A sine-square deformation approach to quantum critical points in one-dimensional systems

We propose a method to determine the quantum phase boundaries of one-dimensional systems using sine-square deformation (SSD). Based on the proposition, supported by several exactly solved cases though not proven in full generality, that ``if a one-dimensional system is gapless, then the expectation value of any local observable in the ground state of the Hamiltonian with SSD exhibits translational symmetry in the thermodynamic limit,’’ we determine the quantum critical point as the location where a local observable becomes site-independent, identified through finite-size scaling analysis. As case studies, we consider two models: the antiferromagnetic Ising chain in mixed transverse and longitudinal magnetic fields with nearest-neighbor and long-range interactions. We calculate the ground state of these Hamiltonians with SSD using the density-matrix renormalization-group algorithm and evaluate the local transverse magnetization. For the nearest-neighbor model, we show that the quantum critical point can be accurately estimated by our procedure with systems of up to 84 sites, or even smaller, in good agreement with results from the literature. For the long-range model, we find that the phase boundary between the antiferromagnetic and paramagnetic phases is slightly shifted relative to the nearest-neighbor case, leading to a reduced region of antiferromagnetic order. Moreover, we propose an experimental procedure to implement the antiferromagnetic J_1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> - J_2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> Ising couplings with SSD using Rydberg atom arrays in optical tweezers, which can be achieved within a very good approximation. We show that our protocol can control the ratio J_2/J_1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mi>/</mml:mi> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> over a wide range, and realize a well-approximated SSD of the nearest-neighbor model. Because multiple independent scaling conditions naturally emerge, our approach enables precise determination of quantum critical points and possibly even the extraction of additional critical phenomena, such as critical exponents, from relatively small system sizes.