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arXiv:2512.14149 · SciPost Physics

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

Yuki Miyazaki, Shiori Tanigawa, Giacomo Marmorini, Nobuo Furukawa, Daisuke Yamamoto

cond-mat.str-elquant-phSubmitted Dec 16, 2025• Announced Dec 17, 2025• 0 citations
v1 · Dec 16, 2025v2 · Dec 18, 2025v3 · May 25, 2026
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Abstract

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 $J_1$-$J_2$ Ising couplings with SSD using Rydberg atom arrays in optical tweezers, which can be achieved within a very good approximation. 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.

Journal reference
SciPost Phys. 20, 174 (2026)
DOI
10.21468/SciPostPhys.20.6.174
Comments
23 pages, 11 figures

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•Mar 1, 2020•0 citations
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 J1J_1J1​-J2J_2J2​ Ising couplings with SSD using Rydberg atom arrays in optical tweezers, which can be achieved within a very good approximation. 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.
We study solvable spin chains, one-dimensional massless Dirac fermions and conformal field theories (CFTs) with sine-square deformation (SSD), in which the Hamiltonian density is modulated by the function f ( x ) = sin 2 (π x /ℓ), where x is the position and ℓ is the length of the system. For the XY chain and the transverse field Ising chain at criticality, it is shown that the ground state of an open system with SSD is identical to that of a uniform chain with periodic boundary conditions. The same holds for the massless Dirac fermions with SSD, corresponding to the continuum limit of the gapless XY chain. For general CFTs, we find that the Hamiltonian of a system with SSD has an expression in terms of the generators of the Virasoro algebra. This allows us to show that the vacuum state is an exact eigenstate of the sine-square deformed Hamiltonian. Furthermore, for a restricted class of CFTs associated with affine Lie (Kac–Moody) algebras, including c = 1 Gaussian CFT, we prove that the vacuum is an exact ground state of the deformed Hamiltonian. This explains why the SSD has succeeded in suppressing boundary effects in one-dimensional critical systems, as observed in previous numerical studies.
We study free fermion systems with the sine-square deformation (SSD) in which the energy scale of local Hamiltonians is modified according to the scaling function $f(x)={\mathrm{sin}}^{2}[\frac{\ensuremath{\pi}}{L}(x\ensuremath{-}\frac{1}{2})]$, where xxx is the position of the local Hamiltonian and LLL is the length of the system in the xxx direction. It has been revealed that when applied to one-dimensional (1D) critical systems, the SSD realizes the translationally-invariant ground state which is the same as that of the uniform periodic system. In this paper, we propose a simple theory to explain how the SSD maintains the translational invariance in the ground-state wave function. In particular, for a certain 1D system with SSD, it is shown that the ground state is exactly identical to the Fermi sea of the uniform periodic chain. We also apply the SSD to 2D systems and show that the SSD is able to suppress the boundary modulations from the open edges extremely well, demonstrating that the SSD works in any dimension and in any direction.
The scaling of the transition temperature into an ordered phase close to a quantum critical point as well as the order parameter fluctuations inside the quantum critical region provide valuable information about universal properties of the underlying quantum critical point. Here, we employ quantum Monte Carlo simulations to examine these relations in detail for two-dimensional quantum systems that exhibit a finite-temperature Ising-transition line in the vicinity of a quantum critical point that belongs to the universality class of either (i) the three-dimensional Ising model for the case of the quantum Ising model in a transverse magnetic field on the square lattice or (ii) the chiral Ising transition for the case of a half-filled system of spinless fermions on the honeycomb lattice with nearest-neighbor repulsion. While the first case allows large-scale simulations to assess the scaling predictions to a high precision in terms of the known values for the critical exponents at the quantum critical point, for the later case, we extract values of the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$, related to the order parameter fluctuations, which we discuss in relation to other recent estimates from ground-state quantum Monte Carlo calculations as well as analytical approaches.
We develop a simple and unbiased numerical method to obtain the uniform susceptibility of quantum many-body systems. When a Hamiltonian is spatially deformed by multiplying it with a sine-square function that smoothly decreases from the system center toward the edges, the size-scaling law of the excitation energy is drastically transformed to a rapidly converging one. Then, the local magnetization at the system center becomes nearly size independent; the one obtained for the deformed Hamiltonian of a system length as small as $L\ensuremath{\sim}10$ provides the value obtained for the original uniform Hamiltonian of $L\ensuremath{\sim}100$. This allows us to evaluate a bulk magnetic susceptibility by using the magnetization at the center by existing numerical solvers without any approximation, parameter tuning, or size-scaling analysis. We demonstrate that the susceptibilities of the spin-1/2 antiferromagnetic Heisenberg chain and square lattice obtained by our scheme at $L\ensuremath{\sim}10$ agree within ${10}^{\ensuremath{-}3}$ with exact analytical and numerical solutions for $L=\ensuremath{\infty}$ down to temperatures of 0.1 times the coupling constant. We apply this method to the spin-1/2 kagome lattice Heisenberg antiferromagnet which is of prime interest in the search for spin liquids.
The critical properties of the one-dimensional spin-1/2 transverse Ising model in the presence of a longitudinal magnetic field were studied by the quantum fidelity method. We used exact diagonalization to obtain the ground-state energies and corresponding eigenvectors for lattice sizes up to 24 spins. The maximum of the fidelity susceptibility was used to locate the various phase boundaries present in the system. The type of dominant spin ordering for each phase was identified by examining the corresponding ground-state eigenvector. For a given antiferromagnetic nearest-neighbor interaction J_{2}, we calculated the fidelity susceptibility as a function of the transverse field B_{x} and the longitudinal field B_{z}. The phase diagram in the (B_{x},B_{z})-plane shows three phases. These findings are in contrast with the published literature that claims that the system has only two phases. For B_{x}<1, we observed an antiferromagnetic phase for small values of B_{z} and a paramagnetic phase for large values of B_{z}. For B_{x}>1 and low B_{z}, we found a disordered phase that undergoes a second-order phase transition to a paramagnetic phase for large values of B_{z}.
Ising model has been successful in describing ferromagnetism and its phase transition to paramagnet. At the critical point, the free energy density function and its derivatives diverge. Their behaviour near the critical point are described by power-laws with associated critical exponents. In many cases, the critical exponents can be determined from analytic solutions via conventional renormalization groups methods or from Monte Carlo simulations. However, for quantum many-body systems, very few are tractable to analytical solutions. The quantum many-body wavefunction belongs to large dimensional Hilbert space that increases exponentially with system size. If the Hamiltonian is gapped and only local interaction is considered, then the wavefunction can be efficiently truncated. Tensor Network formalism provides a scheme to truncate the less important degrees of freedom via Singular Value Decomposition (SVD) of the density matrix. In this study, we investigated the thermodynamic properties and phase transition of one-dimensional transverse-field quantum Ising model (1D-tQIM) under the finite-size effect and random coupling strength. Starting with Matrix Product States (MPS) as a wavefunction ansatz, the Density Matrix Renormalization Group algorithm is applied to the MPS. The variational algorithm, which iteratively performs SVDs and truncation at each bond, approximates the ground state MPS wavefunction. All quantum observables are calculated from the contraction of the resultant ground state MPS. Although theoretically, divergence at critical points only happen in an infinite system, one can obtain the critical exponents through simulation of finite-size 1D-tQIM. Using the analytic solution as a benchmark, we compared the finite-size effects of the system using finite-size scaling analysis and MPS methods. The critical exponents of 1D-tQIM are independently calculated and compared with the analytical results. Thermodynamic quantities such as magnetization, susceptibility and correlation function are calculated for system sizes of 20, 40, 60, 80, 100 and 120 spins. We determined the respective critical exponents: