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arXiv:1108.2973 · Physical Review B

Sine-square deformation of free fermion systems in one and higher dimensions

Isao Maruyama, Hosho Katsura, Toshiya Hikihara

Published Oct 28, 2011• 55 citations
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Abstract

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 $x$ is the position of the local Hamiltonian and $L$ is the length of the system in the $x$ 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.

DOI
10.1103/physrevb.84.165132

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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)=sin2[\ensuremathπL(x\ensuremath−12)]f(x)={\mathrm{sin}}^{2}[\frac{\ensuremath{\pi}}{L}(x\ensuremath{-}\frac{1}{2})]f(x)=sin2[L\ensuremathπ​(x\ensuremath−21​)], 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.
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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.
Sine-square deformation (SSD) is a treatment proposed in quantum systems, which spatially modifies a Hamiltonian, gradually decreasing the local energy scale from the center of the system toward the edges by a sine-squared envelope function. It is known to serve as a good boundary condition as well as to provide physical quantities reproducing those of the infinite-size systems. We apply the SSD to one- and two-dimensional classical Ising models. Based on the analytical calculations and Monte Carlo simulations, we find that the classical SSD system is regarded as an extended canonical ensemble of a local subsystem, each characterized by its own effective temperature. This effective temperature is defined by normalizing the system temperature by the deformed local energy scale. A single calculation for a given system temperature provides a set of physical quantities of various temperatures that quantitatively reproduces well those of the uniform system.
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 one-dimensional quantum critical spin systems with sine-square deformation, in which the energy scale in the Hamiltonian at position xxx is modified by the function ${f}_{x}=\mathrm{sin}{}^{2}[\frac{\ensuremath{\pi}}{L}(x\ensuremath{-}\frac{1}{2})]$, where LLL is the length of the system. By investigating the entanglement entropy, spin correlation functions, and wave-function overlap, we show that the sine-square deformation changes the topology of the geometrical connection of the ground state drastically: Although the system apparently has open edges, the sine-square deformation links those ends and realizes the periodic ground state at the level of the wave function. Our results propose a method to control the topology of quantum states by energy-scale deformation.
We study the sine-square deformed quantum XY chain with open boundary conditions in which the interaction strength at the position x in the chain of length L is proportional to the function . The model can be mapped onto a free spinless fermion model with site-dependent hopping amplitudes and on-site potentials via the Jordan–Wigner transformation. Although the single-particle eigenstates of this system cannot be obtained in closed form, it is shown that the many-body ground state is identical to that of the uniform XY chain with periodic boundary conditions. This proves a conjecture of Hikihara and Nishino (2011 Phys. Rev. B 83 060414) based on numerical evidence.