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Quantum tricriticality of incommensurate phase induced by quantum domain walls in frustrated Ising magnetism
by Zheng Zhou, Dong-Xu Liu, Zheng Yan, Yan Chen, Xue-Feng Zhang
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Submission summary
| Authors (as registered SciPost users): | Xue-Feng Zhang · Zheng Zhou |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2005.11133v2 (pdf) |
| Date submitted: | 2022-08-05 02:54 |
| Submitted by: | Zhou, Zheng |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Incommensurability plays a critical role in many strongly correlated systems. In some cases, the origin of such exotic order can be theoretically understood in the framework of 1d line-like topological excitations known as “quantum strings”. Here we study an extended transverse field Ising model on a triangular lattice. Using the large scale quantum Monte Carlo simulations, we find that the spatial anisotropy can stabilize an incommensurate phase out of the commensurate clock order. Our results for the structure factor and the string density exhibit a linear relationship between incommensurate ordering wave vector and the density of quantum strings, which is reminiscent of hole density in under-doped cuprate superconductors. When introducing the next-nearest-neighbor interaction, we observe a quantum tricritical point out of the incommensurate phase. After carefully analyzing the the ground state energies within different string topological sectors, we conclude that this tricriticality is non-trivially caused by effective long-range inter-string interactions with two competing terms following different power-law decaying behaviors B/r^α − C/r^γ.
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Reports on this Submission
Report #2 by Anonymous (Referee 3) on 2022-9-9 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2005.11133v2, delivered 2022-09-09, doi: 10.21468/SciPost.Report.5672
Report
The authors have studied the ground state phases of the transverse field Ising model with nearest and next nearest neighbor interactions on an anisotropic triangular model at weak transverse fields. They develop a description of the ground state order in terms of strings separating domains of columnar AFM order. Each string behaves as a S=1/2 XY chain with long-range interaction. The central result of the work is the detection of an incommensurate phase and the appearance of a tricritical point when NNN interactions are introduced.
The problem is interesting and the results are intriguing. However, in the opinion of the present referee, the manuscript can be improved before publication. In particular, can the authors address the following?
1. How is the sign problem avoided in SSE? Previous QMC studies (13, 33, https://www.nature.com/articles/s41467-020-14907-8) all use special trick with path integral / world line QMC to avoid the sign problem where the coupling in the imaginary time direction is shown to be ferromagnetic. Sandvik's work (PhysRevE 68 056701 (2003) using SSE on TFIM with long range interactions use a bipartite lattice. Since the authors are probably the first to use SSE for TFIM on a triangular lattice, a discussion on the method would greatly add to the manuscript.
2. In the phase diagram 4(c), can the authors discuss the stripe phase? It appears to be deep inside the incommensurate phase.
3. How do the authors conclude that V(r) decays faster than linear? Doesn't the hard core constraint generalize to fast decaying interaction in the presence of quantum fluctuation (h > 0)?
4. Does "vibration" of the string refer to the process defined in Fig. 2(c)?
5. It is not clear how the authors arrive at the functional dependence of the energy of the strings.
6. Should the width of the plateaus reduce to zero in the thermodynamic limit? What about the stripe phase? Does it have any finite-J_x extent in the thermodynamic limit?
To conclude, the authors have introduced an interesting problem. If they can address the above issues satisfatorily, the manuscript can be considered for publication.
Report #1 by Anonymous (Referee 4) on 2022-9-2 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2005.11133v2, delivered 2022-09-02, doi: 10.21468/SciPost.Report.5625
Report
In the paper, Zhou et al. studied the ground state of the triangular-lattice quantum Ising model with spatial anisotropy and NNN interactions $J’$, focusing on the quantum tricriticality induced by effective interactions between quantum strings. The quantum Ising model, or transverse-field Ising model, has been extensively studied for many years, but recently it attracts lots of attention in relation to the development of artificial quantum simulators with trapped ions, Rydberg atom arrays, etc. Based on the numerical QMC calculations, the authors investigated here the $J’ $ dependence of the energy of strings, assuming that it is given in the form with two different algebraic-decaying terms. As a results, it is concluded that “the tricriticality of the incommensurate phase is caused by the competing of effective long-range inter-string interactions with different power exponents.”
The paper is scientifically sounds and well written. However, I would recommend the manuscript for publication only after the points listed below are properly addressed.
--- page 4: In the first sentence, the authors state that all the bonds connecting parallel spins are aligned along the x-direction, forming stripe phase, for $J_x < J$. However, the actual ground state can be incommensurate phase when $J_x/J$ is larger than a certain critical value between 0 and 1. This is confusing.
--- page 10: The authors state that the leading order approximation of $B$ and $C$ (namely, constant and linear term, respectively) is good enough. However, according to the inset of Fig. 6(c), the liner term should also be considered for $B$, like $B \sim B_0 + B_1 J’$.
--- Although I understood that the energy can be well approximated in the form of Eq. (10) (except for the above-mentioned point), I could not get the idea of why it yielded the conclusion that the effective inter-string interaction can be written in the form $B/r^\alpha – C/r^\gamma$ (what is $r$?).
--- In the sketch of the phase diagram in Fig. 1, the tricritical point is located just on the vertical axis. However, it is not the case, is it? This is somewhat misleading. Also, using the effective energy functional Eq. (10) with numerically fitted values of coefficients, I think that the authors can estimate the location of the tricritical point ($J_x$ and $J’$) in a more quantitative way. It should help to improve the quality and completeness of the paper.
Author: Zheng Zhou on 2022-10-05 [id 2878]
(in reply to Report 1 on 2022-09-02)
We thank the referee for his positive evaluation as well as the critical comments that have helped us improve the manuscript. In the following we give a point-by-point reply to the comments. A more properly formatted version of the reply in LaTeX can be found attached in the PDF file.
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In the second paragraph of Sec. 2, we start by considering the classical Ising limit h=0, i.e., the vertical axis of Fig. 4(c). Whereas the incommensurate phase only emerges at finite transverse field, at h=0 all the J_x<J region belongs to the stripe phase by optimising the energy classically. We start considering the quantum case only from the third paragraph, which is after the position the referee mentions. In the revised manuscript, we make clarification by replacing 'in the anisotropic case J_x<J' by 'in the classical anisotropic case J_x<J and h=0'
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The referee is correct that for an accurate description, it is better to include a linear term in B. However, approximating B to a constant suffices to reproduce the qualitative result. On one hand, unlike C which changes sign with J', the J'-dependence of B is not other drastic. On the other hand, the B-term interaction originates from the hinder of motion when strings are nearby; this mechanism does not depend on the presence of J'; by contrast, the C-term interaction directly comes from the NNN coupling. By approximating B=B_0 and C=C_1J', one can find that the stripe-incommensurate phase transition is first-order when J'>0 and continuous when J'<0, and there is therefore a multicritical point at J'=0.
To clarify, in the revised manuscript, we add some related discussion: 'We also note that for a more accurate description, B(J') should be approximated as B(J')\approx B_0+B_1J' to take into account the fact that the NNN coupling modifies the vibration of string, but this modification does not change the scenario qualitatively and can be therefore considered as a higher-order term.'
- Firstly, r is the distance between two nearby strings. To calculate the average energy of quantum string, we replace r by its average value \bar{r}=L\rho_{QS}.
Regarding the interaction energy ansatz V(r)=B/r^\alpha-C/r^\gamma, we here give a more detailed explanation. There are two mechanisms of string interaction: the first, denoted V_h(r), is a repulsion from the hinder of motion when strings are nearby; the second, denoted V_{J'}(r), comes from the fact that the insertion of single string produces energy cost 3J'/2 per string length, while two adjacent strings cost energy 2J' per string length, which is different than two individual strings; therefore we can write V(r)=V_h(r)+V_{J'}(r). Originally, these two mechanisms all act in short-range; the vibration of strings then turns these short-range interactions into long-range ones. The exact determination of the form of V_{J'}(r) and V_h(r) is a difficult question. In a former work [3], the repulsion V_h(r) from motion hinder has been determined to follow a power-law behaviour in the incommensurate phase a similar model (hardcore Bose-Hubbard model) by fitting the jumping points of the plateaux. As an extension, it is most natural and simplest to assume that V_{J'} also follow a power law. We therefore write down the ansatz V(r)=B/r^\alpha-C/r^\gamma. As an evidence, this ansatz fits well with our numerical result (Fig. 6(c)), as explained in the main text.
To clarify, we have also added the above paragraph to the main text, added the sentence 'where r is the distance between two nearby strings' to the Conclusion section, and deleted the exact expression from the abstract to avoid confusion.
The tricritical point in fact locates on the vertical axis J'=0 exactly. The numerical evidence is provided in Fig. 5(b) inset. We determine the tricritical point at finite size as the J' value where the first plateau disappears, i.e., E(\rho_{QS}=0)=E(\rho_{QS}=2/L)=E(\rho_{QS}=4/L), and performed a finite size scaling. The result shows that the tricritical point should lie on the vertical axis J'_c=0.
Intuitively, the tricriticality between first order and continuous transition line is driven by the string interaction mediated by J' (C-term) changing from attractive to repulsive. As C-term is mediated by the NNN coupling J', it is natural that the tricritical point locates at the zero-point of this coupling.
In the manuscript, it is also mentioned that 'Moving towards the thermodynamic limit, this point is found to gradually approach J'=0 in Fig. 5(b) inset'
Author: Zheng Zhou on 2022-10-05 [id 2880]
(in reply to Report 2 on 2022-09-09)We thank the referee for his positive evaluation as well as the critical comments that have helped us improve the manuscript. In the following we give a point-by-point reply to the comments. A more properly formatted version of the reply in LateX can be found attached in the PDF file.
1 . The transverse field Ising model does not have sign problem regardless of the lattice, as one can always add a constant to eliminate the negative matrix elements. E.g., the AFM TFIM
H=J\sum_{<ij>}S_i^zS_j^z-\Gamma\sum_iS_i^x
can be rewritten in the following by adding a constant energy shift
H=J\sum_{<ij>}(S_i^zS_j^z-1/4)-\Gamma\sum_i(S_i^x+1/2)\\
=-\sum_{<ij>}H^d_{ij}-\sum_iH^o_i,
where the local operators and their matrix elements are
H^d_{ij}=J(-S_i^zS_j^z+1/4)
<↑↓|H^d_{ij}|↑↓>=<↓↑|H^d_{ij}|↓↑>=J/2,(others)=0
H^o_i=\Gamma(S_i^x+1/2)
<↑|H^o_i|↑>=<↑|H^o_i|↓>=<↓|H^o_i|↑>=<↓|H^o_i|↓>=\Gamma/2
All the matrix elements are non-negative; there is therefore no sign problem. In addition, some reference [Phys. Rev. B 98, 174433 (2018)] also applies SSE to non-bipartite lattices; the original reference (Ref. [48]) does not restrict the simulation to bipartite lattice.
2 . The stripe phase is actually not incommensurate, as its Bragg peak situates at the commensurate M point in the momentum space (Fig. 3(a)i). The real-space spin configuration in the stripe phase has parallel spins in the same row and alternating spin-ups and spin-downs in adjacent rows (Fig. 2(a)). Unlike the incommensurate phase which only appears in the presence of quantum term h, the stripe phase optimises the energy in the classical limit h=0 at J_x<J. The reason we choose the stripe state as the reference state is that any local operation acting upon this state violates the local constraint, hence there are no low-energy excitations within the stripe bulk.
We have also added a brief discussion 'The ordering momentum is commensurate. The spin alignment alternates each row in the real space configuration [Fig. 2(a)]. This phase extends to J_x\to-\infty and persists at the classical limit h=0.' to Section 3, Paragraph 2.
3 . We provide a detailed description on how we determine the form of the interaction between strings in the Reply 5. The power of the interaction is determined by fitting. In the presence of quantum fluctuation, the hardcore constraint generates a power-law dynamic interaction.
4 . Yes. In the revised manuscript, we have added the reference to Fig. 2(c) when mentioning the 'vibration'.
5 . As explained in Sec. 2, the energy of the string consists two parts. The first is the energy of single quantum string (Eq. (2)), which consists of the energy gap \Delta and the kinetic energy obtained from mapping the string configuration to an effective XY chain; the second is the interaction between adjacent strings V(\bar{r}), where V(r) is the interaction energy between adjacent strings, r is the distance between two nearby strings, and \bar{r}=L\rho_{QS} is its average value.
Regarding the interaction energy ansatz V(r)=B/r^\alpha-C/r^\gamma, we here give a more detailed explanation. There are two mechanisms of string interaction: the first, denoted V_h(r), is a repulsion from the hinder of motion when strings are nearby; the second, denoted V_{J'}(r), comes from the fact that the insertion of single string produces energy cost 3J'/2 per string length, while two adjacent strings cost energy 2J' per string length, which is different than two individual strings; therefore we can write V(r)=V_h(r)+V_{J'}(r). Originally, these two mechanisms all act in short-range; the vibration of strings then turns these short-range interactions into long-range ones. The exact determination of the form of V_{J'}(r) and V_h(r) is a difficult question. In a former work [3], the repulsion V_h(r) from motion hinder has been determined to follow a power-law behaviour in the incommensurate phase a similar model (hardcore Bose-Hubbard model) by fitting the jumping points of the plateaux. As an extension, it is most natural and simplest to assume that V_{J'} also follow a power law. We therefore write down the ansatz V(r)=B/r^\alpha-C/r^\gamma. As an evidence, this ansatz fits well with our numerical result (Fig. 6(c)), as explained in the main text.
To clarify, we have also added the above paragraph to the main text, added the sentence 'where r is the distance between two nearby strings' to the Conclusion section, and deleted the exact expression from the abstract to avoid confusion.
6 . The width of the plateaux scales to zero in the thermodynamic limit. As an evidence, in the manuscript, we did the finite size scaling for a specific plateau (\rho_{QS}=2/3) and find that the ends of the plateau extrapolate to the same point [Fig. 4(b)]. The stripe phase will remain a finite region in the thermodynamic limit. As an intuitive argument, when the anisotropy is much stronger than the transverse field J-J_x\gg h, the stripe phase is always favoured. The estimation of the stripe-incommensurate critical point J_{x,c}=J-2h/\pi+{O}(h^2) applies also to the thermodynamic limit.
To clarify, we have also added a sentence 'As demonstrated in finite-size scaling (Fig. 4(b)) of the plateau corresponding to the clock phase, the width of each plateaux scales to zero the string density \rho_{QS} should follow the continuous function (7) in the thermodynamic limit.' to the related discussion.
We hope that with these changes made, the Referee could agree that the manuscript is suitable for further consideration for SciPost Phys.
Attachment:
Reply_2.pdf