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Measurement-induced phase transition for free fermions above one dimension

Published 21 Sep 2023 in quant-ph and cond-mat.dis-nn | (2309.12405v3)

Abstract: A theory of the measurement-induced entanglement phase transition for free-fermion models in $d>1$ dimensions is developed. The critical point separates a gapless phase with $\ell{d-1} \ln \ell$ scaling of the second cumulant of the particle number and of the entanglement entropy and an area-law phase with $\ell{d-1}$ scaling, where $\ell$ is a size of the subsystem. The problem is mapped onto an SU($R$) replica non-linear sigma model in $d+1$ dimensions, with $R\to 1$. Using renormalization-group analysis, we calculate critical indices in one-loop approximation justified for $d = 1+ \epsilon$ with $\epsilon \ll 1$. Further, we carry out a numerical study of the transition for a $d=2$ model on a square lattice, determine numerically the critical point, and estimate the critical index of the correlation length, $\nu \approx 1.4$.

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Rev. Lett. 42, 673 (1979). Kramer and MacKinnon [1993] B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . F. Wegner, Four-loop-order β𝛽\betaitalic_β-function of nonlinear σ𝜎\sigmaitalic_σ-models in symmetric spaces, Nuclear Physics B 316, 663 (1989). Abrahams et al. [1979] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979). Kramer and MacKinnon [1993] B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. 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Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] .
  37. I. Burmistrov, K. Tikhonov, I. Gornyi, and A. Mirlin, Entanglement entropy and particle number cumulants of disordered fermions, Annals of Physics (NY) 383, 140 (2017). Evers and Mirlin [2008] F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys.  80, 1355 (2008). Hikami [1981] S. Hikami, Three-loop β𝛽\betaitalic_β-functions of non-linear σ𝜎\sigmaitalic_σ models on symmetric spaces, Physics Letters B 98, 208 (1981). Wegner [1989] F. Wegner, Four-loop-order β𝛽\betaitalic_β-function of nonlinear σ𝜎\sigmaitalic_σ-models in symmetric spaces, Nuclear Physics B 316, 663 (1989). Abrahams et al. [1979] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979). Kramer and MacKinnon [1993] B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys.  80, 1355 (2008). Hikami [1981] S. Hikami, Three-loop β𝛽\betaitalic_β-functions of non-linear σ𝜎\sigmaitalic_σ models on symmetric spaces, Physics Letters B 98, 208 (1981). Wegner [1989] F. Wegner, Four-loop-order β𝛽\betaitalic_β-function of nonlinear σ𝜎\sigmaitalic_σ-models in symmetric spaces, Nuclear Physics B 316, 663 (1989). Abrahams et al. [1979] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979). Kramer and MacKinnon [1993] B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . S. Hikami, Three-loop β𝛽\betaitalic_β-functions of non-linear σ𝜎\sigmaitalic_σ models on symmetric spaces, Physics Letters B 98, 208 (1981). Wegner [1989] F. Wegner, Four-loop-order β𝛽\betaitalic_β-function of nonlinear σ𝜎\sigmaitalic_σ-models in symmetric spaces, Nuclear Physics B 316, 663 (1989). Abrahams et al. [1979] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979). Kramer and MacKinnon [1993] B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . F. Wegner, Four-loop-order β𝛽\betaitalic_β-function of nonlinear σ𝜎\sigmaitalic_σ-models in symmetric spaces, Nuclear Physics B 316, 663 (1989). Abrahams et al. [1979] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979). Kramer and MacKinnon [1993] B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979). Kramer and MacKinnon [1993] B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] .
  38. F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys.  80, 1355 (2008). Hikami [1981] S. Hikami, Three-loop β𝛽\betaitalic_β-functions of non-linear σ𝜎\sigmaitalic_σ models on symmetric spaces, Physics Letters B 98, 208 (1981). Wegner [1989] F. Wegner, Four-loop-order β𝛽\betaitalic_β-function of nonlinear σ𝜎\sigmaitalic_σ-models in symmetric spaces, Nuclear Physics B 316, 663 (1989). Abrahams et al. [1979] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979). Kramer and MacKinnon [1993] B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . S. Hikami, Three-loop β𝛽\betaitalic_β-functions of non-linear σ𝜎\sigmaitalic_σ models on symmetric spaces, Physics Letters B 98, 208 (1981). Wegner [1989] F. Wegner, Four-loop-order β𝛽\betaitalic_β-function of nonlinear σ𝜎\sigmaitalic_σ-models in symmetric spaces, Nuclear Physics B 316, 663 (1989). Abrahams et al. [1979] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979). Kramer and MacKinnon [1993] B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . F. Wegner, Four-loop-order β𝛽\betaitalic_β-function of nonlinear σ𝜎\sigmaitalic_σ-models in symmetric spaces, Nuclear Physics B 316, 663 (1989). Abrahams et al. [1979] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979). Kramer and MacKinnon [1993] B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979). Kramer and MacKinnon [1993] B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] .
  39. S. Hikami, Three-loop β𝛽\betaitalic_β-functions of non-linear σ𝜎\sigmaitalic_σ models on symmetric spaces, Physics Letters B 98, 208 (1981). Wegner [1989] F. Wegner, Four-loop-order β𝛽\betaitalic_β-function of nonlinear σ𝜎\sigmaitalic_σ-models in symmetric spaces, Nuclear Physics B 316, 663 (1989). Abrahams et al. [1979] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979). Kramer and MacKinnon [1993] B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . F. Wegner, Four-loop-order β𝛽\betaitalic_β-function of nonlinear σ𝜎\sigmaitalic_σ-models in symmetric spaces, Nuclear Physics B 316, 663 (1989). Abrahams et al. [1979] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979). Kramer and MacKinnon [1993] B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979). Kramer and MacKinnon [1993] B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] .
  40. F. Wegner, Four-loop-order β𝛽\betaitalic_β-function of nonlinear σ𝜎\sigmaitalic_σ-models in symmetric spaces, Nuclear Physics B 316, 663 (1989). Abrahams et al. [1979] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979). Kramer and MacKinnon [1993] B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979). Kramer and MacKinnon [1993] B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] .
  41. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979). Kramer and MacKinnon [1993] B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] .
  42. B. Kramer and A. MacKinnon, Localization: theory and experiment, Reports on Progress in Physics 56, 1469 (1993). Slevin and Ohtsuki [1997] K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] .
  43. K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett. 78, 4083 (1997). Di Francesco et al. [1997] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] .
  44. P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer New York, NY, 1997). Gross [1976] D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] .
  45. D. J. Gross, Applications of the renormalization group to high-energy physics, in Methods in Field Theory (North-Holland Publishing Co., 1976) pp. 141–250. Wen [1992] X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] .
  46. X.-G. Wen, Scaling theory of conserved current and universal amplitudes at anisotropic critical points, Phys. Rev. B 46, 2655 (1992). Chahine and Buchhold [2023] K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] . K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] .
  47. K. Chahine and M. Buchhold, Entanglement phases, localization and multifractality of monitored free fermions in two dimensions (2023), arXiv:2309.12391 [cond-mat.str-el] .
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