How universal is the relation between sign problem and phase transition

The mystery of the infamous sign problem in quantum Monte Carlo simulations mightily restricts applications of the method in fermionic and frustrated systems. A recent work [Science 375 , 417 (2022)] made a remarkable breakthrough in the sign problem and suggested a positive connection between the sign and phase transition. How general this argument is can be crucial in various fields related to many-body systems, such as condensed matter physics, quantum chemistry, and nuclear physics. Based on universal analyses of typical examples and numerical simulations from different approaches, we discuss when and how studying the sign can provide helpful information on phase transitions in general systems independent of specific models and algorithms. While our results support that the notorious sign offers new angles in exploring quantum many-body problems, we also notice that taking advantage of the sign can even be as challenging as neutralizing the sign problem itself in unknown systems.

[1]  Z. Meng,et al.  Thermodynamic Characteristic for a Correlated Flat-Band System with a Quantum Anomalous Hall Ground State. , 2022, Physical review letters.

[2]  R. Scalettar,et al.  Universality and Critical Exponents of the Fermion Sign Problem , 2022, 2207.09026.

[3]  R. Scalettar,et al.  Bilayer Hubbard model: Analysis based on the fermionic sign problem , 2022, Physical Review B.

[4]  Z. Meng,et al.  Sign Problem in Quantum Monte Carlo Simulation , 2022, 2204.08777.

[5]  Xiao Yan Xu,et al.  Fermion sign bounds theory in quantum Monte Carlo simulation , 2021, Physical Review B.

[6]  T. Devereaux,et al.  Thermodynamics of correlated electrons in a magnetic field , 2021, Communications Physics.

[7]  R. Scalettar,et al.  Quantum critical points and the sign problem , 2021, Science.

[8]  R. Scalettar,et al.  Deconvolving the components of the sign problem , 2021, Physical Review B.

[9]  A. Vishwanath,et al.  Fermionic Monte Carlo Study of a Realistic Model of Twisted Bilayer Graphene , 2021, Physical Review X.

[10]  F. Assaad,et al.  Valence-bond solid to antiferromagnet transition in the two-dimensional Su-Schrieffer-Heeger model by Langevin dynamics , 2021, Physical Review B.

[11]  Wen-Yuan Liu,et al.  Gapless quantum spin liquid and global phase diagram of the spin-1/2 J1-J2 square antiferromagnetic Heisenberg model. , 2020, Science bulletin.

[12]  R. Scalettar,et al.  Hamming Distance and the onset of quantum criticality , 2021, Physical Review B.

[13]  Y. Nomura,et al.  Dirac-Type Nodal Spin Liquid Revealed by Refined Quantum Many-Body Solver Using Neural-Network Wave Function, Correlation Ratio, and Level Spectroscopy , 2020, Physical Review X.

[14]  B. Clark,et al.  Mitigating the Sign Problem through Basis Rotations. , 2019, Physical review letters.

[15]  A. Vishwanath,et al.  Nematic topological semimetal and insulator in magic-angle bilayer graphene at charge neutrality , 2019, Physical Review Research.

[16]  N. Regnault,et al.  Twisted bilayer graphene. III. Interacting Hamiltonian and exact symmetries , 2020, Physical Review B.

[17]  P. Sacramento,et al.  Energy bonds as correlators for long-range symmetry-protected topological models and models with long-range topological order , 2020, Physical Review B.

[18]  F. Becca,et al.  Gapless spin liquid and valence-bond solid in the J1 - J2 Heisenberg model on the square lattice: Insights from singlet and triplet excitations , 2020, 2005.12941.

[19]  Z. Ringel,et al.  Intrinsic sign problem in fermionic and bosonic chiral topological matter , 2020, 2005.05566.

[20]  Adam D. Smith,et al.  Intrinsic sign problems in topological quantum field theories , 2020, 2005.05343.

[21]  S. Wessel,et al.  Reduction of the sign problem near T=0 in quantum Monte Carlo simulations , 2020, 2003.13356.

[22]  M. Katsnelson.  Twisted bilayer graphene , 2020, The Physics of Graphene.

[23]  Fu-Chun Zhang,et al.  Correlated insulating phases of twisted bilayer graphene at commensurate filling fractions: A Hartree-Fock study , 2020, 2001.02476.

[24]  Aaram J. Kim,et al.  Alleviating the sign problem in quantum Monte Carlo simulations of spin-orbit-coupled multiorbital Hubbard models , 2019, Physical Review B.

[25]  Milad Marvian,et al.  Hardness and Ease of Curing the Sign Problem for Two-Local Qubit Hamiltonians , 2019, SIAM J. Comput..

[26]  Ingo Roth,et al.  Easing the Monte Carlo sign problem , 2019, Science Advances.

[27]  Roger G. Melko,et al.  Wave-function positivization via automatic differentiation , 2019, 1906.04654.

[28]  Milad Marvian,et al.  On the computational complexity of curing non-stoquastic Hamiltonians , 2019, Nature Communications.

[29]  G. Guo,et al.  Reply to “Comment on ‘Gapless spin liquid ground state of the spin- 12J1−J2 Heisenberg model on square lattices' ” , 2018, Physical Review B.

[30]  A. Sandvik,et al.  Critical Level Crossings and Gapless Spin Liquid in the Square-Lattice Spin-1/2 J_{1}-J_{2} Heisenberg Antiferromagnet. , 2017, Physical review letters.

[31]  Zohar Ringel,et al.  Quantized gravitational responses, the sign problem, and quantum complexity , 2017, Science Advances.

[32]  Stefan Wessel,et al.  Efficient Quantum Monte Carlo simulations of highly frustrated magnets: the frustrated spin-1/2 ladder , 2017, 1704.01973.

[33]  B. Svistunov,et al.  Polynomial complexity despite the fermionic sign , 2017, 1703.10141.

[34]  R. Rossi.  Determinant Diagrammatic Monte Carlo Algorithm in the Thermodynamic Limit. , 2016, Physical review letters.

[35]  R. Scalettar,et al.  Characterizing the three-orbital Hubbard model with determinant quantum Monte Carlo , 2016, Physical Review B.

[36]  Yusuke Nomura,et al.  Negative sign problem in continuous-time quantum Monte Carlo: optimal choice of single-particle basis for impurity problems , 2015, 1508.06741.

[37]  M. B. Hastings,et al.  How quantum are non-negative wavefunctions? , 2015, 1506.08883.

[38]  R. Scalettar,et al.  Geometry dependence of the sign problem in quantum Monte Carlo simulations , 2015, 1501.02832.

[39]  S. C. Pieper,et al.  Quantum Monte Carlo methods for nuclear physics , 2014, 1412.3081.

[40]  Wei Zhu,et al.  Plaquette ordered phase and quantum phase diagram in the spin-1/2 J(1)-J(2) square Heisenberg model. , 2013, Physical review letters.

[41]  A. Parola,et al.  Direct evidence for a gapless $Z_2$ spin liquid by frustrating Néel antiferromagnetism , 2013, 1304.2630.

[42]  R. Mondaini,et al.  Finite-size effects in transport data from quantum Monte Carlo simulations , 2011, 1107.0230.

[43]  L. Balents,et al.  Spin Liquid Ground State of the Spin-1/2 Square $J_1$-$J_2$ Heisenberg Model , 2011, 1112.2241.

[44]  L. Mitas,et al.  Applications of quantum Monte Carlo methods in condensed systems , 2010, 1010.4992.

[45]  R. Bistritzer,et al.  Moiré bands in twisted double-layer graphene , 2010, Proceedings of the National Academy of Sciences.

[46]  A. Sandvik.  Computational Studies of Quantum Spin Systems , 2010, 1101.3281.

[47]  H. Evertz,et al.  World-line and Determinantal Quantum Monte Carlo Methods for Spins, Phonons and Electrons , 2008 .

[48]  R. Needs,et al.  Quantum Monte Carlo simulations of solids , 2001 .

[49]  L. Capriotti,et al.  Spontaneous plaquette dimerization in the J1-J2 heisenberg model , 1999, Physical review letters.

[50]  C. Hamer,et al.  Dimer order with striped correlations in the J 1 -J 2 Heisenberg model , 1999, cond-mat/9904064.

[51]  D. Ceperley.  Path integrals in the theory of condensed helium , 1995 .

[52]  N. Hatano,et al.  Representation basis in quantum Monte Carlo calculations and the negative-sign problem , 1992 .

[53]  White,et al.  Sign problem in the numerical simulation of many-electron systems. , 1990, Physical review. B, Condensed matter.

[54]  Dagotto,et al.  Phase diagram of the frustrated spin-1/2 Heisenberg antiferromagnet in 2 dimensions. , 1989, Physical review letters.

[55]  White,et al.  Numerical study of the two-dimensional Hubbard model. , 1989, Physical review. B, Condensed matter.

[56]  Chandra,et al.  Possible spin-liquid state at large S for the frustrated square Heisenberg lattice. , 1988, Physical review. B, Condensed matter.

[57]  Inui,et al.  Doping dependence of antiferromagnetic correlations in high-temperature superconductors. , 1988, Physical review. B, Condensed matter.

[58]  S. Miyashita,et al.  Monte Carlo Simulation of Quantum Heisenberg Magnets on the Triangular Lattice , 1986 .