Nature of quantum spin liquids of the S=1/2 Heisenberg antiferromagnet on the triangular lattice: A parallel DMRG study

arXiv:2203.10216 · cond-mat.str-el · Submitted 2022-03-19 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Nature of quantum spin liquids of the S=1/2 Heisenberg antiferromagnet on the triangular lattice".

Mira: Here's the gist of the paper. The gist The QSL phase of the J1-J2-J3 model is a critical spin liquid with one gapless mode and quasi-long-range dimer-dimer correlation but exponentially decaying spin-spin correlation.<ref:2203.10216#pg6> Model and Method The study employs density-matrix renormalization group (DMRG) method to investigate the ground-state properties of the spin-1/2 Heisenberg antiferromagnet on the triangular lattice with nearest-, next-nearest-, and third-neighbor interactions by using density-matrix renormalization group (DMRG) method<ref:2203.10216#pg4>.

Kai: The lattice geometry used in simulations involves cylinders with circumference Ly and length Lx, where Ly and Lx are the number of sites in the e2 and e1 directions, respectively<ref:2203.10216#pg5> The researchers focus on two typical sets of parameters: J1 = 1 as an energy unit, with J2 = 0.11 for the J1-J2 model and J3 = 0.3 and J3 = 0.15 for the J1-J2-J3 model<ref:2203.10216#pg6> They perform both real- and complex-value DMRG simulations, keeping up to m = 9000 SU(2) states (equivalent m = 36000 U(1) states) in each DMRG block<ref:2203.10216#pg6>.

Mira: To enhance numerical efficiency on wide cylinders, an efficient operator-level parallel DMRG scheme with SU(2) spin rotational symmetry was developed, which can improve the numerical efficiency by O(Ly) times<ref:2203.10216#pg6>.

Kai: This scheme allows for keeping up to m = 9000 SU(2) states in the complex-value DMRG simulation to obtain accurate results<ref:2203.10216#pg6>.

Title and authors: Kai: So, to start with the paper "Nature of quantum spin liquids of the S=one/two Heisenberg antiferromagnet on the triangular lattice: A parallel DMRG study," they're looking at how tuning those interaction ratios, J1, J2, and J3 completely changes what kind of quantum spin liquid you get <ref:2203.10216#pg1,Nature of quantum spin liquids of the S=1/2 Heisenberg antiferromagnet>.

Mira: Right. They use this parallel DMRG approach to look at the ground state properties of that spin-one/two antiferromagnetic Heisenberg model on a triangular lattice with nearest, next-nearest, and third-neighbor interactions <ref:2203.10216#pg1>.

Lev: And they are focusing on how those three interaction terms—J1, J2, and J3—dictate whether you get a gapped quantum spin liquid phase or this new critical spin liquid phase <ref:2203.10216#pg6>.

Kai: They tested specific parameter sets for that, like setting J1 = one with J2 = zero point one one for the first model, and then they looked at the J3 interaction with values of zero point three and zero point one five <ref:2203.10216#pg6>.

Mira: The main point they highlight is that the J1-J2 model leads to a gapped quantum spin liquid, which is characterized by the absence of a gapless mode and short-range spin-spin and dimer-dimer correlations <ref:2203.10216#pg6>.

Lev: For that J1-J2 case, they show that the spin correlation F(r) decays exponentially with a finite correlation length ξs, and the dimer-dimer correlation Da(r) also decays exponentially with a finite length ξa <ref:2203.10216#pg7>.

Kai: That means for the J1-J2 model, you have these short-range correlations everywhere that decay exponentially <ref:2203.10216#pg6>. But when they introduce J3, that fundamentally alters the physics, leading to a new type of spin liquid <ref:2203.10216#pg6>.

Mira: That new phase is called a critical quantum spin liquid and it has one gapless mode that's independent of the cylinder circumference <ref:2203.10216#pg6>.

Lev: And for that critical spin liquid, they find quasi-long-range dimer correlations, decaying as a power law, which is Da(r) ∼ r − Ka with some finite exponent Ka <ref:2203.10216#pg8>.

The paper's summary: Kai: Now we look at what they summarize about the work in "Nature of quantum spin liquids of the S=one/two Heisenberg antiferromagnet on the triangular lattice: A parallel DMRG study <ref:2203.10216#pg1,Nature of quantum spin liquids of the S=1/2 Heisenberg antiferromagnet>." They're pointing out a few things they suggest for future work and interpretation <ref:2203.10216#pg3>.

Mira: They stress that we need to be careful about how we extract quantities like the central charge c because it can be dramatically overestimated if you don't properly account for the boundary and finite-size effects <ref:2203.10216#pg3>.

Lev: They specifically mention that for a given cylinder of length Lx, they should remove some data points from both open ends, which they call delta B data points, to get results that are much closer to the long cylinder limit <ref:2203.10216#pg3>.

Kai: That means reducing boundary effect by removing a few data points close to the open ends of the cylinders can give more reliable results that are much closer to what you'd expect in the long cylinder limit <ref:2203.10216#pg3>.

Mira: And they emphasize that these finite-size and boundary effects need to be taken into account in numerical simulations, even when using those large m = nine thousand SU(two) states <ref:2203.10216#pg6>.

Lev: From my view, for running this on real hardware, that means the overhead lands on how much you need to scale up your cylinder size to minimize those boundary effects <ref:2203.10216#pg6>.

The paper's improvements: Kai: So, looking at the suggestions for improvement in "Nature of quantum spin liquids of the S=one/two Heisenberg antiferromagnet on the triangular lattice: A parallel DMRG study," they confirm that tuning those interaction ratios, J1, J2, and J3 changes what we get <ref:2203.10216#pg6>.

Mira: The main implication for us is understanding the distinct phases based on those interaction ratios, showing how tuning J3 can shift the system from one type of QSL to another <ref:2203.10216#pg6>.

Lev: For implementation, it means that if you are aiming for that critical phase, you need to focus your efforts on accurately simulating those quasi-long-range dimer correlations which decay as a power law <ref:2203.10216#pg8>.

Kai: We saw how the study highlights the sensitivity of things like the central charge to boundary effects, and how that can affect our interpretation when we’re not in the long cylinder limit <ref:2203.10216#pg6>.

Mira: This work confirms that tuning J3 fundamentally alters the physics on this lattice, showing us a richer landscape of possible quantum spin liquid states <ref:2203.10216#pg6>.

Lev: Anyway, it’s a solid paper that gives us concrete benchmarks for what to look for in these critical phases and where the limitations lie <ref:2203.10216#pg8>.

Kai: So that’s our take on this piece of work regarding the "Nature of quantum spin liquids of the S=one/two Heisenberg antiferromagnet on the triangular lattice: A parallel DMRG study" <ref:2203.10216#pg1,Nature of quantum spin liquids of the S=1/2 Heisenberg antiferromagnet>.

Conclusion: Kai: To wrap up on "Nature of quantum spin liquids of the S=one/two Heisenberg antiferromagnet on the triangular lattice: A parallel DMRG study," they've shown us how tuning those interaction ratios, J1, J2, and J3 completely changes what kind of quantum spin liquid you get <ref:2203.10216#pg1,Nature of quantum spin liquids of the S=1/2 Heisenberg antiferromagnet>.

Mira: Right. They’ve got this clear distinction between a gapped QSL phase for the J1-J2 model and this new critical spin liquid when you introduce that third interaction term <ref:2203.10216#pg6>.

Lev: And from my side, it's interesting how the authors have to keep up with those large system sizes, like Lx = forty-eight and Ly between six and twelve, just to get those good numbers <ref:2203.10216#pg5>.

Kai: Exactly. The J1-J2 part is consistent with a gapped QSL, meaning no gapless mode at all, and the spin correlations decay exponentially <ref:2203.10216#pg6>.

Mira: And for that gapped case, they show both the spin-spin correlation F(r) and the dimer-dimer correlation Da(r) decay exponentially with finite lengths <ref:2203.10216#pg7>. That’s short range everywhere.

Lev: Which means if we were trying to build a real device based on that J1-J2 model, you'd just be looking at something stable with short correlation lengths <ref:2203.10216#pg6>.

Kai: But when they add J3, things get interesting because they find this new critical spin liquid phase <ref:2203.10216#pg6>.

Mira: That new phase has one gapless mode that doesn't depend on the cylinder circumference <ref:2203.10216#pg6>.

Lev: And they find this critical spin liquid has quasi-long-range dimer correlations, decaying as a power law, which is Da(r) ∼ r minus Ka with some finite exponent Ka <ref:2203.10216#pg8>. That power law decay is what defines the criticality here.

Kai: So, the big picture for someone listening to this show is that adding just one more interaction term can flip you from a fully gapped system to something with a gapless mode and long-range dimer correlations <ref:2203.10216#pg6>.

Mira: And they also looked at the chiral correlations C(r), which are finite, but that value drops notably as Lx increases, suggesting that chiral order might vanish in the true two-dimensional limit <ref:2203.10216#pg8>.

Lev: It’s a bit of a warning for us about how much we rely on those boundary conditions when trying to extrapolate to the infinite system size, which is what they are struggling with <ref:2203.10216#pg3>.

Kai: So the study "Nature of quantum spin liquids of the S=one/two Heisenberg antiferromagnet on the triangular lattice: A parallel DMRG study" gives us two very different pictures depending on the J3 interaction <ref:2203.10216#pg6>.

Mira: And it really shows that in these complex quantum systems, a small change in parameters can lead to a whole new type of physics <ref:2203.10216#pg6>.

Lev: It makes me think about how we design error-correction codes; sometimes you need to adjust the code structure significantly when you move from one regime to another <ref:2203.10216#pg8>.

Kai: Yeah, so next time we talk about spin liquids, remember that tuning those couplings is the real lever for changing the fundamental nature of the ground state.

ShanghaiTech University · Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory and Stanford University

cond-mat.str-el

Submitted: 2022-03-19

Updated: 2022-03-19

Comments: 4 pages, 5 figures + supplemental material

Journal ref: Phys. Rev. B 107, L140411 (2023)

DOI: 10.1103/PhysRevB.107.L140411

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 79/100

The gist: The gist The QSL phase of the J1-J2-J3 model is a critical spin liquid with one gapless mode and quasi-long-range dimer-dimer correlation but exponentially decaying spin-spin

Key concepts

Density-Matrix Renormalization Group (DMRG): A numerical method used to find the ground state of quantum systems, like magnets. It works by systematically breaking down a large system into smaller pieces and calculating the most important parts, allowing researchers to study complex interactions on lattices.
QSL (Quantum Spin Liquid): A type of exotic magnetic state where spins do not order at low temperatures, behaving like a liquid. These states are characterized by unique properties like having a gapless mode or specific types of correlations that decay in unusual ways over distance.
Gapless Mode: An excitation in the system that requires zero energy to create. In magnetic systems, this often corresponds to a continuous spectrum of excitations, which is characteristic of certain critical phases where correlations persist over long distances.
Quasi-long-range Correlation: A type of correlation where the relationship between two distant parts of the system does not decay completely but decays slowly, following a power law rather than an exponential decay. This indicates some form of order that persists over large distances.

Summary

The gist The QSL phase of the J1-J2-J3 model is a critical spin liquid with one gapless mode and quasi-long-range dimer-dimer correlation but exponentially decaying spin-spin correlation.<ref:2203.10216#pg6>

Model and Method

The study employs density-matrix renormalization group (DMRG) method to investigate the ground-state properties of the spin-1/2 Heisenberg antiferromagnet on the triangular lattice with nearest-, next-nearest-, and third-neighbor interactions by using density-matrix renormalization group (DMRG) method<ref:2203.10216#pg4>. The lattice geometry used in simulations involves cylinders with circumference Ly and length Lx, where Ly and Lx are the number of sites in the e2 and e1 directions, respectively<ref:2203.10216#pg5> The researchers focus on two typical sets of parameters: J1 = 1 as an energy unit, with J2 = 0.11 for the J1-J2 model and J3 = 0.3 and J3 = 0.15 for the J1-J2-J3 model<ref:2203.10216#pg6> They perform both real- and complex-value DMRG simulations, keeping up to m = 9000 SU(2) states (equivalent m = 36000 U(1) states) in each DMRG block<ref:2203.10216#pg6>. To enhance numerical efficiency on wide cylinders, an efficient operator-level parallel DMRG scheme with SU(2) spin rotational symmetry was developed, which can improve the numerical efficiency by O(Ly) times<ref:2203.10216#pg6>. This scheme allows for keeping up to m = 9000 SU(2) states in the complex-value DMRG simulation to obtain accurate results<ref:2203.10216#pg6>.

J1-J2 Model Results

The results suggest that the J1-J2 spin liquid is consistent with a gapped QSL, which is characterized by the absence of gapless mode and short-range spin-spin and dimer-dimer correlations<ref:2203.10216#pg6>. This gapped QSL phase is characterized by a fully gapped excitation spectrum and all the correlations, including the spin-spin, dimer-dimer and scalar chiral-chiral correlations, are short-ranged which decay exponentially at long distances<ref:2203.10216#pg6>. The absence of gapless mode suggests that all correlations are short-ranged<ref:2203.10216#pg6>. For the J1-J2 model, the spin-spin correlation function F(r) decays exponentially at long distances and can be well fitted by an exponential function F(r) ∼ e − r/ξs with finite correlation length ξs shown in the inset of Fig.3(a)<ref:2203.10216#pg7>. Similarly, the dimer-dimer correlation Da(r) decays exponentially as Da(r) ∼ e − r/ξA as shown in Fig.3(b) with finite correlation length ξa<ref:2203.10216#pg7>.

J1-J2-J3 Model Results

In the presence of J3 interaction, a new type of QSL, dubbed critical spin liquid, emerges in the J1-J2-J3 model<ref:2203.10216#pg6>. This new phase has a single gapless mode which is independent of the circumference of the cylinders<ref:2203.10216#pg6>. While both spin-spin and scalar chiral-chiral correlations are short-ranged, dimer-dimer correlations are quasi-long-ranged which decays as a power-law at long distances<ref:2203.10216#pg6>. The dimer-dimer correlation function Da(r) decays as a power-law at long distances as Da(r) ∼ r − Ka with a finite exponent Ka, for instance, Ka ∼ 1.8 on Ly = 8 cylinder as shown in Fig.5(b)<ref:2203.10216#pg8>. The absence of gapless mode suggests that all correlations are short-ranged<ref:2203.10216#pg6>.

Correlation Functions and Symmetry

The scalar chiral-chiral correlation function C(r) remains finite on all cylinders even we keep up to m = 9000 SU(2) states (equivalent m = 36000 U(1) states)<ref:2203.10216#pg6>. However, our results show that C(r) decreases notably with the increase of Lx which vanishes in the long cylinder limit Lx = ∞ on all Ly = 6 − 10 cylinders after the finite-size scaling as shown in Fig.5(c)<ref:2203.10216#pg8>. This indicates a possibly vanishing chiral order in the two-dimensional limit<ref:2203.10216#pg8>. The study also tests the possibility of TRS breaking, and our results are consistent with the absence of long-range spin scalar chiral order and the QSL phase of the J1-J2-J3 model preserves the TRS<ref:2203.10216#pg8>.

Summary and Discussion

The ground state of the J1-J2-J3 model is consistent with a critical spin liquid with a single gapless mode<ref:2203.10216#pg6>. The results suggest that the QSL phase of the J1-J2 Heisenberg model is consistent with a gapped spin liquid which is characterized by the absence of gapless spin mode, short-range spin-spin and dimer-dimer correlations<ref:2203.10216#pg6>. In the presence of finite J3 interaction, a new critical spin liquid phase emerges which has one gapless mode and quasi-long-range dimer-dimer correlation but exponentially decaying spin-spin correlation<ref:2203.10216#pg6>. The striking behavior of the central charge that is prominent on cylinder geometry is that its value can be notably affected by both the boundary and finite-size effects<ref:2203.10216#pg6>. Reduced boundary effect by removing a few data points close to the open ends of the cylinders can provide more reliable results that are much closer to that in the long cylinder limit<ref:2203.10216#pg6>. The study emphasizes the perceptible effect of the finitesize and boundary effects which need to be taken into account in the numerical simulations<ref:2203.10216#pg6>.

Acknowledgments

We would like to thank Steven Kivelson, Thomas Devereaux, Dong-Ning Sheng, Shou-Shu Gong and Hong Yao for insightful discussions<ref:2203.10216#pg6>. This work was supported by the Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under Contract DE-AC02-76SF00515<ref:2203.10216#pg6>. Some of the computing for this project was performed on the Sherlock cluster<ref:2203.10216#pg6>. We would like to thank Stanford University and the Stanford Research Computing Center for providing computational resources and support that contributed to these research results<ref:2203.10216#pg6>.

References

[1] L. Balents, Nature 464, 199 (2010)<ref:2203.10216#pg6>

[3] C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T. Senthil, Science 367, eaay0668 (2020)<ref:2203.10216#pg6>

[4] P. W. Anderson, Science 235, 1196 (1987)<ref:2203.10216#pg6>

[5] V. J. Emery, Phys. Rev. Lett. 58, 2794 (1987)<ref:2203.10216#pg6>

[6] P. A. Lee, N. Nagaosa, and X. G. Wen, Rev Mod Phys 78, 17 (2006)<ref:2203.10216#pg6>

[8] E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Rev Mod Phys 87, 457 (2015)<ref:2203.10216#pg6>

[9] T. Jolicoeur, E. Dagotto, E. Gagliano, and S. Bacci, Phys Rev B 42, 4800 (1990)<ref:2203.

Improvements for AI systems

Weighting of QSL Phases based on Interaction Ratios: The AI system can now accurately predict whether a given set of exchange couplings (J1, J2, J3) corresponds to a gapped QSL or a critical spin liquid by analyzing the resulting central charge and correlation functions (Our results suggest that the QSL phase of the J1-J2 Heisenberg model is consistent with a gapped QSL versus a new type of QSL, dubbed critical spin liquid).
Detection of Critical Spin Liquid Character: The AI system can identify the presence of a single gapless mode (In the presence of J3 interaction, we find that a new critical QSL with a single gapless mode emerges) by analyzing the central charge and boundary effects (the extracted c on Ly = 8 cylinders in the long Lx limit is also reasonably close to 1).
Characterization of Correlations: The AI system can distinguish between short-range and quasi-long-range correlations by analyzing decay rates, specifically identifying short-ranged spin correlations (exponential decay) versus quasi-long-ranged dimer correlations (power law decay, e.g., dimer-dimer correlation decays as a power-law at long distances as Da(r) ∼ r − Ka).
Sensitivity to Finite Size and Boundary Effects: The AI system can assess the reliability of numerical results by quantifying the impact of boundary conditions and finite system size, recognizing that reduced boundary effect by removing a few data points close to the open ends of the cylinders can provide more reliable results that are much closer to that in the long cylinder limit.
Predictive Modeling for Complex QSLs: The AI system can extrapolate behavior for distant parameter regimes, as demonstrated by observing the similar critical behavior with one gapless mode at this new point when testing parameters like J2 = 0.36 and J3 = 0.24 against the primary results.
Identification of Time-Reversal Symmetry Preservation: The AI system can verify the preservation of time-reversal symmetry by analyzing chiral correlations, noting that C(Lx/4) decreases quickly with the increase of Lx and drops to zero when Lx is long enough, providing evidence that the QSL phase of the J1-J2-J3 model preserves the TRS.

Sources

Classification and Monte Carlo study of symmetric $Z_{2}$ spin liquids on the triangular lattice