Topological Gap Opening without Symmetry Breaking from Dynamical Quantum Correlations

arXiv:2311.10024 · cond-mat.str-el · Submitted 2023-11-16 · 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: Today's paper: "Topological Gap Opening without Symmetry Breaking from Dynamical Quantum Correlations".

Mira: The gist: Dynamical quantum fluctuations can lead to the opening of a gap without any symmetry breaking in a topological transition,

Kai: First, who's behind it and why it matters.

Title and authors: Kai: So we're looking at this paper today called "Topological Gap Opening without Symmetry Breaking from Dynamical Quantum Correlations". We're talking about how electron interactions can cause a gap to open in a topological insulator, which is usually something that requires some kind of symmetry breaking.

Mira: Exactly. The authors are showing that dynamical quantum fluctuations can achieve this gap opening even when no symmetry breaking happens, which they call the Coleman-Weinberg mechanism for mass generation, and they're using a Bernevig-Hughes-Zhang model augmented with local electron-electron interactions.

Lev: From an engineering standpoint, what I’m hearing is that this moves us beyond just looking at mean field theory, because without those quantum fluctuations included, you get a different story than what the paper is describing.

Kai: Right. And the paper specifically investigates how electron-electron interactions drive the formation of a spontaneous mass for otherwise gapless electrons right at a topological transition point.

Mira: That's the core idea—they start with a non-interacting model where you have a continuous transition between a Quantum Spin Hall Insulator and something trivial when the bare mass M is less than two, but there's always that gapless Dirac state right at M equals two.

Lev: So, what they really found is that adding these interactions changes the character of that transition. They show that because of the quantum fluctuations in different channels, this transition can turn discontinuous for sufficiently large interactions.

Kai: That means we can predict when a continuous transition becomes first-order just by looking at how strong those interactions are, which is a big deal for modeling real materials.

Mira: And they connect this to the Gross-Neveu quantum critical behavior observed near that quantum critical endpoint, with an estimated critical exponent beta of approximately one <ref:2311.10024#pg1>.

Lev: So for anyone building real hardware, this means we can predict the overhead associated with these fluctuations if we try to run simulations in that regime.

Kai: They also identify this gap opening as a condensed matter analog of the Coleman-Weinberg mechanism, where one-loop quantum fluctuations lead to a mass without any symmetry breaking at the boundary line.

Mira: And they show how this manifests in terms of orbital polarization, which evolves smoothly across the transition point when interactions are weak, but then becomes discontinuous beyond a critical interaction value g c T for any given bare mass M.

Lev: That's important because it tells us that the boundary line itself isn't always continuous like mean field theory suggests; it can jump to a first-order line.

Kai: So, if you were driving and needed to know how much interaction strength would make a transition suddenly jump, this paper gives you that metric based on these quantum fluctuations.

Mira: The critical behavior near the Quantum Critical Point is in the Gross-Neveu universality class with an estimated critical exponent beta of about one which suggests some very specific scaling behavior right at that point <ref:2311.10024#pg1>.

Lev: That beta value tells us how sensitive the system's response is to changes in coupling strength when it's near that critical point.

Kai: And the model they use, involving the Hamiltonian H zero k =

M − 2t((kx) + (ky)): three + lambda (kx) thirty-one + lambda (ky) two sets up the non-interacting starting point nicely <ref:2311.10024#pg2,k = M − 2t(\cos(kx) + \cos(ky>.

Mira: They then introduce a generic local interaction term, hinted at by i = - gN squared i - gT squared T z i - gS squared S z i - gR squared R z i, where those terms describe the orbital polarization components <ref:2311.10024#pg1>.

Lev: The challenge for real hardware is incorporating all these different channels—the particle-hole channels, the orbital polarization fluctuations—into a simulation that captures this non-trivial behavior.

Kai: They show that by going beyond mean-field theory and including quantum fluctuations in these different channels, the topological transition point becomes discontinuous for sufficiently large interactions.

Mira: That discontinuity is what they link to the presence of a massive Dirac fermion at the transition point, which is consistent with a spontaneous symmetric mass generation process.

Lev: So, if you were trying to design an error correction scheme for this system, knowing that it can exhibit first-order behavior under strong coupling helps you understand where your simple approximations might fail.

Kai: The paper suggests this mechanism can be applied to other models for topological phase transitions, with a Lifshitz transition in interacting electronic systems being a natural example they point toward.

Mira: And the authors conclude by summarizing that this process of spontaneous mass generation happens through a condensed matter analog of the Coleman-Weinberg mechanism where one-loop quantum fluctuations lead to a mass without symmetry breaking.

Lev: For future work, I see them looking at how this applies to other systems, which is good because it shows the underlying physics isn't just stuck on this specific model.

Kai: So, that’s what we’ve got from "Topological Gap Opening without Symmetry Breaking from Dynamical Quantum Correlations" today—a way to get a gap opening through quantum fluctuations without symmetry breaking.

The paper's summary: Kai: So, we're talking about how these electron interactions can actually open up a gap in a topological insulator without needing any symmetry breaking at all, which is pretty wild.

Mira: Exactly what the paper is showing here—the authors are using dynamical quantum fluctuations in different channels to drive this gap opening, which they compare to the Coleman-Weinberg mechanism we see with mass generation.

Lev: That means we're looking at a scenario where you get a mass for your Dirac fermions just because of these quantum fluctuations, not because the fundamental symmetry of the system is broken.

Kai: Right. They start with this non-interacting setup where you have this continuous transition between different topological phases when the bare mass M is less than two.

Mira: But when you add those electron-electron interactions, especially those local ones they include, things get interesting; they show that the quantum fluctuations make that transition discontinuous for strong enough interactions.

Lev: That discontinuity is what really matters for us on the hardware side because it means we can't just assume a smooth evolution of the phase boundary.

Kai: The paper shows this isn't just a mean-field effect; by including those quantum fluctuations in the particle-hole channels and orbital polarization, you get this first-order line ending at a critical endpoint.

Mira: And at that endpoint, they find Gross-Neveu quantum critical behavior, which means we can estimate universal scaling exponents like beta equals one near that transition point.

Lev: If we try to run this on real hardware with error correction, knowing there's a sharp jump in the phase boundary due to interaction strength changes is crucial for figuring out where your overhead will land.

Kai: It suggests that the effect of these interactions is actually related to spontaneous mass generation, which they call a condensed matter analog of the Coleman-Weinberg mechanism.

Mira: So, this isn't just about changing a bare mass term; it’s one-loop quantum fluctuations creating a gap for the Dirac fermions right at the boundary line without breaking any symmetries.

Lev: That implies that in some interacting systems, you can get a robust topological state even when you don't have the symmetry breaking terms explicitly present in your Hamiltonian.

Kai: And they connect this to other systems too, suggesting this mechanism could apply to other topological phase transitions like Lifshitz transitions in interacting electronic systems.

Mira: It gives us a way to think about how interactions can fundamentally alter the nature of a topological transition, moving it from continuous to discontinuous based on coupling strength.

Lev: So the takeaway for us is that we need to account for these multi-channel quantum fluctuations when modeling real materials because they can cause phase boundaries to behave in unexpected ways.

The paper's improvements: Tom: So, we're talking about how these authors suggest ways to make this model more accurate than just what they showed in this paper.

Kai: They are suggesting that for the next step, we need to look at incorporating more of those different channels—the particle-hole and orbital polarization ones—more rigorously.

Mira: That makes sense because the whole point is that the fluctuations in those other channels are what really make the transition discontinuous when you have strong interactions.

Lev: From an error correction standpoint, if we can model how those different channels couple to each other, it tells us exactly where your noise will be landing on the physical hardware.

Kai: They're also pushing towards applying this mechanism to other specific models that aren't just the Bernevig-Hughes-Zhang setup they used as a starting point.

Mira: That’s the big picture—they want to show this isn't just some fluke for this one material; it’s a general idea for other topological phase transitions, like those Lifshitz transitions in interacting electronic systems.

Lev: If we can generalize the model, it means we can start predicting how these fluctuation effects behave when you move to more complex lattice structures on a real chip.

Kai: They are also looking at how the quantum critical behavior changes as the interaction strength gets even stronger beyond that critical value g c T.

Mira: That’s important because it shows that even after you've hit the point where things become first-order, there’s still a way to characterize what happens near that endpoint using Gross-Neveu scaling.

Lev: It gives us a better roadmap for simulation; instead of just stopping at the critical point, we can see how the scaling behavior evolves as we push the coupling even further into the strong regime.

Kai: So, they are really trying to give us a more complete picture of this transition—from the initial continuous line all the way to where it becomes discontinuous and what happens right at that boundary.

Mira: The implication is that understanding this mechanism means we can design topological materials where you intentionally engineer these quantum fluctuation effects to control the phase transition character.

Lev: It shifts the focus from just finding a gap to actively using interaction-driven fluctuations to tune the system's fundamental topological nature.

Conclusion: Kai: So we're wrapping up this discussion on "Topological Gap Opening without Symmetry Breaking from Dynamical Quantum Correlations" by these authors.

Mira: Basically, they showed that you can get a gap opening in a topological insulator just through dynamical quantum fluctuations, without any of the usual symmetry breaking terms being there in the starting Hamiltonian.

Lev: That means the system generates a mass for its Dirac fermions because of these one-loop corrections to the physics.

Kai: They confirmed this is really analogous to what we see in condensed matter, like with the Coleman-Weinberg mechanism for generating mass without symmetry breaking.

Mira: The main result is that if you include fluctuations in all those different particle-hole and orbital polarization channels, you can get a first-order transition where mean field theory would have said it should be smooth.

Lev: For error correction, this means we need to model the coupling between those channels because they drive the discontinuity in the phase boundary.

Kai: This work opens up possibilities for applying this idea to other systems, like Lifshitz transitions in interacting electronic systems, which is a pretty big scope.

Mira: It suggests that interaction effects can fundamentally change the character of a topological transition based on how strong those interactions are.

Lev: If we can apply this generalized approach, it helps us predict the overhead for modeling these more complex topological phases in real hardware.

Kai: So, to sum up, this paper proves that fluctuations aren't just noise; they’re a mechanism that can create mass where symmetry isn't broken.

Mira: It’s a condensed matter model of how one-loop quantum corrections lead to mass generation without the need for explicit symmetry breaking terms.

Lev: We still have to figure out the exact scaling behavior near that Gross-Neveu critical point, but this gives us a solid baseline for where those exponents might land.

Kai: We’ll keep watching how this concept of fluctuation-driven mass generation plays out as we look at other topological systems.

Mira: Next week, we're diving into the paper on Gibbs-state phases beyond microscopic symmetry breaking and seeing if that idea holds up in different topological settings.

Lev: And I want to talk about how these fluctuation effects impact transport properties in low-dimensional systems next week.

F. Paoletti, L. Fanfarillo, M. Capone, A. Amaricci

International School for Advanced Studies (SISSA) · CNR-ISC · CNR-IOM

cond-mat.str-el

Submitted: 2023-11-16

Updated: 2023-11-16

Comments: 6 pages, 2 figures

Journal ref: Phys. Rev. B 109, 075148 (2024)

DOI: 10.1103/PhysRevB.109.075148

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 77/100

The gist: The gist: Dynamical quantum fluctuations can lead to the opening of a gap without any symmetry breaking in a topological transition, which is analogous to the Coleman-Weinberg mechanism for mass

Key concepts

Topological Phase Transition (TQPT): A transition between two distinct topological phases of a quantum system. In this context, it usually involves the closing and reopening of the energy gap, which can lead to a gapless state at the transition point without necessarily breaking fundamental symmetries.
Spontaneous Symmetry Breaking (SSB): The standard expectation is that opening a gap in these transitions requires SSB to occur. However, this paper demonstrates that dynamical quantum fluctuations allow a gap to open while preserving the original symmetries of the model, leading to a different outcome.
Coleman-Weinberg Mechanism: This is a concept from particle physics where mass generation occurs through one-loop quantum fluctuations rather than explicit symmetry breaking. The authors identify their condensed matter gap opening as an analog to this mechanism, showing how quantum effects create a mass for Dirac fermions without needing SSB.
Gross-Neveu Critical Behavior: This describes the critical behavior observed near the quantum critical endpoint where the transition line becomes first-order. The system exhibits Gross-Neveu universality class behavior, characterized by specific scaling laws and a critical exponent beta approximately equal to 1.

Terminology

Symmetry Breaking — 3× in this paper · explained in 4 episodes
Phase Transition — 1× in this paper · explained in 2 episodes
Phase Transitions — 1× in this paper · explained in 2 episodes
Quantum fluctuations — 1× in this paper · explained in 2 episodes
Topological Phase Transition — 1× in this paper · explained in 2 episodes

Summary

The gist: Dynamical quantum fluctuations can lead to the opening of a gap without any symmetry breaking in a topological transition, which is analogous to the Coleman-Weinberg mechanism for mass generation

Mechanism of Gap Opening

The paper investigates how electron-electron interactions can drive the formation of a spontaneous mass for otherwise gapless electrons at a topological transition The standard phenomenology for this to occur requires a spontaneous symmetry breaking (SSB) The authors solve a Bernevig-Hughes-Zhang model augmented with local electron-electron interactions that preserve some symmetries of the model Without interactions, the non-interacting model features a continuous topological transition between a QSHI for M 2 through the formation of a gapless Dirac state at M = 2

Role of Quantum Fluctuations

The authors go beyond mean-field theory by using a variational approach including quantum fluctuations not only of the orbital polarization, but also in the other particle-hole channels They demonstrate a new scenario in which a gap opens at the TQPT without breaking any of the symmetries of the model Specifically, they show that the quantum fluctuations in the different channels make the TQPT discontinuous for sufficiently large interactions This first-order line ends in a critical endpoint where Gross-Neveu quantum critical behavior is observed as a function of coupling strength

Connection to Mass Generation

The mechanism revealed is reminiscent of the Coleman-Weinberg (CW) theory of mass generation The authors identify the gap opening as a condensed matter analog of the Coleman-Weinberg mechanism of mass generation This process involves one-loop quantum fluctuations leading to a mass without symmetry breaking The finite gap found above the QCP corresponds to giving a mass to the Dirac fermions at the boundary line, which is consistent with a spontaneous symmetric mass generation process

Phase Transition Character

The presence of interactions can change the character of the transition, making it necessarily discontinuous despite symmetries protecting the topological phase are preserved The fluctuation-corrected results show that while mean-field theory always describes a continuous transition, in the corrected theory the boundary line is continuous up to a critical value g cT of the interaction beyond which it becomes of first order This reflects in the behavior of the free energy near the TQPT point in the intermediate to strong coupling regime, i.e., g T > g cT

Critical Behavior and Susceptibility

The critical behavior near the QCP falls in the Gross-Neveu universality class with an estimated critical exponent beta ≃ 1 The uniform orbital susceptibility XT reads XT = -2gT χTT(0) / (1 - gT χTT(0) + λ) This quantity accounts for the contributions of all the other channels of the interaction through the expression of Σ(iωn), see Eq. (6) The minimum of the denominator in Eq. (8) approaches zero when we reach the QCP, as shown in Fig. 2(a)

Conclusion and Implications

The work demonstrates the crucial role of fluctuations in the different local particle-hole channels to qualitatively change the nature of the TQPT Within a mean-field, interactions only lead to a renormalization of the bare mass of the model so that the TQPT has the same character as in non-interacting systems The effect is intrinsically related to a spontaneous gap opening (mass formation) for otherwise gapless Dirac nodes at TQPT point without any symmetry breaking This process of spontaneous mass generation takes place through a condensed matter analog of the CW mechanism in which one-loop quantum fluctuations lead to a mass without symmetry breaking The authors expect that our mechanism can be applied also to other models for topological phase transitions, and a natural example is that of Lifshitz transitions in interacting electronic systems

--- Page 5 ---

The orbital polarization smoothly evolves across the topological transition Coherently with the above scenario, a direct inspection of the MF free energy shows only one minimum for every value of gT and M The first-order line ends in a critical endpoint, where we show a Gross-Neveu quantum critical behavior as a function of the relevant coupling strength As we shall discuss in the following, the mechanism we revealed is reminiscent of the Coleman-Weinberg (CW) theory of mass generation

--- Page 6 ---

In Fig. 1(a) we show the evolution of the orbital polarization, obtained from the self-consistent value of the bosonic field T z = ∆¯ T /gT The behavior at the transition point g∗T(M) changes qualitatively according to the value of the bare mass M For a value close to M = 2, i.e., the non-interacting transition point, the orbital polarization is continuous with respect to the increasing interaction gT This corresponds to a smooth modification of the BI into a non-trivial insulator through the formation of a gapless state at the TQPT Starting from a farther point, the orbital polarization displays a critical behavior at the transition characterized by a divergent susceptibility ∂M T z Beyond this point, for any value of M, the orbital polarization is characterized by a discontinuous evolution across the topological transition This is in stark contrast with the continuous behavior obtained in MF for the same value of the mass, see Fig. 1(a), and it agrees with previous results obtained via Dynamical Mean-Field Theory (DMFT)

--- Page 4 ---

The first-order line ends in a critical endpoint, where we show a Gross-Neveu quantum critical behavior as a function of the relevant coupling strength As we shall discuss in the following, the mechanism we revealed is reminiscent of the Coleman-Weinberg (CW) theory of mass generation

--- Page 3 ---

The starting point of our analysis is to rewrite the partition function of the interacting model (1) in terms of an effective problem coupled to space- and time-dependent (real) bosonic fields ∆q = ∆a=N,T,S,R q, by performing an Hubbard–Stratonovich transformation The partition function reads: Z ≡ e − βF = R D∆ e − βNF [∆] in terms of the free energy functional F[∆] = X aq ∆a q 2 2ga - 1 βN Trln(−G−1 k q) (3) where q = (q, iνm), k = (k, iωn), with q, k wave vectors in the first Brillouin zone and νm, ωn the bosonic and fermionic Matsubara frequencies, β is the inverse temperature and N the total number of sites

--- Page 2 ---

The model is invariant under time-reversal T and inversion P symmetries, U(1) spin rotation around the z axis In the following we set our energy unit so that 2t = 1 and focus on the regime of two electrons per site, i.e.

Improvements for AI systems

Bold header: Spontaneous Mass Generation Mechanism

This mechanism describes the gap opening for Dirac electrons at the boundary of a topological insulator through dynamical quantum fluctuations in the different channels, allowing AI systems to model non-trivial topological transitions where a gap opens without symmetry breaking, analogous to the Coleman-Weinberg mechanism.

Bold header: Discontinuous Topological Transition Modeling

The system can now predict when a continuous transition becomes first-order, as shown by the quantum fluctuations in the different channels make the TQPT discontinuous for sufficiently large interactions, enabling AI to distinguish between continuous and discontinuous topological phase transitions based on interaction strength.

Bold header: Quantum Critical Behavior Analysis

AI can now characterize the critical point using a Gross-Neveu quantum critical behavior near the transition, allowing for the prediction of universal scaling exponents like a critical exponent β ≃ 1 in certain regimes where a QCP separates continuous and discontinuous regimes.

Bold header: Spontaneous Symmetry Breaking Analogs

The system can apply the concept of spontaneous mass generation to other physical phenomena, such as Lifshitz transitions in interacting electronic systems, enabling the AI to generalize topological phase transition predictions beyond the standard Bernevig-Hughes-Zhang model.

Abstract

Topological phase transitions are typically associated with the formation of gapless states. Spontaneous symmetry breaking can lead to a gap opening thereby obliterating the topological nature of the system. Here we highlight a completely different destiny for a topological transition in presence of interaction. Solving a Bernevig-Hughes-Zhang model with local interaction, we show that dynamical quantum fluctuations can lead to the opening of a gap without any symmetry breaking. As we vary the interaction and the bare mass of the model, the continuous gapless topological transition turns into a first-order one, associated with the presence of massive Dirac fermion at the transition point showing a Gross-Neveu critical behaviour near the quantum critical endpoint. We identify the gap opening as a condensed matter analog of the Coleman-Weinberg mechanism of mass generation.