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

Maruyama, Isao; Katsura, Hosho; Hikihara, Takashi

doi:10.1103/physrevb.84.165132

Cited by 54 publications

(73 citation statements)

References 23 publications

(24 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This condition implies that the chiral Hamiltonian, a nearest-neighbor tight-binding model in reciprocal space, does not contain terms with momentum transfer across the Fermi points (for details, see Ref. [10]). The corresponding magnetic field reads h = −J cos(Nπ/L), with N being the number of up spins.…”

Section: Xy Chain With Ssdmentioning

confidence: 99%

“…Similar to the chiral deformation of the lattice fermion systems [10], H ± is expressed in terms of the sum of nearest-neighbor hoppings in momentum space. More importantly, H ± does not include the momentum transfer across the Fermi point (see Fig.…”

Section: Left Rightmentioning

confidence: 99%

“…As discussed in Refs. [9,10], one can make |0 an exact eigenstate by fine-tuning the magnetic field (chemical potential for spinless fermions) so that ǫ 0 (k ∓ δ/2) = 0 when the Fermi point is in between k and k ∓ δ (see Fig. 1 (b)).…”

Section: Xy Chain With Ssdmentioning

confidence: 99%

“…Motivated by this, a rigorous proof of the correspondence for the spin- XY chain, which is equivalent to the system of 1D free fermions, was given by the present author [9]. A simpler and more physical explanation for the success of the SSD in free-fermion systems was given [10], which allows us to extend the idea of the SSD from one dimension to two and higher dimensions. ‡ Criticality is not a necessary condition for the success of the SSD.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Sine-square deformation of solvable spin chains and conformal field theories

Katsura¹

2012

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

Abstract. 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 (πx/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.

show abstract

Section: Xy Chain With Ssdmentioning

confidence: 99%

Section: Left Rightmentioning

confidence: 99%

Section: Xy Chain With Ssdmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Sine-square deformation of solvable spin chains and conformal field theories

Katsura¹

2012

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

show abstract

“…To overcome this difficulty, here we use the ground canonical analysis with the sine-square deformation (SSD) that has been developed recently [25][26][27][28]. The SSD deforms the original Hamiltonian of Eq.…”

mentioning

confidence: 99%

Multiple magnetization plateaus and magnetic structures in theS=12Heisenberg model on the checkerboard lattice

Morita

Shibata

2016

Phys. Rev. B

View full text Add to dashboard Cite

We study the ground state of S = 1/2 Heisenberg model on the checkerboard lattice in a magnetic field by the density matrix renormalization group (DMRG) method with the sine-square deformation. We obtain magnetization plateaus at M/Msat =0, 1/4, 3/8, 1/2, and 3/4 where Msat is the saturated magnetization. The obtained 3/4 plateau state is consistent with the exact result, and the 1/2 plateau is found to have a four-spin resonating loop structure similar to the six-spin loop structure of the 1/3 plateau of the kagome lattice. Different four-spin loop structures are obtained in the 1/4 and 3/8 plateaus but no corresponding states exist in the kagome lattice. The 3/8 plateau has a unique magnetic structure of three types of four-spin local quantum states in a 4 √ 2 × 2 √ 2 magnetic unit cell with a 16-fold degeneracy.Frustrated quantum spin systems exhibit unusual quantum phenomena such as the formations of valence bond solid (VBS), spin-nematic, and quantum spin liquid (QSL) as a result of competing quantum fluctuations [1,2]. Application of an external magnetic field complicates the situation further with the increase in uniaxial anisotropy, and it sometimes leads to a phase transition to a stable quantum state with a jump or cusp in the magnetization curve at zero temperature. Recently observed 1/3 magnetization plateau in the triangular lattice of S = 1/2 quantum spins [3-5] is a typical example, in which up-up-down spin structure emerges with a finite excitation gap [6][7][8][9]. Similar 1/2 plateau is obtained in the J 1 -J 2 square lattice [9-11] and novel multiple magnetization plateaus are realized in the Shastry-Sutherland lattice [12][13][14][15]. In the kagome lattice, 0, 1/9, 1/3, 5/9, and 7/9 plateaus are predicted by the DMRG method with the sine-square deformation [16].The checkerboard lattice shown in Fig. 1 is another example of frustrated system and referred to as twodimensional pyrochlore lattice. Its classical ground state at zero magnetic field is obtained when the magnetic moments of four spins connected by diagonal interactions are canceled as shown in Fig. 2. This means the presence of a macroscopic degeneracy in the ground state as in the cases of the kagome lattice and the J 1 -J 2 square lattice at J 2 /J 1 = 0.5. In contrast, S = 1/2 quantum spins on the checkerboard lattice have stable quantum ground state as in the kagome lattice [16][17][18] and Shastry-Sutherland lattice [12][13][14][15]. Indeed previous studies on the checkerboard lattice indicate that the ground state is a plaquette valence-bond crystal (PVBC) with a large spin singlettriplet gap, ∆ st ≈ 0.6J [19,20] and the 3/4 plateau is realized just before the saturation magnetic field [21] as in the case of the 7/9 plateau in the kagome lattice [22].Although the ideal checkerboard lattice with only one exchange energy J has not been synthesized, similar model substances are present [23,24]. The checkerboard lattice is obtained by replacing the position of anions and cations in the CuO 2 -plane structure. When the nearest-neig...

show abstract

Conformal Floquet dynamics with a continuous drive protocol

Das

Ghosh

Sengupta

2021

J. High Energ. Phys.

View full text Add to dashboard Cite

We study the properties of a conformal field theory (CFT) driven periodically with a continuous protocol characterized by a frequency ωD. Such a drive, in contrast to its discrete counterparts (such as square pulses or periodic kicks), does not admit exact analytical solution for the evolution operator U. In this work, we develop a Floquet perturbation theory which provides an analytic, albeit perturbative, result for U that matches exact numerics in the large drive amplitude limit. We find that the drive yields the well-known heating (hyperbolic) and non-heating (elliptic) phases separated by transition lines (parabolic phase boundary). Using this and starting from a primary state of the CFT, we compute the return probability (Pn), equal (Cn) and unequal (Gn) time two-point primary correlators, energy density(En), and the mth Renyi entropy ($$ {S}_n^m $$ S n m ) after n drive cycles. Our results show that below a crossover stroboscopic time scale nc, Pn, En and Gn exhibits universal power law behavior as the transition is approached either from the heating or the non-heating phase; this crossover scale diverges at the transition. We also study the emergent spatial structure of Cn, Gn and En for the continuous protocol and find emergence of spatial divergences of Cn and Gn in both the heating and non-heating phases. We express our results for $$ {S}_n^m $$ S n m and Cn in terms of conformal blocks and provide analytic expressions for these quantities in several limiting cases. Finally we relate our results to those obtained from exact numerics of a driven lattice model.

show abstract

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

Cited by 54 publications

References 23 publications

Sine-square deformation of solvable spin chains and conformal field theories

Sine-square deformation of solvable spin chains and conformal field theories

Multiple magnetization plateaus and magnetic structures in theS=12Heisenberg model on the checkerboard lattice

Conformal Floquet dynamics with a continuous drive protocol

Contact Info

Product

Resources

About