 (blue wing) | 8.3 | 47.6 $\pm$ 15.0 | -194 $\pm$ 53 | 417 $\pm$ 141 | 8.5 $\pm$ 2.7
J1114+1939 | – | < 48.9 | – | – | < 6.9
J1116+2200 | 10.9 | 63.4 $\pm$ 5.7 | 53 $\pm$ 26 | 606 $\pm$ 65 | 6.7 $\pm$ 0.6
J1222-0007 | – | < 26.7 | – | – | < 4.2
J1316+1753 | – | < 11.8 | – | – | < 2.4
J1356+1026 | – | < 14.5 | – | – | < 0.5
J1430+1339 | 9.3 | 17.5 $\pm$ 1.4 | 33 $\pm$ 19 | 532 $\pm$ 49 | 0.7 $\pm$ 0.1
J1518+1403 | 12.2 | 20.5 $\pm$ 2.1 | -34 $\pm$ 20 | 412 $\pm$ 50 | 2.1 $\pm$ 0.2
Table 8: Properties of the CO(3-2) line emission observed with APEX. We present the integrated S/N, the integrated flux from the single with Gaussian line fit, the velocity offset of the peak compared to $v$ = 0 km $\rm s^{-1}$ (defined from the SDSS redshift), line width ($W_{80})$ and the line luminosity. Upper limits are 3$\sigma$ upper limits based on the median rms in the spectra and using the average line profile from other transitions of the same target, or from others with CO(3-2) detections. Those with no data shown do not have the required data to present. Name | S/N | SCO(6-5) | CO(6-5) $V_{50}$ | CO(6-5) $W_{80}$ | $\rm L^{\prime}_{CO(6-5)}$
---|---|---|---|---|---
|  | (Jy km $\rm s^{-1}$) | (km $\rm s^{-1}$) | (km $\rm s^{-1}$) | (1x109 $\times$ K km $\rm s^{-1}$ pc2)
J0945+1737 | – | < 117.4 | – | – | < 2.6
J1010+0612 | 2.2 | < 59.2 | – | – | < 0.8
J1100+0846 | 1.5 | < 47.5 | – | – | < 0.8
J1430+1339 | 6.7 | 42.9 $\pm$ 18.7 | 126 $\pm$ 121 | 669 $\pm$ 337 | 0.4 $\pm$ 0.2
Table 9: Properties of the CO(6-5) line emission observed with APEX. We present the integrated S/N, the integrated flux from the single with Gaussian line fit, the velocity offset of the peak compared to $v$ = 0 km $\rm s^{-1}$ (defined from the SDSS redshift), line width ($W_{80})$ and the line luminosity. Upper limits are 3$\sigma$ upper limits based on the median rms in the spectra and using the average line profile from other transitions of the same target, or from others with CO(6-5) detections. Those with no data shown do not have the required data to present. Name | S/N | SCO(7-6) | CO(7-6) $V_{50}$ | CO(7-6) $W_{80}$ | $\rm L^{\prime}_{CO(7-6)}$
---|---|---|---|---|---
|  | (Jy km $\rm s^{-1}$) | (km $\rm s^{-1}$) | (km $\rm s^{-1}$) | (1x109 $\times$ K km $\rm s^{-1}$ pc2)
J1108+0659 | – | < 129.8 | – | – | < 4.2
J1116+2200 | – | < 187.1 | – | – | < 3.8
J1518+1403 | – | < 58.2 | – | – | < 1.5
Table 10: Properties of the CO(7-6) line emission observed with APEX. We present the integrated S/N, the integrated flux from the single with Gaussian line fit, the velocity offset of the peak compared to $v$ = 0 km $\rm s^{-1}$ (defined from the SDSS redshift), line width ($W_{80})$ and the line luminosity. Upper limits are 3$\sigma$ upper limits based on the median rms in the spectra and using the average line profile from other transitions of the same target, or from others with or CO(7-6) detections. Those with no data shown do not have the required data to present. Name | Emission Line | 30" $V_{50}$ | 30" $W_{80}$ | 3" $V_{50}$ | 3" $W_{80}$ | # of components
---|---|---|---|---|---|---
|  | (km $\rm s^{-1}$) | (km $\rm s^{-1}$) | (km $\rm s^{-1}$) | (km $\rm s^{-1}$) |
J0909+1052 | [O III]4959 | -34 $\pm$ 3 | 318 $\pm$ 8 | -38 $\pm$ 1 | 402 $\pm$ 3 | 1
J0945+1737 | [O III]5007 | 63 $\pm$ 2 | 734 $\pm$ 33 | 39 $\pm$ 1 | 798 $\pm$ 43 | 2
J0958+1439 | [O III]5007 | 16 $\pm$ 7 | 736 $\pm$ 44 | 18 $\pm$ 4 | 788 $\pm$ 25 | 2
J1000+1242 | [O III]5007 | -95 $\pm$ 4 | 673 $\pm$ 5 | -41 $\pm$ 6 | 808 $\pm$ 8 | 2
J1010+0612 | [O III]5007 | -17 $\pm$ 7 | 1216 $\pm$ 95 | -28 $\pm$ 14 | 1406 $\pm$ 55 | 2
J1010+1413 | [O III]5007 | -76 $\pm$ 7 | 1333 $\pm$ 30 | -14 $\pm$ 4 | 1440 $\pm$ 28 | 2
J1016+0028 | [O III]5007 | -130 $\pm$ 6 | 658 $\pm$ 8 | -41 $\pm$ 4 | 598 $\pm$ 20 | 3
J1055+1102 | [O III]5007 | -12 $\pm$ 1 | 473 $\pm$ 13 | -17 $\pm$ 1 | 473 $\pm$ 6 | 2
J1100+0846 | [O III]5007 | -5 $\pm$ 12 | 926 $\pm$ 45 | 0 $\pm$ 12 | 913 $\pm$ 49 | 2
J1108+0659 | $\rm H\beta$ | 40 $\pm$ 19 | 598 $\pm$ 36 | -47 $\pm$ 5 | 653 $\pm$ 11 | 2
J1114+1939 | [O III]5007 | 30 $\pm$ 4 | 616 $\pm$ 17 | 29 $\pm$ 2 | 648 $\pm$ 7 | 2
J1116+2200 | [O III]5007 | -35 $\pm$ 3 | 521 $\pm$ 43 | -57 $\pm$ 1 | 456 $\pm$ 18 | 2
J1222-0007 | $\rm H\beta$ | -79 $\pm$ 34 | 593 $\pm$ 51 | -54 $\pm$ 3 | 831 $\pm$ 12 | 3
J1316+1753 | [O III]5007 | -27 $\pm$ 4 | 1117 $\pm$ 56 | 29 $\pm$ 30 | 1152 $\pm$ 52 | 3
J1356+1026 | [O III]5007 | 75 $\pm$ 30 | 807 $\pm$ 188 | -10 $\pm$ 33 | 891 $\pm$ 111 | 3
J1430+1339 | [O III]5007 | 83 $\pm$ 15 | 661 $\pm$ 52 | 40 $\pm$ 14 | 796 $\pm$ 63 | 2
J1518+1403 | [O III]5007 | 0 $\pm$ 2 | 569 $\pm$ 64 | 1 $\pm$ 1 | 518 $\pm$ 36 | 2
Table 11: Table presenting the MUSE line profile data, extracted from 30 and 3
arcsec apertures used as a comparison to the molecular gas presented in this
work. Here we show the emission line used as a tracer for the ionised gas, the
velocity offset ($V_{50}$) and line width ($W_{80}$). Also shown is the number
of components present in the line (and therefore how many Gaussian components
we fit to analyse the line profile).

##  Appendix B Spectra

Here we present all spectra for the targets in this paper including the
multiple CO transition data and ionised gas observations from MUSE in Figures
13 – 27. Top panels present the CO data available for that target and in every
case we also present MUSE spectra for the [O III] line extracted from a 30
arcsec diameter aperture in the final panel. In all cases, solid black lines
denote fits to the data. In some cases there are multiple components and as
such, dotted black lines denote the different components that make the total
fit to the spectra. In some CO(2-1) spectra we also show the fit from higher
resolution ALMA observations (solid orange line labelled RA+22, Ramos Almeida
et al., 2022). Shaded grey regions represent the 1$\sigma$ level. For further
information about what is presented in the spectra plots (e.g. lines, fits to
the spectra etc.) see Figure 3 in the main body of the paper.

Figure 11: J0909+1052, non-detections across all three CO transitions. All
shown with 100km $\rm s^{-1}$ bins. MUSE [O III] observations show a single
with Gaussian.

Figure 12: J0945+1737 spectra across four CO transitions. All with 100 km $\rm
s^{-1}$ bins. Detection in CO(2-1) and low S/N detection in CO(1-0), non-
detection in CO(3-2) (although tentative signs of a line at S/N = 2.7) or
CO(6-5). MUSE [O III] observations show a two component fit with a narrow and
broad component.

Figure 13: J0958+1439 detection in CO(1-0) and low S/N detection in CO(3-2).
Non-detection in CO(2-1). MUSE [O III] observations show a two component fit
with a narrow and broad component.

Figure 14: J1000+1242, detections in CO(1-0) and CO(2-1) with a low S/N
detection in CO(3-2). MUSE [O III] observations show a broad single with
Gaussian line profile.

Figure 15: J1010+0612 spectra across three CO transitions, with no CO(1-0)
observations available. All with 100 km $\rm s^{-1}$ bins. Detection in
CO(3-2) and a low S/N detection in CO(2-1) which matches the two component
line profile identified in Ramos Almeida et al. 2022 (spectra shown in orange,
labelled RA+22) for which we see tentative signs here. MUSE [O III]
observations show a two component fit with a narrow and broad component.

Figure 16: J1010+1413, detections in both CO(2-1) and CO(3-2). CO(1-0) not
observed. MUSE [O III] observations show broad single with Gaussian line
profile.

Figure 17: J1016+0028, low S/N detection in CO(1-0) and non detections in
CO(2-1) and CO(3-2). MUSE [O III] observations show a two component line
profile.

Figure 18: J1055+1102 only low S/N detection in CO(1-0), non detections in
CO(2-1) and CO(3-2). MUSE [O III] observations show a two component fit with a
narrow and broad component.

Figure 19: J1100+0846 spectra across four transitions. CO(1-0) with 50 km $\rm
s^{-1}$ bins and CO(2-1) and CO(3-2) with 25 km $\rm s^{-1}$ bins. Detection
in all four transitions, all with same double peak profile. The same double
peaked profile was found by Ramos Almeida et al. 2022 CO(2-1) observations
(spectra shown in orange, labelled RA+22). MUSE [O III] observations show a
two component fit with a narrow and broad component.

Figure 20: J1108+0659 spectra across four CO transitions. First three CO
transitions plotted with 50 km $\rm s^{-1}$ bins and CO(7-6) plotted with 100
km $\rm s^{-1}$ bins due to a lack of detection. Blue wing in the first three
CO transitions indicating a potential molecular outflow component. There is
also potentially a small velocity shift in this outflow component across the
first three CO transitions. We find a non-detection in CO(7-6). MUSE [O III]
observations showing a similar blue wing to the CO data.

Figure 21: J1114+1939, only detection in CO(2-1). Non-detections in CO(1-0)
and CO(3-2). MUSE [O III] observations show a two component fit with a narrow
and broad component, with the broad component seemingly blueshifted
potentially indicating an outflow.

Figure 22: J1116+2200 spectra across four CO transitions. The first three
transitions with 50 km $\rm s^{-1}$ bins and the CO(7-6) with 100 km $\rm
s^{-1}$ bins. Detection of a broad line in the first three transitions. Non-
detection in CO(7-6). MUSE [O III] observations show a single with Gaussian
line profile.

Figure 23: J1222-0007 non-detections in all three CO transitions. MUSE [O III]
observations show a two component fit to the data.

Figure 24: J1316+1753, low S/N detection in CO(2-1) and non-detection in
CO(1-0) and CO(3-2). MUSE [O III] observations show a three component fit to
the data. The low S/N detection in our CO(2-1) data seems to match the middle
component present in the MUSE [O III] spectra.

Figure 25: J1356 spectra across two CO transitions with no CO(1-0) data
available. All set to 100 km $\rm s^{-1}$ bin widths. Non-detection in both
transitions. Also shown is the detection from Ramos Almeida et al. 2022
detection in CO(2-1) (labelled RA+22 N in orange and RA+22 in green, where
orange represents the nucleus only and green is the full line). We use the
RA+22 data in our analysis as well lack a detection in CO(2-1)

. MUSE [O III] observations show a blue wing component.

Figure 26: J1430+1339 spectra across three CO transitions, no observations in
CO(1-0). CO(2-1) and CO(6-5) with 100 km $\rm s^{-1}$ bins, CO(3-2) with 26 km
$\rm s^{-1}$ bins (ACA data). For CO(2-1) data from Ramos Almeida et al. 2022
was also available, shown here in orange (labelled RA+22) and showing an
almost identical spectra to our APEX data. Detections across all CO
transitions observed. MUSE [O III] observations show a two component fit
including a blue wing.

Figure 27: J1518+1403 spectra across four CO transitions. All with 50 km $\rm
s^{-1}$ bins (except for CO(7-6) which is 100 km $\rm s^{-1}$ due to lack of a
detection). Detections in both CO(1-0) and CO(3-2) and a low S/N detection in
CO(2-1). MUSE [O III] observations show a two component fit with a narrow and
broad component.

##  Appendix C Analysis of the CO extent

In Figure 28 we present the contours of the ACA CO(1-0) data, plotted over
optical images of the galaxies and the aperture in which the spectra were
taken. Highlighting that the 30 arcsec aperture is a useful one to make sure
we are obtaining accurate total flux values.

Figure 28: Contour plots for the 9 targets with an integrated S/N $>$ 3
detection in the ACA CO(1-0) data, where red contours show the 2 $\sigma$
levels and upwards. The yellow circle denotes the beam size of the CO(2-1)
observations, which is also the aperture used to extract the CO(1-0) spectra.
The background images are rgb images from the DESI Legacy Imaging Survey in
the (z, r, g) bands.


# Simulating Chiral Ppin Liquids with Projected Entangled-Pair States

Juraj Hasik<EMAIL_ADDRESS>Laboratoire de Physique Théorique, C.N.R.S. and
Université de Toulouse, 31062 Toulouse, France  Institute for Theoretical
Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The
Netherlands  Maarten Van Damme  Department of Physics and Astronomy,
University of Ghent, Krijgslaan 281, 9000 Gent, Belgium  Didier Poilblanc
Laboratoire de Physique Théorique, C.N.R.S. and Université de Toulouse, 31062
Toulouse, France  Laurens Vanderstraeten<EMAIL_ADDRESS>Department of Physics and Astronomy, University of Ghent, Krijgslaan 281, 9000
Gent, Belgium

###### Abstract

Doubts have been raised on the representation of chiral spin liquids
exhibiting topological order in terms of projected entangled pair states
(PEPSs). Here, starting from a simple spin-1/2 chiral frustrated Heisenberg
model, we show that a faithful representation of the chiral spin liquid phase
is in fact possible in terms of a generic PEPS upon variational optimization.
We find a perfectly chiral gapless edge mode and a rapid decay of correlation
functions at short distances consistent with a bulk gap, concomitant with a
gossamer long-range tail originating from a PEPS bulk-edge correspondence. For
increasing bond dimension, (i) the rapid decrease of spurious features – SU(2)
symmetry breaking and long-range tails in correlations – together with (ii) a
faster convergence of the ground state energy as compared to state-of-the-art
cylinder matrix-product state simulations involving far more variational
parameters, prove the fundamental relevance of the PEPS ansatz for simulating
systems with chiral topological order.

_Introduction—_ A chiral spin liquid (CSL) is a strongly correlated two-
dimensional quantum spin system that serves as the lattice analog of a
fractional quantum Hall system [1, 2, 3]. It hosts a disordered ground state,
and inherits many of the characteristic features of fractional quantum Hall
physics, such as fractionalized excitations and protected chiral edge modes.
The CSL was originally proposed on the level of a trial wave function, and
later parent Hamiltonians were found [4, 5, 6] that stabilize a CSL ground
state. Although these are not expected to describe real materials, in recent
years many simpler model Hamiltonians were found that seem to realize CSL
physics [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. In these works, the use
of performant numerical techniques such as variational matrix-product state
(MPS) methods on cylindrical geometries, exact diagonalization on finite
clusters or variational Monte Carlo approaches was crucial for finding
evidence of CSL features. Accurate numerical simulations are instrumental in
relating theoretical descriptions of CSL physics to real materials that can be
probed in experiments; for example, a few materials were found that realize
the triangular-lattice Hubbard model [19, 20, 21, 22, 23, 24], which possibly
exhibits an emergent CSL phase between the metallic and the ordered insulating
phases [25, 26, 27].

In recent years, projected entangled-pair states (PEPSs) [28] have
demonstrated a remarkable ability to simulate the ground states of strongly
correlated lattice models in two dimensions [29, 30, 31, 32, 33]. This class
of tensor-network states can be formulated [34] and variationally optimized
[35, 36] directly in the thermodynamic limit. The control parameters are the
bond dimension $D$, controlling the amount of entanglement in the PEPS wave
function, and the environment bond dimension $\chi$, which is introduced when
contracting the PEPS approximately with boundary MPS [28, 37, 38] or the
corner-transfer matrix renormalization group (CTMRG) [39, 40, 41, 42, 43, 44].
When simulating a critical model, a variational PEPS wave function typically
exhibits a finite correlation length, which provides a length scale that can
be used to extrapolate variational PEPS results [45, 46, 47, 48]. In the case
of nonchiral spin liquids, this procedure has proven very powerful [32].

The case of _chiral_ systems is less straightforward: Whereas nonchiral
topological order can be described by PEPSs in a natural way [49, 50, 51] –
for example, the resonating valence bond state on the kagome lattice is a
simple example of gapped $\mathbb{Z}_{2}$ spin liquid and is written as a PEPS
with bond dimension $D=3$ [52] – capturing a gapped chiral topological wave
function seems to be much harder. The Gaussian fermionic PEPS wave functions
that were proposed exhibit algebraic decay of correlations [53, 54, 55], a
feature that was _proven_ necessary for representing chiral free-fermionic
states as PEPSs [56], see also [57]. By Gutzwiller projecting two of these
Gaussian PEPSs, a CSL wave function can be constructed, exhibiting an
entanglement spectrum with the degeneracy pattern of a chiral conformal field
theory. Subsequently, a class of SU(2) invariant PEPSs was proposed that
exhibits chiral entanglement spectra [58, 59, 60]. Motivated by the properties
of these trial wavefunctions, microscopic CSL Hamiltonians were successfully
studied with PEPS, where both SU(2) and spatial symmetries were imposed on the
PEPS tensors [61, 62]; recently, this approach was even extended to study
SU($N$) CSLs [63, 33]. In all these cases, however, the PEPS exhibits a decay
of correlations slower than exponential, i.e. an _infinite_ correlation
length, which indicates an obstruction for simulating gapped chiral
topological phases with PEPSs, in line with the no-go theorem for the free-
fermionic case. This, in turn, makes these types of simulations very
challenging numerically, all of which has led to the view that PEPSs are ill-
suited for simulating chiral models.

In this paper, we want to disprove this view by performing an unconstrained
variational PEPS simulation of an SU(2) CSL model on the square lattice.
Comparing our results with state-of-the-art MPS simulations on the cylinder,
we find that the PEPS simulation provides very accurate results on the bulk
properties _and_ the topological edge properties of the CSL.

_Model and methods—_ We investigate the model Hamiltonian,

$H=J_{1}\sum_{\braket{ij}}\vec{S}_{i}\cdot\vec{S}_{j}+J_{2}\sum_{\braket{\braket{ij}}}\vec{S}_{i}\cdot\vec{S}_{j}\\\
+i\lambda\sum_{\braket{ijkl}_{p}}\left(P_{ijkl}-P_{ijkl}^{-1}\right),$ (1)

on an infinite square lattice, where $\vec{S}_{i}$ are the spin-1/2 operators
at site $i$, the first two sums are over nearest and next-nearest neighbors
and the last sum is over all four-site plaquettes with $P$ the cyclic
permutation operator. This model was constructed by taking a long-ranged
parent Hamiltonian [64] for the Kalmeyer-Laughlin (KL) state, and truncating
the interaction terms [6]; it was shown numerically that the ground state
stays in the same KL phase for a large choice of parameters. Note that the
last term explicitly breaks the time-reversal and reflection symmetries
separately, such that a CSL ground state is stabilized. Here, we take the
parameters $J_{1}=2\cos(0.06\pi)\cos(0.14\pi)$,
$J_{2}=2\cos(0.06\pi)\sin(0.14\pi)$ and $\lambda=2\sin(0.06\pi)$,
approximately identified as the point for which the ground state has maximal
overlap with the KL state [6]. This choice of parameters was also considered
in Ref. 61, where a variational family of SU(2)-invariant PEPSs was used and
the chiral nature of the ground state was confirmed.

A variational PEPS wave function can be represented as

$\ket{\Psi(A)}=\;\vbox{\hbox{\includegraphics[scale={0.32},page=1]{./diagrams.pdf}}}\;,$
(2)

parametrized by a single tensor $A$ directly in the thermodynamic limit; the
four virtual legs have bond dimension $D$ and are contracted, whereas the
physical legs correspond to the spins in the lattice. We impose rotation and
reflection-conjugation symmetry on the tensor

$\;\vbox{\hbox{\includegraphics[scale={0.32},page=2]{./diagrams.pdf}}}\;=\;\vbox{\hbox{\includegraphics[scale={0.32},page=3]{./diagrams.pdf}}}\;,\quad\;\vbox{\hbox{\includegraphics[scale={0.32},page=4]{./diagrams.pdf}}}\;=\;\vbox{\hbox{\includegraphics[scale={0.32},page=3]{./diagrams.pdf}}}\;.$
(3)

The tensor $A$ can be optimized by a variational energy minimization; here, we
use gradient optimization [36], where the energy is evaluated by CTMRG
contraction and the energy gradient by automatic differentiation [65, 66]. It
is the use of automatic differentiation that makes the unconstrained
optimization possible, since its reverse mode evaluates gradient at only
slightly higher cost, a multiplicative factor of $\mathcal{O}(1)$, than the
energy itself irrespective of the number of variational parameters. The CTMRG
contraction induces a second control parameter, the environment bond dimension
$\chi$. In our further calculations with the optimized PEPS, we use the CTMRG
and variational uniform MPS (VUMPS) algorithms interchangeably for finding
approximate environments, as these were recently shown to be equivalent [38].

Additionally, we simulate this model on infinite cylinders with circumference
$L_{y}$ using infinite MPS that are wound around the cylinder in a snakelike
fashion. We have exploited SU(2) symmetry in the MPS simulation: because
continuous symmetries necessarily remain unbroken on a finite-$L_{y}$
cylinder, imposing this symmetry on the MPS is not a restriction. We use the
VUMPS algorithm to find an optimal MPS approximation for the ground state [67,
68]. In our simulations, we go up to circumference $L_{y}=10$; with a
truncation error of $\epsilon=10^{-5}$ we find that all reported observables
have converged with a total bond dimension around $D_{\mathrm{MPS}}=15000$.

Figure 1: Energetics. Main: The ground state energy per site as a function of
bond dimension $D$ for PEPSs (blue circles) and SU(2)-symmetric PEPS of Ref.
[61] (yellow crosses), or $L_{y}/2$ for cylinder-MPS (red circles). Inset: the
variational PEPS energy compared to the value $e=-1$. Figure 2: Long-range
properties of PEPS CSL. Left: scaling of the real space spin-spin correlation
function $\langle\vec{S}_{r0}\cdot\vec{S}_{00}\rangle$ with bond dimension $D$
at largest considered environment dimension $\chi$. Right: scaling of
connected part of the real space spin-spin correlation functions
$\langle\vec{S}_{r0}\cdot\vec{S}_{00}\rangle-m^{2}$ for $D=5$ PEPS with
$\chi$. For $D=5$ PEPS $m^{2}=1\times 10^{-7}$. Figure 3: Static structure
factor. The momentum-resolved correlation function [Eq. (4)] for the $D=5$
PEPS for a path through the Brillouin zone.

_Bulk properties—_ Let us first look at the variational energies that are
obtained with both methods. In Fig. 1 we plot the ground state energies. For
$D=4$, we find a significantly better value of the energy than the
SU(2)-invariant PEPS with the same bond dimension [61]. As the inset shows,
the variational energy converges exponentially with $D$, which is in stark
contrast with PEPS approximations for critical models, where a power-law
extrapolation with the correlation length is typically needed [46, 45]. We
have also plotted the energy density as obtained from the (converged)
cylinder-MPS simulations, where the finite circumference leads to an
undershooting of the energy. From this comparison, it is clear that PEPSs give
more accurate values for the ground state energy than the cylinder-MPS
approach – certainly given the fact that the largest MPS in our simulations
contains around 70 million variational parameters [incorporating the SU(2)
symmetry], compared to only around 2600 for the $D=6$ PEPS.

Surprisingly, we find that the energy converges to the value of $e=-1$. Given
that the parameters in the model were not fine-tuned or chosen in a natural
way, this feature is really peculiar and we did not find any satisfying
explanation for this observation.

Let us then look at the magnetization given by
$m^{2}=|\langle\vec{S}\rangle|^{2}$, which is equivalent to the spin-spin
correlation function in the long-distance limit
$m^{2}=\lim_{r\to\infty}\langle\vec{S}_{r0}\cdot\vec{S}_{00}\rangle$. In PEPS
simulations of models at (or close to) criticality, order parameters are
typically overestimated due to finite-$D$ effects, and a proper scaling of the
order parameter is needed to get accurate results [46, 45, 32]. Here, however,
we find surprisingly small values of the magnetization already for small $D$:
for $D=3$ we find a magnetization of $m^{2}=4\times 10^{-6}$, which further
decreases down to $m^{2}=6\times 10^{-9}$ at $D=6$. This shows that the SU(2)
symmetry of the CSL is realized in our PEPS description with extremely high
accuracy.

Next, in Fig. 2 we have plotted the structure of the correlations in the PEPS.
The spin-spin correlation functions along a horizontal row in the lattice
(left panel) show an extended regime of exponential decay with a short
correlation length $\xi_{\mathrm{bulk}}=0.49$, consistent with a bulk gap.
With increasing bond dimension $D$ this regime extends in range and the long-
range order $m^{2}$ sharply decreases. However, for all $D$ considered, after
the initial bulk-gapped regime the tails of the correlation functions decay in
a very slow fashion. The data for the connected spin-spin correlations of the
$D=5$ PEPS (right panel) show these tails attaining large correlation lengths:
for the largest $\chi$ considered $\xi_{\mathrm{tail}}(\chi=400)=103$ and an
estimate based on scaling of the finite-$\chi$ data suggests
$\xi_{\mathrm{tail}}(\chi\to\infty)\approx 450$ at least. In a true CSL there
are no such features. Here, as we argue below, these tails are an artefact
induced by the PEPS representation of a CSL. Nevertheless, their magnitude,
$O(10^{-7})$ as indicated by the intercept with the $r=0$ axis, makes their
contribution to spin-spin correlations minuscule. For comparison, we perform
similar analysis on both SU(2)-invariant and unconstrained $D=3$ PEPS in the
Appendix.

Finally, in Fig. 3 we examine the momentum-resolved structure factor

$s(q_{x},q_{y})=\sum_{mn}\mathrm{e}^{i(q_{x}m+q_{y}n)}\braket{\vec{S}_{mn}\cdot\vec{S}_{00}},$
(4)

which we evaluate using the methods from Ref. 38. We plot this quantity
through a path of the Brillouin zone for different PEPS bond dimensions,
showing a smooth behavior at the $X$ point and a vanishing value at the
$\Gamma$ point – exactly what one would expect for a gapped spin liquid (long-
range order would manifest itself as a divergence at the $X$ point). In
addition, the curve has nicely converged as a function of PEPS bond dimension
$D$. Note that this is the first instance where a spin liquid for a spin-$1/2$
system on a square lattice was found numerically by single-site PEPS, without
imposing the SU(2) symmetry explicitly.

_Entanglement spectrum—_ As one of the defining characteristics of a chiral
topological phase, we now investigate the entanglement spectrum of our
variational PEPS ground states. Following the PEPS bulk-boundary
correspondence [69], the entanglement Hamiltonian $H_{\mathrm{ent}}$ can be
inferred directly from the fixed point $\rho$ of the PEPS transfer matrix as
$\rho=\mathrm{e}^{-H_{\mathrm{ent}}}$. The fixed-point equation for PEPS
transfer matrix, defined by a column (or row) of double-layer tensors $a$ with
bond dimension $D^{2}$, reads

$\;\vbox{\hbox{\includegraphics[scale={0.15},page=1]{./tmdiags.pdf}}}\;=\;\vbox{\hbox{\includegraphics[scale={0.15},page=2]{./tmdiags.pdf}}}\;,\quad\;\vbox{\hbox{\includegraphics[scale={0.15},page=3]{./tmdiags.pdf}}}\;\propto\;\vbox{\hbox{\includegraphics[scale={0.15},page=4]{./tmdiags.pdf}}}\;,$
(5)

where the tensor $T$ defines the boundary MPS approximation with bond
dimension $\chi$ of the leading eigenvector of the PEPS transfer matrix.
Numerically, we take the boundary MPS, fold it open and interpret it as a
matrix-product operator (MPO) approximation for $\rho$. We have found above
that $\rho$ has infinite correlation length in the infinite-$\chi$ limit,
suggesting that $H_{\mathrm{ent}}$ has long-range interaction terms. Indeed, a
thermal density operator for a local or exponentially decaying Hamiltonian
could be efficiently represented as a MPO with finite bond dimension [70].

First, we compute the low-lying spectrum of $H_{\mathrm{ent}}$ for the
infinite system: we first find the leading eigenvector of $\rho$ as an
infinite MPS with the VUMPS algorithm, and then apply a variational excitation
ansatz with fixed momentum to capture the excited states of $\rho$ [71, 68]. A
third control parameter enters here, the bond dimension $\chi^{\prime}$ of
this infinite MPS. In Fig. 4 we plot the results of this procedure for the
$D=5$ PEPS. The lower-left panel shows the typical sawtooth form of a chiral
spectrum in the full Brillouin zone, but the enlarged spectrum in the top
panel around the origin shows that we also find a very steep mode. As the
bottom-right panel shows, this additional mode gets steeper as the MPO’s bond
dimension $\chi$ is increased. This finding agrees with the fact that at
finite $\chi$ the MPO has finite correlation length, and that
$H_{\mathrm{ent}}$ is not truly long-range; therefore, the spectrum cannot be
perfectly chiral [72] at finite $\chi$. Only in the infinite-$\chi$ limit can
we expect to find a perfectly chiral entanglement spectrum.

Second, we compute the entanglement spectrum on a finite system of $N$ sites
with periodic boundary conditions by placing the MPO approximation for $\rho$
on a finite ring. Since computing the spectrum exactly scales as
$\mathcal{O}(D^{N})$, it becomes quickly unfeasible for large $D$ or $N$. As
an alternative, we use open-boundary MPS methods for finding the low-lying
spectrum of MPOs on a ring [73]; here, again, the bond dimension
$\chi^{\prime}$ enters as a control parameter. In Fig. 5 we show the
entanglement spectrum of the $D=5$ PEPS confined on rings of different sizes,
clearly showing the chiral branch with the degeneracies that are expected for
the $\mathrm{SU}(2)_{1}$ Wess-Zumino-Witten conformal field theory. Moreover,
we find the degeneracies of the SU(2) multiplets, again indicating that the
PEPS is an SU(2)-invariant state up to high precision. In the left panel, we
compare the spectra with the entanglement spectra as obtained from the MPS
ground states on the cylinder; here, we use the usual prescription [74] for
assigning momentum quantum numbers to the MPS entanglement spectrum. The two
figures agree nicely, but the PEPS methodology allows us to reach much larger
system sizes with moderate computational cost.

Figure 4: Entanglement spectrum in thermodynamic limit. We show results for
the entanglement spectrum of the $D=5$ PEPS, using the quasiparticle ansatz.
Top: a close-up for momenta around the origin, evaluated with $\chi=64$ and
$\chi^{\prime}=128$, nicely showing the two branches. Left: the dispersion in
the full Brillouin zone. Right: an even narrower close-up around the origin
for three different values of $\chi$ and $\chi^{\prime}=128$.

Figure 5: Entanglement spectrum for finite system. Left: cylinder-MPS results
for three different $L_{y}$. Right: spectrum of the MPO representation for the
entanglement Hamiltonian $\rho$ confined on a ring of two different sizes $N$,
as computed with the method in Ref. 73 with $\chi=32$ and $\chi^{\prime}=80$.
Here we show the four first values of the momentum. We do not show the
degeneracies of the SU(2) multiplets (for the cylinder-MPS these are exact;
for the PEPS we find degeneracies up to the third or fourth digit).

_Conclusions—_ This work shows that CSLs exhibiting topological order can be
represented faithfully in terms of generic bosonic PEPSs upon variational
optimization. The suspicion of an obstruction [56] stems directly from the
PEPS bulk-edge correspondence: the existence of a perfectly gapless chiral
edge mode implies that the (one-dimensional) entanglement Hamiltonian [69, 52]
is long-range which, in turn, implies an infinite correlation length in the
bulk. Nevertheless, our work shows that this feature does not prohibit us from
variationally optimizing these PEPSs efficiently, and that the divergence of
the correlation length manifests in a form of an artefact: an irrelevant
vanishing long-range tail in bulk correlation functions. The rapid exponential
convergence of the energy (much faster than the convergence of cylinder-MPS
with circumference) and the vanishing of the magnetization are consistent with
the expected bulk gap. Similarly, the expected topological properties in form
of the chiral edge spectrum are obtained with high accuracy. The precise
mechanism that gives rise to long-range tail artefact in PEPS description of
CSL and its systematic dependence on bond dimension $D$ needs further
investigation.

Although the case of an SU($2$) Abelian CSL is considered here, we believe
that chiral PEPSs can more broadly faithfully represent Abelian and non-
Abelian SU($N$) CSL, $N\geq 2$, as suggested by calculations involving spin-
symmetric PEPS [62, 33]. We can also tackle cases of bosonic fractional Chern
insulators such as the Harper-Hofstadter model [75], which can be realized in
atomic simulators [76, 77]. Finally, our results indicate that fermionic PEPSs
[78, 79] would be extremely well suited to describe electronic systems with
chiral topological order like interacting Chern insulators at fractional
filling [80, 81] or chiral spin liquids close to a metal-insulator transition
[25, 26, 27].

Finally, going beyond ground-state properties, our work opens up the prospect
of simulating bulk and edge excitation spectra [82], real-time dynamics, or
finite-temperature properties [83] of chiral topological phases with PEPSs.

_Acknowledgments—_ We acknowledge inspiring discussions with Norbert Schuch
and Frank Verstraete. We thank the Centro de Ciencias de Benasque Pedro
Pascual, where this work was initiated. This work was supported by the
Research Foundation Flanders, and the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement no. 101001604). D. P. acknowledges support by the TNTOP
ANR-18-CE30-0026-01 grant awarded by the French Research Council. This work
was granted access to the HPC resources of CALMIP center under the allocation
2017-P1231.

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## Appendix: Long-range data for D=3 SU(2)-invariant PEPS

Figure 6: Comparison of D=3 SU(2)-symmetric (left) and unconstrained PEPS
(right). Top: Connected part of the real space spin-spin correlation
functions. Unconstrained D=3 PEPS retain finite $m^{2}=4\times 10^{-6}$.
Center: Scaling of the correlation length by subleading gap in the transfer
matrix $E$. For SU(2)-symmetric PEPS subleading gap is
$\delta_{17}=-log|\lambda_{7}/\lambda_{1}|$. For unconstrained PEPS it is
$\delta_{14}=-log|\lambda_{4}/\lambda_{1}|$ instead. Bottom: Leading part of
the transfer matrix spectrum as a function of the environment dimension
$\chi$.

The structure of spin-spin correlations
$\langle\vec{S}_{r0}\cdot\vec{S}_{00}\rangle$ of PEPS, represented by 2D
tensor network with spin operators $\vec{S}$ inserted into two double-layer
tensors at distance $r$, can be analyzed by reduction to 1D tensor network. We
make use of fixed-point of PEPS transfer matrix of Eq. 5 to obtain a network
analogous to 3-leg ladder

$\vbox{\hbox{\includegraphics[scale={0.16}]{./ss-
full.pdf}}}\approx\vbox{\hbox{\includegraphics[scale={0.16}]{./ss-
bmps.pdf}}}.$ (6)

The central $r$ rungs of this ladder define $\chi D^{2}\times\chi D^{2}$
transfer matrix $E$ raised to $r$-th power. The spectrum $\lambda_{i}$ of
transfer matrix $E$ resolved as $E=\sum_{i}\lambda_{i}|r_{i}\rangle\langle
l_{i}|$ then determines the decay of the spin-spin correlations, while the
leading eigenvectors $|r_{0}\rangle,\langle l_{0}|$ define right and left
boundary of this 3-leg ladder network.

In this appendix we present the scaling analysis of data for D=3
SU(2)-invariant PEPS ansatz, employed in [61] and compare it with the result
for unconstrained D=3 PEPS. The real-space spin-spin correlations $\langle
S_{r0}\cdot\vec{S}_{00}\rangle$, scaling of the correlation length $\xi$, and
the leading part of the spectrum of transfer matrix $E$ are presented in Fig.
6. The real-space spin-spin correlations show two distinct regimes analogous
to the unrestricted PEPS ansatz: Up to a range of roughly 10 sites a fast
exponential decay, with $\xi_{\mathrm{bulk}}=0.54$ for D=3 SU(2) ansatz, then
crossing over into regime of extremely slow decay with strong finite-$\chi$
dependence. Unlike unconstrained PEPS with finite $\xi\approx 23$, the tail of
the SU(2)-symmetric ansatz shows genuine algebraic decay in the
$\chi\to\infty$ limit. This feature is revealed by the analysis of the
spectrum of the transfer matrix $E$ and decay of the tail of spin-spin
correlations. In particular, we scale finite-$\chi$ data in two different
ways: First, by extracting the spin-spin correlation length $\xi_{SS}$ from
the tails of spin-spin correlations and performing a power-law fit with
respect to the environment dimension $\chi$. Second, by scaling the
correlation length $1/\xi_{TM}$ given by the leading gap in the transfer
matrix $E$ against the subleading gap and again performing power-law fit as
suggested by Ref. [84]. Note that for SU(2)-symmetric PEPS the data suggest
$\lambda_{7}$ to define subleading gap for larger $D^{2}/\chi$ and we thus use
it for the purpose of the scaling.