Hi,

its a very good discussion to start.

And it is related to the previously asked question.

Actually it is totally observation based. The time of observation of macroscopic property spin or orbital degree of freedom. Please see this and add please more explanation

thank you

https://www.researchgate.net/post/Can_any_ferromagnetic_order_break_the_time_reversal_symmetry

No, because it involves an even number of fermions. Thus you haven't degenerate Kramers pairs associated to the T inversion operator for this magnetic order.

Sorry, I want to say that you have degenerate Kramers pairs associated to this T inversion operator. as no magnetic order would exist.

@ Dear Prof. Daniel Baldomir. First of all thank you for nice reply. But I did not get your point clearly. Are you saying that AFM state is a set of degenerate Kramers pairs and so in AFM state time-revers symmetry remains preserved.

Dear Aga,

What I'm saying is that the AFM order doesn't prevent you to have degenerate Kramers pairs due to have time inversion symmetry. That is not possible in a ferromagnetic order where M is not zero.

Hello (again) Daniel,

I am not sure I can follow you. When you write "the AFM order doesn't prevent you to have degenerate Kramers pairs", the question that immediately comes to my mind is: "where doesn't the AFM order prevent... ?"

Plase take a look at the question Abdul Ahad already mentioned (https://www.researchgate.net/post/Can_any_ferromagnetic_order_break_the_time_reversal_symmetry). There, Behnam Farid took a different approach to state time reversal symmetry is actually broken in (or in presence of?) an AFM.

What is your opinion?

Hi Kai,

Time reversal operator is not related with equilibrium (thermodynamic or not) as you accept in the previous thread. Time reversal just protets the topological conditions of the material calculated with the Chern number as associated to the Z2 cyclic group. This leads directly to the degeneration of states and the formation of Kramers pairs with opposite chiral currents. One important feature is that there is absence of backscattering for the impurities or other. Never the equilibrium is there as a condition.

I suppose that you know very well all these conditions for the interesting Nature that you have published on the Topological Insulators, where I didn't understand quite well how you avoided the breaking of time reversibility when the magnetic field B was present.

Hi again Daniel, still thinking about this one.

When you talk about an even number of Fermions, then I guess you have the magnetic unit cell in mind. No questions here. But is that sufficient? Does it address the point?

Consider CoO, which locally is d7, below TN (tetragonally distorted rocksalt structure). Now, given the AFM state of the crystal (and holding the moments fixed), for any local Co ion, the time reversed states are not degenerate, are they? What does it tell w.r.t. the question of the OP? [The same would be true for moniclinic CuO with spin 1/2 (d9 config), I think.]

I would also think of systems with AFM/FM interfaces, where materials are chosen such that TN > TC. Cool through TN (with an external bias field for a macroscopic effect, my guess is that the same is valid without bias on the length scale of AFM domains), then take off the field and I am sure that the FM (if thin enough) will be magnetically polarized and that the Curie temperature will rise.

Does this have anything to do with your statements(s)? Still thinking...

Dear Kai,

The ground state configuration of the ion Co- is 2d84s2 , i.e. 3F4 assuming they are isolated atoms. Would you consider AFM order from this information?

Daniel,

Co in CoO is Co2+, and therfore 3d74s0 in formal notation. And this IS an antiferromagnet. Ni in NiO is 3d8 by the way, and also an antiferromagnet (and quite a famous one...). So this is, where my example and question were situated.

For a single Co- ion I would expet a high spin config, as you write it, but the notion of order is irrelevant, isn't it? Probably I am not getting at what you're trying to ask me.

Through google search I find few more answers to this question and here I am sharing them. In support of time reversal breaking

Through google search I find few more answers to this question and here I am sharing them. In support of Time Reversal Invariance.

Dear @Aga shahee sir,

What you arr talking about time reversal is very basic thing. Now here the main concern is to find the energy degeneracy or the Kramers degeneracy role in symmetry breaking. I can say.. Basically the if you include thr external magnetic field in the Hamiltonian then definitely the time symmetry got broken. But the field produced by system itself like an atom, symmetry got preserved. Because of vector potential which is not invariant under tume reversal.

With respect to the second quote I would actually tend to not agree. For an even number of magnetic sites I agree that Stot=0. But does that make it a singlet? Let me symbolically write in one dimension (U="up", D="down"):

|AF1>= U D U D U D U D

|AF2>= D U D U D U D U

Both have total spin 0. Still, these are two (degenerate) states. There may be a sizeable anisotropy barrier between them, suppressing fluctuation.

Makes me also think of really small antiferromagnetic nanostructures that were studied at IMB (Almaden) in Andreas Heinrich's group. If I remember right, it took only little more than 10 magnetic sites to be stable and thus able to stay in an identifiable magnetic state (at very low temperature, of course, with a spin polarized STM I guess). I can't recall whether there was an even-odd alternation, but actually I would be really surprised if there was. Actually it says in a comment 12 Fe sites was the minimum size.

Article Bistability in Atomic-Scale Antiferromagnets

Dear Abdul Ahad

You are right my question is a basic question. Even it is basic, it is still what I am not able to understand and am concern about it. I need someone to clear my understanding.

@ Dear Prof. Kai Fauth Thank you. Through your latest answer things are getting clear to me.

See also: https://www.google.de/url?sa=t&rct=j&q=&esrc=s&source=web&cd=8&cad=rja&uact=8&ved=2ahUKEwjtkaLjv87fAhUJZVAKHX2HDaQQFjAHegQIARAC&url=http%3A%2F%2Fphys.org%2Fpdf245595412.pdf&usg=AOvVaw1DPMi5644p7kDOOrOT3Gs5

But I understand your question. For a local probe (at the atomic scale) like that STM I would say yes, time symmetry is broken. The conditions for spin up and down tunneling are not degenerate.

Now what I don't know is what about a delocalized probe, "sensing" all those atoms at a time with equal weight.

So suppose, those iron atoms would do the same thing when deposited onto a TI surface...

I'll try a theoretician or two to visit this thread.

I'm loving it!

Dear Kai,

From my humble point of view you, when you consider the magnetism of the ions the physics can be quite different of the collective magnetic order. For applying the time reversal operator T you only need to count the spins (spin-orbit is not necessary) assuming the orbitals L are quenched or integers. In the case of considering antiferromagnetism following a simple Weiss model (independently of the degenate antiferromagnetic ground state: A,C,E and G) you can count clearly the spin lattices (also without taking into account the Hyperfine field). It is true that we can complicate all these things with many subtleties as the existance of antiferromagnetic domains, but my answer was only for the antiferromagnetic order in its ground state.

Going to your ion Co+2 the crystal field is quite high respect to the one that I have written for you Co+1. This ion has the fine structure is given by 4F9/2 and I don't know what is the extra knowledge that you want to obtain respect to T symmetry. This was my question, are you interested in "antiferromagnetism" close to the Neel temperature ? What is the necessety of complicate so simple definition of the T operator?

Following

I try to summarize from my point of view:

the original question was whether (or not) the presence of AFM order breaks time reversal symmetry. As you can see above Dr Aga Shahee has colleted pro an con arguments on the web. The con argument put forward that with an even number of magnetic sites in the AFM the resultant state is a singlet an that would suffice to say that time reversal symmetry was not broken.

Daniel Baldomir even suggested that it be sufficient to have an even number of eletrons (I assume that's meant to be per unit cell) to know that time reversal symmetry be not broken.

My memories brought me back to a brilliant talk of Andreas Heinrich I heard quite a few years ago about these cute antiferromagnetic Iron nanostructures. I indicated the references a couple posts up. These could be used as "magnetic storage elements", being written by electric pulses and stably read out by spin polarized STM at low temperature (which means that the sp-STM current is different for up and down electrons, which in turn signifies that these are not degenerate).

That, for me implies that in this case, symbolically written, the atomic sequence

UDUDUDUDUDUD and the sequence

DUDUDUDUDUDU, while degenerate and of total Spin zero (well, the z component, certainly) do not form a singlet. [I would actually guess that they are related by time inversion - each local moment being taken to its negative]. Their stability in the experiment implies a sufficiently large energy barrier and sufficiently small magnetic tunneling amplitude.

All of this together gave me the feeling that "at least locally" time reversal symmetry must be broken.

Behnam came up with the notion of the staggered magnetization as the relevant order parameter etc. and the necessity to tell the story of sublattices which (I think) expressed neatly what I was unable to cast in organized words.

Daniel, on the other side, noticed the distinction between the single ion state and the ordered state physics, but I haven't quite been able to integrate that in my picture.

Whatever:

The one question that is really open to me is one that may or may not actually interest the OP (sorry, Dr Aga Shahee, if this is not the case). Put such a nanostructure or some other AFM (mono-) layer on the surface of an otherwise topological isulator. What will happen to this guy now?! Would finite size, commensurability or 2D lattice symmetry play a role?

Special thanks to Prof D. Khomskii

An email reply from Prof. Daniel Khomskii

Does Anti-ferromagnetic ordering breaks the time-revers symmetry? Discussion between Prof. Ronald Fisch and Prof. Jose Rodriguez

Dear Prof. Kai Fauth

Thank you for sharing such a beautiful work.

Going through referred work on anti-ferromagnetic Iron nano-structures published in Science "Bistability in atomic-scale antiferromagnets," Science 335, 196-199 (2012) , it is clear that as few as six Fe atoms periodically arranged exhibit stable Néel states and this NEEL state breaking time reversal symmetry. While Identical chains built from Mn atoms do not show the magnetic bistability and preserves time reversal symmetry. According to them the key difference between Fe and Mn chains is the strength of the magnetic anisotropy, which is ~50 times stronger in Fe than in Mn on this surface.

What I understand from this long discussion is that any magnetic long range order with # 0 breaks time reversal symmetry. There are some interesting states, such as spin liquids where time reversal symmetry remains preserved.

I am also very thankful to Prof. Behnam Farid and Prof. Daniel Baldomir for interesting discussion.

Regards

Aga Shahee

Dear Aga,

Can you please share your mentioned science paper pdf file here?

Many Thanks in advance.

Dear Gokaran Shukla

Below is the PDF of mentioned science paper

Finally, the conclusion again matches with the initial ideas that M = N. is zero than it breaks thr time invariance as simply understandable by Maxwell equation that charge is invariant but but not the charge density.

Yeahhh

Any comment.

And one more thing.. Can we observe this time invariance in experiment except superconductivity (don't understand honestly).

Thank you

What is clear is that the reverse time operator is a Quantum Mechanical concept quite well defined and also that in antiferromagnetism we have a even number of spins, and therefore there is a time inversion symmetry. In Quantum Mechanics doesn't exist anything as antiferromagnetic state and less its transformation under T operator, in such a case this would be needed to be defined which would be difficult to understand its necessety if QM allows to do straghtforwardly.

Dear Aga,

When you say:

According to them the key difference between Fe and Mn chains is the strength of the magnetic anisotropy, which is ~50 times stronger in Fe than in Mn on this surface.

This is not the main difference between these atoms Fe and Mn. The main difference is that one is ferromagnetic while the other is antiferromagnetic, i.e. the constants of coupling for one is positive while for the other is negative. There are also other key details in the nanostructure of the Science which would take a lot of time to discuss and which are very well known for people working with so small number of atoms, we have found many interesting results as an Ag ferromagnetic material instead of being diamagnetic. If you want you can find it in my papers here in RG.

Dear Prof. Daniel Baldomir

Thanks for sharing your teams interesting work about Ag ferromagnetic material. I will surely go though it.

Actually I quoted what they have written.

Paragraph-4 from Science 2012 paper without any change " In assemblies of just a few magnetic atoms,the atomic spins often couple to form quantum superposition states (20). For AFM coupling, this results in a singlet ground state, characterized by a wave function in which all spins populate opposing spin states equally (21). In contrast, we find that isolated AFM structures with as few as six Fe atoms exhibit stable Néel states, in which the spin orientation alternates between neighboring atoms. These states can be well described by the classical Ising model (22), in which the spins always point along one axis. Spin-polarized STM images of a linear eight-atom chain (Fig. 1, B to E) can clearly distinguish the two Néel states. The spin-polarized STM tip forms a magnetic tunnel junction in which the conductance lternates between high (parallel alignment of tip and sample spins) and low (antiparallel alignment) as the tip passes from atom to atom along the chain (12, 13, 23). Identical chains built from Mn atoms do not show the magnetic bistability and do not exhibit a spin-polarized contrast along the chain (fig. S2). A key difference between Fe and Mn chains is the strength of the magnetic anisotropy, which is ~50 times stronger in Fe than in Mn on this surface (19). The strong easy-axis anisotropy of Fe evidently stabilizes the two Néel states as observable magnetic states".

Regards

Aga

I agree with D. Baldomir and D. Khomskii in that if the state is a singlet, such as the antiferromagnetic one, there is no breaking of time reversal symmetry. One should also emphasize that although the FM state is essentially a classical, diagonal, order, antiferromagnetism is essentially a quantum off-diagonal order. In this quantum AFM order, as K. Fauth has pointed out, the two possible Neel orders are degenerate and hence even for a sublattice =0. I think that this fact was missed by B. Farid. Finally, I would add the recommend, in addition to more well-known books on quantum magnetism suzh as D. C. Mattis', the book "Spin Systems" by W. J. Caspers.

Dear Aga,

Thank you for sharing this interesting discussion!

In brief, I agree with Prof. Khomskii and Prof. Farid points of view.

Dear Oleg,

Can I know in what are you agreeing with respect to Prof. Khomskii and Prof. Farid?

Are you also in agreement with Prof. Kai when he put me a post that you have above?

Hello (again) Daniel,

I am not sure I can follow you. When you write "the AFM order doesn't prevent you to have degenerate Kramers pairs", the question that immediately comes to my mind is: "where doesn't the AFM order prevent... ?"

Plase take a look at the question Abdul Ahad already mentioned (https://www.researchgate.net/post/Can_any_ferromagnetic_order_break_the_time_reversal_symmetry). There, Behnam Farid took a different approach to state time reversal symmetry is actually broken in (or in presence of?) an AFM.

What is your opinion?

Dear Daniel Baldomir, Kai Fauth Oleg I. Utesov @José A. Riera

Electronic Dirac points are protected by time-reversal (T) symmetry and I think same should be case with Dirac magnons. The Dirac magnons has been experimentally observed in AFM Cu3TeO6. Does it mean that time-reversal (T) symmetry is protected by Dirac magnons in AFM phase of Cu3TeO6?

Dear Daniel,

I agree with the opinion, that any type of magnetic ordering with nonzero average site spins can only exist, if the system has not time-reversal symmetry in its magnetic symmetry group. If it has, then =- and =0. Thus, Neel ordering breaks time-reversal symmetry, while singlet ground state of antiferromagnet Hamiltonian doesn't.

Dear Oleg,

Dear Oleg,

My question is not related with your answer. But I'll try to make some comments for using this post in a more useful form than to just repeating arguments.

1. Although it is not difficult to realize that the number of spins able to produce magnetization M different to zero is even we can do it in a more pedantic form. The ground state of the S =1/2 Heisenberg AFM with an even number N of spins is a singlet a as Prof. José A. Riera. Therefore the time reversal symmetry is well defined for an antisymmetric magntic order.

2. You finish with the sentence:

Thus, Neel ordering breaks time-reversal symmetry, while singlet ground state of antiferromagnet Hamiltonian doesn't.

Why do you introduce Nèel ordering ( I suppose you are speaking about a Néel state) when this is a big problem not solved for the moment: there is a clear inconsistency of the Néel state with the antiferromagnetic Heisenberg Hamiltonian. Do you ignore this or it is just a form to confuse an answer to this question? Why?

Dear Aga,

Your sentence:

Electronic Dirac points are protected by time-reversal (T) symmetry and I think same should be case with Dirac magnons

is not true in general. The Dirac points are only protected topologically by T reversal symmetry in a topological insulator.

Dear Prof. Daniel Baldomir

I am very new to this field and as per my knowledge, Electronic Dirac points are observed only in topological Insulators and Graphene . And in both systems, Dirac points are protected by time-reversal (T) symmetry. I may be wrong?

Dear Aga,

You are right but notice this is not related with "Dirac magnons" that you have mentioned in your previous post and question.

Dear Prof Daniel Baldomir

That is what I want to clear (does Dirac magnons too preserve time reverse symmert?). Thanks for your comment reply. I will be very thankful, if you briefly explain who Dirac magnons allows breaking of time reverse symmetry.

Regards

Aga

Dear Aga,

This issue is not so simple to answer you (at least for me), because there are several conditions to follow more than the simple counting of spins usual in Quantum Mechanics. But I can give you the basic idea that I hope is assumed without showing exactly the details of its proof. It was shown that the Dirac magnon Hamiltonian preserves time-reversal symmetry defined with the sublattice pseudo spins and the Dirac points are robust against magnon-magnon interactions in the case of the graphene. And also the Dirac points are gapped if time-reversal symmetry is broken introducing a Dzyaloshinskii-Moriya (DM) interaction among the near-neighbours.

Speaking about in more general configurations as in geometrically frustrated Kagome lattices would be much more difficult to do it, at least for me and it is a field of knowledge full of controversy.

Dear Daniel,

as far as I understand the topic, we discuss antiferromagnetic ordering. I agree with you that the singlet ground state of the antiferromagnet does not break time reversal symmetry. But what I call an "antiferromagnetic ordering" is Neel state, which breaks time reversal symmetry, because we have nonzero magnetization in each sublattice. Even though it is not the ground state of the system it can stay intact for a long time, much longer than the time of measurements and is evidently seen in e.g. neutron experiments.

Dear Oleg I. Utesov

I fully agree with you.

My main question was about the anti-ferromagnetic ordered states like Neel state, other collinear and non-collinear spin states, which are evident in neutron diffraction experiments. Even though they are not eigenvalue state but they exit and have been experimentally observed.

Dear Oleg and Aga,

Let me to try to answer you. First of all the question is so simple as:

Does anti-ferromagnetic ordering breaks the time-revers symmetry?

The answer is NO. The antiferromagnetic order doesn't break the time reverse symmetry (which by other hand is a very simple concept belonging to the Lorentz group contained in the invariant CPT theorem). Thus this is a simple concept with a simple answer for writing in the blackboard in lecture of magnetism.

Now Oleg, has written in a previous post that:

Thus, Neel ordering breaks time-reversal symmetry, while singlet ground state of antiferromagnet Hamiltonian doesn't.

My question without answer is why he introduced Neel ordering if the question was other? And Aga says something surprising too: his answer was exactly about Neél ordering. My suggestion to Aga is that he ought change the question or at least to add this new point.

I take this opportunity to give the answer anyway. Antiferromagnetic order is time-reversal symmetric while a Neél state isn't. The reason of the T breaking for the Neél state has been thought due due to the high degeneracy of the fundamental state of AFM but nowadays is known that its reason is a quantum decoherency breaking the initial symmetry. Therefore the interactions are not so important as the quantum entanglement between the system and its environment.

I hope that I have helped to answer the question, its changes and future changes that perhaps I am not going to follow.

Dear Daniel Baldomir

Truly, I have written the question in a very simple way and that may have misleading many in this discussion. I am extremely sorry for that.

The main of this question was to understand about time reversal symmetry in different experimental observed AFM order states " Neél state, other collinear and non-collinear spin states". Even, I did not have written question is much appropriate way. Dear Prof. Daniel Baldomir " in the body of the question I have mentioned about Neel and other AFM orders. I am very much thankful to you and others senior scientist involved in this discussion to clearing many important things related to this topic.

Special thanks to you dear Prof. Daniel Baldomir " for a wonderful discussion and much explained recent comment. It have deepen my understanding about this topic and clarify many of my droughts.

With Best Regards

Aga

Daniel, could you give a pointer to some piece(s) of literature to consult for improving my understanding of your standpoints? Thanks!!

Hi Kai,

One good and clear reference is a course that Subir Sachdev has given some years ago but which is still interesting in distinguishing these subtle concepts

http://qpt.physics.harvard.edu/c10.pdf

and the more modern problems in this issue within a gauge context theories

https://arxiv.org/pdf/1808.04826.pdf

https://arxiv.org/pdf/1806.02837.pdf

https://arxiv.org/pdf/1708.04626.pdf

https://arxiv.org/pdf/1705.06289.pdf

Following

Dear Prof. Dr Aga Shahee

To my knowlegde the expression for the free Gibbs energy in antiferromagnets is given approximately by the following expression Eantif (*), and it has to be time reversal invariant always---pp. 25 & 170 in [1] & also pp 167 in [2].

Eantif = a M1.M2 - 1/2 b [(M1.n)2 + (M2.n)2] - (M1 . M2).H (*) where a is the exchange constant & b is the anisotropy constants, n is the anisotropic axis, H the external field & M1 & M2 the magnetization vectors which are given by the sum of all magnetic dipoles inside the sublattices of the antiferromagnet (*) pp. 250 of [1]

I quote Profs. Kaganov & Tsukernik book pp. 25 & 170-171:

"...The energy cannot change sign under time reversal (in these cases energy is said to be invariant under time reversal) This is clear from the expression for the ellergy of a free particle E = mv2/2. Under the reversal: t ---> - t & the sign of the velocity v is reversed, while that of v2 is not..."

[1] M.I. Kaganov & V. M. Tsukernik, "The Nature of Magnetism" Science for everyone, Mir-Moscow, 1995.

https://archive.org/details/TheNatureOfMagnetismKaganov/mode/2up

You can also check:

[2] Eletrodynamics of continuous media by Acad. L. Landau & E. Lifshitz, ch V-#48 pp 167, eq 48.2, Pergamon 1984. They use the phi thermodynamic potential free energy.

CC. Prof. Behnam Farid